awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hc : ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
mpr : ∀ (p : α) (hp : Prime p) (hpa : p ∣ a) (hpb : p ∣ b), False
⊢ False

Tactic:
obtain ⟨p, hp, hpa, hpb⟩ := exists_prime_dvd_of_not_inf_one ha0 hb0 hg

Premises used: Associates.exists_prime_dvd_of_not_inf_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hc : ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
p : α
hp : Prime p
hpa : p ∣ a
hpb : p ∣ b
⊢ False

Tactic:
hc hpa hpb hp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha0 : a ≠ 0
hb0 : b ≠ 0
hg : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hc : ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
p : α
hp : Prime p
hpa : p ∣ a
hpb : p ∣ b
⊢ False

Tactic:
exact hc hpa hpb hp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
rw :
  p =
    match (motive := FactorSet α → Associates α)
      (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
        WithTop (Multiset { a : Associates α // Irreducible a })) with
    | none => 0
    | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod
⊢ p.factors.prod =
    FactorSet.prod
      (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
        WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
rw [factors_prod, FactorSet.prod.eq_def]

Premises used: Associates.factors_prod, Associates.FactorSet.prod.eq_def
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
dsimp : p = Multiset.prod {p}
⊢ p =
    match (motive := FactorSet α → Associates α)
      (↑{Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp} :
        WithTop (Multiset { a : Associates α // Irreducible a })) with
    | none => 0
    | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod

Tactic:
dsimp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
rw rw₁ : p = p
⊢ p = Multiset.prod {p}

Tactic:
rw [prod_singleton]

Premises used: Multiset.prod_singleton
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
rw :
  p ^ k =
    match (motive := FactorSet α → Associates α)
      (↑(replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp)) :
        WithTop (Multiset { a : Associates α // Irreducible a })) with
    | none => 0
    | some s => (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a)) s).prod
⊢ (p ^ k).factors.prod =
    FactorSet.prod
      (↑(replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp)) :
        WithTop (Multiset { a : Associates α // Irreducible a }))

Tactic:
rw [Associates.factors_prod, FactorSet.prod.eq_def]

Premises used: Associates.factors_prod, Associates.FactorSet.prod.eq_def
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p : Associates α
hp : Irreducible p
k : ℕ
rw : p ^ k = (replicate k (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α)).prod
rw₁ : p ^ k = (↑(Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) : Associates α) ^ k
rw₂ rw₃ : p ^ k = p ^ k
⊢ p ^ k =
    (Multiset.map (Subtype.val (p := fun (a : Associates α) ↦ Irreducible a))
        (replicate k (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp))).prod

Tactic:
rw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk]

Premises used: Subtype.coe_mk, Multiset.prod_replicate, Multiset.map_replicate
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
m p : Associates α
h₁ : m ≠ 0
h₂ : Irreducible p
k : ℕ
rcases :
  ∀ (m : α) (hm : m ≠ 0) (h₁ : Ne (α := Associates α) (Associates.mk m) 0),
    p ^ k ≤ Associates.mk m ↔
      k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) (factors (Associates.mk m))
⊢ p ^ k ≤ m ↔ k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors

Tactic:
rcases Associates.exists_non_zero_rep h₁ with ⟨m, hm, rfl⟩

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)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Ne (α := Associates α) (Associates.mk m) 0
⊢ Nontrivial α

Tactic:
have := nontrivial_of_ne _ _ hm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Ne (α := Associates α) (Associates.mk m) 0
this : Nontrivial α
h h₃ : m ≠ 0
⊢ p ^ k ≤ Associates.mk m ↔
    k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) (factors (Associates.mk m))

Tactic:
rw [bcount.eq_def, factors_mk, Multiset.le_count_iff_replicate_le, ← factors_le,
    factors_prime_pow, factors_mk, WithTop.coe_le_coe]

Premises used: Associates.factors_mk, Associates.factors_le, Associates.factors_prime_pow, Associates.bcount.eq_def, Multiset.le_count_iff_replicate_le, WithTop.coe_le_coe
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : DecidableEq (Associates α)
p : Associates α
h₂ : Irreducible p
k : ℕ
m : α
hm : m ≠ 0
h₁ : Ne (α := Associates α) (Associates.mk m) 0
this : Nontrivial α
⊢ m ≠ 0

Tactic:
assumption
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
h : Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod (factors 1)) (FactorSet.prod 0)
⊢ Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0

Tactic:
apply FactorSet.unique

Premises used: Associates.FactorSet.unique
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
h : Eq.{u_1 + 1} (α := Associates α) 1 (FactorSet.prod 0)
⊢ Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod (factors 1)) (FactorSet.prod 0)

Tactic:
rw [Associates.factors_prod]

Premises used: Associates.factors_prod
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
⊢ Eq.{u_1 + 1} (α := Associates α) 1 (FactorSet.prod 0)

