awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a p : Associates α
hp : Irreducible p
inst_4 : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≠ 0
hα : Nontrivial α
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
have h₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry).factors
Tactic: convert! count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a p : Associates α
hp : Irreducible p
inst_4 : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≠ 0
hα : Nontrivial α
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors
have h₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry * p.count p.factors
Tactic: rw [count_pow hp.ne_zero hp, count_self hp, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a p : Associates α
hp : Irreducible p
inst_4 : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≠ 0
hα : Nontrivial α
h₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ * p.count p.factors
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors
have h₂ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry * 1
Tactic: rw [count_pow hp.ne_zero hp, count_self hp, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
inst_3 : (p : Associates α) → Decidable (Irreducible p)
a p : Associates α
hp : Irreducible p
inst_4 : (n : ℕ) → Decidable (a ∣ p ^ n)
n : ℕ
h : a ∣ p ^ n
hph : p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≠ 0
hα : Nontrivial α
h₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ * p.count p.factors
h₂ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ * 1
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors
have h₃ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry
Tactic: rw [count_pow hp.ne_zero hp, count_self hp, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have a₁ : Nontrivial α
Tactic: nontriviality α
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
⊢ ∃ (d : Associates α), a = d ^ k
have pos : ∀ (hk0 : k = 0), ∃ (d : Associates α), a = d ^ k
Tactic: by_cases hk0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
pos : ∀ (hk0 : k = 0), ∃ (d : Associates α), a = d ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have neg : ∀ (hk0 : ¬k = 0), ∃ (d : Associates α), a = d ^ k
Tactic: by_cases hk0 : k = 0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : k = 0
⊢ ∃ (d : Associates α), a = d ^ k
have h₁ : a = HPow.hPow (α := Associates α) 1 k
Tactic: use 1
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : k = 0
⊢ a = HPow.hPow (α := Associates α) 1 k
have h₁ : ∀ (h : a * b = 1), Nontrivial α → ∀ (hk0 : k = 0), a = 1
Tactic: rw [hk0, pow_zero] at h ⊢
Cleared hypotheses: h, h_1, hk0
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : ¬k = 0
⊢ ∃ (d : Associates α), a = d ^ k
have neg : ∀ (p : Associates α) (hp : Irreducible p), k ∣ p.count a.factors
Tactic: refine is_pow_of_dvd_count ha fun p hp ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
⊢ k ∣ p.count a.factors
have neg : k ∣ p.count (a * b).factors
Tactic: apply dvd_count_of_dvd_count_mul hb hp hab
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
⊢ k ∣ p.count (a * b).factors
have neg : k ∣ p.count (c ^ k).factors
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
h_1 : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
⊢ k ∣ p.count (c ^ k).factors
have apply : c ≠ 0
Tactic: apply dvd_count_pow _ hp
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h_1 : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
h : a * b = HPow.hPow (α := Associates α) 0 k
⊢ False
have h₁ : a * b = 0
Tactic: rw [zero_pow hk0] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have nontriviality : Subsingleton α → ∃ (d : Associates α), a = d ^ k
Tactic: classical
  nontriviality α
  by_cases hk0 : k = 0
  · use 1
    rw [hk0, pow_zero] at h ⊢
    apply (mul_eq_one.1 h).1
  · refine is_pow_of_dvd_count ha fun p hp ↦ ?_
    apply dvd_count_of_dvd_count_mul hb hp hab
    rw [h]
    apply dvd_count_pow _ hp
    rintro rfl
    rw [zero_pow hk0] at h
    cases mul_eq_zero.mp h <;> contradiction
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
nontriviality : Subsingleton α → ∃ (d : Associates α), a = d ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have nontriviality₁ : Nontrivial α → ∃ (d : Associates α), a = d ^ k
Tactic: classical
  nontriviality α
  by_cases hk0 : k = 0
  · use 1
    rw [hk0, pow_zero] at h ⊢
    apply (mul_eq_one.1 h).1
  · refine is_pow_of_dvd_count ha fun p hp ↦ ?_
    apply dvd_count_of_dvd_count_mul hb hp hab
    rw [h]
    apply dvd_count_pow _ hp
    rintro rfl
    rw [zero_pow hk0] at h
