Analysis I, Section 7.4: Rearrangement of series

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

Rearrangement of non-negative or absolutely convergent series.

namespace Chapter7

theorem Series.sum_eq_sum (b:ℕ → ℝ) {N:ℤ} (hN: N ≥ 0) : ∑ n ∈ .Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ .Iic N.toNat, b n := byb:ℕ → ℝN:ℤhN:N ≥ 0⊢ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n
      convert Finset.sum_image (g := Int.ofNat) (byb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Set.InjOn Int.ofNat ↑?m.42 simpAll goals completed! 🐙)
      ext xh.e'_2.h.hb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ x ∈ Finset.Icc 0 N ↔ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat); simph.e'_2.h.hb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N ↔ ∃ a ≤ N.toNat, ↑a = x; constructorh.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = xh.e'_2.h.h.mprb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ (∃ a ≤ N.toNat, ↑a = x) → 0 ≤ x ∧ x ≤ N
      .h.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = x intro ⟨ _, _ ⟩h.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ ∃ a ≤ N.toNat, ↑a = x; use x.toNathb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll goals completed! 🐙

Proposition 7.4.1

theorem Series.converges_of_permute_nonneg {a:ℕ → ℝ} (ha: (a:Series).nonneg) (hconv: (a:Series).converges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n) : Series).converges ∧ (a:Series).sum = (fun n ↦ a (f n) : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  -- This proof is written to follow the structure of the original text.
  set af : ℕ → ℝ := fun n ↦ a (f n)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have haf : (af:Series).nonneg := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    intro na:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤ⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0; by_cases h : n ≥ 0pos._@._hyg.513a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0neg._@._hyg.513a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 <;>pos._@._hyg.513a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0neg._@._hyg.513a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 simp [h, af]All goals completed! 🐙
    specialize ha (f n.toNat)pos._@._hyg.513a:ℕ → ℝhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)n:ℤh:n ≥ 0ha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑(f n.toNat) ≥ 0⊢ 0 ≤ a (f n.toNat); grindAll goals completed! 🐙
  set S := (a:Series).partiala:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partial⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set T := (af:Series).partiala:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partial⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hSmono : Monotone S := Series.partial_of_nonneg haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hTmono : Monotone T := Series.partial_of_nonneg hafa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set L := iSup Sa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set L' := iSup Ta:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hSBound : ∃ Q, ∀ N, S N ≤ Q := (converges_of_nonneg_iff ha).mp hconva:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  suffices: (∃ Q, ∀ M, T M ≤ Q) ∧ L = L'a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sumthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604⊢ (∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'
  .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum have Ssum : L = (a:Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum = L; apply sum_of_convergesha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesTo L; simp [convergesTo, L]ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ Filter.Tendsto { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial Filter.atTop
  (nhds (iSup S))
      apply tendsto_atTop_isLUB hSmono (isLUB_csSup _ _)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ (Set.range S).Nonemptya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ BddAbove (Set.range S)
      .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ (Set.range S).Nonempty use (S 0)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ S 0 ∈ Set.range S; aesopAll goals completed! 🐙
      choose Q hQ using hSBounda:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Q:ℝhQ:∀ (N : ℤ), S N ≤ Q⊢ BddAbove (Set.range S); use Qha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Q:ℝhQ:∀ (N : ℤ), S N ≤ Q⊢ Q ∈ upperBounds (Set.range S); simp [upperBounds, hQ]All goals completed! 🐙
    have Tsum : L' = (af:Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum = L'; apply sum_of_convergesha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'; simp [convergesTo, L']ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ Filter.Tendsto { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial Filter.atTop
  (nhds (iSup T))
      apply tendsto_atTop_isLUB hTmono (isLUB_csSup _ _)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ (Set.range T).Nonemptya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ BddAbove (Set.range T)
      .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ (Set.range T).Nonempty use (T 0)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912⊢ T 0 ∈ Set.range T; aesopAll goals completed! 🐙
      choose Q hQ using this.1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912Q:ℝhQ:∀ (M : ℤ), T M ≤ Q⊢ BddAbove (Set.range T); use Qha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912Q:ℝhQ:∀ (M : ℤ), T M ≤ Q⊢ Q ∈ upperBounds (Set.range T); simp [upperBounds, hQ]All goals completed! 🐙
    simp [←Ssum, ←Tsum, this.2, converges_of_nonneg_iff haf]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604this:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:_fvar.20577 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.20912Tsum:_fvar.20708 = { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.sum := ?_mvar.122064⊢ ∃ M, ∀ (N : ℤ), { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial N ≤ M
    convert this.1All goals completed! 🐙
  have hTL (M:ℤ) : T M ≤ L := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    by_cases hM : M ≥ 0pos._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0⊢ T M ≤ Lneg._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:¬M ≥ 0⊢ T M ≤ L
    swapneg._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:¬M ≥ 0⊢ T M ≤ Lpos._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0⊢ T M ≤ L
    .neg._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:¬M ≥ 0⊢ T M ≤ L have hM' : M < 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
      simp [T, Series.partial, hM']neg._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:¬M ≥ 0hM':_fvar.247761 < 0 := ?_mvar.247887⊢ 0 ≤ L
      convert le_ciSup (f := S) ?_ (-1)neg._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:¬M ≥ 0hM':_fvar.247761 < 0 := ?_mvar.247887⊢ BddAbove (Set.range S)
      simp [BddAbove, Set.Nonempty, upperBounds, hSBound]All goals completed! 🐙
    set Y := Finset.Iic M.toNatpos._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)⊢ T M ≤ L
    have hN : ∃ N, ∀ m ∈ Y, f m ≤ N := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      use (Y.image f).sup idha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)⊢ ∀ m ∈ Y, f m ≤ (Finset.image f Y).sup id; intro m hmha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)m:ℕhm:m ∈ Y⊢ f m ≤ (Finset.image f Y).sup id
      apply Finset.le_sup (f := id)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)m:ℕhm:m ∈ Y⊢ f m ∈ Finset.image f Y; grindAll goals completed! 🐙
    choose N hN using hNpos._@._hyg.1627a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ T M ≤ L
    calc
      _ = ∑ m ∈ Y, af m := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ T M = ∑ m ∈ Y, af m simp [T, Series.partial, af]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (∑ n ∈ Finset.Icc 0 M, if 0 ≤ n then a (f n.toNat) else 0) = ∑ n ∈ Y, a (f n); exact sum_eq_sum af hMAll goals completed! 🐙
      _ = ∑ n ∈ f '' Y, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ m ∈ Y, af m = ∑ n ∈ (f '' ↑Y).toFinset, a n symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ (f '' ↑Y).toFinset, a n = ∑ m ∈ Y, af m; convert Finset.sum_image (bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ Set.InjOn ?m.631 ↑?m.630 solve_by_elim [hf.injective]All goals completed! 🐙); simpAll goals completed! 🐙
      _ ≤ ∑ n ∈ .Iic N, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ (f '' ↑Y).toFinset, a n ≤ ∑ n ∈ Finset.Iic N, a n
        apply Finset.sum_le_sum_of_subset_of_nonnegha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset ⊆ Finset.Iic Nhfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∀ i ∈ Finset.Iic N, i ∉ (f '' ↑Y).toFinset → 0 ≤ a i
        ·ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset ⊆ Finset.Iic N intro _ _ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ Na✝¹:ℕa✝:a✝¹ ∈ (f '' ↑Y).toFinset⊢ a✝¹ ∈ Finset.Iic N; aesopAll goals completed! 🐙
        intro i _ _hfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ Ni:ℕa✝¹:i ∈ Finset.Iic Na✝:i ∉ (f '' ↑Y).toFinset⊢ 0 ≤ a i; specialize ha ihfa:ℕ → ℝhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ Ni:ℕa✝¹:i ∈ Finset.Iic Na✝:i ∉ (f '' ↑Y).toFinsetha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑i ≥ 0⊢ 0 ≤ a i; aesopAll goals completed! 🐙
      _ = S N := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ Finset.Iic N, a n = S ↑N simp [S, Series.partial]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ Finset.Iic N, a n = ∑ n ∈ Finset.Icc 0 ↑N, if 0 ≤ n then a n.toNat else 0; symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (∑ n ∈ Finset.Icc 0 ↑N, if 0 ≤ n then a n.toNat else 0) = ∑ n ∈ Finset.Iic N, a n; apply sum_eq_sum (N:=N) aa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ↑N ≥ 0; positivityAll goals completed! 🐙
      _ ≤ L := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ S ↑N ≤ L apply le_ciSup _ (N:ℤ)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604M:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic (Int.toNat _fvar.247761)N:ℕhN:∀ m ∈ Y, f m ≤ N⊢ BddAbove (Set.range S); simp [BddAbove, Set.Nonempty, upperBounds, hSBound]All goals completed! 🐙
