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 p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
q : α
hq : q ∈ UniqueFactorizationMonoid.factors a
hpq : p ~ᵤ q
⊢ Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk q) ⋯ =
    Subtype.mk (p := Irreducible) (Associates.mk p) ⋯

Tactic:
exact Subtype.ext (Eq.symm (mk_eq_mk_iff_associated.mpr hpq))

Premises used: Eq.symm, Subtype.ext
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    p ∣ a
mpr :
  p ∣ a →
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯)
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) ↔
    p ∣ a

Tactic:
constructor
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    p ∣ a

Tactic:
rw [← mk_dvd_mk]

Premises used: Associates.mk_dvd_mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp : a ≠ 0
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯) →
    Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)

Tactic:
apply dvd_of_mem_factors'

Premises used: Associates.dvd_of_mem_factors'
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ a ≠ 0

Tactic:
apply ha0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ p ∣ a →
    Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a)
      (Subtype.mk (p := Irreducible) (Associates.mk p) ⋯)

Tactic:
apply mem_factors'_of_dvd ha0 hp

Premises used: Associates.mem_factors'_of_dvd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
rw :
  Membership.mem (γ := FactorSet α) (↑(factors' a) : WithTop (Multiset { a : Associates α // Irreducible a }))
    (Associates.mk p)
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p)

Tactic:
rw [factors_mk _ ha0]

Premises used: Associates.factors_mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Membership.mem (γ := FactorSet α) (↑(factors' a) : WithTop (Multiset { a : Associates α // Irreducible a }))
    (Associates.mk p)

Tactic:
mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd)

Premises used: Iff.mpr, Associates.mem_factorSet_some, Associates.mem_factors'_of_dvd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a
⊢ Membership.mem (γ := FactorSet α) (↑(factors' a) : WithTop (Multiset { a : Associates α // Irreducible a }))
    (Associates.mk p)

Tactic:
exact mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd)

Premises used: Iff.mpr, Associates.mem_factorSet_some, Associates.mem_factors'_of_dvd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp : Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) → p ∣ a
mpr : p ∣ a → Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p)
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) ↔ p ∣ a

Tactic:
constructor
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
mp :
  Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) →
    Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) → p ∣ a

Tactic:
rw [← mk_dvd_mk]

Premises used: Associates.mk_dvd_mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p) →
    Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a)

Tactic:
apply dvd_of_mem_factors

Premises used: Associates.dvd_of_mem_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
⊢ p ∣ a → Membership.mem (γ := FactorSet α) (factors (Associates.mk a)) (Associates.mk p)

Tactic:
apply mem_factors_of_dvd ha0 hp

Premises used: Associates.mem_factors_of_dvd
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
contrapose :
  ∀ (hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0),
    Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
⊢ Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0

Tactic:
contrapose h with hf
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
rw : Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod 0) 1
⊢ Eq (α := Associates α) (FactorSet.prod (factors (Associates.mk a) ⊓ factors (Associates.mk b))) 1

Tactic:
rw [hf]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
⊢ Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod 0) 1

Tactic:
Multiset.prod_zero

Premises used: Multiset.prod_zero, Multiset.map, List.map, Multiset.zero, Multiset.ofList, Associates.FactorSet.prod
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : Eq (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
⊢ Eq.{u_1 + 1} (α := Associates α) (FactorSet.prod 0) 1

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
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
⊢ Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0

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
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (factors (Associates.mk a) ⊓ factors (Associates.mk b)) 0
⊢ Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0

Tactic:
rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz

Premises used: Associates.factors_mk, WithTop.coe_inf
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
obtain :
  ∀ (p0 : Associates α) (p0_irr : Irreducible p0)
    (p0_mem :
      Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ⊓ factors' b)
        (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)),
    ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b

Tactic:
obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)

Premises used: Iff.mpr, mt, WithTop.coe_eq_coe, Multiset.exists_mem_of_ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ⊓ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
⊢ Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)

Tactic:
rw [Multiset.inf_eq_inter] at p0_mem

Premises used: Multiset.inf_eq_inter
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
⊢ ∃ (p : α), Associates.mk p = p0

Tactic:
Quot.exists_rep p0

Premises used: Quot.exists_rep, Quotient
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p0 : Associates α
p0_irr : Irreducible p0
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) p0 p0_irr)
obtain :
  ∀ (p : α) (p0_irr : Irreducible (M := Associates α) (Associates.mk p))
    (p0_mem :
      Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
        (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)),
    ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b

Tactic:
obtain ⟨p, rfl⟩ : ∃ p, Associates.mk p = p0 := Quot.exists_rep p0

Premises used: Quot.exists_rep, rfl
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b

Tactic:
⟨p, ?_, ?_, ?_⟩

Premises used: And.intro, Exists.intro
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_1 : Prime p
refine_2 : p ∣ a
refine_3 : p ∣ b
⊢ ∃ (p : α), Prime p ∧ p ∣ a ∧ p ∣ b

Tactic:
refine ⟨p, ?_, ?_, ?_⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_1 : Irreducible (M := Associates α) (Associates.mk p)
⊢ Prime p

Tactic:
rw [← UniqueFactorizationMonoid.irreducible_iff_prime, ← irreducible_mk]

Premises used: Associates.irreducible_mk, UniqueFactorizationMonoid.irreducible_iff_prime
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ Irreducible (M := Associates α) (Associates.mk p)

Tactic:
p0_irr
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ Irreducible (M := Associates α) (Associates.mk p)

Tactic:
exact p0_irr
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_2 : LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)
⊢ p ∣ a

Tactic:
apply dvd_of_mk_le_mk

Premises used: Associates.dvd_of_mk_le_mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
apply : a ≠ 0
⊢ LE.le (α := Associates α) (Associates.mk p) (Associates.mk a)

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

Premises used: Associates.dvd_of_mem_factors', And.left, Iff.mp, Multiset.mem_inter
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ a ≠ 0

Tactic:
apply ha
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
refine_3 : LE.le (α := Associates α) (Associates.mk p) (Associates.mk b)
⊢ p ∣ b

Tactic:
apply dvd_of_mk_le_mk

Premises used: Associates.dvd_of_mk_le_mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
apply : b ≠ 0
⊢ LE.le (α := Associates α) (Associates.mk p) (Associates.mk b)

Tactic:
apply dvd_of_mem_factors' (Multiset.mem_inter.mp p0_mem).right

Premises used: Associates.dvd_of_mem_factors', And.right, Iff.mp, Multiset.mem_inter
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Ne (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
hz : Ne (α := FactorSet α) (↑(factors' a ⊓ factors' b) : WithTop (Multiset { a : Associates α // Irreducible a })) 0
p : α
p0_irr : Irreducible (M := Associates α) (Associates.mk p)
p0_mem :
  Membership.mem (γ := Multiset { a : Associates α // Irreducible a }) (factors' a ∩ factors' b)
    (Subtype.mk (p := fun (a : Associates α) ↦ Irreducible a) (Associates.mk p) p0_irr)
⊢ b ≠ 0

Tactic:
apply hb
-- 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
mp : Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 → ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
mpr : (∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d) → Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1
⊢ Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 ↔ ∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d

Tactic:
constructor
-- 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
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
⊢ False

Tactic:
(Associates.prime_mk.mpr hp).not_unit (isUnit_of_dvd_one ?_)

Premises used: Iff.mpr, Prime.not_unit, Associates.prime_mk, isUnit_of_dvd_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
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
mp : Dvd.dvd (α := Associates α) (Associates.mk p) 1
⊢ False

Tactic:
refine (Associates.prime_mk.mpr hp).not_unit (isUnit_of_dvd_one ?_)

Premises used: Iff.mpr, Prime.not_unit, Associates.prime_mk, isUnit_of_dvd_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
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
mp : Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a ⊓ Associates.mk b)
⊢ Dvd.dvd (α := Associates α) (Associates.mk p) 1

Tactic:
rw [← hg]
-- 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
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
⊢ Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a ⊓ Associates.mk b)

Tactic:
le_inf (mk_le_mk_of_dvd ha) (mk_le_mk_of_dvd hb)

Premises used: le_inf, Associates.mk_le_mk_of_dvd
-- 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
p : α
ha : p ∣ a
hb : p ∣ b
hp : Prime p
⊢ Dvd.dvd (α := Associates α) (Associates.mk p) (Associates.mk a ⊓ Associates.mk b)

Tactic:
exact le_inf (mk_le_mk_of_dvd ha) (mk_le_mk_of_dvd hb)

Premises used: le_inf, Associates.mk_le_mk_of_dvd
-- 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
mpr : ¬Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1 → ¬∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d
⊢ (∀ {d : α}, d ∣ a → d ∣ b → ¬Prime d) → Eq (α := Associates α) (Associates.mk a ⊓ Associates.mk b) 1

Tactic:
contrapose