awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.BigOperators (goal)
Goal:
α : Type u_1
β : Type u_2
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
inst_5 : Archimedean α
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
har1 : |r| < 1
m : ℕ
hmn : n.succ ≤ m
hr : 0 < r
k : ℕ
hk : m = k + n.succ
inr :
  ∀ (n : ℕ) (h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) (m : ℕ) (hmn : n.succ ≤ m) (hk : m = 0 + n.succ),
    abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
⊢ abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
have inr₁ : ∀ (k : ℕ)
  (ih :
    ∀ (n : ℕ) (h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) (m : ℕ) (hmn : n.succ ≤ m) (hk : m = k + n.succ),
      abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m)
  (n : ℕ) (h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) (m : ℕ) (hmn : n.succ ≤ m) (hk : m = k + 1 + n.succ),
  abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
Tactic: induction k generalizing m n with
  | zero =>
    rw [hk, Nat.zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel_right₀]
    positivity
  | succ k ih =>
    have kn : k + n.succ ≥ n.succ := by
      rw [← zero_add n.succ]; exact add_le_add (Nat.zero_le _) (by simp)
    rw [hk, Nat.succ_add, pow_succ r, ← mul_assoc]
    refine
      le_trans (by rw [mul_comm] <;> exact h _ (Nat.le_of_succ_le kn))
        (mul_le_mul_of_nonneg_right ?_ hr0)
    exact ih _ h _ (by simp) rfl
Cleared hypotheses: n, h, m, hmn, k, hk
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f : CauSeq β abv
⊢ Eq (α := Cauchy abv) (mk f) 0 ↔ f.LimZero
have this : Eq (α := Cauchy abv) (mk f) 0 ↔ (f - 0).LimZero
Tactic: have : mk f = 0 ↔ LimZero (f - 0) := Quotient.eq
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : Ring β
abv : β → α
inst_4 : IsAbsoluteValue abv
f : CauSeq β abv
this : Eq (α := Cauchy abv) (mk f) 0 ↔ (f - 0).LimZero
⊢ Eq (α := Cauchy abv) (mk f) 0 ↔ f.LimZero
have rwa : ∀ (this : Eq (α := Cauchy abv) (mk f) 0 ↔ f.LimZero), Eq (α := Cauchy abv) (mk f) 0 ↔ f.LimZero
Tactic: rwa [sub_zero] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
⊢ Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
have this : f.LimZero ↔ g.LimZero
Tactic: have := limZero_congr fg
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
⊢ Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
have pos : ∀ (hf : f.LimZero),
  Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
Tactic: by_cases hf : LimZero f
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
pos :
  ∀ (hf : f.LimZero),
    Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h))
      (mk (if h : g.LimZero then 0 else g.inv h))
⊢ Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
have neg : ∀ (hf : ¬f.LimZero),
  Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
Tactic: by_cases hf : LimZero f
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
⊢ Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
have hg : ¬g.LimZero
Tactic: have hg := mt this.2 hf
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
⊢ Eq.{u_2 + 1} (α := Cauchy abv) (mk (if h : f.LimZero then 0 else f.inv h)) (mk (if h : g.LimZero then 0 else g.inv h))
have neg : Eq (α := Cauchy abv) (mk (f.inv sorry)) (mk (g.inv sorry))
Tactic: simp only [hf, dite_false, hg]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk f) = 1
Tactic: have If : mk (inv f hf) * mk f = 1 := mk_eq.2 (inv_mul_cancel hf)
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk f) = 1
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Tactic: have Ig : mk (inv g hg) * mk g = 1 := mk_eq.2 (inv_mul_cancel hg)
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk f) = 1
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
