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)
a b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : p.count b.factors = 0
neg : p.count a.factors = p.count (a * b).factors
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

Tactic:
apply Or.intro_right

Premises used: Or.intro_right
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb0 : p.count b.factors = 0
neg : p.count a.factors = p.count a.factors + p.count b.factors
neg₁ : p.count a.factors = p.count a.factors + 0
neg₂ neg₃ : p.count a.factors = p.count a.factors
⊢ p.count a.factors = p.count (a * b).factors

Tactic:
rw [count_mul ha hb hp, hb0, add_zero]

Premises used: Associates.count_mul, add_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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
pos : ∀ (ha : a = 0), p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors
neg : ∀ (ha : ¬a = 0), p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors
⊢ p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors

Tactic:
by_cases ha : a = 0
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : a = 0
⊢ p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors

Tactic:
simp [ha]
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
pos : ∀ (hb : b = 0), p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors
neg : ∀ (hb : ¬b = 0), p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors
⊢ p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors

Tactic:
by_cases hb : b = 0
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : b = 0
⊢ p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors

Tactic:
simp [hb]
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
neg :
  p.count a.factors + p.count b.factors = p.count a.factors ∨ p.count a.factors + p.count b.factors = p.count b.factors
⊢ p.count (a * b).factors = p.count a.factors ∨ p.count (a * b).factors = p.count b.factors

Tactic:
rw [count_mul ha hb hp]

Premises used: Associates.count_mul
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
neg :
  ∀ (ha0 : p.count a.factors = 0),
    p.count a.factors + p.count b.factors = p.count a.factors ∨
      p.count a.factors + p.count b.factors = p.count b.factors
neg₁ :
  ∀ (hb0 : p.count b.factors = 0),
    p.count a.factors + p.count b.factors = p.count a.factors ∨
      p.count a.factors + p.count b.factors = p.count b.factors
⊢ p.count a.factors + p.count b.factors = p.count a.factors ∨ p.count a.factors + p.count b.factors = p.count b.factors

Tactic:
rcases count_of_coprime ha hb hab hp with ha0 | hb0

Premises used: Associates.count_of_coprime
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
ha0 : p.count a.factors = 0
neg : p.count a.factors + p.count b.factors = p.count b.factors
⊢ p.count a.factors + p.count b.factors = p.count a.factors ∨ p.count a.factors + p.count b.factors = p.count b.factors

Tactic:
apply Or.intro_right

Premises used: Or.intro_right
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
ha0 : p.count a.factors = 0
neg : 0 + p.count b.factors = p.count b.factors
neg₁ neg₂ : p.count b.factors = p.count b.factors
⊢ p.count a.factors + p.count b.factors = p.count b.factors

Tactic:
rw [ha0, zero_add]

Premises used: zero_add
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
hb0 : p.count b.factors = 0
neg : p.count a.factors + p.count b.factors = p.count a.factors
⊢ p.count a.factors + p.count b.factors = p.count a.factors ∨ p.count a.factors + p.count b.factors = p.count b.factors

Tactic:
apply Or.intro_left

Premises used: Or.intro_left
-- 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 b p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hb : ¬b = 0
hb0 : p.count b.factors = 0
neg : p.count a.factors + 0 = p.count a.factors
neg₁ neg₂ : p.count a.factors = p.count a.factors
⊢ p.count a.factors + p.count b.factors = p.count a.factors

Tactic:
rw [hb0, add_zero]

Premises used: add_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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
pos : ∀ (ha : a = 0), k ∣ p.count a.factors
neg : ∀ (ha : ¬a = 0), k ∣ p.count a.factors
⊢ k ∣ p.count a.factors

Tactic:
by_cases ha : a = 0
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : a = 0
⊢ k ∣ p.count a.factors

Tactic:
simpa [*] using habk
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
neg : ∀ (hz : p.count a.factors = 0), k ∣ p.count a.factors
neg₁ : ∀ (h : p.count b.factors = 0), k ∣ p.count a.factors
⊢ k ∣ p.count a.factors

Tactic:
rcases count_of_coprime ha hb hab hp with hz | h

Premises used: Associates.count_of_coprime
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
hz : p.count a.factors = 0
neg : k ∣ 0
⊢ k ∣ p.count a.factors

Tactic:
rw [hz]
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
hz : p.count a.factors = 0
⊢ k ∣ 0

Tactic:
dvd_zero k

Premises used: dvd_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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
hz : p.count a.factors = 0
⊢ k ∣ 0

Tactic:
exact dvd_zero k

Premises used: dvd_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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
h : p.count b.factors = 0
⊢ k ∣ p.count a.factors + 0

Tactic:
rw [count_mul ha hb hp, h] at habk

Premises used: Associates.count_mul
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
h : p.count b.factors = 0
habk₁ : k ∣ p.count a.factors + 0
⊢ ¬a = 0

Tactic:
rw [count_mul ha hb hp, h] at habk

Premises used: Associates.count_mul
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
h : p.count b.factors = 0
habk₁ : k ∣ p.count a.factors + 0
ha₁ : ¬a = 0
⊢ p.count b.factors = 0

