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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : b = 0
pos : Associated (HMul.hMul (α := α) ((normalize : (a : α) → α) a : α) (lcm a 0)) (a * 0)
pos₁ : Associated (HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) a : α) (0 : α)) (a * 0)
pos₂ : Associated 0 (a * 0)
⊢ Associated (HMul.hMul (α := α) ((normalize : (a : α) → α) a : α) (lcm a b)) (a * b)
have pos₃ : Associated.{u_1} (M := α) 0 0
Tactic: rw [h_1, eq_zero_of_zero_dvd (dvd_lcm_right _ _), mul_zero, mul_zero]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
⊢ Associated (Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = lcm a b * x) ⋯ * lcm a b) (a * b)
have neg : Associated ((normalize : (a : α) → α) (a * b)) (a * b)
Tactic: rw [mul_comm, ← Classical.choose_spec (exists_gcd a b)]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
⊢ Eq (α := α)
    ((normalize : (a : α) → α)
        (if a = 0 then (normalize : (a : α) → α) b
          else
            if b = 0 then (normalize : (a : α) → α) a
            else Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = lcm a b * x) ⋯ :
          α) :
      α)
    (if a = 0 then (normalize : (a : α) → α) b
      else
        if b = 0 then (normalize : (a : α) → α) a
        else Classical.choose (p := fun (x : α) ↦ (normalize : (a : α) → α) (a * b) = lcm a b * x) ⋯ :
      α)
have dsimp : (if a = 0 then b * (↑(normUnit b) : α)
    else
      if b = 0 then a * (↑(normUnit a) : α)
      else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry) *
    (↑(normUnit
          (if a = 0 then b * (↑(normUnit b) : α)
          else
            if b = 0 then a * (↑(normUnit a) : α)
            else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) :
      α) =
  if a = 0 then b * (↑(normUnit b) : α)
  else
    if b = 0 then a * (↑(normUnit a) : α)
    else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry
Tactic: dsimp [normalize]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
⊢ (if a = 0 then b * (↑(normUnit b) : α)
      else
        if b = 0 then a * (↑(normUnit a) : α)
        else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) *
      (↑(normUnit
            (if a = 0 then b * (↑(normUnit b) : α)
            else
              if b = 0 then a * (↑(normUnit a) : α)
              else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) :
        α) =
    if a = 0 then b * (↑(normUnit b) : α)
    else
      if b = 0 then a * (↑(normUnit a) : α)
      else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
have pos : ∀ (h : a = 0), b * (↑(normUnit b) : α) * (↑(normUnit (b * (↑(normUnit b) : α))) : α) = b * (↑(normUnit b) : α)
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 : NormalizationMonoid α
inst_3 : 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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
pos : ∀ (h : a = 0), b * (↑(normUnit b) : α) * (↑(normUnit (b * (↑(normUnit b) : α))) : α) = b * (↑(normUnit b) : α)
⊢ (if a = 0 then b * (↑(normUnit b) : α)
      else
        if b = 0 then a * (↑(normUnit a) : α)
        else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) *
      (↑(normUnit
            (if a = 0 then b * (↑(normUnit b) : α)
            else
              if b = 0 then a * (↑(normUnit a) : α)
              else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) :
        α) =
    if a = 0 then b * (↑(normUnit b) : α)
    else
      if b = 0 then a * (↑(normUnit a) : α)
      else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
have pos₁ : ∀ (h : ¬a = 0) (h_1 : b = 0),
  a * (↑(normUnit a) : α) * (↑(normUnit (a * (↑(normUnit a) : α))) : α) = a * (↑(normUnit a) : α)
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 : NormalizationMonoid α
inst_3 : 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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
pos : ∀ (h : a = 0), b * (↑(normUnit b) : α) * (↑(normUnit (b * (↑(normUnit b) : α))) : α) = b * (↑(normUnit b) : α)
pos₁ :
  ∀ (h : ¬a = 0) (h_1 : b = 0),
    a * (↑(normUnit a) : α) * (↑(normUnit (a * (↑(normUnit a) : α))) : α) = a * (↑(normUnit a) : α)
⊢ (if a = 0 then b * (↑(normUnit b) : α)
      else
        if b = 0 then a * (↑(normUnit a) : α)
        else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) *
      (↑(normUnit
            (if a = 0 then b * (↑(normUnit b) : α)
            else
              if b = 0 then a * (↑(normUnit a) : α)
              else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) :
        α) =
    if a = 0 then b * (↑(normUnit b) : α)
    else
      if b = 0 then a * (↑(normUnit a) : α)
      else Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
have neg : ∀ (h : ¬a = 0) (h_1 : ¬b = 0),
  Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α) =
    Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry
Tactic: split_ifs with h h_1
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
⊢ Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α) =
    Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
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
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α) =
    Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
have neg : lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α)) =
  lcm a b * Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry
Tactic: apply mul_left_cancel₀ h0
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ lcm a b *
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    lcm a b * Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯
have neg : lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α)) =
  (normalize : (a : α) → α) (a * b)
Tactic: refine _root_.trans ?_ (Classical.choose_spec (exists_gcd a b))
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
| lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α))
have a₁ : lcm a b
Tactic: congr
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
a₁ : lcm a b
| lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α))
have a₂ : Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
  (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α)
Tactic: congr
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
| lcm a b
have a₁ : ((normalize : (a : α) → α) (lcm a b) : α)
Tactic: rw [← normalize_lcm a b]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
| lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α))
have a₁ : ((normalize : (a : α) → α) (lcm a b) : α)
Tactic: congr
        rw [← normalize_lcm a b]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
a₁ : ((normalize : (a : α) → α) (lcm a b) : α)
| lcm a b *
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α))
have a₂ : Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
  (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α)
Tactic: congr
        rw [← normalize_lcm a b]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ lcm a b *
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    (normalize : (a : α) → α) (a * b)
have neg : HMul.hMul (α := α) ((normalize : (a : α) → α) (lcm a b) : α)
    (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry *
      (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry)) : α)) =
  (normalize : (a : α) → α) (a * b)
Tactic: conv_lhs =>
        congr
        rw [← normalize_lcm a b]
-- 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
⊢ HMul.hMul (α := α) ((normalize : (a : α) → α) (lcm a b) : α)
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    (normalize : (a : α) → α) (a * b)
have neg : HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (lcm a b) : α)
    ((normalize : (a : α) → α)
        (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry) :
      α) =
  (normalize : (a : α) → α) (a * b)
Tactic: rw [← normalize_apply, ← normalize.map_mul,
        ← Classical.choose_spec (exists_gcd a b), 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
neg :
  HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (lcm a b) : α)
      ((normalize : (a : α) → α)
          (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) :
        α) =
    (normalize : (a : α) → α) (a * b)
⊢ HMul.hMul (α := α) ((normalize : (a : α) → α) (lcm a b) : α)
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    (normalize : (a : α) → α) (a * b)
have neg₁ : Eq (α := α)
  ((normalize : (a : α) → α)
      (lcm a b * Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) sorry) :
    α)
  ((normalize : (a : α) → α) (a * b) : α)
Tactic: rw [← normalize_apply, ← normalize.map_mul,
        ← Classical.choose_spec (exists_gcd a b), 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
neg :
  HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (lcm a b) : α)
      ((normalize : (a : α) → α)
          (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) :
        α) =
    (normalize : (a : α) → α) (a * b)
neg₁ :
  Eq (α := α)
    ((normalize : (a : α) → α)
        (lcm a b * Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) :
      α)
    ((normalize : (a : α) → α) (a * b) : α)
⊢ HMul.hMul (α := α) ((normalize : (a : α) → α) (lcm a b) : α)
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    (normalize : (a : α) → α) (a * b)
have neg₂ : Eq (α := α) ((normalize : (a : α) → α) ((normalize : (a : α) → α) (a * b) : α) : α)
  ((normalize : (a : α) → α) (a * b) : α)
Tactic: rw [← normalize_apply, ← normalize.map_mul,
        ← Classical.choose_spec (exists_gcd a b), 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
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
normalize_lcm : ∀ (a b : α), (normalize : (a : α) → α) (lcm a b) = lcm a b
exists_gcd : ∀ (a b : α), lcm a b ∣ (normalize : (a : α) → α) (a * b) := ⋯
a b : α
h : ¬a = 0
h_1 : ¬b = 0
h0 : lcm a b ≠ 0
neg :
  HMul.hMul (α := α) (β := α) ((normalize : (a : α) → α) (lcm a b) : α)
      ((normalize : (a : α) → α)
          (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) :
        α) =
    (normalize : (a : α) → α) (a * b)
neg₁ :
  Eq (α := α)
    ((normalize : (a : α) → α)
        (lcm a b * Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯) :
      α)
    ((normalize : (a : α) → α) (a * b) : α)
neg₂ :
  Eq (α := α) ((normalize : (a : α) → α) ((normalize : (a : α) → α) (a * b) : α) : α)
    ((normalize : (a : α) → α) (a * b) : α)
⊢ HMul.hMul (α := α) ((normalize : (a : α) → α) (lcm a b) : α)
      (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯ *
        (↑(normUnit (Classical.choose (p := fun (x : α) ↦ a * b * (↑(normUnit (a * b)) : α) = lcm a b * x) ⋯)) : α)) =
    (normalize : (a : α) → α) (a * b)
have neg₃ : Eq (α := α) ((normalize : (a : α) → α) (a * b) : α) ((normalize : (a : α) → α) (a * b) : α)
Tactic: rw [← normalize_apply, ← normalize.map_mul,
        ← Classical.choose_spec (exists_gcd a b), normalize_idem]
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
G₀ : Type u_2
inst : CommGroupWithZero G₀
inst_1 : DecidableEq G₀
a b : G₀
⊢ Associated (HMul.hMul (α := G₀) (if a = 0 ∧ b = 0 then 0 else 1) (if a = 0 ∨ b = 0 then 0 else 1)) (a * b)
have pos : a = 0 ∧ b = 0 → a = 0 ∨ b = 0 → Associated (HMul.hMul (α := G₀) 0 0) (a * b)
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.Algebra.GCDMonoid.Basic (goal)
Goal:
α : Type u_1
G₀ : Type u_2
inst : CommGroupWithZero G₀
inst_1 : DecidableEq G₀
a b : G₀
pos : a = 0 ∧ b = 0 → a = 0 ∨ b = 0 → Associated (HMul.hMul (α := G₀) 0 0) (a * b)
⊢ Associated (HMul.hMul (α := G₀) (if a = 0 ∧ b = 0 then 0 else 1) (if a = 0 ∨ b = 0 then 0 else 1)) (a * b)
have neg : a = 0 ∧ b = 0 → ¬(a = 0 ∨ b = 0) → Associated (HMul.hMul (α := G₀) 0 1) (a * b)
Tactic: split_ifs