Tactic: have Ig' : mk g * mk (inv g hg) = 1 := mk_eq.2 (mul_inv_cancel hg)
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk f) = 1
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have If₁ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Tactic: rw [mk_eq.2 fg, ← Ig] at If
Cleared hypotheses: If, Ig, Ig'
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₂ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) * mk (g.inv hg) = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
neg₂ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₃ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) * mk (g.inv hg) = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
neg₂ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
neg₃ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₄ : mk (g.inv hg) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
neg₂ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
neg₃ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
neg₄ : mk (g.inv hg) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₅ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) 1 = mk (g.inv sorry)
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f g : CauSeq β abv
fg : f ≈ g
this : f.LimZero ↔ g.LimZero
hf : ¬f.LimZero
hg : ¬g.LimZero
If : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) = HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g)
Ig : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) = 1
Ig' : HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = 1
neg : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) 1 = mk (g.inv ⋯)
neg₁ : mk (f.inv hf) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
neg₂ : HMul.hMul (α := Cauchy abv) (mk (f.inv hf)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
neg₃ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) (mk g) * mk (g.inv hg) = mk (g.inv ⋯)
neg₄ : mk (g.inv hg) * HMul.hMul (α := Cauchy abv) (mk g) (mk (g.inv hg)) = mk (g.inv ⋯)
neg₅ : HMul.hMul (α := Cauchy abv) (mk (g.inv hg)) 1 = mk (g.inv ⋯)
⊢ Eq (α := Cauchy abv) (mk (f.inv ⋯)) (mk (g.inv ⋯))
have neg₆ : Eq (α := Cauchy abv) (mk (g.inv hg)) (mk (g.inv sorry))
Tactic: rw [← mul_one (mk (inv f hf)), ← Ig', ← mul_assoc, If, mul_assoc, Ig', mul_one]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
⊢ Eq (α := CauSeq β abv) (if h : LimZero (const abv 0) then 0 else inv (const abv 0) h) (const abv 0)
have rw : Eq (α := CauSeq β abv) 0 (const abv 0)
Tactic: rw [dif_pos]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
rw : Eq (α := CauSeq β abv) 0 (const abv 0)
⊢ Eq (α := CauSeq β abv) (if h : LimZero (const abv 0) then 0 else inv (const abv 0) h) (const abv 0)
have hc : LimZero (const abv 0)
Tactic: rw [dif_pos]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
⊢ Eq (α := CauSeq β abv) (if h : LimZero (const abv 0) then 0 else inv (const abv 0) h) (const abv 0)
have rw : Eq (α := CauSeq β abv) 0 (const abv 0)
Tactic: rw [dif_pos] <;> [rfl; exact zero_limZero]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
rw : Eq (α := CauSeq β abv) 0 (const abv 0)
⊢ Eq (α := CauSeq β abv) (if h : LimZero (const abv 0) then 0 else inv (const abv 0) h) (const abv 0)
have hc : LimZero (const abv 0)
Tactic: rw [dif_pos] <;> [rfl; exact zero_limZero]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f.LimZero
⊢ (if h : f.LimZero then 0 else f.inv h) = f.inv hf
have rw : f.inv hf = f.inv hf
Tactic: rw [dif_neg]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f : CauSeq β abv
hf : Ne (α := Cauchy abv) ⟦f⟧ 0
⊢ HMul.hMul (α := Cauchy abv) (⟦f⟧)⁻¹ ⟦f⟧ = 1
have hf₁ : ¬f.LimZero
Tactic: simp only [mk_eq_mk, ne_eq, mk_eq_zero] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f : CauSeq β abv
hf : ¬f.LimZero
⊢ HMul.hMul (α := Cauchy abv) (⟦f⟧)⁻¹ ⟦f⟧ = 1
