awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommRing α
inst_1 : NormalizedGCDMonoid α
a b c d : α
hd : b - c = a * d
e : α
he : c = @gcd _ CommSemiring.toCommMonoidWithZero _ a c * e
f : α
hf : a = @gcd _ CommSemiring.toCommMonoidWithZero _ a c * f
h : b = @gcd _ CommSemiring.toCommMonoidWithZero _ a c * e + @gcd _ CommSemiring.toCommMonoidWithZero _ a c * (f * d)
h₁ : b = c + @gcd _ CommSemiring.toCommMonoidWithZero _ a c * (f * d)
h₂ : b = c + @gcd _ CommSemiring.toCommMonoidWithZero _ a c * f * d
h₃ : b = c + a * d
h₄ : b = c + (b - c)
h₅ : b = c + b - c
h₆ : b = b + c - c
⊢ b = @gcd _ CommSemiring.toCommMonoidWithZero _ a c * (e + f * d)
have h₇ : b = b
Tactic: rw [mul_add, ← he, ← mul_assoc, ← hf, ← hd, ← add_sub_assoc, add_comm c b, add_sub_cancel_right]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommRing α
inst_1 : NormalizedGCDMonoid α
a b c : α
h : a ∣ b - c
⊢ @gcd _ CommSemiring.toCommMonoidWithZero _ b a = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
have rw : @gcd _ CommSemiring.toCommMonoidWithZero _ a b = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
Tactic: rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommRing α
inst_1 : NormalizedGCDMonoid α
a b c : α
h : a ∣ b - c
rw : @gcd _ CommSemiring.toCommMonoidWithZero _ a b = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
⊢ @gcd _ CommSemiring.toCommMonoidWithZero _ b a = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
have rw₁ : @gcd _ CommSemiring.toCommMonoidWithZero _ a b = @gcd _ CommSemiring.toCommMonoidWithZero _ a c
Tactic: rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommRing α
inst_1 : NormalizedGCDMonoid α
a b c : α
h : a ∣ b - c
rw : @gcd _ CommSemiring.toCommMonoidWithZero _ a b = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
rw₁ : @gcd _ CommSemiring.toCommMonoidWithZero _ a b = @gcd _ CommSemiring.toCommMonoidWithZero _ a c
⊢ @gcd _ CommSemiring.toCommMonoidWithZero _ b a = @gcd _ CommSemiring.toCommMonoidWithZero _ c a
have rw₂ : @gcd _ CommSemiring.toCommMonoidWithZero _ a c = @gcd _ CommSemiring.toCommMonoidWithZero _ a c
Tactic: rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ (if a * b = 0 then 1
    else
      Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
        ⋯) =
    (if a = 0 then 1
      else Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) *
      if b = 0 then 1
      else Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯
have simp_rw : (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b))) sorry) :
    α) =
  (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) sorry) : α) *
    (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) sorry) : α)
Tactic: simp_rw [if_neg (mul_ne_zero ha hb), if_neg ha, if_neg hb, Units.ext_iff, Units.val_mul]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
this :
  a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
⊢ (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b))) ⋯) :
      α) =
    (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α)
have apply : a * b *
    (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
          sorry) :
      α) =
  a * b *
    ((↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) sorry) : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) sorry) : α))
Tactic: apply mul_left_cancel₀ (mul_ne_zero ha hb) _
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b))) ⋯) :
      α) =
    (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α)
have suffices : a * b *
    (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
          sorry) :
      α) =
  a * (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) sorry) : α) *
    (b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) sorry) : α))
Tactic: suffices a * b * ↑(Classical.choose (associated_map_mk hinv (a * b))) =
        a * ↑(Classical.choose (associated_map_mk hinv a)) *
        (b * ↑(Classical.choose (associated_map_mk hinv b))) by
      apply mul_left_cancel₀ (mul_ne_zero ha hb) _
      simpa only [mul_assoc, mul_comm, mul_left_comm] using this
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
have rw : a * b *
    (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
          sorry) :
      α) =
  HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
