awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
pos : ∀ (ha : a = 0), a ^ n ∣ b ^ n ↔ a ∣ b
⊢ a ^ n ∣ b ^ n ↔ a ∣ b
have neg : ∀ (ha : ¬a = 0), a ^ n ∣ b ^ n ↔ a ∣ b
Tactic: by_cases ha : a = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
⊢ a ^ n ∣ b ^ n ↔ a ∣ b
have neg : a ^ n ∣ b ^ n → a ∣ b
Tactic: refine ⟨?_, fun h ↦ pow_dvd_pow_of_dvd h n⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
⊢ a ^ n ∣ b ^ n → a ∣ b
have neg : (∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)) →
  ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b
Tactic: rw [dvd_iff_emultiplicity_le (pow_ne_zero n ha), dvd_iff_emultiplicity_le ha]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (hypothesis)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
⊢ emultiplicity p a ≤ emultiplicity p b
have this : emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
Tactic: have := H p hp
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
this : emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
⊢ emultiplicity p a ≤ emultiplicity p b
have neg : ∀ (this : (↑n : ℕ∞) * emultiplicity p a ≤ emultiplicity p (b ^ n)), emultiplicity p a ≤ emultiplicity p b
Tactic: rwa [emultiplicity_pow hp, emultiplicity_pow hp,
    ENat.mul_le_mul_left_iff (by exact_mod_cast hn) (ENat.coe_ne_top _)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
this : emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
neg : ∀ (this : (↑n : ℕ∞) * emultiplicity p a ≤ emultiplicity p (b ^ n)), emultiplicity p a ≤ emultiplicity p b
⊢ emultiplicity p a ≤ emultiplicity p b
have neg₁ : ∀ (this : (↑n : ℕ∞) * emultiplicity p a ≤ (↑n : ℕ∞) * emultiplicity p b), emultiplicity p a ≤ emultiplicity p b
Tactic: rwa [emultiplicity_pow hp, emultiplicity_pow hp,
    ENat.mul_le_mul_left_iff (by exact_mod_cast hn) (ENat.coe_ne_top _)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
a b : R
n : ℕ
hn : n ≠ 0
ha : ¬a = 0
H : ∀ (p : R), Prime p → emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
p : R
hp : Prime p
this : emultiplicity p (a ^ n) ≤ emultiplicity p (b ^ n)
neg : ∀ (this : (↑n : ℕ∞) * emultiplicity p a ≤ emultiplicity p (b ^ n)), emultiplicity p a ≤ emultiplicity p b
neg₁ : ∀ (this : (↑n : ℕ∞) * emultiplicity p a ≤ (↑n : ℕ∞) * emultiplicity p b), emultiplicity p a ≤ emultiplicity p b
⊢ emultiplicity p a ≤ emultiplicity p b
have neg₂ : ∀ (this : emultiplicity p a ≤ emultiplicity p b), emultiplicity p a ≤ emultiplicity p b
Tactic: rwa [emultiplicity_pow hp, emultiplicity_pow hp,
    ENat.mul_le_mul_left_iff (by exact_mod_cast hn) (ENat.coe_ne_top _)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
α : Type u_3
M : Type u_4
inst_2 : CommMonoid M
inst_3 : Subsingleton.{u_2 + 1} Rˣ
f : α → M
g : α → R
hgi : Function.Injective g
hg : ∀ (s : α), Irreducible (g s)
r : R
hr : r ≠ 0
⊢ Function.HasFiniteMulSupport.{u_3, u_4} (M := M) fun (s : α) ↦ f s ^ multiplicity (g s) r
have simp : Function.HasFiniteMulSupport.{u_3, u_4} (M := M) fun (s : α) ↦ f s ^ Multiset.count (g s) (normalizedFactors r)
Tactic: simp only [multiplicity_eq_count_normalizedFactors (hg _) hr, normalize_eq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity (goal)
Goal:
R : Type u_2
inst : CommMonoidWithZero R
inst_1 : UniqueFactorizationMonoid R
α : Type u_3
M : Type u_4
inst_2 : CommMonoid M
inst_3 : Subsingleton.{u_2 + 1} Rˣ
f : α → M
g : α → R
hgi : Function.Injective g
hg : ∀ (s : α), Irreducible (g s)
r : R
hr : r ≠ 0
⊢ Function.HasFiniteMulSupport.{u_3, u_4} (M := M) fun (s : α) ↦ f s ^ multiplicity (g s) r
have simp : Function.HasFiniteMulSupport.{u_3, u_4} (M := M) fun (s : α) ↦ f s ^ Multiset.count (g s) (normalizedFactors r)
Tactic: classical
  simp only [multiplicity_eq_count_normalizedFactors (hg _) hr, normalize_eq]
  fun_prop
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
⊢ WellFounded (α := ℕ) DvdNotUnit
have refine : ∀ {a b : ℕ},
  DvdNotUnit a b →
    LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) a)
      ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
Tactic: refine RelHomClass.wellFounded
      (⟨fun x : ℕ => if x = 0 then (⊤ : ℕ∞) else x, ?_⟩ : DvdNotUnit →r (· < ·)) wellFounded_lt
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
a b : ℕ
h : DvdNotUnit a b
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) a)
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
