-- Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
theorem is_pow_of_dvd_count {a : Associates α}
    (ha : a ≠ 0) {k : ℕ} (hk : ∀ p : Associates α, Irreducible p → k ∣ count p a.factors) :
    ∃ b : Associates α, a = b ^ k

Proof state:
case h.h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ (Associates.mk a0).factors = (FactorSet.prod ↑u ^ k).factors

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

Result:
case h.h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ ↑(k • u) = k • ↑u

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (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
a✝ : Nontrivial α
⊢ ∃ b, a = b ^ k

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

Result:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count (Associates.mk a0).factors
⊢ ∃ b, Associates.mk a0 = b ^ k

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
⊢ ∃ b, Associates.mk a0 = b ^ k

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

Result:
case intro.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ ∃ b, Associates.mk a0 = b ^ k

Proof state:
case h.h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ ↑(k • u) = k • ↑u

Tactic:
exact WithBot.coe_nsmul u k

Result:
no goals

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count (Associates.mk a0).factors
⊢ ∃ b, Associates.mk a0 = b ^ k

Tactic:
rw [factors_mk a0 hz] at hk

Result:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
⊢ ∃ b, Associates.mk a0 = b ^ k

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
p : { a // Irreducible a }
⊢ k ∣ Multiset.count p (factors' a0)

Tactic:
have pp : p = ⟨p.val, p.2⟩ := by simp only [Subtype.coe_eta]

Result:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
p : { a // Irreducible a }
pp : p = ⟨↑p, ⋯⟩
⊢ k ∣ Multiset.count p (factors' a0)

Proof state:
case intro.intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ ∃ b, Associates.mk a0 = b ^ k

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

Result:
case h
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
u : Multiset { a // Irreducible a }
hu : factors' a0 = k • u
⊢ Associates.mk a0 = FactorSet.prod ↑u ^ k

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (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
⊢ ∃ b, a = b ^ k

Tactic:
nontriviality α

Result:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (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
a✝ : Nontrivial α
⊢ ∃ b, a = b ^ k

Proof state:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
⊢ ∃ b, Associates.mk a0 = b ^ k

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

Result:
case intro.intro
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
k : ℕ
a✝ : Nontrivial α
a0 : α
hz : a0 ≠ 0
ha : Associates.mk a0 ≠ 0
hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count ↑(factors' a0)
hk' : ∀ p ∈ factors' a0, k ∣ Multiset.count p (factors' a0)
⊢ ∃ b, Associates.mk a0 = b ^ k