Tactic:
Multiset.prod_zero

Premises used: Multiset.prod_zero, Multiset.foldr, Multiset.prod, Associates.FactorSet.prod, List.foldr
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
⊢ Eq.{u_1 + 1} (α := Associates α) 1 (FactorSet.prod 0)

Tactic:
exact Multiset.prod_zero

Premises used: Multiset.prod_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
zero : Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0
⊢ (HPow.hPow (β := ℕ) a 0).factors = HSMul.hSMul (α := ℕ) 0 a.factors

Tactic:
rw [zero_nsmul, pow_zero]

Premises used: pow_zero, zero_nsmul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
⊢ Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0

Tactic:
factors_one

Premises used: Associates.factors_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
⊢ Eq.{u_1 + 1} (α := FactorSet α) (factors 1) 0

Tactic:
exact factors_one

Premises used: Associates.factors_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
n : ℕ
h : (a ^ n).factors = n • a.factors
succ : (a ^ n * a).factors = (n + 1) • a.factors
succ₁ : (a ^ n * a).factors = n • a.factors + a.factors
succ₂ : (a ^ n).factors + a.factors = n • a.factors + a.factors
succ₃ succ₄ : n • a.factors + a.factors = n • a.factors + a.factors
⊢ (a ^ (n + 1)).factors = (n + 1) • a.factors

Tactic:
rw [pow_succ, succ_nsmul, factors_mul, h]

Premises used: Associates.factors_mul, pow_succ, succ_nsmul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
a : Associates α
k : ℕ
zero : (HPow.hPow (β := ℕ) a 0).factors = HSMul.hSMul (α := ℕ) 0 a.factors
succ : ∀ (n : ℕ) (h : (a ^ n).factors = n • a.factors), (a ^ (n + 1)).factors = (n + 1) • a.factors
⊢ (a ^ k).factors = k • a.factors

Tactic:
induction k with
  | zero => rw [zero_nsmul, pow_zero]; exact factors_one
  | succ n h => rw [pow_succ, succ_nsmul, factors_mul, h]

Premises used: Associates.factors_mul, pow_succ, pow_zero, zero_nsmul, Associates.factors_one, succ_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)
m p : Associates α
h₁ : m ≠ 0
h₂ : Irreducible p
k : ℕ
rw :
  p ^ k ≤ m ↔
    k ≤
      (if hp : Irreducible p then bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p hp) else 0)
        m.factors
rw₁ : p ^ k ≤ m ↔ k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors
rw₂ :
  k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors ↔
    k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors
h₃ : Irreducible p
rw₃ :
  k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors ↔
    k ≤ bcount (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p h₂) m.factors
h₄ : Irreducible p
⊢ p ^ k ≤ m ↔ k ≤ p.count m.factors

Tactic:
rw [count, dif_pos h₂, prime_pow_le_iff_le_bcount h₁]

Premises used: Associates.count, dif_pos, Associates.prime_pow_le_iff_le_bcount
-- 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)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
rw : 0 < p.count m.factors → p ≤ m
⊢ p.count m.factors ≠ 0 → p ≤ m

Tactic:
rw [← pos_iff_ne_zero]

Premises used: pos_iff_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)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
h : 0 < p.count m.factors
rw : HPow.hPow (β := ℕ) p 1 ≤ m
⊢ p ≤ m

Tactic:
rw [← pow_one p]

Premises used: pow_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)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
h : 0 < p.count m.factors
apply : 1 ≤ p.count m.factors
⊢ HPow.hPow (β := ℕ) p 1 ≤ m

Tactic:
apply (prime_pow_dvd_iff_le h0 hp).2

Premises used: Iff.mpr, Associates.prime_pow_dvd_iff_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)
m p : Associates α
h0 : m ≠ 0
hp : Irreducible p
h : 0 < p.count m.factors
⊢ 1 ≤ p.count m.factors

Tactic:
simpa only
-- 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 : α
ha0 : a ≠ 0
hp : Irreducible p
rw :
  count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔
    LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔ p ∣ a

Tactic:
rw [← Associates.mk_le_mk_iff_dvd]

Premises used: Associates.mk_le_mk_iff_dvd
-- 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 : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔
    LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)

Tactic:
⟨fun h =>
      Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0)
        (Associates.irreducible_mk.mpr hp) h,
      fun h => ?_⟩

Premises used: Iff.mpr, Associates.le_of_count_ne_zero, Associates.irreducible_mk, Iff.intro, Associates.mk_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 : α
ha0 : a ≠ 0
hp : Irreducible p
h :
  ∀ (h : LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)),
    count (Associates.mk p) (factors (Associates.mk a)) ≠ 0
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0 ↔
    LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)