have zero : ∀ (h : DvdNotUnit 0 b),
  LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) 0)
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
Tactic: rcases a with - | a
Cleared hypotheses: a, h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
a b : ℕ
h : DvdNotUnit a b
zero :
  ∀ (h : DvdNotUnit 0 b),
    LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) 0)
      ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) a)
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
have succ : ∀ (a : ℕ) (h : DvdNotUnit (a + 1) b),
  LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
Tactic: rcases a with - | a
Cleared hypotheses: a, h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
b : ℕ
h : DvdNotUnit 0 b
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) 0)
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
have zero : False
Tactic: exfalso
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
b : ℕ
h : DvdNotUnit 0 b
⊢ False
have zero : ∀ (h : DvdNotUnit 0 b), False
Tactic: revert h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
b a : ℕ
h : DvdNotUnit (a + 1) b
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
have succ : ∀ (h : DvdNotUnit (a + 1) 0),
  LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) 0)
Tactic: cases b
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
b a : ℕ
h : DvdNotUnit (a + 1) b
succ :
  ∀ (h : DvdNotUnit (a + 1) 0),
    LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
      ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) 0)
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) b)
have succ₁ : ∀ (n : ℕ) (h : DvdNotUnit (a + 1) (n + 1)),
  LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (n + 1))
Tactic: cases b
Cleared hypotheses: b, h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (hypothesis)
Goal:
a n : ℕ
h : DvdNotUnit (a + 1) (n + 1)
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (n + 1))
have h1 : a + 1 ∣ n + 1
Tactic: obtain ⟨h1, h2⟩ := dvd_and_not_dvd_iff.2 h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (hypothesis)
Goal:
a n : ℕ
h : DvdNotUnit (a + 1) (n + 1)
h1 : a + 1 ∣ n + 1
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (n + 1))
have h2 : ¬n + 1 ∣ a + 1
Tactic: obtain ⟨h1, h2⟩ := dvd_and_not_dvd_iff.2 h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
a n : ℕ
h : DvdNotUnit (a + 1) (n + 1)
h1 : a + 1 ∣ n + 1
h2 : ¬n + 1 ∣ a + 1
⊢ LT.lt (α := ℕ∞) ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (a + 1))
    ((fun (x : ℕ) ↦ (if x = 0 then ⊤ else (↑x : ℕ∞) : ℕ∞)) (n + 1))
have succ : a + 1 < n + 1
Tactic: simp only [succ_ne_zero, cast_lt, if_false]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
a n : ℕ
h : DvdNotUnit (a + 1) (n + 1)
h1 : a + 1 ∣ n + 1
h2 : ¬n + 1 ∣ a + 1
⊢ a + 1 < n + 1
have succ : ∀ (con : a + 1 = n + 1), n + 1 ∣ a + 1
Tactic: refine lt_of_le_of_ne (Nat.le_of_dvd (Nat.succ_pos _) h1) fun con => h2 ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
a n : ℕ
h : DvdNotUnit (a + 1) (n + 1)
h1 : a + 1 ∣ n + 1
h2 : ¬n + 1 ∣ a + 1
con : a + 1 = n + 1
⊢ n + 1 ∣ a + 1
have succ : n + 1 ∣ n + 1
Tactic: rw [con]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
n : ℕ
⊢ normalizedFactors (n + 1) = (↑(n + 1).primeFactorsList : Multiset ℕ)
have rw : Multiset.Rel (fun (x1 x2 : ℕ) ↦ Associated x1 x2) (normalizedFactors (n + 1)) (↑(n + 1).primeFactorsList : Multiset ℕ)
Tactic: rw [← Multiset.rel_eq, ← associated_eq_eq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
n : ℕ
⊢ Multiset.Rel (fun (x1 x2 : ℕ) ↦ Associated x1 x2) (normalizedFactors (n + 1)) (↑(n + 1).primeFactorsList : Multiset ℕ)
have apply : Associated (normalizedFactors (n + 1)).prod (↑(n + 1).primeFactorsList : Multiset ℕ).prod
Tactic: apply UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor _
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
n : ℕ
apply : Associated (normalizedFactors (n + 1)).prod (↑(n + 1).primeFactorsList : Multiset ℕ).prod
⊢ Multiset.Rel (fun (x1 x2 : ℕ) ↦ Associated x1 x2) (normalizedFactors (n + 1)) (↑(n + 1).primeFactorsList : Multiset ℕ)
have apply₁ : ∀ x ∈ (↑(n + 1).primeFactorsList : Multiset ℕ), Irreducible x
Tactic: apply UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor _
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
n : ℕ
