awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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_1 : 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
h : Associates.mk a0 = FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k

Tactic:
use FactorSet.prod (u : FactorSet α)

Premises used: WithTop.some, Associates.FactorSet.prod
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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_1 : 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
h :
  factors (Associates.mk a0) =
    (FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k).factors
⊢ Associates.mk a0 = FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k

Tactic:
apply eq_of_factors_eq_factors

Premises used: Associates.eq_of_factors_eq_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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_1 : 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
h :
  (↑(k • u) : WithTop (Multiset { a : Associates α // Irreducible a })) =
    k • (↑u : WithTop (Multiset { a : Associates α // Irreducible a }))
⊢ factors (Associates.mk a0) =
    (FactorSet.prod (↑u : WithTop (Multiset { a : Associates α // Irreducible a })) ^ k).factors

Tactic:
rw [pow_factors, prod_factors, factors_mk a0 hz, hu]

Premises used: Associates.pow_factors, Associates.factors_mk, Associates.prod_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ (↑(k • u) : WithTop (Multiset { a : Associates α // Irreducible a })) =
    k • (↑u : WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
WithBot.coe_nsmul u k

Premises used: WithTop.some, WithBot.coe_nsmul, WithTop
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ (↑(k • u) : WithTop (Multiset { a : Associates α // Irreducible a })) =
    k • (↑u : WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
exact WithBot.coe_nsmul u k

Premises used: WithBot.coe_nsmul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ Nontrivial α

Tactic:
nontriviality α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ p ^ n ≠ 0

Tactic:
have hph := pow_ne_zero n hp.ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 ≠ 0

Tactic:
have ha := ne_zero_of_dvd_ne_zero hph h
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
apply : ∀ (p_1 : Associates α), Irreducible p_1 → p_1.count a.factors = p_1.count (p ^ p.count a.factors).factors
⊢ a = p ^ p.count a.factors

Tactic:
apply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)

Premises used: Associates.eq_of_eq_counts, pow_ne_zero, Irreducible.ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 ≤ q.count (p ^ n).factors

Tactic:
count_le_count_of_le hph hq h

Premises used: Associates.count_le_count_of_le
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : 0 = q.count (p ^ n).factors
⊢ q.count a.factors ≤ 0

Tactic:
convert! count_le_count_of_le hph hq h

Premises used: Associates.count_le_count_of_le
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : q.count (p ^ n).factors = 0
⊢ 0 = q.count (p ^ n).factors

Tactic:
symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
h₃ h₄ : Eq (α := ℕ) 0 0
⊢ q.count (p ^ n).factors = 0

Tactic:
rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]

Premises used: Associates.count_pow, MulZeroClass.mul_zero, Irreducible.ne_zero, Associates.count_eq_zero_of_ne
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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), 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]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
rw : q.count a.factors = p.count a.factors * q.count p.factors
⊢ q.count a.factors = q.count (p ^ p.count a.factors).factors

Tactic:
rw [count_pow hp.ne_zero hq]

Premises used: Associates.count_pow, Irreducible.ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg : ∀ (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

Tactic:
by_cases h : q = p
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
pos₂ pos₃ : p.count a.factors = p.count a.factors
⊢ q.count a.factors = p.count a.factors * q.count p.factors

Tactic:
rw [h, count_self hp, mul_one]

Premises used: Associates.count_self, mul_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg₂ neg₃ : Eq (α := ℕ) 0 0
⊢ q.count a.factors = p.count a.factors * q.count p.factors

Tactic:
rw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq h]

Premises used: MulZeroClass.mul_zero, Associates.count_eq_zero_of_ne
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
a₂ : p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count a.factors

Tactic:
apply le_antisymm

Premises used: le_antisymm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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

Tactic:
Nat.find_le ⟨1, ?_⟩

Premises used: Nat.find_le, Exists.intro, OfNat.ofNat
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : p ^ p.count a.factors = a * 1
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≤ p.count a.factors

Tactic:
refine Nat.find_le ⟨1, ?_⟩

Premises used: Nat.find_le
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : p ^ p.count a.factors = a
⊢ p ^ p.count a.factors = a * 1

Tactic:
rw [mul_one]

Premises used: mul_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : a = p ^ p.count a.factors
⊢ p ^ p.count a.factors = a

Tactic:
symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ a = p ^ p.count a.factors

Tactic:
eq_pow_count_factors_of_dvd_pow hp h

Premises used: Associates.eq_pow_count_factors_of_dvd_pow
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ a = p ^ p.count a.factors

Tactic:
exact eq_pow_count_factors_of_dvd_pow hp h

Premises used: Associates.eq_pow_count_factors_of_dvd_pow
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ ≠ 0

Tactic:
have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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) ⋯
a₂ : ∀ (hα : Nontrivial 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) ⋯

Tactic:
rcases subsingleton_or_nontrivial α with hα | hα

Premises used: subsingleton_or_nontrivial
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 : Subsingleton α
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯

Tactic:
simp [eq_iff_true_of_subsingleton] at hph

Premises used: eq_iff_true_of_subsingleton
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 ≤ p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors

Tactic:
count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)

Premises used: Associates.count_le_count_of_le, Exists.intro, Nat.find_spec
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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) ⋯ = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors
⊢ p.count a.factors ≤ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯

Tactic:
convert! count_le_count_of_le hph hp (@Nat.find_spec (fun n => a ∣ p ^ n) _ ⟨n, h⟩)

Premises used: Associates.count_le_count_of_le, Exists.intro, Nat.find_spec
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
h₃ h₄ : Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯
⊢ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯ = p.count (p ^ Nat.find (p := fun (n : ℕ) ↦ a ∣ p ^ n) ⋯).factors

Tactic:
rw [count_pow hp.ne_zero hp, count_self hp, mul_one]

Premises used: Associates.count_self, Associates.count_pow, Irreducible.ne_zero, mul_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
⊢ Nontrivial α

Tactic:
nontriviality α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg : ∀ (hk0 : ¬k = 0), ∃ (d : Associates α), a = d ^ k
⊢ ∃ (d : Associates α), a = d ^ k

Tactic:
by_cases hk0 : k = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
h₁ : a = HPow.hPow (α := Associates α) 1 k
⊢ ∃ (d : Associates α), a = d ^ k

Tactic:
use 1
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
h₁ : ∀ (h : a * b = 1), Nontrivial α → ∀ (hk0 : k = 0), a = 1
⊢ a = HPow.hPow (α := Associates α) 1 k

Tactic:
rw [hk0, pow_zero] at h ⊢

Premises used: pow_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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 = 1
h_1 : Nontrivial α
hk0 : k = 0
⊢ a = 1

Tactic:
apply (mul_eq_one.1 h).1

Premises used: mul_eq_one, And.left, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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

Tactic:
is_pow_of_dvd_count ha fun p hp ↦ ?_

Premises used: Associates.is_pow_of_dvd_count
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg : ∀ (p : Associates α) (hp : Irreducible p), k ∣ p.count a.factors
⊢ ∃ (d : Associates α), a = d ^ k

Tactic:
refine is_pow_of_dvd_count ha fun p hp ↦ ?_

Premises used: Associates.is_pow_of_dvd_count
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg : k ∣ p.count (a * b).factors
⊢ k ∣ p.count a.factors

Tactic:
apply dvd_count_of_dvd_count_mul hb hp hab

Premises used: Associates.dvd_count_of_dvd_count_mul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
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
neg : k ∣ p.count (c ^ k).factors
⊢ k ∣ p.count (a * b).factors

Tactic:
rw [h]