  have hTbound : ∃ Q, ∀ M, T M ≤ Q := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum use LAll goals completed! 🐙
  simp [hTbound]thisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508⊢ L = L'
  have hSL' (N:ℤ) : S N ≤ L' := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    by_cases hN : N ≥ 0pos._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0⊢ S N ≤ L'neg._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:¬N ≥ 0⊢ S N ≤ L'
    swapneg._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:¬N ≥ 0⊢ S N ≤ L'pos._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0⊢ S N ≤ L'
    .neg._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:¬N ≥ 0⊢ S N ≤ L' have hN' : N < 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
      simp [S, Series.partial, hN']neg._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:¬N ≥ 0hN':_fvar.469256 < 0 := ?_mvar.469382⊢ 0 ≤ L'
      convert le_ciSup (f := T) ?_ (-1)neg._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:¬N ≥ 0hN':_fvar.469256 < 0 := ?_mvar.469382⊢ BddAbove (Set.range T)
      simp [BddAbove, Set.Nonempty, upperBounds, hTbound]All goals completed! 🐙
    set X := Finset.Iic N.toNatpos._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)⊢ S N ≤ L'
    have hM : ∃ M, ∀ n ∈ X, ∃ m, f m = n ∧ m ≤ M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      use (X.preimage f (Set.injOn_of_injective hf.1)).sup idha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)⊢ ∀ n ∈ X, ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id
      intro n hnha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)n:ℕhn:n ∈ X⊢ ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id; choose m hm using hf.2 nha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)n:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id
      refine ⟨ _, hm, ?_ ⟩ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)n:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ m ≤ (X.preimage f ⋯).sup id
      apply Finset.le_sup (f := id)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)n:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ m ∈ X.preimage f ⋯
      simp [Finset.mem_preimage, hm, hn]All goals completed! 🐙
    choose M hM using hMpos._@._hyg.2454a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ M⊢ S N ≤ L'
    have sum_eq_sum (b:ℕ → ℝ) {N:ℤ} (hN: N ≥ 0)
      : ∑ n ∈ .Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ .Iic N.toNat, b n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      convert Finset.sum_image (g := Int.ofNat) (bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Set.InjOn Int.ofNat ↑?m.1144 simpAll goals completed! 🐙)
      ext xh.e'_2.h.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ x ∈ Finset.Icc 0 N ↔ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat); simph.e'_2.h.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N ↔ ∃ a ≤ N.toNat, ↑a = x; constructorh.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = xh.e'_2.h.h.mpra:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ (∃ a ≤ N.toNat, ↑a = x) → 0 ≤ x ∧ x ≤ N
      .h.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = x intro ⟨ _, _ ⟩h.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ ∃ a ≤ N.toNat, ↑a = x; use x.toNatha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := ?_mvar.14799S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := fun M => @?_mvar.247769 MhTbound:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.468508N✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll goals completed! 🐙
    calc
      _ = ∑ n ∈ X, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ S N = ∑ n ∈ X, a n simp [S, sum_eq_sum, hN, X]All goals completed! 🐙
      _ = ∑ n ∈ ((Finset.Iic M).filter (f ·∈ X)).image f, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∑ n ∈ X, a n = ∑ n ∈ Finset.image f ({x ∈ Finset.Iic M | f x ∈ X}), a n
        congre_sa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ X = Finset.image f ({x ∈ Finset.Iic M | f x ∈ X}); exte_s.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕ⊢ a✝ ∈ X ↔ a✝ ∈ Finset.image f ({x ∈ Finset.Iic M | f x ∈ X}); simpe_s.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕ⊢ a✝ ∈ X ↔ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝; constructore_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕ⊢ a✝ ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝e_s.h.mpra:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕ⊢ (∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝) → a✝ ∈ X
        .e_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕ⊢ a✝ ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝ intro he_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))a✝:ℕh:a✝ ∈ X⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝; obtain ⟨ m, rfl, hm' ⟩ := hM _ he_s.h.mp.intro.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))m:ℕhm':m ≤ Mh:f m ∈ X⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = f m; use mAll goals completed! 🐙
        rintro ⟨ _, ⟨ _, _⟩, rfl ⟩e_s.h.mpr.intro.intro.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))w✝:ℕleft✝:w✝ ≤ Mright✝:f w✝ ∈ X⊢ f w✝ ∈ X; simp_allAll goals completed! 🐙
      _ ≤ ∑ m ∈ .Iic M, af m := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∑ n ∈ Finset.image f ({x ∈ Finset.Iic M | f x ∈ X}), a n ≤ ∑ m ∈ Finset.Iic M, af m
        rw [Finset.sum_image (by solve_by_elim [hf.injective])]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∑ x ∈ Finset.Iic M with f x ∈ X, a (f x) ≤ ∑ m ∈ Finset.Iic M, af m
        apply Finset.sum_le_sum_of_subset_of_nonnegha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ {x ∈ Finset.Iic M | f x ∈ X} ⊆ Finset.Iic Mhfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∀ i ∈ Finset.Iic M, i ∉ {x ∈ Finset.Iic M | f x ∈ X} → 0 ≤ a (f i)
        .ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ {x ∈ Finset.Iic M | f x ∈ X} ⊆ Finset.Iic M aesopAll goals completed! 🐙
        intro i _ _hfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))i:ℕa✝¹:i ∈ Finset.Iic Ma✝:i ∉ {x ∈ Finset.Iic M | f x ∈ X}⊢ 0 ≤ a (f i); specialize haf ihfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))i:ℕa✝¹:i ∈ Finset.Iic Ma✝:i ∉ {x ∈ Finset.Iic M | f x ∈ X}haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq ↑i ≥ 0⊢ 0 ≤ a (f i); aesopAll goals completed! 🐙
      _ = T M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∑ m ∈ Finset.Iic M, af m = T ↑M simp [T, Series.partial, af]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ∑ n ∈ Finset.Iic M, a (f n) = ∑ n ∈ Finset.Icc 0 ↑M, if 0 ≤ n then a (f n.toNat) else 0; symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ (∑ n ∈ Finset.Icc 0 ↑M, if 0 ≤ n then a (f n.toNat) else 0) = ∑ n ∈ Finset.Iic M, a (f n); apply sum_eq_sum afa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ ↑M ≥ 0; positivityAll goals completed! 🐙
      _ ≤ L' := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ T ↑M ≤ L' apply le_ciSup _ (M:ℤ)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => @_fvar.14602 (@_fvar.14605 n)haf:{ m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.nonneg := 
  fun n =>
    if h : n ≥ 0 then
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_true (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_true h))))
            ge_iff_le._simp_1))
        (Chapter7.Series.converges_of_permute_nonneg._proof_1 _fvar.14604 _fvar.14606 n h
          (@_fvar.14603 ↑(@_fvar.14605 n.toNat)))
    else
      of_eq_true
        (Eq.trans
          (Eq.trans
            (congrArg (fun x => x ≥ 0)
              (ite_cond_eq_false (@_fvar.14617 n.toNat) 0 (Eq.trans ge_iff_le._simp_1 (eq_false h))))
            ge_iff_le._simp_1)
          (le_refl._simp_1 0))S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone _fvar.19926 := Chapter7.Series.partial_of_nonneg _fvar.14603hTmono:Monotone _fvar.20097 := Chapter7.Series.partial_of_nonneg _fvar.14800L:ℝ := iSup _fvar.19926L':ℝ := iSup _fvar.20097hSBound:∃ Q, ∀ (N : ℤ), @_fvar.19926 N ≤ Q := (Chapter7.Series.converges_of_nonneg_iff _fvar.14603).mp _fvar.14604hTL:∀ (M : ℤ), @_fvar.20097 M ≤ _fvar.20577 := 
  fun M =>
    if hM : M ≥ 0 then
      let Y := Finset.Iic M.toNat;
      have hN :=
        Exists.intro ((Finset.image _fvar.14605 Y).sup id) fun m hm =>
          Finset.le_sup
            (Chapter7.Series.converges_of_permute_nonneg._proof_3 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
              _fvar.20439 _fvar.20560 _fvar.20877 M hM m hm);
      (fun N hN =>
          Trans.trans
            (Trans.trans
              (Trans.trans
                (Trans.trans
                  (Eq.mpr
                    (id
                      (congr
                        (congrArg Eq
                          (Eq.trans
                            (congrArg (fun x => x.partial M)
                              ((fun m m_1 e_m =>
                                  Eq.rec (motive := fun m_2 e_m =>
                                    ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                      { m := m, seq := seq, vanish := vanish } =
                                        { m := m_2, seq := seq_1,
                                          vanish :=
                                            Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                              (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                    (fun seq seq_1 e_seq =>
                                      Eq.rec (motive := fun seq_2 e_seq =>
                                        ∀ (vanish : ∀ n < m, seq n = 0),
                                          { m := m, seq := seq, vanish := vanish } =
                                            { m := m, seq := seq_2,
                                              vanish :=
                                                Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                  (Eq.refl m) })
                                        (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                    e_m)
                                0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                                (fun n => if 0 ≤ n then @_fvar.14602 (@_fvar.14605 n.toNat) else 0)
