Analysis I, Section 8.3: Uncountable sets

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:

• Uncountable sets.

Some non-trivial API is provided beyond what is given in the textbook in order connect these notions with existing summation notions.

namespace Chapter8 theorem Uncountable.iff (X:Type) : Uncountable X ↔ ¬ AtMostCountable X := byX:Type⊢ Uncountable X ↔ ¬AtMostCountable X rw [AtMostCountable.iff, uncountable_iff_not_countable]All goals completed! 🐙theorem Uncountable.equiv {X Y: Type} (hXY : EqualCard X Y) : Uncountable X ↔ Uncountable Y := byX:TypeY:TypehXY:EqualCard X Y⊢ Uncountable X ↔ Uncountable Y simp [Uncountable.iff, AtMostCountable.equiv hXY]All goals completed! 🐙 open Real in /-- Corollary 8.3.4 -/ theorem Uncountable.real : Uncountable ℝ := by⊢ Uncountable ℝ -- This proof is written to follow the structure of the original text. set a : ℕ → ℝ := fun n ↦ (10:ℝ)^(-(n:ℝ))a:ℕ → ℝ := fun n => 10 ^ (-↑n)⊢ Uncountable ℝ set f : Set ℕ → ℝ := fun A ↦ ∑' n:A, a na:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑n⊢ Uncountable ℝ have hsummable (A: Set ℕ) : Summable (fun n:A ↦ a n) := by⊢ Uncountable ℝ apply Summable.subtype (f := a)hfa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕ⊢ Summable a convert summable_geometric_of_lt_one (?_:0 ≤ (1/10:ℝ)) ?_ using 2 with nh.e'_5.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ a n = (1 / 10) ^ nhf.convert_1a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕ⊢ 0 ≤ 1 / 10hf.convert_2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕ⊢ 1 / 10 < 1 <;>h.e'_5.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ a n = (1 / 10) ^ nhf.convert_1a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕ⊢ 0 ≤ 1 / 10hf.convert_2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕ⊢ 1 / 10 < 1 try norm_numAll goals completed! 🐙 unfold ah.e'_5.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ 10 ^ (-↑n) = (1 / 10) ^ n rw [one_div_pow, rpow_neg, one_div]h.e'_5.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ (10 ^ ↑n)⁻¹ = (10 ^ n)⁻¹h.e'_5.h.hxa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ 0 ≤ 10; simph.e'_5.h.hxa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nA:Set ℕn:ℕ⊢ 0 ≤ 10; norm_numAll goals completed! 🐙 have h_decomp {A B C: Set ℕ} (hC : C = A ∪ B) (hAB: ∀ n, n ∉ A ∩ B) : ∑' n:C, a n = ∑' n:A, a n + ∑' n:B, a n := by⊢ Uncountable ℝ convert Summable.tsum_union_disjoint ?_ ?_ ?_h.e'_2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ ∑' (n : ↑C), a ↑n = ∑' (x : ↑(A ∪ B)), a ↑xconvert_6a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ T2Space ℝconvert_7a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ ContinuousAdd ℝconvert_10a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Disjoint A Bconvert_11a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Summable (a ∘ Subtype.val)convert_12a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Summable (a ∘ Subtype.val) <;>h.e'_2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ ∑' (n : ↑C), a ↑n = ∑' (x : ↑(A ∪ B)), a ↑xconvert_6a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ T2Space ℝconvert_7a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ ContinuousAdd ℝconvert_10a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Disjoint A Bconvert_11a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Summable (a ∘ Subtype.val)convert_12a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Summable (a ∘ Subtype.val) first | infer_instanceconvert_12a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ Summable (a ∘ Subtype.val) | try apply hsummableAll goals completed! 