Tactic:
refine
    ⟨fun h =>
      Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0)
        (Associates.irreducible_mk.mpr hp) h,
      fun 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)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
h : LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)
⊢ 1 ≤ count (Associates.mk p) (factors (Associates.mk a))

Tactic:
rw [← pow_one (Associates.mk p),
    Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero.mpr ha0)
      (Associates.irreducible_mk.mpr hp)] at h

Premises used: Iff.mpr, Associates.mk, Associates.prime_pow_dvd_iff_le, Associates.irreducible_mk, pow_one, Associates.mk_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 : α
ha0 : a ≠ 0
hp : Irreducible p
h : 1 ≤ count (Associates.mk p) (factors (Associates.mk a))
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0

Tactic:
(zero_lt_one.trans_le h).ne'

Premises used: LT.lt.ne', zero_lt_one, LT.lt.trans_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 : α
ha0 : a ≠ 0
hp : Irreducible p
h : 1 ≤ count (Associates.mk p) (factors (Associates.mk a))
⊢ count (Associates.mk p) (factors (Associates.mk a)) ≠ 0

Tactic:
exact (zero_lt_one.trans_le h).ne'

Premises used: LT.lt.ne', zero_lt_one, LT.lt.trans_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)
inst_4 : Nontrivial α
p : Associates α
hp : Irreducible p
⊢ p.count p.factors = 1

Tactic:
simp [factors_self hp, Associates.count_some hp]

Premises used: Associates.count_some, Associates.factors_self
-- 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 q : Associates α
hp : Irreducible p
hq : Irreducible q
h : p ≠ q
⊢ p.count q.factors = 0

Tactic:
not_ne_iff.mp fun h' ↦ h <| associated_iff_eq.mp <| hp.associated_of_dvd hq <|
    le_of_count_ne_zero hq.ne_zero hp h'

Premises used: Associates.le_of_count_ne_zero, Irreducible.associated_of_dvd, not_ne_iff, associated_iff_eq, Iff.mp, 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)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
obtain :
  ∀ (a0 : α) (nza : a0 ≠ 0) (ha : Ne (α := Associates α) (Associates.mk a0) 0),
    p.count (Associates.mk a0 * b).factors = p.count (factors (Associates.mk a0)) + p.count b.factors
⊢ p.count (a * b).factors = p.count a.factors + p.count b.factors

Tactic:
obtain ⟨a0, nza, 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)
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
a0 : α
nza : a0 ≠ 0
ha : Ne (α := Associates α) (Associates.mk a0) 0
obtain :
  ∀ (b0 : α) (nzb : b0 ≠ 0) (hb : Ne (α := Associates α) (Associates.mk b0) 0),
    p.count (HMul.hMul (α := Associates α) (Associates.mk a0) (Associates.mk b0)).factors =
      p.count (factors (Associates.mk a0)) + p.count (factors (Associates.mk b0))
⊢ p.count (Associates.mk a0 * b).factors = p.count (factors (Associates.mk a0)) + p.count b.factors

Tactic:
obtain ⟨b0, nzb, rfl⟩ := exists_non_zero_rep hb

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

Tactic:
rw [or_iff_not_imp_left, ← Ne]

Premises used: Ne, Classical.or_iff_not_imp_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 : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
contrapose! : ∀ (hcb : p.count b.factors ≠ 0), ∃ (d : Associates α), d ∣ a ∧ d ∣ b ∧ Prime d
⊢ p.count b.factors = 0

Tactic:
contrapose! hab with hcb
-- 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
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
hcb : p.count b.factors ≠ 0
⊢ ∃ (d : Associates α), d ∣ a ∧ d ∣ b ∧ Prime d

Tactic:
⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb,
    UniqueFactorizationMonoid.irreducible_iff_prime.mp hp⟩

Premises used: Associates.le_of_count_ne_zero, UniqueFactorizationMonoid.irreducible_iff_prime, And.intro, Exists.intro, Iff.mp
-- 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
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
hcb : p.count b.factors ≠ 0
⊢ ∃ (d : Associates α), d ∣ a ∧ d ∣ b ∧ Prime d

Tactic:
exact ⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb,
    UniqueFactorizationMonoid.irreducible_iff_prime.mp hp⟩
-- 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
pos : ∀ (ha : a = 0), p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors
neg : ∀ (ha : ¬a = 0), p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * 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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : a = 0
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * 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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
neg : ∀ (hz : p.count a.factors = 0), p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors
neg₁ : ∀ (hb0 : p.count b.factors = 0), p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

Tactic:
rcases count_of_coprime ha hb hab hp with hz | 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 : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
ha : ¬a = 0
hz : p.count a.factors = 0
neg neg₁ : ¬p.count a.factors = 0 → p.count a.factors = p.count (a * b).factors
⊢ p.count a.factors = 0 ∨ p.count a.factors = p.count (a * b).factors

Tactic:
tauto