⊢ Associated (normalizedFactors (n + 1)).prod (↑(n + 1).primeFactorsList : Multiset ℕ).prod
have rw : Associated (normalizedFactors (n + 1)).prod n.succ
Tactic: rw [Multiset.prod_coe, Nat.prod_primeFactorsList n.succ_ne_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
s : Multiset ℕ
h : ∀ x ∈ s, Irreducible x
⊢ normalizedFactors s.prod = s
have rw : Multiset.Rel (fun (x1 x2 : ℕ) ↦ Associated x1 x2) (normalizedFactors s.prod) s
Tactic: rw [← Multiset.rel_eq, ← associated_eq_eq]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
s : Multiset ℕ
h : ∀ x ∈ s, Irreducible x
⊢ Multiset.Rel (fun (x1 x2 : ℕ) ↦ Associated x1 x2) (normalizedFactors s.prod) s
have apply : s.prod ≠ 0
Tactic: apply UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor h
    (prod_normalizedFactors _)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Nat (goal)
Goal:
s : Multiset ℕ
h : ∀ x ∈ s, Irreducible x
⊢ s.prod ≠ 0
have rw : 0 ∉ s
Tactic: rw [Ne, Multiset.prod_eq_zero_iff]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Subsingleton.{u_2 + 1} Mˣ
x : M
⊢ factors x = Multiset.map (⇑normalize : (a : M) → M) (factors x)
have h : ∀ x_1 ∈ factors x, Eq.{u_2 + 1} (α := (a : M) → M) (⇑normalize : (a : M) → M) id
Tactic: convert! (Multiset.map_id (factors x)).symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Subsingleton.{u_2 + 1} Mˣ
x x_1 : M
h : x_1 ∈ factors x
⊢ Eq.{u_2 + 1} (α := (a : M) → M) (⇑normalize : (a : M) → M) id
have h : ∀ (p : M), (normalize : (a : M) → M) p = id p
Tactic: ext p
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (normalizedFactors a).prod ~ᵤ a
have rw : Multiset.prod
    (Multiset.map (⇑normalize : (a : α) → α)
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry)) ~ᵤ
  a
Tactic: rw [normalizedFactors, factors, dif_neg ane0]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod
      (Multiset.map (⇑normalize : (a : α) → α)
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯)) ~ᵤ
    a
have refine : Multiset.prod
    (Multiset.map (⇑normalize : (a : α) → α)
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry)) ~ᵤ
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry).prod
Tactic: refine Associated.trans ?_ (Classical.choose_spec (exists_prime_factors a ane0)).2
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Multiset.prod
      (Multiset.map (⇑normalize : (a : α) → α)
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯)) ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod
have rw : Eq (α := Associates α)
  (Multiset.prod
    (Multiset.map (Associates.mk ∘ (⇑normalize : (a : α) → α))
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry)))
  (Multiset.prod
    (Multiset.map Associates.mk (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry)))
Tactic: rw [← Associates.mk_eq_mk_iff_associated, ← Associates.prod_mk, ← Associates.prod_mk,
    Multiset.map_map]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Eq (α := Associates α)
    (Multiset.prod
      (Multiset.map (Associates.mk ∘ (⇑normalize : (a : α) → α))
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯)))
    (Multiset.prod
      (Multiset.map Associates.mk (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯)))
have e_a : Eq.{u_1 + 1} (α := α → Associates α) (Associates.mk ∘ (⇑normalize : (a : α) → α)) Associates.mk
Tactic: congr 2
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ Eq.{u_1 + 1} (α := α → Associates α) (Associates.mk ∘ (⇑normalize : (a : α) → α)) Associates.mk
have e_a : ∀ (x : α), Eq (α := Associates α) ((Associates.mk ∘ (⇑normalize : (a : α) → α)) x) (Associates.mk x)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
x : α
⊢ Eq (α := Associates α) ((Associates.mk ∘ (⇑normalize : (a : α) → α)) x) (Associates.mk x)
have e_a : Eq (α := Associates α) (Associates.mk ((normalize : (a : α) → α) x) : Associates α) (Associates.mk x : Associates α)
Tactic: rw [Function.comp_apply, Associates.mk_normalize]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
x : α
e_a :
  Eq (α := Associates α) (Associates.mk ((normalize : (a : α) → α) x) : Associates α) (Associates.mk x : Associates α)
⊢ Eq (α := Associates α) ((Associates.mk ∘ (⇑normalize : (a : α) → α)) x) (Associates.mk x)
have e_a₁ : Eq (α := Associates α) (Associates.mk x) (Associates.mk x)
Tactic: rw [Function.comp_apply, Associates.mk_normalize]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (normalizedFactors a).prod = (normalize : (a : α) → α) a