🐙 .h.e'_2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ ∑' (n : ↑C), a ↑n = ∑' (x : ↑(A ∪ B)), a ↑x rw [hC]All goals completed! 🐙 rw [Set.disjoint_iff_inter_eq_empty]convert_10a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 AA:Set ℕB:Set ℕC:Set ℕhC:C = A ∪ BhAB:∀ (n : ℕ), n ∉ A ∩ B⊢ A ∩ B = ∅; grindAll goals completed! 🐙 have h_nonneg (A:Set ℕ) : ∑' n:A, a n ≥ 0 := by⊢ Uncountable ℝ simp [a]a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABA:Set ℕ⊢ 0 ≤ ∑' (n : ↑A), (10 ^ ↑n)⁻¹; positivityAll goals completed! 🐙 have h_congr {A B: Set ℕ} (hAB: A = B) : ∑' n:A, a n = ∑' n:B, a n := by⊢ Uncountable ℝ rw [hAB]All goals completed! 🐙 have : Function.Injective f := by⊢ Uncountable ℝ intro A B hABa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f B⊢ A = B; by_contra!a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bthis:A ≠ B⊢ False rw [←Set.symmDiff_nonempty] at thisa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bthis:(symmDiff A B).Nonempty⊢ False apply Nat.min_spec at thisa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bthis:Nat.min (symmDiff A B) ∈ symmDiff A B ∧ ∀ n ∈ symmDiff A B, Nat.min (symmDiff A B) ≤ n⊢ False set n₀ := Nat.min (symmDiff A B)a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)this:n₀ ∈ symmDiff A B ∧ ∀ n ∈ symmDiff A B, n₀ ≤ n⊢ False simp [symmDiff] at thisa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)this:(n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A) ∧ ∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n⊢ False; choose h1 h2 using thisa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n⊢ False wlog h : n₀ ∈ A ∧ n₀ ∉ B generalizing A Binra:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)⊢ Falsea:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nh:n₀ ∈ A ∧ n₀ ∉ B⊢ False .inra:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)⊢ False simp [h] at h1inra:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ False apply this hAB.symminr.h1a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff B A) ∈ B ∧ Nat.min (symmDiff B A) ∉ A ∨ Nat.min (symmDiff B A) ∈ A ∧ Nat.min (symmDiff B A) ∉ Binr.h2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ ∀ (n : ℕ), n ∈ B ∧ n ∉ A ∨ n ∈ A ∧ n ∉ B → Nat.min (symmDiff B A) ≤ ninr.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff B A) ∈ B ∧ Nat.min (symmDiff B A) ∉ A <;>inr.h1a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff B A) ∈ B ∧ Nat.min (symmDiff B A) ∉ A ∨ Nat.min (symmDiff B A) ∈ A ∧ Nat.min (symmDiff B A) ∉ Binr.h2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ ∀ (n : ℕ), n ∈ B ∧ n ∉ A ∨ n ∈ A ∧ n ∉ B → Nat.min (symmDiff B A) ≤ ninr.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff B A) ∈ B ∧ Nat.min (symmDiff B A) ∉ A simp [symmDiff_comm]inr.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff A B) ∈ B ∧ Nat.min (symmDiff A B) ∉ A <;>inr.h1a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff A B) ∈ B ∧ Nat.min (symmDiff A B) ∉ A ∨ Nat.min (symmDiff A B) ∈ A ∧ Nat.min (symmDiff A B) ∉ Binr.h2a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ ∀ (n : ℕ), n ∈ B ∧ n ∉ A ∨ n ∈ A ∧ n ∉ B → Nat.min (symmDiff A B) ≤ ninr.