Tactic:
rw [count_mul ha hb hp, h] at habk

Premises used: Associates.count_mul
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count a.factors + 0
ha : ¬a = 0
h : p.count b.factors = 0
⊢ k ∣ p.count a.factors

Tactic:
habk

Premises used: Associates.count, dite
-- 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 b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count a.factors + 0
ha : ¬a = 0
h : p.count b.factors = 0
⊢ k ∣ p.count a.factors

Tactic:
exact habk
-- 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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
zero : p.count (factors 1) = 0 * p.count a.factors
zero₁ : p.count 0 = 0 * p.count a.factors
zero₂ : p.count 0 = 0
zero₃ zero₄ : Eq (α := ℕ) 0 0
⊢ p.count (HPow.hPow (β := ℕ) a 0).factors = 0 * p.count a.factors

Tactic:
rw [pow_zero, factors_one, zero_mul, count_zero hp]

Premises used: MulZeroClass.zero_mul, Associates.count_zero, pow_zero, Associates.factors_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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
n : ℕ
h : p.count (a ^ n).factors = n * p.count a.factors
succ : p.count a.factors + n * p.count a.factors = (n + 1) * p.count a.factors
⊢ p.count (a ^ (n + 1)).factors = (n + 1) * p.count a.factors

Tactic:
rw [pow_succ', count_mul ha (pow_ne_zero _ ha) hp, h]

Premises used: pow_ne_zero, Associates.count_mul, pow_succ'
-- 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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
n : ℕ
h : p.count (a ^ n).factors = n * p.count a.factors
⊢ p.count a.factors + n * p.count a.factors = (n + 1) * p.count a.factors

Tactic:
ring
-- 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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
k : ℕ
zero : p.count (HPow.hPow (β := ℕ) a 0).factors = 0 * p.count a.factors
succ :
  ∀ (n : ℕ) (h : p.count (a ^ n).factors = n * p.count a.factors),
    p.count (a ^ (n + 1)).factors = (n + 1) * p.count a.factors
⊢ p.count (a ^ k).factors = k * p.count a.factors

Tactic:
induction k with
  | zero => rw [pow_zero, factors_one, zero_mul, count_zero hp]
  | succ n h => rw [pow_succ', count_mul ha (pow_ne_zero _ ha) hp, h]; ring

Premises used: MulZeroClass.zero_mul, Associates.count_zero, pow_ne_zero, pow_zero, Associates.count_mul, Associates.factors_one, pow_succ'
-- 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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
k : ℕ
rw : k ∣ k * p.count a.factors
⊢ k ∣ p.count (a ^ k).factors

Tactic:
rw [count_pow ha hp]

Premises used: Associates.count_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)
inst_4 : Nontrivial α
a : Associates α
ha : a ≠ 0
p : Associates α
hp : Irreducible p
k : ℕ
⊢ k ∣ k * p.count a.factors

Tactic:
apply dvd_mul_right

Premises used: dvd_mul_right
-- 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 : Associates α
ha : a ≠ 0
k : ℕ
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count a.factors
⊢ 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)
a : Associates α
ha : a ≠ 0
k : ℕ
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count a.factors
h : Nontrivial α
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
⊢ ∃ (b : Associates α), a = b ^ k

Tactic:
obtain ⟨a0, hz, rfl⟩ := exists_non_zero_rep ha

Premises used: Associates.exists_non_zero_rep, rfl
-- 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 (Associates.mk a0))
⊢ ∀ (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

Premises used: Associates.factors_mk
-- 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 }))
p : { a : Associates α // Irreducible a }
⊢ p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯

Tactic:
simp only [Subtype.coe_eta]

Premises used: Subtype.coe_eta
-- 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 }))
p : { a : Associates α // Irreducible a }
⊢ p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯

Tactic:
have pp : p = ⟨p.val, p.2⟩ := by simp only [Subtype.coe_eta]
-- 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 }))
p : { a : Associates α // Irreducible a }
pp : p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯
rw : k ∣ (↑p : Associates α).count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))
⊢ k ∣ Multiset.count p (factors' a0)

Tactic:
rw [pp, ← count_some p.2]

Premises used: Associates.count_some, Subtype.property
-- 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 }))
p : { a : Associates α // Irreducible a }
pp : p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯
⊢ k ∣ (↑p : Associates α).count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
hk p.val p.2

Premises used: Subtype.val, Subtype.property
-- 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 }))
p : { a : Associates α // Irreducible a }
pp : p = Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (↑p : Associates α) ⋯
⊢ k ∣ (↑p : Associates α).count (↑(factors' a0) : WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
exact hk p.val p.2

Premises used: Subtype.val, Subtype.property
-- 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 }))
⊢ ∀ (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
-- 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)
obtain :
  ∀ (u : Multiset { a : Associates α // Irreducible a }) (hu : factors' a0 = k • u),
    ∃ (b : Associates α), Associates.mk a0 = b ^ k
⊢ ∃ (b : Associates α), Associates.mk a0 = b ^ k

Tactic:
obtain ⟨u, hu⟩ := Multiset.exists_smul_of_dvd_count _ hk'

Premises used: Multiset.exists_smul_of_dvd_count