Analysis I, Section 7.5: The root and ratio tests

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:

A point that is only implicitly stated in the text is that for the root and ratio tests, the lim inf and lim sup should be interpreted within the extended reals. The Lean formalizations below make this point more explicit.

namespace Chapter7open Filter Real EReal

Theorem 7.5.1(a) (Root test). A technical condition is needed to ensure the limsup is finite.

theorem Series.root_test_pos {s : Series} (h : atTop.limsup (fun n ↦ ((|s.seq n|^(1/(n:ℝ)):ℝ):EReal)) < 1) : s.absConverges := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges -- This proof is written to follow the structure of the original text. s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1⊢ s.absConverges have hpos : 0 ≤ α' := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges apply le_limsup_of_frequently_le (Frequently.of_forall _) (s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun n => ↑(|s.seq n| ^ (1 / ↑n)) All goals completed! 🐙) s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1x✝:ℤ⊢ 0 ≤ ↑(|s.seq x✝| ^ (1 / ↑x✝)); All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291⊢ s.absConverges have hαα' : α' = α := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291⊢ ↑α = α'; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291⊢ α' ≠ ⊤s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291⊢ α' ≠ ⊥ s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291⊢ α' ≠ ⊤ s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hpos:0 ≤ _fvar.1291 := ?_mvar.1416α:ℝ := EReal.toReal _fvar.1291h:α' = ⊤⊢ 1 ≤ α'; All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)h:α' < 1α:ℝ := EReal.toReal _fvar.1291hpos:α' = ⊥⊢ α' < 0; All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hpos:0 ≤ ↑αh:↑α < 1hαα':α' = ↑α⊢ s.absConverges; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ α⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2⊢ s.absConverges have hε : 0 < ε := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2⊢ α < 1; All goals completed! 🐙 have hε' : α' < (α+ε:ℝ) := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462⊢ α < α + ε; All goals completed! 🐙 have hα : α + ε < 1 := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146⊢ α + (1 - α) / 2 < 1; All goals completed! 🐙 have hα' : 0 < α + ε := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 have := eventually_lt_of_limsup_lt hε' (s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun n => ↑(|s.seq n| ^ (1 / ↑n)) All goals completed! 🐙) s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934this:∃ a, ∀ b ≥ a, ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)⊢ s.absConverges; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ s.absConverges have (n:ℤ) (hn: n ≥ N) : |s.seq n| ≤ (α + ε)^n := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges have : n ≥ N' := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 have npos : 0 < n := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := ?_mvar.16862npos:0 < _fvar.16724 := ?_mvar.17078hN:↑(|s.seq n| ^ (1 / ↑n)) < ↑(α + ε)⊢ |s.seq n| ≤ (α + ε) ^ n s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := ?_mvar.16862npos:0 < _fvar.16724 := ?_mvar.17078hN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ |s.seq n| ≤ (α + ε) ^ n calc _ = (|s.seq n|^(1/(n:ℝ)))^n := s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ |s.seq n| = (|s.seq n| ^ (1 / ↑n)) ^ n s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ |s.seq n| = |s.seq n| ^ (1 / ↑n * ↑n) s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ |s.seq n| ^ (1 / ↑n * ↑n) = |s.seq n|; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ 1 / ↑n * ↑n = 1; All goals completed! 🐙 _ ≤ _ := s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ (|s.seq n| ^ (1 / ↑n)) ^ n ≤ (α + ε) ^ n convert pow_le_pow_left₀ (s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ 0 ≤ |s.seq n| ^ (1 / ↑n) All goals completed! 🐙) (le_of_lt hN) n.toNat all_goals s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤN:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)n:ℤhn:n ≥ Nthis:_fvar.16724 ≥ _fvar.16471 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_1 _fvar.16471 _fvar.16724 _fvar.16732 anpos:0 < _fvar.16724 := Decidable.byContradiction fun a => Chapter7.Series.root_test_pos._proof_2 _fvar.16471 _fvar.16724 _fvar.16732 _fvar.16863 ahN:|s.seq n| ^ (1 / ↑n) < α + ε⊢ n = ↑n.toNat; All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ s.absConverges have hNk : N = s.m + k := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 have hgeom : (fun n ↦ (α+ε) ^ n : Series).converges := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).converges⊢ s.absConverges have : (s.from N).absConverges := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).converges⊢ (s.from N).m = ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).ms:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).converges⊢ ∀ n ≥ (s.from N).m, |(s.from N).seq n| ≤ ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).seq n s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).converges⊢ (s.from N).m = ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).m s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).converges⊢ max s.m N = max N 0; All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:n ≥ (s.from N).m⊢ |(s.from N).seq n| ≤ ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).seq n; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:s.m ≤ n ∧ N ≤ n⊢ |(s.from N).seq n| ≤ ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).seq n have hn' : n ≥ 0 := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop < 1⊢ s.absConverges All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:s.m ≤ n ∧ N ≤ nhn':_fvar.53974 ≥ 0 := ?_mvar.57342⊢ |s.seq n| ≤ (α + ε) ^ n.toNat s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:s.m ≤ n ∧ N ≤ nhn':_fvar.53974 ≥ 0 := ?_mvar.57342⊢ (α + ε) ^ n.toNat = (α + ε) ^ n; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:s.m ≤ n ∧ N ≤ nhn':_fvar.53974 ≥ 0 := ?_mvar.57342⊢ (α + ε) ^ n = (α + ε) ^ n.toNat; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesn:ℤhn:s.m ≤ n ∧ N ≤ nhn':_fvar.53974 ≥ 0 := ?