                                (funext fun n =>
                                  ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 (@_fvar.14605 n.toNat)))
                                    fun a => Eq.refl 0)
                                (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                            (Finset.sum_congr (Eq.refl (Finset.Icc 0 M)) fun x a =>
                              Eq.refl (if 0 ≤ x then @_fvar.14602 (@_fvar.14605 x.toNat) else 0))))
                        (Finset.sum_congr (Eq.refl Y) fun x a => Eq.refl (@_fvar.14602 (@_fvar.14605 x)))))
                    (Chapter7.Series.sum_eq_sum _fvar.14617 hM))
                  (Eq.symm
                    (Eq.mpr
                      (eq_of_heq
                        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                              (fun h =>
                                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                                  (fun e_2 h =>
                                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1))
                                      e'_3
                                      (fun h =>
                                        Eq.ndrec (motive := fun a' =>
                                          ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                      (Eq.refl a'_1) (HEq.refl e'_3))
                                  (Eq.symm h) e'_2)
                              (Eq.refl a') (HEq.refl e'_2))
                          ℝ (∑ n ∈ (_fvar.14605 '' ↑Y).toFinset, @_fvar.14602 n)
                          (∑ x ∈ Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y, @_fvar.14602 x)
                          (Finset.sum_congr
                            (of_eq_true
                              (Eq.trans
                                (congrArg (fun x => x = Finset.image (fun ⦃x₁⦄ => @_fvar.14605 x₁) Y)
                                  (Eq.trans (Set.toFinset_image _fvar.14605 ↑Y)
                                    (congrArg (Finset.image _fvar.14605) (Finset.toFinset_coe Y))))
                                (eq_self (Finset.image _fvar.14605 Y))))
                            fun x a => Eq.refl (@_fvar.14602 x))
                          (∑ m ∈ Y, @_fvar.14617 m) (∑ x ∈ Y, @_fvar.14602 (@_fvar.14605 x))
                          (Eq.refl (∑ m ∈ Y, @_fvar.14617 m))))
                      (Finset.sum_image fun ⦃x₁⦄ a ⦃x₂⦄ a => Function.Bijective.injective _fvar.14606))))
                (Finset.sum_le_sum_of_subset_of_nonneg
                  (fun ⦃a⦄ a_1 =>
                    Eq.mpr (id Finset.mem_Iic._simp_1)
                      (Exists.casesOn
                        (Eq.mp
                          (congrArg Chapter7.Series.converges
                            ((fun m m_1 e_m =>
                                Eq.rec (motive := fun m_2 e_m =>
                                  ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m_2, seq := seq_1,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                                  (fun seq seq_1 e_seq =>
                                    Eq.rec (motive := fun seq_2 e_seq =>
                                      ∀ (vanish : ∀ n < m, seq n = 0),
                                        { m := m, seq := seq, vanish := vanish } =
                                          { m := m, seq := seq_2,
                                            vanish :=
                                              Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                (Eq.refl m) })
                                      (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                  e_m)
                              0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                              (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                              (funext fun n =>
                                ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                              (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                          _fvar.14604)
                        fun w h =>
                        Exists.casesOn
                          (Eq.mp
                            (congrArg Exists
                              (funext fun Q =>
                                forall_congr fun N =>
                                  congrArg (fun x => x.partial N ≤ Q)
                                    ((fun m m_1 e_m =>
                                        Eq.rec (motive := fun m_2 e_m =>
                                          ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                            { m := m, seq := seq, vanish := vanish } =
                                              { m := m_2, seq := seq_1,
                                                vanish :=
                                                  Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                                    (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                    e_m })
                                          (fun seq seq_1 e_seq =>
                                            Eq.rec (motive := fun seq_2 e_seq =>
                                              ∀ (vanish : ∀ n < m, seq n = 0),
                                                { m := m, seq := seq, vanish := vanish } =
                                                  { m := m, seq := seq_2,
                                                    vanish :=
                                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                                        (Eq.refl m) })
                                              (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                                          e_m)
                                      0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                                      (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                                      (funext fun n =>
                                        ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a =>
                                          Eq.refl 0)
                                      (Chapter7.Series.instCoe._proof_1 _fvar.14602))))
                            _fvar.20877)
                          fun w_1 h_1 =>
                          Exists.casesOn
                            (Eq.mp
                              (Eq.trans
                                (Eq.trans
                                  (congrArg (fun x => a ∈ x)
                                    (Eq.trans
                                      (Eq.trans
                                        (Set.toFinset_congr (congrArg (Set.image _fvar.14605) (Finset.coe_Iic M.toNat)))
                                        (Set.toFinset_image _fvar.14605 (Set.Iic M.toNat)))
                                      (congrArg (Finset.image _fvar.14605) (Set.toFinset_Iic M.toNat))))
                                  Finset.mem_image._simp_1)
                                (congrArg Exists
                                  (funext fun a_2 =>
                                    congrArg (fun x => x ∧ @_fvar.14605 a_2 = a)
                                      (Eq.trans Finset.mem_Iic._simp_1
                                        (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM)))))))
                              a_1)
                            fun w_2 h_2 =>
                            And.casesOn h_2 fun left right =>
                              right ▸
                                of_eq_true
                                  ((fun m a =>
                                      eq_true
                                        (Eq.mp
                                          (forall_congr fun m =>
                                            implies_congr
                                              (Eq.trans Finset.mem_Iic._simp_1
                                                (Int.le_toNat._simp_1 (id (Eq.mp ge_iff_le._simp_1 hM))))
                                              (Eq.refl (@_fvar.14605 m ≤ N)))
                                          hN m a))
                                    w_2 left)))
                  fun i a a =>
                  id
                    (Eq.mp
                      (Eq.trans
                        (congrArg (fun x => x ≥ 0)
                          (ite_cond_eq_true (@_fvar.14602 (↑i).toNat) 0
                            (Eq.trans ge_iff_le._simp_1 (Nat.cast_nonneg._simp_1 i))))
                        ge_iff_le._simp_1)
                      (@_fvar.14603 ↑i))))
              (Eq.mpr
                (id
                  (congrArg (Eq (∑ n ∈ Finset.Iic N, @_fvar.14602 n))
                    (Eq.trans
                      (congrArg (fun x => x.partial ↑N)
                        ((fun m m_1 e_m =>
                            Eq.rec (motive := fun m_2 e_m =>
                              ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                                { m := m, seq := seq, vanish := vanish } =
                                  { m := m_2, seq := seq_1,
                                    vanish :=
                                      Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                        (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                              (fun seq seq_1 e_seq =>
                                Eq.rec (motive := fun seq_2 e_seq =>
                                  ∀ (vanish : ∀ n < m, seq n = 0),
                                    { m := m, seq := seq, vanish := vanish } =
                                      { m := m, seq := seq_2,
                                        vanish :=
                                          Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                            (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq)
                                            (Eq.refl m) })
                                  (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                              e_m)