have simp : Eq (α := Cauchy abv) (mk (f.inv sorry * f)) 1
Tactic: simp only [mk_eq_mk, hf, not_false_eq_true, inv_mk, mk_mul]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (hypothesis)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f : CauSeq β abv
hf : Ne (α := Cauchy abv) ⟦f⟧ 0
⊢ HMul.hMul (α := Cauchy abv) ⟦f⟧ (⟦f⟧)⁻¹ = 1
have hf₁ : ¬f.LimZero
Tactic: simp only [mk_eq_mk, ne_eq, mk_eq_zero] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : Cauchy abv
f : CauSeq β abv
hf : ¬f.LimZero
⊢ HMul.hMul (α := Cauchy abv) ⟦f⟧ (⟦f⟧)⁻¹ = 1
have simp : Eq (α := Cauchy abv) (mk (f * f.inv sorry)) 1
Tactic: simp only [mk_eq_mk, hf, not_false_eq_true, inv_mk, mk_mul]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : β
⊢ Eq (α := CauSeq β abv) (const abv x⁻¹) (if h : LimZero (const abv x) then 0 else inv (const abv x) h)
have pos : ∀ (h : LimZero (const abv x)), Eq (α := CauSeq β abv) (const abv x⁻¹) 0
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : β
pos : ∀ (h : LimZero (const abv x)), Eq (α := CauSeq β abv) (const abv x⁻¹) 0
⊢ Eq (α := CauSeq β abv) (const abv x⁻¹) (if h : LimZero (const abv x) then 0 else inv (const abv x) h)
have neg : ∀ (h : ¬LimZero (const abv x)), const abv x⁻¹ = inv (const abv x) h
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : β
⊢ Eq (α := CauSeq β abv) (const abv x⁻¹) (if h : LimZero (const abv x) then 0 else inv (const abv x) h)
have pos : ∀ (h : LimZero (const abv x)), Eq (α := CauSeq β abv) (const abv x⁻¹) 0
Tactic: split_ifs with h <;>
    [simp only [const_limZero.1 h, GroupWithZero.inv_zero, const_zero]; rfl]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
x : β
pos : ∀ (h : LimZero (const abv x)), Eq (α := CauSeq β abv) (const abv x⁻¹) 0
⊢ Eq (α := CauSeq β abv) (const abv x⁻¹) (if h : LimZero (const abv x) then 0 else inv (const abv x) h)
have neg : ∀ (h : ¬LimZero (const abv x)), const abv x⁻¹ = inv (const abv x) h
Tactic: split_ifs with h <;>
    [simp only [const_limZero.1 h, GroupWithZero.inv_zero, const_zero]; rfl]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
q : ℚ≥0
⊢ (↑q : Cauchy abv) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
have simp_rw : ofRat (↑q : β) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
Tactic: simp_rw [← ofRat_nnratCast, NNRat.cast_def, ofRat_div, ofRat_natCast]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
q : ℚ≥0
simp_rw : ofRat (↑q : β) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
⊢ (↑q : Cauchy abv) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
have simp_rw₁ : ofRat (HDiv.hDiv (α := β) (↑q.num : β) (↑q.den : β)) =
  HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
Tactic: simp_rw [← ofRat_nnratCast, NNRat.cast_def, ofRat_div, ofRat_natCast]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
q : ℚ≥0
simp_rw : ofRat (↑q : β) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
simp_rw₁ :
  ofRat (HDiv.hDiv (α := β) (↑q.num : β) (↑q.den : β)) =
    HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
⊢ (↑q : Cauchy abv) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
have simp_rw₂ : HDiv.hDiv (α := Cauchy abv) (ofRat (↑q.num : β)) (ofRat (↑q.den : β)) =
  HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
Tactic: simp_rw [← ofRat_nnratCast, NNRat.cast_def, ofRat_div, ofRat_natCast]
-- Lean 4 proof step from Mathlib.Algebra.Order.CauSeq.Completion (goal)
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
β : Type u_2
inst_3 : DivisionRing β
abv : β → α
inst_4 : IsAbsoluteValue abv
q : ℚ
⊢ (↑q : Cauchy abv) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
have rw : ofRat (↑q : β) = HDiv.hDiv (α := Cauchy abv) (↑q.num : Cauchy abv) (↑q.den : Cauchy abv)
Tactic: rw [← ofRat_ratCast, Rat.cast_def, ofRat_div, ofRat_natCast, ofRat_intCast]