_mvar.57342⊢ n = ↑n.toNat; All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825⊢ s.abs.converges s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825⊢ (s.abs.from (s.abs.m + ↑k)).converges; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825⊢ s.abs.from (s.abs.m + ↑k) = (s.from N).abs; s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825⊢ s.m + ↑k = max s.m N ∧ (fun n => if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = fun n => if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0; refine ⟨ s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := Eq.mpr (id (Eq.trans (div_pos_iff_of_pos_right._simp_1 (of_eq_true Nat.ofNat_pos._simp_1)) one_lt_div'._simp_4)) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a))))hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := Eq.mpr (id (congrArg (fun _a => _a < ↑(_fvar.4877 + _fvar.8335)) _fvar.6576)) (Eq.mpr (id (congrArg (fun _a => _a) (propext EReal.coe_lt_coe_iff))) (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))))hα:_fvar.4877 + _fvar.8335 < 1 := id (lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.zero_mul (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.7965) (Eq.mp (congrArg (fun _a => _a ≤ 0) (CancelDenoms.sub_subst rfl (CancelDenoms.add_subst rfl (CancelDenoms.div_subst (CancelDenoms.sub_subst rfl rfl) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_div (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNNRat.to_isNat (Mathlib.Meta.NormNum.isNNRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)))) (Eq.refl (Nat.mul 2 1)) (Eq.refl 2)))))) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))))))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le a) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl false)))))))hα':0 < _fvar.4877 + _fvar.8335 := lt_of_not_ge fun a => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (Rat.rawCast (Int.negOfNat 1) 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (Rat.rawCast (Int.negOfNat 1) 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.div_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.atom_pf _fvar.4877) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul _fvar.4877 (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.div_pf (Mathlib.Tactic.Ring.inv_single (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (NNRat.rawCast 1 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + 0))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsRat.to_raw_eq (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Tactic.Ring.mul_zero (_fvar.4877 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (NNRat.rawCast 1 2 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.4877 ^ Nat.rawCast 1 * Rat.rawCast (Int.negOfNat 1) 2 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_gt (NNRat.rawCast 1 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNNRat.to_raw_eq (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)) (Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.neg_zero (Mathlib.Tactic.Ring.add_pf_add_zero (NNRat.rawCast 1 2 + (_fvar.4877 ^ Nat.rawCast 1 * NNRat.rawCast 1 2 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.4877 (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.IsRat.to_isInt (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl ((Int.negOfNat 1).mul (Int.negOfNat 1))) (Eq.refl 2)))))) (Eq.refl (Int.negOfNat 1)) (Eq.refl 2)))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.IsRat.to_isNNRat (Mathlib.Meta.NormNum.isRat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))) (Mathlib.Meta.NormNum.IsRat.of_raw ℝ (Int.negOfNat 1) 2 (Mathlib.Meta.NormNum.IsRat.den_nz (Mathlib.Meta.NormNum.isRat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.to_isRat (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))) (Mathlib.Meta.NormNum.IsNNRat.to_isRat (Mathlib.Meta.NormNum.IsNNRat.of_raw ℝ 1 2 (Mathlib.Meta.NormNum.IsNNRat.den_nz (Mathlib.Meta.NormNum.isNNRat_inv_pos (Mathlib.Meta.NormNum.IsNat.to_isNNRat (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)))))) (Eq.refl ((Int.negOfNat 1).mul (Int.ofNat 1))) (Eq.refl 2)))) (Eq.refl (Int.ofNat 1)) (Eq.refl 2)))))) (Eq.refl (Int.ofNat 0)) (Eq.refl 4))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_le_of_neg (Mathlib.Tactic.Linarith.sub_nonpos_of_le _fvar.8142) (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.8463)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le a)))N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)k:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)⊢ s.m + ↑k = max s.m N All goals completed! 🐙, ?_ ⟩ s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤ⊢ (if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.m⊢ (if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:¬n ≥ s.m⊢ (if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.m⊢ (if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:¬n ≥ s.m⊢ (if s.m + ↑k ≤ n then if s.m ≤ n then |s.seq n| else 0 else 0) = if s.m ≤ n ∧ N ≤ n then |if s.m ≤ n ∧ N ≤ n then s.seq n else 0| else 0 All goals completed! 🐙 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:n ≥ N⊢ (if s.m + ↑k ≤ n then |s.seq n| else 0) = if N ≤ n then |if N ≤ n then s.seq n else 0| else 0s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:¬n ≥ N⊢ (if s.m + ↑k ≤ n then |s.seq n| else 0) = if N ≤ n then |if N ≤ n then s.seq n else 0| else 0 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:n ≥ N⊢ (if s.m + ↑k ≤ n then |s.seq n| else 0) = if N ≤ n then |if N ≤ n then s.seq n else 0| else 0s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:¬n ≥ N⊢ (if s.m + ↑k ≤ n then |s.seq n| else 0) = if N ≤ n then |if N ≤ n then s.seq n else 0| else 0 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:¬n ≥ N⊢ s.m + ↑k ≤ n → s.seq n = 0 s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:n ≥ N⊢ n < s.m + ↑k → 0 = |s.seq n|s:Seriesα':EReal := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)α:ℝ := EReal.toReal _fvar.1291hαα':α' = ↑αh:α < 1hpos:0 ≤ αε:ℝ := (1 - _fvar.4877) / 2hε:0 < _fvar.8335 := ?_mvar.8462hε':_fvar.1291 < ↑(_fvar.4877 + _fvar.8335) := ?_mvar.12146hα:_fvar.4877 + _fvar.8335 < 1 := ?_mvar.12437hα':0 < _fvar.4877 + _fvar.8335 := ?_mvar.14934N':ℤhN:∀ b ≥ N', ↑(|s.seq b| ^ (1 / ↑b)) < ↑(α + ε)N:ℤ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)this✝:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := fun n hn => @?_mvar.16849 n hnk:ℕ := failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)hNk:failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation) := ?_mvar.40555hgeom:({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.from ({ m := 0, seq := fun n => if n ≥ 0 then (fun n => (α + ε) ^ n) n.toNat else 0, vanish := ⋯ }.m + ↑N.toNat)).convergesthis:(Chapter7.Series.from _fvar.669 _fvar.16568).abs.converges := ?_mvar.44825n:ℤhnm:n ≥ s.mhn:¬n ≥ N⊢ s.m + ↑k ≤ n → s.seq n = 0 All goals completed! 🐙