    (b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) sorry) : α))
Tactic: rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      map_mul, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
rw :
  a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
⊢ a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
have rw₁ : (f : (a : Associates α) → α) (Associates.mk (a * b)) =
  HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
    (b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) sorry) : α))
Tactic: rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      map_mul, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
rw :
  a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
rw₁ :
  (f : (a : Associates α) → α) (Associates.mk (a * b)) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
⊢ a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
have rw₂ : (f : (a : Associates α) → α) (Associates.mk (a * b)) =
  HMul.hMul (α := α) (β := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
    ((f : (a : Associates α) → α) (Associates.mk b) : α)
Tactic: rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      map_mul, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
rw :
  a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
rw₁ :
  (f : (a : Associates α) → α) (Associates.mk (a * b)) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
rw₂ :
  (f : (a : Associates α) → α) (Associates.mk (a * b)) =
    HMul.hMul (α := α) (β := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      ((f : (a : Associates α) → α) (Associates.mk b) : α)
⊢ a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
have rw₃ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (a * b)) : α)
  ((f : (a : Associates α) → α) (HMul.hMul (α := Associates α) (Associates.mk a) (Associates.mk b)) : α)
Tactic: rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      map_mul, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
a b : α
ha : a ≠ 0
hb : b ≠ 0
rw :
  a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
rw₁ :
  (f : (a : Associates α) → α) (Associates.mk (a * b)) =
    HMul.hMul (α := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
rw₂ :
  (f : (a : Associates α) → α) (Associates.mk (a * b)) =
    HMul.hMul (α := α) (β := α) ((f : (a : Associates α) → α) (Associates.mk a) : α)
      ((f : (a : Associates α) → α) (Associates.mk b) : α)
rw₃ :
  Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (a * b)) : α)
    ((f : (a : Associates α) → α) (HMul.hMul (α := Associates α) (Associates.mk a) (Associates.mk b)) : α)
⊢ a * b *
      (↑(Classical.choose (p := fun (x : αˣ) ↦ a * b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (a * b)))
            ⋯) :
        α) =
    a *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ a * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk a)) ⋯) :
          α) *
      (b *
        (↑(Classical.choose (p := fun (x : αˣ) ↦ b * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk b)) ⋯) : α))
have rw₄ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (a * b)) : α)
  ((f : (a : Associates α) → α) (Associates.mk (a * b)) : α)
Tactic: rw [map_mk_unit_aux hinv a, map_mk_unit_aux hinv (a * b), map_mk_unit_aux hinv b, ←
      map_mul, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
⊢ (if (↑u : α) = 0 then 1
    else
      Classical.choose (p := fun (x : αˣ) ↦ (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α)))
        ⋯) =
    u⁻¹
have a : Nontrivial α
Tactic: nontriviality α
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
⊢ (if (↑u : α) = 0 then 1
    else
      Classical.choose (p := fun (x : αˣ) ↦ (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α)))
        ⋯) =
    u⁻¹
have simp_rw : (↑(Classical.choose (p := fun (x : αˣ) ↦ (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α)))
        sorry) :
    α) =
  (↑u⁻¹ : α)
Tactic: simp_rw [if_neg (Units.ne_zero u), Units.ext_iff]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
⊢ (↑(Classical.choose (p := fun (x : αˣ) ↦ (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α)))
          ⋯) :
      α) =
    (↑u⁻¹ : α)
have apply : (↑u : α) *
    (↑(Classical.choose (p := fun (x : αˣ) ↦
          (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) sorry) :