have trans : (normalizedFactors a).prod = (normalize : (a : α) → α) (normalizedFactors a).prod
Tactic: trans normalize (normalizedFactors a).prod
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
trans : (normalizedFactors a).prod = (normalize : (a : α) → α) (normalizedFactors a).prod
⊢ (normalizedFactors a).prod = (normalize : (a : α) → α) a
have trans₁ : Eq (α := α) ((normalize : (a : α) → α) (normalizedFactors a).prod : α) ((normalize : (a : α) → α) a : α)
Tactic: trans normalize (normalizedFactors a).prod
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (normalizedFactors a).prod = (normalize : (a : α) → α) (normalizedFactors a).prod
have rw : Eq (α := α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a)) : α)
  ((normalize : (a : α) → α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a))) : α)
Tactic: rw [normalizedFactors, ← map_multiset_prod, normalize_idem]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
rw :
  Eq (α := α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a)) : α)
    ((normalize : (a : α) → α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a))) : α)
⊢ (normalizedFactors a).prod = (normalize : (a : α) → α) (normalizedFactors a).prod
have rw₁ : Eq (α := α) ((normalize : (a : α) → α) (factors a).prod : α)
  ((normalize : (a : α) → α) ((normalize : (a : α) → α) (factors a).prod : α) : α)
Tactic: rw [normalizedFactors, ← map_multiset_prod, normalize_idem]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
rw :
  Eq (α := α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a)) : α)
    ((normalize : (a : α) → α) (Multiset.prod (Multiset.map (⇑normalize : (a : α) → α) (factors a))) : α)
rw₁ :
  Eq (α := α) ((normalize : (a : α) → α) (factors a).prod : α)
    ((normalize : (a : α) → α) ((normalize : (a : α) → α) (factors a).prod : α) : α)
⊢ (normalizedFactors a).prod = (normalize : (a : α) → α) (normalizedFactors a).prod
have rw₂ : Eq (α := α) ((normalize : (a : α) → α) (factors a).prod : α) ((normalize : (a : α) → α) (factors a).prod : α)
Tactic: rw [normalizedFactors, ← map_multiset_prod, normalize_idem]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
⊢ ∀ (x : α),
    Membership.mem (γ := Multiset α)
        (Multiset.map (α := α) (⇑normalize : (a : α) → α)
          (if h : a = 0 then 0 else Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
        x →
      Prime x
have pos : ∀ (ane0 : a = 0) (x : α),
  Membership.mem (γ := Multiset α) (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x) 0) x → Prime x
Tactic: split_ifs with ane0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
pos :
  ∀ (ane0 : a = 0) (x : α),
    Membership.mem (γ := Multiset α) (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x) 0) x → Prime x
⊢ ∀ (x : α),
    Membership.mem (γ := Multiset α)
        (Multiset.map (α := α) (⇑normalize : (a : α) → α)
          (if h : a = 0 then 0 else Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
        x →
      Prime x
have neg : ∀ (ane0 : ¬a = 0) (x : α),
  Membership.mem (γ := Multiset α)
      (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry))
      x →
    Prime x
Tactic: split_ifs with ane0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
x : α
hx :
  Membership.mem (γ := Multiset α)
    (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
    x
⊢ Prime x
have neg : ∀ (y : α) (hy : y ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry)
  (hx :
    Membership.mem (α := α) (γ := Multiset α)
      (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
        (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) sorry))
      ((normalize : (a : α) → α) y : α)),
  Prime (M := α) ((normalize : (a : α) → α) y : α)
Tactic: rcases Multiset.mem_map.1 hx with ⟨y, ⟨hy, rfl⟩⟩
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : NormalizationMonoid α
inst_2 : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
y : α
hy : y ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯
hx :
  Membership.mem (α := α) (γ := Multiset α)
    (Multiset.map (fun (x : α) ↦ (normalize : (a : α) → α) x)
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯))
    ((normalize : (a : α) → α) y : α)
⊢ Prime (M := α) ((normalize : (a : α) → α) y : α)
have neg : Prime y
Tactic: rw [(normalize_associated _).prime_iff]