awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
have pos : ∀ (h : b = 0),
  LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card : ℕ∞))
Tactic: by_cases h : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
pos :
  ∀ (h : b = 0),
    LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
      (if h : b = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
have neg : ∀ (h : ¬b = 0),
  LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card : ℕ∞))
Tactic: by_cases h : b = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
have neg : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card <
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card
Tactic: rw [dif_neg h, Nat.cast_lt]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ c ≠ 0
have refine : ∀ (con : c = 0), b = 0
Tactic: refine mt (fun con => ?_) h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
con : c = 0
⊢ b = 0
have rw : a * c = 0
Tactic: rw [b_eq, con, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
con : c = 0
rw : a * c = 0
⊢ b = 0
have rw₁ : a * 0 = 0
Tactic: rw [b_eq, con, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
con : c = 0
rw : a * c = 0
rw₁ : a * 0 = 0
⊢ b = 0
have rw₂ : Eq.{u_1 + 1} (α := α) 0 0
Tactic: rw [b_eq, con, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card
have cne0 : c ≠ 0
Tactic: have cne0 : c ≠ 0 := by
          refine mt (fun con => ?_) h
          rw [b_eq, con, mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card
have neg : ∀ (con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry = 0), c ~ᵤ 1
Tactic: calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
neg : ∀ (con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0), c ~ᵤ 1
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card
have neg₁ : ∀
  x ∈
    Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry +
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry,
  Prime x
Tactic: calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
neg : ∀ (con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0), c ~ᵤ 1
neg₁ :
  ∀
    x ∈
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯,
    Prime x
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card
have neg₂ : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry +
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry).prod ~ᵤ
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).prod
Tactic: calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
⊢ c ~ᵤ 1
have h₁ : 1 = (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry).prod
Tactic: convert! (Classical.choose_spec (pf c cne0)).2.symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
⊢ 1 = (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod
have h₁ : Eq.{u_1 + 1} (α := α) 1 (Multiset.prod 0)
Tactic: rw [con, Multiset.prod_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
h₁ : Eq.{u_1 + 1} (α := α) 1 (Multiset.prod 0)
⊢ 1 = (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod
have h₂ : Eq.{u_1 + 1} (α := α) 1 1
Tactic: rw [con, Multiset.prod_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
x : α
hadd :
  x ∈
    Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
⊢ Prime x
have hadd₁ : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry ∨
  x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry
Tactic: rw [Multiset.mem_add] at hadd
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
x : α
hadd :
  x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ ∨
    x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
⊢ Prime x
have neg : ∀ (h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry), Prime x
Tactic: rcases hadd with h | h
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
x : α
hadd :
  x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ ∨
    x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
neg : ∀ (h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯), Prime x
⊢ Prime x
have neg₁ : ∀ (h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry), Prime x
Tactic: rcases hadd with h | h
Cleared hypotheses: hadd
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
have neg : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).prod *
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry).prod ~ᵤ
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).prod
Tactic: rw [Multiset.prod_add]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
have trans : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).prod *
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry).prod ~ᵤ
  a * c
Tactic: trans a * c
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
trans :
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    a * c
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
have trans₁ : a * c ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).prod
Tactic: trans a * c
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    a * c
have h₁ : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).prod ~ᵤ a
Tactic: apply Associated.mul_mul
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
h₁ : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod ~ᵤ a
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    a * c
have h₂ : (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) sorry).prod ~ᵤ c
Tactic: apply Associated.mul_mul
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ a * c ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
have rw : b ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).prod
Tactic: rw [← b_eq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ b ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
have apply : b ≠ 0
Tactic: apply (Classical.choose_spec (pf _ _)).2.symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit
have refine_1 : α → ℕ∞
Tactic: classical
      refine RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h, Nat.cast_lt]
        have cne0 : c ≠ 0 := by
          refine mt (fun con => ?_) h
          rw [b_eq, con, mul_zero]
        calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
        · convert! (Classical.choose_spec (pf c cne0)).2.symm
          rw [con, Multiset.prod_zero]
        · intro x hadd
          rw [Multiset.mem_add] at hadd
          rcases hadd with h | h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption
        · rw [Multiset.prod_add]
          trans a * c
          · apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption
          · rw [← b_eq]
            apply (Classical.choose_spec (pf _ _)).2.symm; assumption
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Basic (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
refine_1 : α → ℕ∞
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit
have refine_2 : ∀ {a b : α},
  DvdNotUnit a b →
    LT.lt (α := ℕ∞)
      (if h : a = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).card : ℕ∞))
      (if h : b = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) sorry).card : ℕ∞))
Tactic: classical
      refine RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt
      · intro a
        by_cases h : a = 0
        · exact ⊤
        exact ↑(Multiset.card (Classical.choose (pf a h)))
      rintro a b ⟨ane0, ⟨c, hc, b_eq⟩⟩
      rw [dif_neg ane0]
      by_cases h : b = 0
      · simp [h, lt_top_iff_ne_top]
      · rw [dif_neg h, Nat.cast_lt]
        have cne0 : c ≠ 0 := by
          refine mt (fun con => ?_) h
          rw [b_eq, con, mul_zero]
        calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)
        · convert! (Classical.choose_spec (pf c cne0)).2.symm
          rw [con, Multiset.prod_zero]
        · intro x hadd
          rw [Multiset.mem_add] at hadd
          rcases hadd with h | h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption
        · rw [Multiset.prod_add]
          trans a * c
          · apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption
          · rw [← b_eq]
            apply (Classical.choose_spec (pf _ _)).2.symm; assumption