      α) =
  (↑u : α) * (↑u⁻¹ : α)
Tactic: apply mul_left_cancel₀ (Units.ne_zero u)
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
⊢ (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    (↑u : α) * (↑u⁻¹ : α)
have rw : (↑u : α) *
    (↑(Classical.choose (p := fun (x : αˣ) ↦
          (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) sorry) :
      α) =
  1
Tactic: rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, map_one]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
rw :
  (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    1
⊢ (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    (↑u : α) * (↑u⁻¹ : α)
have rw₁ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (↑u : α)) : α) (1 : α)
Tactic: rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, map_one]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
rw :
  (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    1
rw₁ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (↑u : α)) : α) (1 : α)
⊢ (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    (↑u : α) * (↑u⁻¹ : α)
have rw₂ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk 1) : α) (1 : α)
Tactic: rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, map_one]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
rw :
  (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    1
rw₁ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (↑u : α)) : α) (1 : α)
rw₂ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk 1) : α) (1 : α)
⊢ (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    (↑u : α) * (↑u⁻¹ : α)
have rw₃ : Eq (α := α) ((f : (a : Associates α) → α) 1 : α) (1 : α)
Tactic: rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, map_one]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
f : Associates α →* α
hinv : Function.RightInverse (α := α) (β := Associates α) (⇑f : (a : Associates α) → α) Associates.mk
u : αˣ
h : Nontrivial α
rw :
  (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    1
rw₁ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk (↑u : α)) : α) (1 : α)
rw₂ : Eq (α := α) ((f : (a : Associates α) → α) (Associates.mk 1) : α) (1 : α)
rw₃ : Eq (α := α) ((f : (a : Associates α) → α) 1 : α) (1 : α)
⊢ (↑u : α) *
      (↑(Classical.choose (p := fun (x : αˣ) ↦
            (↑u : α) * (↑x : α) = (f : (a : Associates α) → α) (Associates.mk (↑u : α))) ⋯) :
        α) =
    (↑u : α) * (↑u⁻¹ : α)
have rw₄ : Eq.{u_1 + 1} (α := α) 1 1
Tactic: rw [Units.mul_inv, map_mk_unit_aux hinv u,
      Associates.mk_eq_mk_iff_associated.2 (associated_one_iff_isUnit.2 ⟨u, rfl⟩),
      Associates.mk_one, map_one]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
gcd : α → α → α
gcd_dvd_left : ∀ (a b : α), gcd a b ∣ a
gcd_dvd_right : ∀ (a b : α), gcd a b ∣ b
dvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b
a b : α
⊢ Associated (gcd a b * if a = 0 then 0 else Classical.choose (p := fun (x : α) ↦ a * b = gcd a b * x) ⋯) (a * b)
have pos : ∀ (a0 : a = 0), Associated (gcd a b * 0) (a * b)
Tactic: split_ifs with a0
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
gcd : α → α → α
gcd_dvd_left : ∀ (a b : α), gcd a b ∣ a
gcd_dvd_right : ∀ (a b : α), gcd a b ∣ b
dvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b
a b : α
pos : ∀ (a0 : a = 0), Associated (gcd a b * 0) (a * b)
⊢ Associated (gcd a b * if a = 0 then 0 else Classical.choose (p := fun (x : α) ↦ a * b = gcd a b * x) ⋯) (a * b)
have neg : ∀ (a0 : ¬a = 0), Associated (gcd a b * Classical.choose (p := fun (x : α) ↦ a * b = gcd a b * x) sorry) (a * b)
Tactic: split_ifs with a0
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
gcd : α → α → α
gcd_dvd_left : ∀ (a b : α), gcd a b ∣ a
gcd_dvd_right : ∀ (a b : α), gcd a b ∣ b
dvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b
a b : α
a0 : a = 0
⊢ Associated (gcd a b * 0) (a * b)
have pos : Associated 0 (a * b)
Tactic: rw [mul_zero, a0, zero_mul]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : DecidableEq α
gcd : α → α → α
gcd_dvd_left : ∀ (a b : α), gcd a b ∣ a
gcd_dvd_right : ∀ (a b : α), gcd a b ∣ b
dvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b
a b : α
a0 : a = 0
pos : Associated 0 (a * b)
⊢ Associated (gcd a b * 0) (a * b)
have pos₁ : Associated 0 (0 * b)
Tactic: rw [mul_zero, a0, zero_mul]