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => @?_mvar.11064 Ah_decomp:∀ {A : Set ℕ} {B : Set ℕ} {C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} {C} hC hAB => @?_mvar.13645 A B C hC hABh_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => @?_mvar.18320 Ah_congr:∀ {A : Set ℕ} {B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A} {B} hAB => @?_mvar.32425 A B hABA:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.32459 _fvar.32462)h2:∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ nthis:∀ ⦃A B : Set ℕ⦄, f A = f B → let n₀ := Nat.min (symmDiff A B); n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ A → (∀ (n : ℕ), n ∈ A ∧ n ∉ B ∨ n ∈ B ∧ n ∉ A → n₀ ≤ n) → n₀ ∈ A ∧ n₀ ∉ B → Falseh:¬(n₀ ∈ A ∧ n₀ ∉ B)h1:n₀ ∈ B ∧ n₀ ∉ A⊢ Nat.min (symmDiff A B) ∈ B ∧ Nat.min (symmDiff A B) ∉ A grindAll goals completed! 🐙 replace h2 {n:ℕ} (hn: n < n₀) : n ∈ A ↔ n ∈ B := by⊢ Uncountable ℝ grindAll goals completed! 🐙 have : (0:ℝ) > 0 := calc _ = f A - f B := bya:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ 0 = f A - f B linarithAll goals completed! 🐙 _ = ∑' n:A, a n - ∑' n:B, a n := rfl _ = (∑' n:{n ∈ A|n ≤ n₀}, a n + ∑' n:{n ∈ A|n > n₀}, a n) - (∑' n:{n ∈ B|n ≤ n₀}, a n + ∑' n:{n ∈ B|n > n₀}, a n) := bya:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑A), a ↑n - ∑' (n : ↑B), a ↑n = ∑' (n : ↑{n | n ∈ A ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑n - (∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n) congre_a._@.Init.Prelude._hyg.2150a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑A), a ↑n = ∑' (n : ↑{n | n ∈ A ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑ne_a._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑B), a ↑n = ∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n; all_goals {e_a._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑B), a ↑n = ∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n apply h_decompe_a.hC._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ B = {n | n ∈ B ∧ n ≤ n₀} ∪ {n | n ∈ B ∧ n > n₀}e_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∀ (n : ℕ), n ∉ {n | n ∈ B ∧ n ≤ n₀} ∩ {n | n ∈ B ∧ n > n₀} .e_a.hC._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ B = {n | n ∈ B ∧ n ≤ n₀} ∪ {n | n ∈ B ∧ n > n₀} ext ne_a.hC.h._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕ⊢ n ∈ B ↔ n ∈ {n | n ∈ B ∧ n ≤ n₀} ∪ {n | n ∈ B ∧ n > n₀}; simpe_a.hC.h._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕ⊢ n ∈ B ↔ n ∈ B ∧ n ≤ n₀ ∨ n ∈ B ∧ n₀ < n; grindAll goals completed! 🐙 intro n hne_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕhn:n ∈ {n | n ∈ B ∧ n ≤ n₀} ∩ {n | n ∈ B ∧ n > n₀}⊢ False; simp at hne_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕhn:(n ∈ B ∧ n ≤ n₀) ∧ n ∈ B ∧ n₀ < n⊢ False; linarithAll goals completed! 