awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
cons : ∀ (b : α) (hbg : b ∈ g) (hb : p ~ᵤ b), Multiset.Rel Associated (p ::ₘ f) g
⊢ Multiset.Rel Associated (p ::ₘ f) g

Tactic:
let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
cons : ∀ (this : DecidableEq α), Multiset.Rel Associated (p ::ₘ f) g
⊢ Multiset.Rel Associated (p ::ₘ f) g

Tactic:
haveI := Classical.decEq α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
cons : Multiset.Rel Associated (p ::ₘ f) (b ::ₘ g.erase b)
⊢ Multiset.Rel Associated (p ::ₘ f) g

Tactic:
rw [← Multiset.cons_erase hbg]

Premises used: Multiset.cons_erase
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
q : α
hq : q ∈ f
⊢ q ∈ p ::ₘ f

Tactic:
simp [hq]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
rwa : (p ::ₘ f).prod ~ᵤ b * (g.erase b).prod
rwa₁ : (p ::ₘ f).prod ~ᵤ (b ::ₘ g.erase b).prod
rwa₂ rwa₃ rwa₄ rwa₅ : (p ::ₘ f).prod ~ᵤ g.prod
⊢ p * f.prod ~ᵤ b * (g.erase b).prod

Tactic:
rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]

Premises used: Multiset.prod_cons, Multiset.cons_erase
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ p ∈ p ::ₘ f

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ Multiset.Rel Associated (p ::ₘ f) (b ::ₘ g.erase b)

Tactic:
exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
            (hf p (by simp)).ne_zero))

Premises used: Multiset.Rel.cons, Prime.ne_zero, Associated.of_mul_left, Multiset.mem_of_mem_erase
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
f : Multiset α
empty :
  ∀ {g : Multiset α},
    (∀ (x : α), Membership.mem (γ := Multiset α) 0 x → Prime x) →
      (∀ x ∈ g, Prime x) → Multiset.prod 0 ~ᵤ g.prod → Multiset.Rel Associated 0 g
cons :
  ∀ (p : α) (f : Multiset α)
    (ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g)
    {g : Multiset α},
    (∀ x ∈ p ::ₘ f, Prime x) → (∀ x ∈ g, Prime x) → (p ::ₘ f).prod ~ᵤ g.prod → Multiset.Rel Associated (p ::ₘ f) g
⊢ ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g

Tactic:
induction f using Multiset.induction_on with
  | empty =>
    intro g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_notMem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  | cons p f ih =>
    intro g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
            (hf p (by simp)).ne_zero))

Premises used: Iff.mpr, Multiset.rel_zero_left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
intro : ∀ {f g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
⊢ ∀ {f g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g

Tactic:
classical
  intro f
  induction f using Multiset.induction_on with
  | empty =>
    intro g _ hg h
    exact Multiset.rel_zero_left.2 <|
      Multiset.eq_zero_of_forall_notMem fun x hx =>
        have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm
        (hg x hx).not_unit <|
          isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)
  | cons p f ih =>
    intro g hf hg hfg
    let ⟨b, hbg, hb⟩ :=
      (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|
        hfg.dvd_iff_dvd_right.1 (show p ∣ (p ::ₘ f).prod by simp)
    haveI := Classical.decEq α
    rw [← Multiset.cons_erase hbg]
    exact
      Multiset.Rel.cons hb
        (ih (fun q hq => hf _ (by simp [hq]))
          (fun {q} (hq : q ∈ g.erase b) => hg q (Multiset.mem_of_mem_erase hq))
          (Associated.of_mul_left
            (by rwa [← Multiset.prod_cons, ← Multiset.prod_cons, Multiset.cons_erase hbg]) hb
            (hf p (by simp)).ne_zero))

Premises used: Iff.mpr, Multiset.rel_zero_left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
f g : Multiset α
hf : ∀ x ∈ f, Irreducible x
hg : ∀ x ∈ g, Irreducible x
h : f.prod ~ᵤ g.prod
⊢ Multiset.Rel Associated f g

Tactic:
prime_factors_unique (fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hf x hx))
    (fun x hx => UniqueFactorizationMonoid.irreducible_iff_prime.mp (hg x hx)) h

Premises used: UniqueFactorizationMonoid.irreducible_iff_prime, prime_factors_unique, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
pos : ∀ (hb : b = 0), (factors a).card = (factors b).card
neg : ∀ (hb : ¬b = 0), (factors a).card = (factors b).card
⊢ (factors a).card = (factors b).card

Tactic:
by_cases hb : b = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : b = 0
⊢ (factors a).card = (factors b).card

Tactic:
simp_all
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : ¬b = 0
⊢ a ≠ 0

Tactic:
have ha : a ≠ 0 := h.ne_zero_iff.mpr hb
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : ¬b = 0
ha : a ≠ 0
neg : Multiset.Rel Associated (factors a) (factors b)
neg₁ : α → α → Prop
⊢ (factors a).card = (factors b).card

Tactic:
apply Multiset.card_eq_card_of_rel

Premises used: Multiset.card_eq_card_of_rel
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : ¬b = 0
ha : a ≠ 0
⊢ (factors a).prod ~ᵤ (factors b).prod

Tactic:
(factors_prod ha).trans <| h.trans (factors_prod hb).symm

Premises used: Associated.trans, Associated.symm, UniqueFactorizationMonoid.factors_prod
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
a b : α
h : a ~ᵤ b
hb : ¬b = 0
ha : a ≠ 0
⊢ (factors a).prod ~ᵤ (factors b).prod

Tactic:
exact (factors_prod ha).trans <| h.trans (factors_prod hb).symm

Premises used: Associated.trans, Associated.symm, UniqueFactorizationMonoid.factors_prod
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
haveI : ∀ (this : DecidableEq α), ∃ (p : α), a ~ᵤ p ∧ f = {p}
⊢ ∃ (p : α), a ~ᵤ p ∧ f = {p}