                          0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14602 n.toNat else 0)
                          (fun n => if 0 ≤ n then @_fvar.14602 n.toNat else 0)
                          (funext fun n =>
                            ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14602 n.toNat)) fun a => Eq.refl 0)
                          (Chapter7.Series.instCoe._proof_1 _fvar.14602)))
                      (Finset.sum_congr (Eq.refl (Finset.Icc 0 ↑N)) fun x a =>
                        Eq.refl (if 0 ≤ x then @_fvar.14602 x.toNat else 0)))))
                (Eq.symm (Chapter7.Series.sum_eq_sum _fvar.14602 (Nat.cast_nonneg' N)))))
            (le_ciSup
              (of_eq_true
                (Eq.trans
                  (congrArg Exists
                    (funext fun x =>
                      congrArg (fun x_1 => x ∈ setOf x_1)
                        (funext fun x =>
                          Eq.trans
                            (forall_congr fun ⦃a⦄ =>
                              Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x)))
                                forall_exists_index._simp_1)
                            forall_apply_eq_imp_iff._simp_1)))
                  (eq_true _fvar.20877)))
              ↑N))
        (Classical.choose hN) (Classical.choose_spec hN)
    else
      have hM' :=
        lt_of_not_ge fun a =>
          Mathlib.Tactic.Linarith.lt_irrefl
            (Eq.mp
              (congrArg (fun _a => _a < 0)
                (Mathlib.Tactic.Ring.of_eq
                  (Mathlib.Tactic.Ring.add_congr
                    (Mathlib.Tactic.Ring.add_congr
                      (Mathlib.Tactic.Ring.neg_congr
                        (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1)))))
                          Mathlib.Tactic.Ring.neg_zero))
                      (Mathlib.Tactic.Ring.sub_congr
                        (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf M)
                          (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                          (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.add_pf_add_zero (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                        (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                        (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                          (Mathlib.Tactic.Ring.add_pf_add_zero
                            (Nat.rawCast 1 + (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                      (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.ofNat 0))))
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                    (Mathlib.Tactic.Ring.sub_congr
                      (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                      (Mathlib.Tactic.Ring.atom_pf M)
                      (Mathlib.Tactic.Ring.sub_pf
                        (Mathlib.Tactic.Ring.neg_add
                          (Mathlib.Tactic.Ring.neg_mul M (Nat.rawCast 1)
                            (Mathlib.Tactic.Ring.neg_one_mul
                              (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                                (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                  (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                  (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                  (Eq.refl (Int.negOfNat 1))))))
                          Mathlib.Tactic.Ring.neg_zero)
                        (Mathlib.Tactic.Ring.add_pf_zero_add (M ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero M (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0)))
                  (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
              (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
                (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
                  (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (lt_of_not_ge hM))))
                (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)));
      Eq.mpr
        (id
          (Eq.trans
            (congrArg (fun x => x ≤ _fvar.20577)
              (Eq.trans
                (congrArg (fun x => x.partial M)
                  ((fun m m_1 e_m =>
                      Eq.rec (motive := fun m_2 e_m =>
                        ∀ (seq seq_1 : ℤ → ℝ) (e_seq : seq = seq_1) (vanish : ∀ n < m, seq n = 0),
                          { m := m, seq := seq, vanish := vanish } =
                            { m := m_2, seq := seq_1,
                              vanish :=
                                Eq.ndrec (motive := fun m => ∀ n < m, seq_1 n = 0)
                                  (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) e_m })
                        (fun seq seq_1 e_seq =>
                          Eq.rec (motive := fun seq_2 e_seq =>
                            ∀ (vanish : ∀ n < m, seq n = 0),
                              { m := m, seq := seq, vanish := vanish } =
                                { m := m, seq := seq_2,
                                  vanish :=
                                    Eq.ndrec (motive := fun m => ∀ n < m, seq_2 n = 0)
                                      (Eq.ndrec (motive := fun seq => ∀ n < m, seq n = 0) vanish e_seq) (Eq.refl m) })
                            (fun vanish => Eq.refl { m := m, seq := seq, vanish := vanish }) e_seq)
                        e_m)
                    0 0 (Eq.refl 0) (fun n => if n ≥ 0 then @_fvar.14617 n.toNat else 0)
                    (fun n => if 0 ≤ n then @_fvar.14617 n.toNat else 0)
                    (funext fun n =>
                      ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.14617 n.toNat)) fun a => Eq.refl 0)
                    (Chapter7.Series.instCoe._proof_1 _fvar.14617)))
                (Finset.sum_congr (Finset.Icc_eq_empty_of_lt hM') fun x a =>
                  Eq.refl (if 0 ≤ x then @_fvar.14617 x.toNat else 0))))
            ge_iff_le._simp_1))
        (Eq.mpr
          (eq_of_heq
            ((fun α self self' e'_2 a a' e'_3 a_1 =>
                Eq.casesOn (motive := fun a_2 x => self' = a_2 → e'_2 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_2
                  (fun h =>
                    Eq.ndrec (motive := fun self' => ∀ (e_2 : self = self'), e_2 ≍ Eq.refl self → (a ≤ a_1) ≍ (a' ≤ a_1))
                      (fun e_2 h =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                              (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
                          (Eq.refl a') (HEq.refl e'_3))
                      (Eq.symm h) e'_2)
                  (Eq.refl self') (HEq.refl e'_2))
              ℝ Real.instLE Real.instConditionallyCompleteLinearOrder.toSemilatticeInf.toLE (Eq.refl Real.instLE) 0
              (@_fvar.19926 (-1)) (Eq.refl 0) _fvar.20577))
          (le_ciSup
            (of_eq_true
              (Eq.trans
                (congrArg Exists
                  (funext fun x =>
                    congrArg (fun x_1 => x ∈ setOf x_1)
                      (funext fun x =>
                        Eq.trans
                          (forall_congr fun ⦃a⦄ =>
                            Eq.trans (implies_congr Set.mem_range._simp_1 (Eq.refl (a ≤ x))) forall_exists_index._simp_1)
                          forall_apply_eq_imp_iff._simp_1)))
                (eq_true _fvar.20877)))
            (-1)))hTbound:∃ Q, ∀ (M : ℤ), @_fvar.20097 M ≤ Q := Exists.intro _fvar.20577 _fvar.247770N:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic (Int.toNat _fvar.469256)M:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := 
  fun b {N} hN =>
    Eq.mpr
      (eq_of_heq
        ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                  (fun e_2 h =>
                    Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                      (fun h =>
                        Eq.ndrec (motive := fun a' => ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                          (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                      (Eq.refl a'_1) (HEq.refl e'_3))
                  (Eq.symm h) e'_2)
              (Eq.refl a') (HEq.refl e'_2))
          ℝ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0)
          (∑ x ∈ Finset.image Int.ofNat (Finset.Iic N.toNat), if 0 ≤ x then b x.toNat else 0)
          (Finset.sum_congr
            (Finset.ext fun x =>
              Eq.mpr
                (id
                  (congr (congrArg Iff Finset.mem_Icc._simp_1)
                    (Eq.trans Finset.mem_image._simp_1
                      (congrArg Exists (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Iic._simp_1)))))
                {
                  mp := fun h =>
                    match h with
                    | ⟨left, right⟩ =>
                      Exists.intro x.toNat
                        (Decidable.byContradiction fun a =>
                          Chapter7.Series.converges_of_permute_nonneg._proof_5 _fvar.469256 hN x left right a),
                  mpr :=
                    Chapter7.Series.converges_of_permute_nonneg._proof_6 _fvar.14603 _fvar.14604 _fvar.14606 _fvar.14800
                      _fvar.20439 _fvar.20560 _fvar.20877 _fvar.247770 _fvar.468509 _fvar.469256 _fvar.469329 _fvar.482509
                      _fvar.482513 b hN x })
            fun x a => Eq.refl (if 0 ≤ x then b x.toNat else 0))
          (∑ n ∈ Finset.Iic N.toNat, b n) (∑ x ∈ Finset.Iic N.toNat, if 0 ≤ Int.ofNat x then b (Int.ofNat x).toNat else 0)
          (Finset.sum_congr (Eq.refl (Finset.Iic N.toNat)) fun x a => Eq.refl (b x))))
      (Finset.sum_image
        (of_eq_true
          (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Iic N.toNat))
            (of_eq_true
              (Eq.trans
                (forall_congr fun x =>
                  Eq.trans
                    (forall_congr fun y =>
                      Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                    (implies_true ℕ))
                (implies_true ℕ))))))⊢ BddAbove (Set.range T); simp [BddAbove, Set.Nonempty, upperBounds, hTbound]All goals completed! 🐙
  linarith [ciSup_le hSL', ciSup_le hTL]All goals completed! 🐙