Theorem 7.5.1(b) (Root test)

theorem Series.root_test_neg {s : Series} (h : atTop.limsup (fun n ↦ ((|s.seq n|^(1/(n:ℝ)):ℝ):EReal)) > 1) : s.diverges := s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop > 1⊢ s.diverges -- This proof is written to follow the structure of the original text. apply frequently_lt_of_lt_limsup (s:Seriesh:limsup (fun n => ↑(|s.seq n| ^ (1 / ↑n))) atTop > 1⊢ IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun n => ↑(|s.seq n| ^ (1 / ↑n)) All goals completed! 🐙) at h s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))⊢ ¬Tendsto s.seq atTop (nhds 0) s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:Tendsto s.seq atTop (nhds 0)⊢ False; s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ ε > 0, ∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < ε⊢ False; specialize this 1 (s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ ε > 0, ∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < ε⊢ 1 > 0 All goals completed! 🐙) s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n - 0| < 1⊢ False s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n| < 1⊢ False; s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':?_mvar.158753 := Real.rpow_lt_one ?_mvar.158756 _fvar.158748 ?_mvar.158950⊢ Falses:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n| < 1⊢ 0 ≤ |s.seq n|s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n| < 1⊢ 0 < 1 / ↑n s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':?_mvar.158753 := Real.rpow_lt_one ?_mvar.158756 _fvar.158748 ?_mvar.158950⊢ Falses:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n| < 1⊢ 0 ≤ |s.seq n|s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < ↑(|s.seq n| ^ (1 / ↑n))hs':|s.seq n| < 1⊢ 0 < 1 / ↑n try All goals completed! 🐙 s:Seriesh:∃ᶠ (x : ℤ) in atTop, 1 < ↑(|s.seq x| ^ (1 / ↑x))this:∀ᶠ (b : ℤ) in atTop, |s.seq b - 0| < 1n:ℤhn:n ≥ 1hs:1 < |s.seq n| ^ (1 / ↑n)hs':|s.seq n| ^ (1 / ↑n) < 1⊢ False All goals completed! 🐙

