awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
this :
  gcd a b * Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯ =
    (normalize : (a : α) → α) (a * b)
⊢ Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = gcd a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = gcd a b * x) ⋯)) : α) =
    Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = gcd a b * x) ⋯
have neg : let l := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) sorry;
∀ (this : gcd a b * l = (normalize : (a : α) → α) (a * b)), l * (↑(normUnit l) : α) = l
Tactic: set l := Classical.choose (dvd_normalize_iff.2 ((gcd_dvd_left a b).trans (Dvd.intro b rfl)))
Cleared hypotheses: 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
⊢ l * (↑(normUnit l) : α) = l
have neg : let l := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) sorry;
∀ (this : gcd a 0 * l = (normalize : (a : α) → α) (a * 0)), l * (↑(normUnit l) : α) = l
Tactic: obtain rfl | hb := eq_or_ne b 0
Cleared hypotheses: b, l, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
neg :
  let l := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯;
  ∀ (this : gcd a 0 * l = (normalize : (a : α) → α) (a * 0)), l * (↑(normUnit l) : α) = l
⊢ l * (↑(normUnit l) : α) = l
have neg₁ : ∀ (hb : b ≠ 0), l * (↑(normUnit l) : α) = l
Tactic: obtain rfl | hb := eq_or_ne b 0
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
this : gcd a 0 * l = (normalize : (a : α) → α) (a * 0)
⊢ l * (↑(normUnit l) : α) = l
have this₁ : gcd a 0 = 0 ∨ l = 0
Tactic: rw [mul_zero a, normalize_zero, mul_eq_zero] at this
Cleared hypotheses: 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
this : gcd a 0 = 0 ∨ l = 0
⊢ l * (↑(normUnit l) : α) = l
have neg : ∀ (ha : gcd a 0 = 0), l * (↑(normUnit l) : α) = l
Tactic: obtain ha | hl := this
Cleared hypotheses: 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
this : gcd a 0 = 0 ∨ l = 0
neg : ∀ (ha : gcd a 0 = 0), l * (↑(normUnit l) : α) = l
⊢ l * (↑(normUnit l) : α) = l
have neg₁ : ∀ (hl : l = 0), l * (↑(normUnit l) : α) = l
Tactic: obtain ha | hl := this
Cleared hypotheses: 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
ha : gcd a 0 = 0
⊢ l * (↑(normUnit l) : α) = l
have apply : a = 0
Tactic: apply (a0 _).elim
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
ha : gcd a 0 = 0
⊢ a = 0
have rw : gcd a 0 ∣ a
Tactic: rw [← zero_dvd_iff, ← ha]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
hl : l = 0
⊢ l * (↑(normUnit l) : α) = l
have neg : HMul.hMul (α := α) 0 (↑(normUnit 0) : α) = 0
Tactic: rw [hl, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * 0) = gcd a 0 * x) ⋯
hl : l = 0
neg : HMul.hMul (α := α) 0 (↑(normUnit 0) : α) = 0
⊢ l * (↑(normUnit l) : α) = l
have neg₁ : Eq.{u_1 + 1} (α := α) 0 0
Tactic: rw [hl, zero_mul]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
⊢ gcd a b ≠ 0
have hab : a * b ≠ 0
Tactic: have hab : a * b ≠ 0 := mul_ne_zero a0 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
hab : a * b ≠ 0
⊢ gcd a b ≠ 0
have contrapose : ∀ (hab : gcd a b = 0), a * b = 0
Tactic: contrapose hab
Cleared hypotheses: hab
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
hab : gcd a b = 0
⊢ a * b = 0
have rw : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) (0 : α)
Tactic: rw [← normalize_eq_zero, ← this, hab, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
hab : gcd a b = 0
rw : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) (0 : α)
⊢ a * b = 0
have rw₁ : gcd a b * l = 0
Tactic: rw [← normalize_eq_zero, ← this, hab, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
hab : gcd a b = 0
rw : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) (0 : α)
rw₁ : gcd a b * l = 0
⊢ a * b = 0
have rw₂ : 0 * l = 0
Tactic: rw [← normalize_eq_zero, ← this, hab, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
hab : gcd a b = 0
rw : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) (0 : α)
rw₁ : gcd a b * l = 0
rw₂ : 0 * l = 0
⊢ a * b = 0
have rw₃ : Eq.{u_1 + 1} (α := α) 0 0
Tactic: rw [← normalize_eq_zero, ← this, hab, zero_mul]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
⊢ l * (↑(normUnit l) : α) = l
have h1 : gcd a b ≠ 0
Tactic: have h1 : gcd a b ≠ 0 := by
          have hab : a * b ≠ 0 := mul_ne_zero a0 hb
          contrapose hab
          rw [← normalize_eq_zero, ← this, hab, 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 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
⊢ (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
have rw : Eq (α := α) ((normalize : (a : α) → α) ((normalize : (a : α) → α) (a * b) : α) : α)
  ((normalize : (a : α) → α) (a * b) : α)
Tactic: rw [this, normalize_idem]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
rw :
  Eq (α := α) ((normalize : (a : α) → α) ((normalize : (a : α) → α) (a * b) : α) : α)
    ((normalize : (a : α) → α) (a * b) : α)