Example 7.4.2

theorem declaration uses 'sorry'Series.zeta_2_converges : (fun n:ℕ ↦ 1/(n+1:ℝ)^2 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => 1 / (↑n + 1) ^ 2) n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.permuted_zeta_2_converges :
  (fun n:ℕ ↦ if Even n then 1/(n+2:ℝ)^2 else 1/(n:ℝ)^2 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => if Even n then 1 / (↑n + 2) ^ 2 else 1 / ↑n ^ 2) n.toNat else 0,
    vanish := ⋯ }.converges
    sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.permuted_zeta_2_eq_zeta_2 :
  (fun n:ℕ ↦ if Even n then 1/(n+2:ℝ)^2 else 1/(n:ℝ)^2 : Series).sum = (fun n:ℕ ↦ 1/(n+1:ℝ)^2 : Series).sum := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => if Even n then 1 / (↑n + 2) ^ 2 else 1 / ↑n ^ 2) n.toNat else 0,
      vanish := ⋯ }.sum =
  { m := 0, seq := fun n => if n ≥ 0 then (fun n => 1 / (↑n + 1) ^ 2) n.toNat else 0, vanish := ⋯ }.sum
    sorryAll goals completed! 🐙

Proposition 7.4.3 (Rearrangement of series)