🐙 } _ = ((∑' n:{n ∈ A|n < n₀}, a n + ∑' n:{n ∈ A|n = n₀}, a n) + ∑' n:{n ∈ A|n > n₀}, a n) - ((∑' n:{n ∈ B|n < n₀}, a n + ∑' n:{n ∈ B|n = n₀}, a n) + ∑' n:{n ∈ B|n > n₀}, a n) := bya:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑n - (∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n) = ∑' (n : ↑{n | n ∈ A ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑n - (∑' (n : ↑{n | n ∈ B ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n = n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n) congre_a.e_a._@.Init.Prelude._hyg.2150a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n ≤ n₀}), a ↑n = ∑' (n : ↑{n | n ∈ A ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑ne_a.e_a._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n = ∑' (n : ↑{n | n ∈ B ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n = n₀}), a ↑n; all_goals {e_a.e_a._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ B ∧ n ≤ n₀}), a ↑n = ∑' (n : ↑{n | n ∈ B ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n = n₀}), a ↑n apply h_decompe_a.e_a.hC._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ {n | n ∈ B ∧ n ≤ n₀} = {n | n ∈ B ∧ n < n₀} ∪ {n | n ∈ B ∧ n = n₀}e_a.e_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∀ (n : ℕ), n ∉ {n | n ∈ B ∧ n < n₀} ∩ {n | n ∈ B ∧ n = n₀} .e_a.e_a.hC._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ {n | n ∈ B ∧ n ≤ n₀} = {n | n ∈ B ∧ n < n₀} ∪ {n | n ∈ B ∧ n = n₀} ext ne_a.e_a.hC.h._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕ⊢ n ∈ {n | n ∈ B ∧ n ≤ n₀} ↔ n ∈ {n | n ∈ B ∧ n < n₀} ∪ {n | n ∈ B ∧ n = n₀}; simp [le_iff_lt_or_eq]All goals completed! 🐙 intro n hne_a.e_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕhn:n ∈ {n | n ∈ B ∧ n < n₀} ∩ {n | n ∈ B ∧ n = n₀}⊢ False; simp at hne_a.e_a.hAB._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕhn:(n ∈ B ∧ n < n₀) ∧ n ∈ B ∧ n = n₀⊢ False; linarithAll goals completed! 🐙 } _ = ((∑' n:{n ∈ A|n < n₀}, a n + a n₀) + ∑' n:{n ∈ A|n > n₀}, a n) - ((∑' n:{n ∈ B|n < n₀}, a n + 0) + ∑' n:{n ∈ B|n > n₀}, a n) := bya:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑n + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑n - (∑' (n : ↑{n | n ∈ B ∧ n < n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n = n₀}), a ↑n + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n) = ∑' (n : ↑{n | n ∈ A ∧ n < n₀}), a ↑n + a n₀ + ∑' (n : ↑{n | n ∈ A ∧ n > n₀}), a ↑n - (∑' (n : ↑{n | n ∈ B ∧ n < n₀}), a ↑n + 0 + ∑' (n : ↑{n | n ∈ B ∧ n > n₀}), a ↑n) congr 3e_a.e_a.e_a._@.Init.Prelude._hyg.2150a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑n = a n₀e_a.e_a.e_a._@.Init.Prelude._hyg.2152a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ B ∧ n = n₀}), a ↑n = 0 .e_a.e_a.e_a._@.Init.Prelude._hyg.2150a:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑n = a n₀ calc _ = ∑' n:({n₀}:Set ℕ), a n := bya:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ ∑' (n : ↑{n | n ∈ A ∧ n = n₀}), a ↑n = ∑' (n : ↑{n₀}), a ↑n apply h_congrhABa:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hn⊢ {n | n ∈ A ∧ n = n₀} = {n₀}; ext nhAB.ha:ℕ → ℝ := fun n => 10 ^ (-↑n)f:Set ℕ → ℝ := fun A => ∑' (n : ↑A), @_fvar.10315 ↑nhsummable:∀ (A : Set ℕ), Summable fun n => @_fvar.10315 ↑n := fun A => Summable.subtype (Eq.mpr (eq_of_heq ((fun α β inst inst' e'_3 inst_1 f f' e'_5 => Eq.casesOn (motive := fun a x => inst' = a → e'_3 ≍ x → Summable f ≍ Summable f') e'_3 (fun h => Eq.ndrec (motive := fun inst' => ∀ (e_3 : inst = inst'), e_3 ≍ Eq.refl inst → Summable f ≍ Summable f') (fun e_3 h => Eq.casesOn (motive := fun a x => f' = a → e'_5 ≍ x → Summable f ≍ Summable f') e'_5 (fun h => Eq.ndrec (motive := fun f' => ∀ (e_5 : f = f'), e_5 ≍ Eq.refl f → Summable f ≍ Summable f') (fun e_5 h => HEq.refl (Summable f)) (Eq.symm h) e'_5) (Eq.refl f') (HEq.refl