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 : Associates α
ha : a ≠ 0
k : ℕ
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count a.factors
h : Nontrivial α
⊢ ∃ (b : Associates α), a = b ^ k
have obtain : ∀ (a0 : α) (hz : a0 ≠ 0) (ha : Ne (α := Associates α) (Associates.mk a0) 0)
  (hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count (factors (Associates.mk a0))),
  ∃ (b : Associates α), Associates.mk a0 = b ^ k
Tactic: obtain ⟨a0, hz, rfl⟩ := exists_non_zero_rep ha
Cleared hypotheses: a, ha, hk
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count (factors (Associates.mk a0))
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k
have hk₁ : ∀ (p : Associates α),
  Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
Tactic: rw [factors_mk a0 hz] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
p : { a : Associates α // Irreducible a }
⊢ k ∣ Multiset.count p (factors' a0)
have pp : p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) sorry
Tactic: have pp : p = ⟨p.val, p.2⟩ := by simp only [Subtype.coe_eta]
-- 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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
p : { a : Associates α // Irreducible a }
pp : p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯
⊢ k ∣ Multiset.count p (factors' a0)
have rw : k ∣ (↑p : Associates α).count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
Tactic: rw [pp, ← count_some p.2]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k
have hk' : ∀ (p : { a : Associates α // Irreducible a }),
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) p →
    k ∣ Multiset.count p (factors' a0)
Tactic: have hk' : ∀ p, p ∈ factors' a0 → k ∣ (factors' a0).count p := by
    rintro p -
    have pp : p = ⟨p.val, p.2⟩ := by simp only [Subtype.coe_eta]
    rw [pp, ← count_some p.2]
    exact hk p.val p.2
-- 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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
hk' :
  ∀ (p : { a : Associates α // Irreducible a }),
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) p →
      k ∣ Multiset.count p (factors' a0)
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k
have obtain : ∀ (u : Multiset { a : Associates α // Irreducible a }) (hu : factors' a0 = k • u),
  ∃ (b : Associates α), Associates.mk a0 = b ^ k
Tactic: obtain ⟨u, hu⟩ := Multiset.exists_smul_of_dvd_count _ hk'
-- 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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
hk' :
  ∀ (p : { a : Associates α // Irreducible a }),
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) p →
      k ∣ Multiset.count p (factors' a0)
u : Multiset { a : Associates α // Irreducible a }
hu : factors' a0 = k • u
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k
have h : Associates.mk a0 = FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k
Tactic: use FactorSet.prod (u : FactorSet α)
-- 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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
hk' :
  ∀ (p : { a : Associates α // Irreducible a }),
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) p →
      k ∣ Multiset.count p (factors' a0)
u : Multiset { a : Associates α // Irreducible a }
hu : factors' a0 = k • u
⊢ Associates.mk a0 = FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k
have h : factors (Associates.mk a0) =
  (FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k).factors
Tactic: apply eq_of_factors_eq_factors
-- 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)
k : ℕ
h : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
hk :
  ∀ (p : Associates α),
    Irreducible p → k ∣ p.count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
hk' :
  ∀ (p : { a : Associates α // Irreducible a }),
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a0) p →
      k ∣ Multiset.count p (factors' a0)
u : Multiset { a : Associates α // Irreducible a }
hu : factors' a0 = k • u
⊢ factors (Associates.mk a0) =
    (FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k).factors
have h : (↑(k • u) : WithTop (Multiset { a : Associates α // Irreducible a })) =
  k • (↑u : WithTop (Multiset { a : Associates α // Irreducible a }))
Tactic: rw [pow_factors, prod_factors, factors_mk a0 hz, hu]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
⊢ a = p ^ p.count a.factors
have a₁ : Nontrivial α
Tactic: nontriviality α
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
⊢ a = p ^ p.count a.factors
have hph : p ^ n ≠ 0
Tactic: have hph := pow_ne_zero n hp.ne_zero
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
⊢ a = p ^ p.count a.factors
have ha : a ≠ 0
Tactic: have ha := ne_zero_of_dvd_ne_zero hph 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ a = p ^ p.count a.factors
have apply : ∀ (p_1 : Associates α), Irreducible p_1 → p_1.count a.factors = p_1.count (p ^ p.count a.factors).factors
Tactic: apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ q.count a.factors ≤ 0
have h₁ : 0 = q.count (p ^ n).factors
Tactic: convert! count_le_count_of_le hph hq 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ 0 = q.count (p ^ n).factors
have h₁ : q.count (p ^ n).factors = 0