theorem declaration uses 'sorry'Series.absConverges_of_permute {a:ℕ → ℝ} (ha : (a:Series).absConverges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n):Series).absConverges  ∧ (a:Series).sum = (fun n ↦ a (f n) : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  -- This proof is written to follow the structure of the original text.
  set L := (a:Series).abs.suma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  have hconv := converges_of_absConverges haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  unfold absConverges at haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  have habs : (fun n ↦ |a (f n)| : Series).converges ∧ L = (fun n ↦ |a (f n)| : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    convert converges_of_permute_nonneg (a := fun n ↦ |a n|) _ _ hf using 3h.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs =
  { m := 0, seq := fun n => if n ≥ 0 then |a n.toNat| else 0, vanish := ⋯ }convert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.nonnegconvert_2a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.converges
    .h.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs =
  { m := 0, seq := fun n => if n ≥ 0 then |a n.toNat| else 0, vanish := ⋯ } simph.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ (fun n => if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = fun n => if 0 ≤ n then |a n.toNat| else 0; ext nh.e'_2.h.e'_2.h.e'_1.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤ⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0; by_cases n ≥ 0pos._@._hyg.4113a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0neg._@._hyg.4113a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:¬n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0 <;>pos._@._hyg.4113a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0neg._@._hyg.4113a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:¬n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0 grindAll goals completed! 🐙
    .convert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.nonneg intro nconvert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤ⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0; by_cases h: n ≥ 0pos._@._hyg.4162a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0neg._@._hyg.4162a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 <;>pos._@._hyg.4162a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0neg._@._hyg.4162a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 simp [h]All goals completed! 🐙
    convert ha with nh.e'_1.h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤ⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0; by_cases n ≥ 0pos._@._hyg.4290a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0neg._@._hyg.4290a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:¬n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0 <;>pos._@._hyg.4290a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0neg._@._hyg.4290a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629n:ℤh✝:¬n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0 grindAll goals completed! 🐙
  set L' := (a:Series).suma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  set af : ℕ → ℝ := fun n ↦ a (f n)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  suffices: (af:Series).convergesTo L'a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sumthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'
  .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum simp [sum_of_converges this, absConverges]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.abs.converges
    convert habs.1 with nh.e'_1.h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤ⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0; by_cases n ≥ 0pos._@._hyg.4726a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh✝:n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0neg._@._hyg.4726a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh✝:¬n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0 <;>pos._@._hyg.4726a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh✝:n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0neg._@._hyg.4726a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh✝:¬n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0 grindAll goals completed! 🐙
  simp [convergesTo, LinearOrderedAddCommGroup.tendsto_nhds]thisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)⊢ ∀ (ε : ℝ),
  0 < ε →
    ∃ a,
      ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  intro ε hεthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:?_mvar.846824 := Chapter7.Series.converges_of_absConverges _fvar.846629habs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
      vanish := ⋯ }.converges ∧
  _fvar.846722 =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => |@_fvar.846628 (@_fvar.846630 n)|) n.toNat else 0,
        vanish := ⋯ }.sum := 
  ?_mvar.846963L':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.846630 n)ε:ℝhε:0 < ε⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  rw [converges_iff_tail_decay] at hathisa:ℕ → ℝha:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < ε⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  choose N₁ hN₁ ha using ha _ (half_pos hε)thisa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤhN₁:N₁ ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.mha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε; simp at hN₁thisa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have : ∃ N ≥ N₁, |(a:Series).partial N - L'| < ε/2 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    apply convergesTo_sum at hconva:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    simp [convergesTo, LinearOrderedAddCommGroup.tendsto_nhds] at hconva:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∀ (ε : ℝ),
  0 < ε →
    ∃ a_2,
      ∀ (b : ℤ),
        a_2 ≤ b →
          |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
                { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
            ε⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    choose N hN using hconv _ (half_pos hε)a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∀ (ε : ℝ),
  0 < ε →
    ∃ a_2,
      ∀ (b : ℤ),
        a_2 ≤ b →
          |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
                { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
            εN:ℤhN:∀ (b : ℤ),
  N ≤ b →
    |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
          { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
      ε / 2⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    use max N N₁, (bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∀ (ε : ℝ),
  0 < ε →
    ∃ a_2,
      ∀ (b : ℤ),
        a_2 ≤ b →
          |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
                { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
            εN:ℤhN:∀ (b : ℤ),
  N ≤ b →
    |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
          { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
      ε / 2⊢ max N N₁ ≥ N₁ grindAll goals completed! 🐙); apply hNright.aa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∀ (ε : ℝ),
  0 < ε →
    ∃ a_2,
      ∀ (b : ℤ),
        a_2 ≤ b →
          |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
                { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
            εN:ℤhN:∀ (b : ℤ),
  N ≤ b →
    |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
          { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
      ε / 2⊢ N ≤ max N N₁; grindAll goals completed! 🐙
  choose N hN hN2 using thisthisa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have hNpos : N ≥ 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
  let finv : ℕ → ℕ := Function.invFun fthisa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have : ∃ M, ∀ n ≤ N.toNat, finv n ≤ M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    use ((Finset.Iic (N.toNat)).image finv).sup idha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228⊢ ∀ n ≤ N.toNat, finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    intro n hnha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228n:ℕhn:n ≤ N.toNat⊢ finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    apply Finset.le_sup (f := id)ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228n:ℕhn:n ≤ N.toNat⊢ (if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ) ∈ Finset.image finv (Finset.Iic N.toNat); simp [Finset.mem_image]ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228n:ℕhn:n ≤ N.toNat⊢ ∃ a ≤ N.toNat, finv a = if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ; use n, hnrighta:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228n:ℕhn:n ≤ N.toNat⊢ finv n = if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ; rflAll goals completed! 🐙
  choose M hM using thisthisa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ M⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε; use Mha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ M⊢ ∀ (b : ℤ), ↑M ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε; intro M' hM'ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have hM'_pos : M' ≥ 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
  have why : (Finset.Iic M'.toNat).image f ⊇ .Iic N.toNat := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    sorryAll goals completed! 🐙
  set X : Finset ℕ := (Finset.Iic M'.toNat).image f \ .Iic N.toNatha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := ?_mvar.1194057why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := ?_mvar.1196288X:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have claim : ∑ m ∈ .Iic M'.toNat, a (f m) = ∑ n ∈ .Iic N.toNat, a n + ∑ n ∈ X, a n := calc
    _ = ∑ n ∈ (Finset.Iic M'.toNat).image f , a n := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ ∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n
      symma:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n = ∑ m ∈ Finset.Iic M'.toNat, a (f m); apply Finset.sum_imagea._@.Mathlib.Algebra.BigOperators.Group.Finset.Basic._hyg.692a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ Set.InjOn f ↑(Finset.Iic M'.toNat); solve_by_elim [hf.1]All goals completed! 🐙
    _ = _ := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a n
      convert Finset.sum_union _ using 2h.e'_2.ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ Finset.image f (Finset.Iic M'.toNat) = Finset.Iic N.toNat ∪ Xconvert_7a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ DecidableEq ℕconvert_8a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ Disjoint (Finset.Iic N.toNat) X
      .h.e'_2.ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ Finset.image f (Finset.Iic M'.toNat) = Finset.Iic N.toNat ∪ X simp [X, why]All goals completed! 🐙
      .convert_7a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ DecidableEq ℕ infer_instanceAll goals completed! 🐙
      rw [Finset.disjoint_right]convert_8a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)⊢ ∀ ⦃a : ℕ⦄, a ∈ X → a ∉ Finset.Iic N.toNat; intro n hnconvert_8a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)n:ℕhn:n ∈ X⊢ n ∉ Finset.Iic N.toNat; simp only [X, Finset.mem_sdiff] at hnconvert_8a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)n:ℕhn:n ∈ Finset.image f (Finset.Iic M'.toNat) ∧ n ∉ Finset.Iic N.toNat⊢ n ∉ Finset.Iic N.toNat; tautoAll goals completed! 🐙
  choose q' hq using X.bddAboveha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := ?_mvar.1194057why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := ?_mvar.1196288X:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  Trans.trans ?_mvar.1197196 ?_mvar.1197206q':ℕhq:q' ∈ upperBounds ↑X⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  set q := max q' N.toNatha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := ?_mvar.1184633finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := ?_mvar.1194057why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := ?_mvar.1196288X:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  Trans.trans ?_mvar.1197196 ?_mvar.1197206q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have why2 : X ⊆ Finset.Icc (N.toNat+1) q := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum sorryAll goals completed! 🐙
  have claim2 : |∑ n ∈ X, a n| ≤ ε/2 := calc
    _ ≤ ∑ n ∈ X, |a n| := X.abs_sum_le_sum_abs a
    _ ≤ ∑ n ∈ .Icc (N.toNat+1) q, |a n| := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ X, |a n| ≤ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n|
      apply Finset.sum_le_sum_of_subset_of_nonneg why2a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∀ i ∈ Finset.Icc (N.toNat + 1) q, i ∉ X → 0 ≤ |a i|; simpAll goals completed! 🐙
    _ ≤ ε/2 := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| ≤ ε / 2
      convert ha (N.toNat+1) _ q _h.e'_3a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  |∑ n ∈ Finset.Icc (↑N.toNat + 1) ↑q,
      { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n|convert_1a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ↑N.toNat + 1 ≥ N₁convert_2a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ↑q ≥ N₁ <;>h.e'_3a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  |∑ n ∈ Finset.Icc (↑N.toNat + 1) ↑q,
      { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n|convert_1a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ↑N.toNat + 1 ≥ N₁convert_2a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ↑q ≥ N₁ try omegaAll goals completed! 🐙
      simp [hNpos]h.e'_3a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  |∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0|; rw [abs_of_nonneg (by positivity)]h.e'_3a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  ∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0; symmh.e'_3a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ (∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0) =
  ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n|