⊢ (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
have rw₁ : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) ((normalize : (a : α) → α) (a * b) : α)
Tactic: rw [this, normalize_idem]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
⊢ l * (↑(normUnit l) : α) = l
have h2 : (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
Tactic: have h2 : normalize (gcd a b * l) = gcd a b * l := by rw [this, normalize_idem]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : gcd a b * l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
h2 : (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
⊢ l * (↑(normUnit l) : α) = l
have this₁ : HMul.hMul (α := α) ((normalize : (a : α) → α) (gcd a b) : α) l = (normalize : (a : α) → α) (a * b)
Tactic: rw [← normalize_gcd] at this
Cleared hypotheses: this, hb, h1, h2
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : HMul.hMul (α := α) ((normalize : (a : α) → α) (gcd a b) : α) l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
h2 : (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
⊢ l * (↑(normUnit l) : α) = l
have neg : ∀
  (h2 :
    HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (gcd a b) : α) ((normalize : (a : α) → α) l : α) =
      gcd a b * l),
  l * (↑(normUnit l) : α) = l
Tactic: rwa [normalize.map_mul, normalize_gcd, mul_right_inj' h1] at h2
Cleared hypotheses: h2
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : HMul.hMul (α := α) ((normalize : (a : α) → α) (gcd a b) : α) l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
h2 : (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
neg :
  ∀
    (h2 :
      HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (gcd a b) : α) ((normalize : (a : α) → α) l : α) =
        gcd a b * l),
    l * (↑(normUnit l) : α) = l
⊢ l * (↑(normUnit l) : α) = l
have neg₁ : ∀ (h2 : HMul.hMul (β := α) (gcd a b) ((normalize : (a : α) → α) l : α) = gcd a b * l), l * (↑(normUnit l) : α) = l
Tactic: rwa [normalize.map_mul, normalize_gcd, mul_right_inj' h1] at h2
Cleared hypotheses: h2
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
inst_2 : NormalizationMonoid α
inst_3 : 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
normalize_gcd : ∀ (a b : α), (normalize : (a : α) → α) (gcd a b) = gcd a b
a b : α
a0 : ¬a = 0
l : α := Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = gcd a b * x) ⋯
this : HMul.hMul (α := α) ((normalize : (a : α) → α) (gcd a b) : α) l = (normalize : (a : α) → α) (a * b)
hb : b ≠ 0
h1 : gcd a b ≠ 0
h2 : (normalize : (a : α) → α) (gcd a b * l) = gcd a b * l
neg :
  ∀
    (h2 :
      HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (gcd a b) : α) ((normalize : (a : α) → α) l : α) =
        gcd a b * l),
    l * (↑(normUnit l) : α) = l
neg₁ :
  ∀ (h2 : HMul.hMul (β := α) (gcd a b) ((normalize : (a : α) → α) l : α) = gcd a b * l), l * (↑(normUnit l) : α) = l
⊢ l * (↑(normUnit l) : α) = l
have neg₂ : ∀ (h2 : (normalize : (a : α) → α) l = l), l * (↑(normUnit l) : α) = l
Tactic: rwa [normalize.map_mul, normalize_gcd, mul_right_inj' h1] at h2
Cleared hypotheses: h2
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
⊢ (if a = 0 then b else if b = 0 then a else Classical.choose (p := fun (x : α) ↦ a * b = lcm a b * x) ⋯) ∣ a
have pos : ∀ (h : a = 0), b ∣ a
Tactic: split_ifs with h h_1
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
pos : ∀ (h : a = 0), b ∣ a
⊢ (if a = 0 then b else if b = 0 then a else Classical.choose (p := fun (x : α) ↦ a * b = lcm a b * x) ⋯) ∣ a
have pos₁ : ∀ (h : ¬a = 0) (h_1 : b = 0), a ∣ a
Tactic: split_ifs with h h_1
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
pos : ∀ (h : a = 0), b ∣ a
pos₁ : ∀ (h : ¬a = 0) (h_1 : b = 0), a ∣ a
⊢ (if a = 0 then b else if b = 0 then a else Classical.choose (p := fun (x : α) ↦ a * b = lcm a b * x) ⋯) ∣ a
have neg : ∀ (h : ¬a = 0) (h_1 : ¬b = 0), Classical.choose (p := fun (x : α) ↦ a * b = lcm a b * x) sorry ∣ a
Tactic: split_ifs with h h_1
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
h : a = 0
⊢ b ∣ a
have pos : b ∣ 0
Tactic: rw [h]
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
⊢ False
have h₁ : lcm a b ∣ a * b
Tactic: have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
h_2 : ¬a = 0
h_1 : ¬b = 0
con : lcm a b = 0
h : lcm a b ∣ a * b
⊢ False
have h₁ : a = 0 ∨ b = 0
Tactic: rw [con, zero_dvd_iff, mul_eq_zero] at h
Cleared hypotheses: h
-- 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
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
⊢ Classical.choose (p := fun (x : α) ↦ a * b = lcm a b * x) ⋯ ∣ a
have h0 : lcm a b ≠ 0
Tactic: have h0 : lcm a b ≠ 0 := by
        intro con
        have h := lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
        rw [con, zero_dvd_iff, mul_eq_zero] at h
        cases h
        · exact absurd ‹a = 0› h
        · exact absurd ‹b = 0› h_1