Tactic: symm
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
⊢ q.count (p ^ n).factors = 0
have h₁ : n * q.count p.factors = 0
Tactic: rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
h₁ : n * q.count p.factors = 0
⊢ q.count (p ^ n).factors = 0
have h₂ : n * 0 = 0
Tactic: rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
q : Associates α
hq : Irreducible q
h' : q ≠ p
h₁ : n * q.count p.factors = 0
h₂ : n * 0 = 0
⊢ q.count (p ^ n).factors = 0
have h₃ : Eq (α := ℕ) 0 0
Tactic: rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ ∀ (p_1 : Associates α), Irreducible p_1 → p_1.count a.factors = p_1.count (p ^ p.count a.factors).factors
have eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
Tactic: have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
    Nat.eq_zero_of_le_zero <| by
      convert! count_le_count_of_le hph hq h
      symm
      rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
⊢ q.count a.factors = q.count (p ^ p.count a.factors).factors
have rw : q.count a.factors = p.count a.factors * q.count p.factors
Tactic: rw [count_pow hp.ne_zero hq]
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have pos : ∀ (h : q = p), q.count a.factors = p.count a.factors * q.count p.factors
Tactic: by_cases h : q = p
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
h_1 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
pos : ∀ (h : q = p), q.count a.factors = p.count a.factors * q.count p.factors
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have neg : ∀ (h : ¬q = p), q.count a.factors = p.count a.factors * q.count p.factors
Tactic: by_cases h : q = p
-- 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : q = p
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have pos : p.count a.factors = p.count a.factors * p.count p.factors
Tactic: rw [h, 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : q = p
pos : p.count a.factors = p.count a.factors * p.count p.factors
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have pos₁ : p.count a.factors = p.count a.factors * 1
Tactic: rw [h, 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : q = p
pos : p.count a.factors = p.count a.factors * p.count p.factors
pos₁ : p.count a.factors = p.count a.factors * 1
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have pos₂ : p.count a.factors = p.count a.factors
Tactic: rw [h, 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : ¬q = p
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have neg : q.count a.factors = p.count a.factors * 0
Tactic: rw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : ¬q = p
neg : q.count a.factors = p.count a.factors * 0
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have neg₁ : q.count a.factors = 0
Tactic: rw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq 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)
p a : Associates α
hp : Irreducible p
n : ℕ
h_1 : a ∣ p ^ n
h_2 : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
q : Associates α
hq : Irreducible q
h : ¬q = p
neg : q.count a.factors = p.count a.factors * 0
neg₁ : q.count a.factors = 0
⊢ q.count a.factors = p.count a.factors * q.count p.factors
have neg₂ : Eq (α := ℕ) 0 0
Tactic: rw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq 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
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count a.factors
have a₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry ≤ p.count a.factors
Tactic: apply le_antisymm
-- 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
a₁ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≤ p.count a.factors
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count a.factors
have a₂ : p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry
Tactic: apply le_antisymm
-- 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
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≤ p.count a.factors
have a₁ : p ^ p.count a.factors = a * 1
Tactic: refine Nat.find_le ⟨1, ?_⟩
-- 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
⊢ p ^ p.count a.factors = a * 1
have a₁ : p ^ p.count a.factors = a
Tactic: rw [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
⊢ p ^ p.count a.factors = a
have a₁ : a = p ^ p.count a.factors
Tactic: symm
-- Lean 4 proof step from Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet (hypothesis)
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
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
have hph : p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry ≠ 0
Tactic: have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
-- 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
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
have a₁ : ∀ (hα : Subsingleton α), p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry
Tactic: rcases subsingleton_or_nontrivial α with hα | hα
-- 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
a₁ : ∀ (hα : Subsingleton α), p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
have a₂ : ∀ (hα : Nontrivial α), p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) sorry
Tactic: rcases subsingleton_or_nontrivial α with hα | hα