      convert Finset.sum_image (g := fun (n:ℕ) ↦ (n:ℤ)) (bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorry⊢ Set.InjOn (fun n => ↑n) ↑?m.2320 simpAll goals completed! 🐙) using 2
      ext xh.e'_2.h.ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤ⊢ x ∈ Finset.Icc (N + 1) ↑q ↔ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (N.toNat + 1) q); simph.e'_2.h.ha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q ↔ ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x; constructorh.e'_2.h.h.mpa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q → ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = xh.e'_2.h.h.mpra:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤ⊢ (∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x) → N + 1 ≤ x ∧ x ≤ ↑q
      .h.e'_2.h.h.mpa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q → ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x intro ⟨ _, _ ⟩h.e'_2.h.h.mpa:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤleft✝:N + 1 ≤ xright✝:x ≤ ↑q⊢ ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x; use x.toNatha:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryx:ℤleft✝:N + 1 ≤ xright✝:x ≤ ↑q⊢ (N.toNat + 1 ≤ x.toNat ∧ x.toNat ≤ q) ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll goals completed! 🐙
  calc
    _ ≤ |(af:Series).partial M' - (a:Series).partial N| + |(a:Series).partial N - L'| := abs_sub_le _ _ _
    _ < |(af:Series).partial M' - (a:Series).partial N| + ε/2 := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| <
  |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    ε / 2 gcongrAll goals completed! 🐙
    _ ≤ ε/2 + ε/2 := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    ε / 2 ≤
  ε / 2 + ε / 2
      gcongrbca:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
      { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| ≤
  ε / 2; convert claim2h.e'_3.h.e'_4a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
    { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N =
  ∑ n ∈ X, a n
      simp [Series.partial, sum_eq_sum _ hM'_pos, sum_eq_sum _ hNpos]h.e'_3.h.e'_4a:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ ∑ n ∈ Finset.Iic M'.toNat, af n - ∑ n ∈ Finset.Iic N.toNat, a n = ∑ n ∈ X, a n; grindAll goals completed! 🐙
    _ = ε := bya:ℕ → ℝha✝:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => @_fvar.846628 (@_fvar.1078228 n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:_fvar.1184577 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul _fvar.1078345 (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.1078345 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1078345)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1184577 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (_fvar.1078345 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1078345 (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1184577 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1184577)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1184577 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1184577 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1079286))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1184586))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))finv:ℕ → ℕ := Function.invFun _fvar.1078228M:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:_fvar.1194012 ≥ 0 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.add_congr
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.neg_congr
                    (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                            (Eq.refl (Int.negOfNat 1)))))
                      Mathlib.Tactic.Ring.neg_zero))
                  (Mathlib.Tactic.Ring.sub_congr
                    (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                    (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (↑_fvar.1193903) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (↑_fvar.1193903 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))
                (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑_fvar.1193903)
                  (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.sub_pf
                    (Mathlib.Tactic.Ring.neg_add
                      (Mathlib.Tactic.Ring.neg_mul _fvar.1194012 (Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.neg_one_mul
                          (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                            (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                              (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                              (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                              (Eq.refl (Int.negOfNat 1))))))
                      Mathlib.Tactic.Ring.neg_zero)
                    (Mathlib.Tactic.Ring.add_pf_add_lt (↑_fvar.1193903 ^ Nat.rawCast 1 * Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.add_pf_zero_add
                        (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑_fvar.1193903) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                          (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_zero_add
                      (_fvar.1194012 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.1194012)
                  (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)))
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))
                (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero
                  (Mathlib.Tactic.Ring.add_pf_add_zero
                    (Nat.rawCast 1 + (_fvar.1194012 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                    (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.1194012 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))
        (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
          (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le
            (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one)
              (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ _fvar.1193903)))
            (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.1194015))
          (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr a))))why:Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) ⊇ Finset.Iic (Int.toNat _fvar.1184577) := sorryX:Finset ℕ := Finset.image _fvar.1078228 (Finset.Iic (Int.toNat _fvar.1194012)) \ Finset.Iic (Int.toNat _fvar.1184577)claim:∑ m ∈ Finset.Iic (Int.toNat _fvar.1194012), @_fvar.846628 (@_fvar.1078228 m) =
  ∑ n ∈ Finset.Iic (Int.toNat _fvar.1184577), @_fvar.846628 n + ∑ n ∈ _fvar.1196544, @_fvar.846628 n := 
  failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)q':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max _fvar.1206066 (Int.toNat _fvar.1184577)why2:_fvar.1196544 ⊆ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081 := sorryclaim2:|∑ n ∈ _fvar.1196544, @_fvar.846628 n| ≤ _fvar.1078235 / 2 := 
  Trans.trans
    (Trans.trans (Finset.abs_sum_le_sum_abs _fvar.846628 _fvar.1196544)
      (Finset.sum_le_sum_of_subset_of_nonneg _fvar.1206561
        (of_eq_true
          (Eq.trans
            (forall_congr fun i =>
              Eq.trans
                (implies_congr Finset.mem_Icc._simp_1
                  (Eq.trans (implies_congr (Eq.refl (i ∉ _fvar.1196544)) (one_le_mabs._simp_4 (@_fvar.846628 i)))
                    (implies_true (i ∉ _fvar.1196544))))
                (implies_true (Int.toNat _fvar.1184577 + 1 ≤ i ∧ i ≤ _fvar.1206081)))
            (implies_true ℕ)))))
    (Eq.mpr
      (eq_of_heq
        ((fun α self a a' e'_3 a_1 =>
            Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_3 ≍ x → (a ≤ a_1) ≍ (a' ≤ a_1)) e'_3
              (fun h =>
                Eq.ndrec (motive := fun a' => ∀ (e_3 : a = a'), e_3 ≍ Eq.refl a → (a ≤ a_1) ≍ (a' ≤ a_1))
                  (fun e_3 h => HEq.refl (a ≤ a_1)) (Eq.symm h) e'_3)
              (Eq.refl a') (HEq.refl e'_3))
          ℝ Real.instLE (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
          |∑ n ∈ Finset.Icc (↑(Int.toNat _fvar.1184577) + 1) ↑_fvar.1206081,
              { m := 0, seq := fun n => if n ≥ 0 then @_fvar.846628 n.toNat else 0,
                      vanish := Chapter7.Series.instCoe._proof_1 _fvar.846628 }.abs.seq
                n|
          (Eq.mpr
            (id
              (congrArg (fun x => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = |x|)
                (Finset.sum_congr
                  (congrArg (fun x => Finset.Icc (x + 1) ↑_fvar.1206081)
                    (Eq.trans (Int.ofNat_toNat _fvar.1184577) (sup_of_le_left _fvar.1184634)))
                  fun x a =>
                  dite_congr ge_iff_le._simp_1
                    (fun h =>
                      congrArg abs
                        (ite_congr ge_iff_le._simp_1 (fun a => Eq.refl (@_fvar.846628 x.toNat)) fun a => Eq.refl 0))
                    fun h => Eq.refl 0)))
            (Eq.mpr
              (id
                (congrArg (fun _a => ∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n| = _a)
                  (abs_of_nonneg
                    (Finset.sum_nonneg fun i x =>
                      Mathlib.Meta.Positivity.ite_nonneg✝ (0 ≤ i)
                        (IsAbsoluteValue.abv_nonneg abs (if 0 ≤ i then @_fvar.846628 i.toNat else 0))
                        (Mathlib.Meta.Positivity.nonneg_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))))
              (Eq.symm
                (Eq.mpr
                  (eq_of_heq
                    ((fun α a a' e'_2 a_1 a'_1 e'_3 =>
                        Eq.casesOn (motive := fun a_2 x => a' = a_2 → e'_2 ≍ x → (a = a_1) ≍ (a' = a'_1)) e'_2
                          (fun h =>
                            Eq.ndrec (motive := fun a' => ∀ (e_2 : a = a'), e_2 ≍ Eq.refl a → (a = a_1) ≍ (a' = a'_1))
                              (fun e_2 h =>
                                Eq.casesOn (motive := fun a_2 x => a'_1 = a_2 → e'_3 ≍ x → (a = a_1) ≍ (a = a'_1)) e'_3
                                  (fun h =>
                                    Eq.ndrec (motive := fun a' =>
                                      ∀ (e_3 : a_1 = a'), e_3 ≍ Eq.refl a_1 → (a = a_1) ≍ (a = a'))
                                      (fun e_3 h => HEq.refl (a = a_1)) (Eq.symm h) e'_3)
                                  (Eq.refl a'_1) (HEq.refl e'_3))
                              (Eq.symm h) e'_2)
                          (Eq.refl a') (HEq.refl e'_2))
                      ℝ
                      (∑ x ∈ Finset.Icc (_fvar.1184577 + 1) ↑_fvar.1206081,
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (∑ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081),
                        if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0)
                      (Finset.sum_congr
                        (Finset.ext fun x =>
                          Eq.mpr
                            (id
                              (congr (congrArg Iff Finset.mem_Icc._simp_1)
                                (Eq.trans Finset.mem_image._simp_1
                                  (congrArg Exists
                                    (funext fun a => congrArg (fun x_1 => x_1 ∧ ↑a = x) Finset.mem_Icc._simp_1)))))
                            {
                              mp := fun h =>
                                match h with
                                | ⟨left, right⟩ =>
                                  Exists.intro x.toNat
                                    (Decidable.byContradiction fun a =>
                                      Chapter7.Series.absConverges_of_permute._proof_18 _fvar.1078345 _fvar.1184577
                                        _fvar.1184634 _fvar.1193903 _fvar.1194012 _fvar.1206066 x left right a),
                              mpr :=
                                Chapter7.Series.absConverges_of_permute._proof_19 _fvar.1078225 _fvar.1078229
                                  _fvar.1078231 _fvar.1078232 _fvar.1078235 _fvar.1078236 _fvar.1078345 _fvar.1078357
                                  _fvar.1079286 _fvar.1184577 _fvar.1184586 _fvar.1184590 _fvar.1184634 _fvar.1193903
                                  _fvar.1193907 _fvar.1194012 _fvar.1194015 _fvar.1194058 _fvar.1196289 _fvar.1197274
                                  _fvar.1206066 _fvar.1206069 _fvar.1206561 x })
                        fun x a => Eq.refl (if 0 ≤ x then |if 0 ≤ x then @_fvar.846628 x.toNat else 0| else 0))
                      (∑ n ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081, |@_fvar.846628 n|)
                      (∑ x ∈ Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081,
                        if 0 ≤ ↑x then |if 0 ≤ ↑x then @_fvar.846628 (↑x).toNat else 0| else 0)
                      (Finset.sum_congr (Eq.refl (Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081)) fun x a =>
                        Eq.refl |@_fvar.846628 x|)))
                  (Finset.sum_image
                    (of_eq_true
                      (Set.injOn_of_eq_iff_eq._simp_1 (↑(Finset.Icc (Int.toNat _fvar.1184577 + 1) _fvar.1206081))
                        (of_eq_true
                          (Eq.trans
                            (forall_congr fun x =>
                              Eq.trans
                                (forall_congr fun y =>
                                  Eq.trans (congrArg (fun x_1 => x_1 ↔ x = y) Nat.cast_inj._simp_1) (iff_self (x = y)))
                                (implies_true ℕ))
                            (implies_true ℕ))))))))))
          (_fvar.1078235 / 2)))
      (@_fvar.1078357 (↑(Int.toNat _fvar.1184577) + 1)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_16 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 a)
        (↑_fvar.1206081)
        (Decidable.byContradiction fun a =>
          Chapter7.Series.absConverges_of_permute._proof_17 _fvar.1078345 _fvar.1184577 _fvar.1184586 _fvar.1184634
            _fvar.1193903 _fvar.1194012 _fvar.1206066 a)))⊢ ε / 2 + ε / 2 = ε ringAll goals completed! 🐙