e'_5)) (Eq.symm h) e'_3) (Eq.refl inst') (HEq.refl e'_3)) ℝ ℕ Real.instAddCommGroup.toAddCommMonoid Real.instAddCommMonoid (Eq.refl Real.instAddCommGroup.toAddCommMonoid) PseudoMetricSpace.toUniformSpace.toTopologicalSpace _fvar.10315 (fun n => (1 / 10) ^ n) (funext fun n => Eq.mpr (id (Eq.trans (congrArg (Eq (@_fvar.10315 n)) (Eq.trans (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))) (congrArg (fun x => x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (congrArg (fun x => @_fvar.10315 n = x ^ n) (Mathlib.Meta.NormNum.IsNNRat.to_eq (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) Nat.cast_one (Eq.refl 10))))) (id (Eq.mpr (id (congrArg (fun _a => 10 ^ (-↑n) = _a) (one_div_pow 10 n))) (Eq.mpr (id (congrArg (fun _a => _a = 1 / 10 ^ n) (Real.rpow_neg (Mathlib.Meta.NormNum.isNat_le_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl true)) ↑n))) (Eq.mpr (id (congrArg (fun _a => (10 ^ ↑n)⁻¹ = _a) (one_div (10 ^ n)))) (of_eq_true (Eq.trans (congrArg (fun x => x⁻¹ = (10 ^ n)⁻¹) (Real.rpow_natCast 10 n)) (eq_self (10 ^ n)⁻¹)))))))))) (summable_geometric_of_lt_one (Mathlib.Meta.NormNum.isRat_le_true (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10)))) (Eq.refl true)) (Mathlib.Meta.NormNum.isNNRat_lt_true (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)))) (Eq.refl (Nat.mul 1 1)) (Eq.refl 10))) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Eq.refl true)))) Ah_decomp:∀ {A B C : Set ℕ}, C = A ∪ B → (∀ (n : ℕ), n ∉ A ∩ B) → ∑' (n : ↑C), @_fvar.10315 ↑n = ∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B C} hC hAB => 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 : ↑C), @_fvar.10315 ↑n) (∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) (Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (x : ↑(A ∪ B)), @_fvar.10315 ↑x) hC)) (Eq.refl (∑' (n : ↑(A ∪ B)), @_fvar.10315 ↑n))) (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n) (∑' (x : ↑A), @_fvar.10315 ↑x + ∑' (x : ↑B), @_fvar.10315 ↑x) (Eq.refl (∑' (n : ↑A), @_fvar.10315 ↑n + ∑' (n : ↑B), @_fvar.10315 ↑n)))) (Summable.tsum_union_disjoint (Eq.mpr (id (congrArg (fun _a => _a) (propext Set.disjoint_iff_inter_eq_empty))) (Chapter8.Uncountable.real._proof_2 hAB)) (@_fvar.11065 A) (@_fvar.11065 B))h_nonneg:∀ (A : Set ℕ), ∑' (n : ↑A), @_fvar.10315 ↑n ≥ 0 := fun A => Eq.mpr (id (Eq.trans (congrArg (fun x => tsum x ≥ 0) (funext fun n => congrFun (funext fun n => Eq.trans (Real.rpow_neg_natCast 10 n) (Eq.trans (zpow_neg 10 ↑n) (congrArg Inv.inv (zpow_natCast 10 n)))) ↑n)) ge_iff_le._simp_1)) (tsum_nonneg fun n => le_of_lt (inv_pos_of_pos (pow_pos (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 10)) (Eq.refl (Nat.ble 1 10))) ↑n)))h_congr:∀ {A B : Set ℕ}, A = B → ∑' (n : ↑A), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n := fun {A B} hAB => Eq.mpr (id (congrArg (fun _a => ∑' (n : ↑_a), @_fvar.10315 ↑n = ∑' (n : ↑B), @_fvar.10315 ↑n) hAB)) (Eq.refl (∑' (n : ↑B), @_fvar.10315 ↑n))A:Set ℕB:Set ℕhAB:f A = f Bn₀:ℕ := Chapter8.Nat.min (symmDiff _fvar.41696 _fvar.41697)h1:n₀ ∈ A ∧ n₀ ∉ B ∨ n₀ ∈ B ∧ n₀ ∉ Ah:n₀ ∈ A ∧ n₀ ∉ Bh2:∀ {n : ℕ}, n < _fvar.41699 → (n ∈ _fvar.41696 ↔ n ∈ _fvar.41697) := fun {n} hn => Chapter8.Uncountable.real._proof_8 _fvar.13646 _fvar.18321 _fvar.32426 _fvar.41698 _fvar.41700 _fvar.41701 _fvar.41706 hnn:ℕ⊢ n ∈ {n | n ∈ A ∧ n = n₀} ↔ n ∈ {n₀}; simphAB.ha: