l3lab/ntp-mathlib huggingface dataset:
theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a

Next tactic:
obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a := by
  obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
  

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a

Next tactic:
apply Multiset.mem_pmap.mpr
theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a := by
  obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
  apply Multiset.mem_pmap.mpr; 

Proof state:
case intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ a_1,
    ∃ (h : a_1 ∈ UniqueFactorizationMonoid.factors a),
      { val := Associates.mk a_1, property := (_ : Irreducible (Associates.mk a_1)) } =
        { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }

Next tactic:
use q
theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a := by
  obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
  apply Multiset.mem_pmap.mpr; use q; 

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ ∃ (h : q ∈ UniqueFactorizationMonoid.factors a),
    { val := Associates.mk q, property := (_ : Irreducible (Associates.mk q)) } =
      { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }

Next tactic:
use hq
theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a := by
  obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
  apply Multiset.mem_pmap.mpr; use q; use hq
  

Proof state:
case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ { val := Associates.mk q, property := (_ : Irreducible (Associates.mk q)) } =
    { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }

Next tactic:
exact Subtype.eq (Eq.symm (mk_eq_mk_iff_associated.mpr hpq))
theorem mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a ↔ p ∣ a

Next tactic:
constructor
theorem mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a → p ∣ a

Next tactic:
rw [← mk_dvd_mk]
theorem mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    

Proof state:
case mp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a →
    Associates.mk p ∣ Associates.mk a

Next tactic:
apply dvd_of_mem_factors'
theorem mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    apply dvd_of_mem_factors'
    

Proof state:
case mp.hz
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ a ≠ 0

Next tactic:
apply ha0
theorem mem_factors'_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Subtype.mk (Associates.mk p) ((irreducible_mk _).2 hp) ∈ factors' a ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    apply dvd_of_mem_factors'
    apply ha0
  · 

Proof state:
case mpr
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ p ∣ a → { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } ∈ factors' a

Next tactic:
apply mem_factors'_of_dvd ha0 hp
theorem mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Associates.mk p ∈ factors (Associates.mk a) := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Associates.mk p ∈ factors (Associates.mk a)

Next tactic:
rw [factors_mk _ ha0]
theorem mem_factors_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
    Associates.mk p ∈ factors (Associates.mk a) := by
  rw [factors_mk _ ha0]
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Associates.mk p ∈ ↑(factors' a)

Next tactic:
exact mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd)
theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a

Next tactic:
constructor
theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Associates.mk p ∈ factors (Associates.mk a) → p ∣ a

Next tactic:
rw [← mk_dvd_mk]
theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    

Proof state:
case mp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Associates.mk p ∈ factors (Associates.mk a) → Associates.mk p ∣ Associates.mk a

Next tactic:
apply dvd_of_mem_factors
theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    apply dvd_of_mem_factors
    

Proof state:
case mp.hp
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Irreducible (Associates.mk p)

Next tactic:
exact (irreducible_mk p).mpr hp
theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
    Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by
  constructor
  · rw [← mk_dvd_mk]
    apply dvd_of_mem_factors
    exact (irreducible_mk p).mpr hp
  · 

Proof state:
case mpr
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ p ∣ a → Associates.mk p ∈ factors (Associates.mk a)

Next tactic:
apply mem_factors_of_dvd ha0 hp
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
⊢ factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0

Next tactic:
contrapose! h with hf
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : factors (Associates.mk a) ⊓ factors (Associates.mk b) = 0
⊢ Associates.mk a ⊓ Associates.mk b = 1

Next tactic:
change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : factors (Associates.mk a) ⊓ factors (Associates.mk b) = 0
⊢ FactorSet.prod (factors (Associates.mk a) ⊓ factors (Associates.mk b)) = 1

Next tactic:
rw [hf]
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : factors (Associates.mk a) ⊓ factors (Associates.mk b) = 0
⊢ FactorSet.prod 0 = 1

Next tactic:
exact Multiset.prod_zero
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  

Proof state:
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  

Proof state:
case intro.mk
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem : { val := p0, property := p0_irr } ∈ factors' a ⊓ factors' b
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
rw [Multiset.inf_eq_inter] at p0_mem
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  

Proof state:
case intro.mk
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem : { val := p0, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
  

Proof state:
case intro.mk.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : { val := Associates.mk p, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ ∃ p, Prime p ∧ p ∣ a ∧ p ∣ b

Next tactic:
refine' ⟨p, _, _, _⟩
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
  refine' ⟨p, _, _, _⟩
  · 

Proof state:
case intro.mk.intro.refine'_1
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : { val := Associates.mk p, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ Prime p

Next tactic:
rw [← irreducible_iff_prime, ← irreducible_mk]
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
  refine' ⟨p, _, _, _⟩
  · rw [← irreducible_iff_prime, ← irreducible_mk]
    

Proof state:
case intro.mk.intro.refine'_1
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : { val := Associates.mk p, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ Irreducible (Associates.mk p)

Next tactic:
exact p0_irr
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
  refine' ⟨p, _, _, _⟩
  · rw [← irreducible_iff_prime, ← irreducible_mk]
    exact p0_irr
  · 

Proof state:
case intro.mk.intro.refine'_2
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : { val := Associates.mk p, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ p ∣ a

Next tactic:
apply dvd_of_mk_le_mk
theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by
  have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
    contrapose! h with hf
    change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
    rw [hf]
    exact Multiset.prod_zero
  rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
  obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)
  rw [Multiset.inf_eq_inter] at p0_mem
  obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0
  refine' ⟨p, _, _, _⟩
  · rw [← irreducible_iff_prime, ← irreducible_mk]
    exact p0_irr
  · apply dvd_of_mk_le_mk
    

Proof state:
case intro.mk.intro.refine'_2.a
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
dec_irr : (p : Associates α) → Decidable (Irreducible p)
inst✝ : UniqueFactorizationMonoid α
dec : DecidableEq α
dec' : DecidableEq (Associates α)
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : { val := Associates.mk p, property := p0_irr } ∈ factors' a ∩ factors' b
⊢ Associates.mk p ≤ Associates.mk a

Next tactic:
apply dvd_of_mem_factors' (Multiset.mem_inter.mp p0_mem).left