    cases mul_eq_zero.mp h <;> contradiction
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
nontriviality : Subsingleton α → ∃ (d : Associates α), a = d ^ k
nontriviality₁ : Nontrivial α → ∃ (d : Associates α), a = d ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have pos : Nontrivial α → ∀ (hk0 : k = 0), ∃ (d : Associates α), a = d ^ k
Tactic: classical
  nontriviality α
  by_cases hk0 : k = 0
  · use 1
    rw [hk0, pow_zero] at h ⊢
    apply (mul_eq_one.1 h).1
  · refine is_pow_of_dvd_count ha fun p hp ↦ ?_
    apply dvd_count_of_dvd_count_mul hb hp hab
    rw [h]
    apply dvd_count_pow _ hp
    rintro rfl
    rw [zero_pow hk0] at h
    cases mul_eq_zero.mp h <;> contradiction
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
h : a * b = c ^ k
nontriviality : Subsingleton α → ∃ (d : Associates α), a = d ^ k
nontriviality₁ : Nontrivial α → ∃ (d : Associates α), a = d ^ k
pos : Nontrivial α → ∀ (hk0 : k = 0), ∃ (d : Associates α), a = d ^ k
⊢ ∃ (d : Associates α), a = d ^ k
have neg : Nontrivial α → ∀ (hk0 : ¬k = 0), ∃ (d : Associates α), a = d ^ k
Tactic: classical
  nontriviality α
  by_cases hk0 : k = 0
  · use 1
    rw [hk0, pow_zero] at h ⊢
    apply (mul_eq_one.1 h).1
  · refine is_pow_of_dvd_count ha fun p hp ↦ ?_
    apply dvd_count_of_dvd_count_mul hb hp hab
    rw [h]
    apply dvd_count_pow _ hp
    rintro rfl
    rw [zero_pow hk0] at h
    cases mul_eq_zero.mp h <;> contradiction
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (hypothesis)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
⊢ Fintype { x : M // x ∣ y }
have this : Nontrivial M
Tactic: haveI : Nontrivial M := ⟨⟨y, 0, hy⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this : Nontrivial M
⊢ Fintype { x : M // x ∣ y }
have haveI : (this : NormalizationMonoid M) → Fintype { x : M // x ∣ y }
Tactic: haveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this : NormalizationMonoid M
⊢ Fintype { x : M // x ∣ y }
have haveI : (this : DecidableEq M) → Fintype { x : M // x ∣ y }
Tactic: haveI := Classical.decEq M
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this : DecidableEq M
⊢ Fintype { x : M // x ∣ y }
have haveI : (this : DecidableEq (Associates M)) → Fintype { x : M // x ∣ y }
Tactic: haveI := Classical.decEq (Associates M)
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
⊢ Fintype { x : M // x ∣ y }
have Finset : ∀ (x : M),
  Membership.mem (γ := Finset M)
      (Finset.image (fun (s : Multiset M × Mˣ) ↦ (↑s.2 : M) * s.1.prod)
        (SProd.sprod (β := Finset Mˣ) (normalizedFactors y).powerset.toFinset Finset.univ))
      x ↔
    x ∣ y
Tactic: refine
    Fintype.subtype
      (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
        (s.snd : M) * s.fst.prod)
      fun x => ?_
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ Membership.mem (γ := Finset M)
      (Finset.image (fun (s : Multiset M × Mˣ) ↦ (↑s.2 : M) * s.1.prod)
        (SProd.sprod (β := Finset Mˣ) (normalizedFactors y).powerset.toFinset Finset.univ))
      x ↔
    x ∣ y
have simp : (∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x) ↔ x ∣ y
Tactic: simp only [Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true,
    Multiset.mem_toFinset, Multiset.mem_powerset]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
⊢ (∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x) ↔ x ∣ y
have mp : (∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x) → x ∣ y
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
mp : (∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x) → x ∣ y
⊢ (∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x) ↔ x ∣ y
have mpr : x ∣ y → ∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ (↑s.2 : M) * s.1.prod ∣ y
have mp : s.1.prod ∣ (normalizedFactors y).prod
Tactic: rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (prod_normalizedFactors hy).dvd_iff_dvd_right]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
⊢ x ≠ 0
have refine : ∀ (hx : x = 0), y = 0
Tactic: refine mt (fun hx => ?_) hy
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x = 0
⊢ y = 0
have rwa : ∀ (h : 0 ∣ y) (hx : x = 0), y = 0
Tactic: rwa [hx, zero_dvd_iff] at h
Cleared hypotheses: h, hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x = 0
rwa : ∀ (h : 0 ∣ y) (hx : x = 0), y = 0
⊢ y = 0
have rwa₁ : ∀ (h : y = 0) (hx : x = 0), y = 0
Tactic: rwa [hx, zero_dvd_iff] at h
Cleared hypotheses: h, hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (hypothesis)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
⊢ ∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x