Theorem 7.5.1(c) (Root test) / Exercise 7.5.3

theorem declaration uses 'sorry'Series.root_test_inconclusive: ∃ s:Series, atTop.Tendsto (fun n ↦ |s.seq n|^(1/(n:ℝ))) (nhds 1) ∧ s.diverges := ⊢ ∃ s, Tendsto (fun n => |s.seq n| ^ (1 / ↑n)) atTop (nhds 1) ∧ s.diverges All goals completed! 🐙

Theorem 7.5.1 (Root test) / Exercise 7.5.3

theorem declaration uses 'sorry'Series.root_test_inconclusive' : ∃ s:Series, atTop.Tendsto (fun n ↦ |s.seq n|^(1/(n:ℝ))) (nhds 1) ∧ s.absConverges := ⊢ ∃ s, Tendsto (fun n => |s.seq n| ^ (1 / ↑n)) atTop (nhds 1) ∧ s.absConverges All goals completed! 🐙

Lemma 7.5.2 / Exercise 7.5.1

theorem declaration uses 'sorry'Series.ratio_ineq {c:ℤ → ℝ} (m:ℤ) (hpos: ∀ n ≥ m, c n > 0) : atTop.liminf (fun n ↦ ((c (n+1) / c n:ℝ):EReal)) ≤ atTop.liminf (fun n ↦ ↑((c n)^(1/(n:ℝ)):ℝ)) ∧ atTop.liminf (fun n ↦ (((c n)^(1/(n:ℝ)):ℝ):EReal)) ≤ atTop.limsup (fun n ↦ ↑((c n)^(1/(n:ℝ)):ℝ)) ∧ atTop.limsup (fun n ↦ (((c n)^(1/(n:ℝ)):ℝ):EReal)) ≤ atTop.limsup (fun n ↦ ↑(c (n+1) / c n:ℝ)) := c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ liminf (fun n => ↑(c (n + 1) / c n)) atTop ≤ liminf (fun n => ↑(c n ^ (1 / ↑n))) atTop ∧ liminf (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ∧ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ limsup (fun n => ↑(c (n + 1) / c n)) atTop -- This proof is written to follow the structure of the original text. c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ liminf (fun n => ↑(c (n + 1) / c n)) atTop ≤ liminf (fun n => ↑(c n ^ (1 / ↑n))) atTopc:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun n => ↑(c n ^ (1 / ↑n))c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ IsBoundedUnder (fun x1 x2 => x1 ≥ x2) atTop fun n => ↑(c n ^ (1 / ↑n))c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ limsup (fun n => ↑(c (n + 1) / c n)) atTop c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ liminf (fun n => ↑(c (n + 1) / c n)) atTop ≤ liminf (fun n => ↑(c n ^ (1 / ↑n))) atTopc:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun n => ↑(c n ^ (1 / ↑n))c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ IsBoundedUnder (fun x1 x2 => x1 ≥ x2) atTop fun n => ↑(c n ^ (1 / ↑n))c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ limsup (fun n => ↑(c (n + 1) / c n)) atTop try c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ limsup (fun n => ↑(c (n + 1) / c n)) atTop c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0⊢ liminf (fun n => ↑(c (n + 1) / c n)) atTop ≤ liminf (fun n => ↑(c n ^ (1 / ↑n))) atTop All goals completed! 🐙 c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0L':EReal := Filter.limsup (fun n => ↑(@_fvar.168394 (n + 1) / @_fvar.168394 n)) Filter.atTop⊢ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ L' c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0L':EReal := Filter.limsup (fun n => ↑(@_fvar.168394 (n + 1) / @_fvar.168394 n)) Filter.atTophL:L' = ⊤⊢ limsup (fun n => ↑(c n ^ (1 / ↑n))) atTop ≤ L'c:ℤ → ℝm:ℤhpos:∀ n ≥ m, c n > 0L':EReal := Filter.limsup (fun