Example 7.4.4

noncomputable abbrev Series.a_7_4_4 : ℕ → ℝ := fun n ↦ (-1:ℝ)^n / (n+2)

theorem declaration uses 'sorry'Series.ex_7_4_4_conv : (a_7_4_4 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then a_7_4_4 n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4_sum : (a_7_4_4 : Series).sum > 0 := by⊢ { m := 0, seq := fun n => if n ≥ 0 then a_7_4_4 n.toNat else 0, vanish := ⋯ }.sum > 0 sorryAll goals completed! 🐙

abbrev Series.f_7_4_4 : ℕ → ℕ := fun n ↦ if n % 3 = 0 then 2 * (n/3) else 2*n - (n/3) - 1

theorem declaration uses 'sorry'Series.f_7_4_4_bij : Function.Bijective f_7_4_4 := by⊢ Function.Bijective f_7_4_4 sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4'_conv : (fun n ↦ a_7_4_4 (f_7_4_4 n) :Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a_7_4_4 (f_7_4_4 n)) n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4'_sum : (fun n ↦ a_7_4_4 (f_7_4_4 n) :Series).sum < 0 := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a_7_4_4 (f_7_4_4 n)) n.toNat else 0, vanish := ⋯ }.sum < 0 sorryAll goals completed! 🐙

Exercise 7.4.1

theorem declaration uses 'sorry'Series.absConverges_of_subseries {a:ℕ → ℝ} (ha: (a:Series).absConverges) {f: ℕ → ℕ} (hf: StrictMono f) :
  (fun n ↦ a (f n):Series).absConverges := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:StrictMono f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges sorryAll goals completed! 🐙

Exercise 7.4.2 : reprove Proposition 7.4.3 using Proposition 7.41, Proposition 7.2.14, and expressing Unknown identifier `a`a n as the difference of Unknown identifier `a`sorry + |sorry| : ?m.7a n + |Unknown identifier `a`a n| and |sorry| : ?m.1|Unknown identifier `a`a n|.

theorem declaration uses 'sorry'Series.absConverges_of_permute' {a:ℕ → ℝ} (ha : (a:Series).absConverges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n):Series).absConverges  ∧ (a:Series).sum = (fun n ↦ a (f n):Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum sorryAll goals completed! 🐙

end Chapter7