have hx : x ≠ 0
Tactic: have hx : x ≠ 0 := by
      refine mt (fun hx => ?_) hy
      rwa [hx, zero_dvd_iff] at h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x ≠ 0
⊢ ∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x
have mpr : ∀ (u : Mˣ) (hu : (normalizedFactors x).prod * (↑u : M) = x),
  ∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x
Tactic: obtain ⟨u, hu⟩ := prod_normalizedFactors hx
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finite (goal)
Goal:
α : Type u_1
M : Type u_2
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
inst_2 : Fintype Mˣ
y : M
hy : y ≠ 0
this_1 : Nontrivial M
this_2 : NormalizationMonoid M
this_3 : DecidableEq M
this : DecidableEq (Associates M)
x : M
h : x ∣ y
hx : x ≠ 0
u : Mˣ
hu : (normalizedFactors x).prod * (↑u : M) = x
⊢ ∃ (a : Multiset M × Mˣ), a.1 ≤ normalizedFactors y ∧ (↑a.2 : M) * a.1.prod = x
have mpr : (normalizedFactors x, u).1 ≤ normalizedFactors y
Tactic: refine ⟨⟨normalizedFactors x, u⟩, ?_, (mul_comm _ _).trans hu⟩
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finsupp (goal)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : NormalizationMonoid α
inst_3 : DecidableEq α
x : α
n : ℕ
⊢ factorization (x ^ n) = n • factorization x
have h : ∀ (a : α), Eq (α := ℕ) ((factorization (x ^ n) : (a : α) → ℕ) a : ℕ) ((n • factorization x : (a : α) → ℕ) a : ℕ)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.Finsupp (hypothesis)
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : NormalizationMonoid α
inst_3 : DecidableEq α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : factorization a = factorization b
⊢ a ~ᵤ b
have h₁ : normalizedFactors a = normalizedFactors b
Tactic: simp_rw [factorization, AddEquiv.apply_eq_iff_eq] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid (goal)
Goal:
α_1 : Type u_1
α : Type u_2
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
⊢ Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) ∣ a
have rw : LE.le (α := Associates α) (Associates.mk a ⊓ Associates.mk b) (Associates.mk a)
Tactic: rw [← mk_dvd_mk, Associates.quot_out, congr_fun₂ dvd_eq_le]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid (goal)
Goal:
α_1 : Type u_1
α : Type u_2
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
⊢ Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) ∣ b
have rw : LE.le (α := Associates α) (Associates.mk a ⊓ Associates.mk b) (Associates.mk b)
Tactic: rw [← mk_dvd_mk, Associates.quot_out, congr_fun₂ dvd_eq_le]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid (goal)
Goal:
α_1 : Type u_1
α : Type u_2
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b c : α
hac : a ∣ c
hab : a ∣ b
⊢ a ∣ Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk c ⊓ Associates.mk b)
have rw : a ∣ c ∧ a ∣ b
Tactic: rw [← mk_dvd_mk, Associates.quot_out, congr_fun₂ dvd_eq_le, le_inf_iff,
      mk_le_mk_iff_dvd, mk_le_mk_iff_dvd]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid (goal)
Goal:
α_1 : Type u_1
α : Type u_2
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
⊢ Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) *
      Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊔ Associates.mk b) ~ᵤ
    a * b
have rw : Eq (α := Associates α)
  (Associates.mk
    (Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) *
      Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊔ Associates.mk b)))
  (Associates.mk (a * b))
Tactic: rw [← mk_eq_mk_iff_associated, ← Associates.mk_mul_mk, ← associated_iff_eq, Associates.quot_out,
      Associates.quot_out, mul_comm, sup_mul_inf, Associates.mk_mul_mk]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid (goal)
Goal:
α_1 : Type u_1
α : Type u_2
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
rw :
  Eq (α := Associates α)
    (Associates.mk
      (Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) *
        Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊔ Associates.mk b)))
    (Associates.mk (a * b))
⊢ Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b) *
      Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊔ Associates.mk b) ~ᵤ
    a * b
have rw₁ : HMul.hMul (α := Associates α)
    (Associates.mk (Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊓ Associates.mk b)))
    (Associates.mk (Quot.out (r := (⇑(Associated.setoid α) : α → α → Prop)) (Associates.mk a ⊔ Associates.mk b))) =
  Associates.mk (a * b)
Tactic: rw [← mk_eq_mk_iff_associated, ← Associates.mk_mul_mk, ← associated_iff_eq, Associates.quot_out,
      Associates.quot_out, mul_comm, sup_mul_inf, Associates.mk_mul_mk]