Tactic:
haveI := Classical.decEq α
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
⊢ ∃ (p : α), a ~ᵤ p ∧ f = {p}

Tactic:
@Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1

Premises used: Multiset.induction_on, And.right, And.left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
refine_1 :
  Multiset.prod 0 ~ᵤ a →
    (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}
refine_2 :
  ∀ (a_1 : α) (s : Multiset α),
    (fun (g : Multiset α) ↦ g.prod ~ᵤ a → (∀ b ∈ g, Prime b) → ∃ (p : α), a ~ᵤ p ∧ g = {p}) s →
      (fun (g : Multiset α) ↦ g.prod ~ᵤ a → (∀ b ∈ g, Prime b) → ∃ (p : α), a ~ᵤ p ∧ g = {p}) (a_1 ::ₘ s)
⊢ ∃ (p : α), a ~ᵤ p ∧ f = {p}

Tactic:
refine @Multiset.induction_on _
    (fun g => (g.prod ~ᵤ a) → (∀ b ∈ g, Prime b) → ∃ p, a ~ᵤ p ∧ g = {p}) f ?_ ?_ pfa.2 pfa.1

Premises used: Multiset.induction_on, And.right, And.left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
h : Multiset.prod 0 ~ᵤ a
⊢ (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}

Tactic:
(ha.not_isUnit (associated_one_iff_isUnit.1 (Associated.symm h))).elim

Premises used: False.elim, associated_one_iff_isUnit, Irreducible.not_isUnit, Associated.symm, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
h : Multiset.prod 0 ~ᵤ a
⊢ (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}

Tactic:
exact (ha.not_isUnit (associated_one_iff_isUnit.1 (Associated.symm h))).elim

Premises used: False.elim, associated_one_iff_isUnit, Irreducible.not_isUnit, Associated.symm, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h_1 : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
h : a ~ᵤ p ∧ p ::ₘ s = {p}
⊢ ∃ (p_1 : α), a ~ᵤ p_1 ∧ p ::ₘ s = {p_1}

Tactic:
use p
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
by_contra : ∀ (hs0 : ¬s = 0), False
⊢ s = 0

Tactic:
by_contra hs0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
obtain : ∀ (q : α) (hq : q ∈ s), False
⊢ False

Tactic:
obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0

Premises used: Multiset.exists_mem_of_ne_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ q ∈ p ::ₘ s

Tactic:
simp [hq]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
apply : IsUnit q
⊢ False

Tactic:
apply (hs q (by simp [hq])).2.1

Premises used: And.right, And.left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ IsUnit q

Tactic:
(ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_

Premises used: Or.resolve_left, Irreducible.isUnit_or_isUnit
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
refine_1 : a = p * (↑u : α) * (s.erase q).prod * q
refine_2 : ¬IsUnit (p * (↑u : α) * (s.erase q).prod)
⊢ IsUnit q

Tactic:
refine (ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_

Premises used: Or.resolve_left, Irreducible.isUnit_or_isUnit
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
refine_1 : (↑u : α) * (p ::ₘ s).prod = (↑u : α) * (p * s.prod)
⊢ a = p * (↑u : α) * (s.erase q).prod * q

Tactic:
rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]

Premises used: Multiset.prod_cons, mul_assoc, mul_right_comm, Multiset.cons_erase, mul_comm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ (↑u : α) * (p ::ₘ s).prod = (↑u : α) * (p * s.prod)

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
apply : ¬IsUnit p
⊢ ¬IsUnit (p * (↑u : α) * (s.erase q).prod)

Tactic:
apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)

Premises used: mt, isUnit_of_mul_isUnit_left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : ¬s = 0
q : α
hq : q ∈ s
⊢ ¬IsUnit p

Tactic:
apply (hs p (Multiset.mem_cons_self _ _)).2.1

Premises used: And.right, And.left, Multiset.mem_cons_self
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
⊢ s = 0

Tactic:
have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by simp [hq])).2.1
      refine (ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_
      · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]
        simp
      apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)
      apply (hs p (Multiset.mem_cons_self _ _)).2.1
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
h_1 : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ (p : α), a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : s = 0
h :
  (1 ~ᵤ a →
      (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}) →
    ∀ (hu : p * (↑u : α) = a) (hs : ∀ b ∈ p ::ₘ 0, Prime b) (hs0 : True), a ~ᵤ p ∧ p ::ₘ 0 = {p}
⊢ a ~ᵤ p ∧ p ::ₘ s = {p}

Tactic:
simp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at *

Premises used: Multiset.prod_zero, Multiset.prod_cons, mul_one
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
u : αˣ
h :
  1 ~ᵤ a → (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}
hu : p * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ 0, Prime b
hs0 : True
⊢ a ~ᵤ p ∧ p ::ₘ 0 = {p}

Tactic:
⟨Associated.symm ⟨u, hu⟩, rfl⟩

Premises used: Associated.symm, And.intro, Exists.intro, rfl
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
u : αˣ
h :
  1 ~ᵤ a → (∀ (b : α), Membership.mem (γ := Multiset α) 0 b → Prime b) → ∃ (p : α), a ~ᵤ p ∧ Eq (α := Multiset α) 0 {p}
hu : p * (↑u : α) = a
hs : ∀ b ∈ p ::ₘ 0, Prime b
hs0 : True
⊢ a ~ᵤ p ∧ p ::ₘ 0 = {p}

Tactic:
exact ⟨Associated.symm ⟨u, hu⟩, rfl⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
eif : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
h : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : a * b = 0
ha0 : a = 0
⊢ p ∣ a ∨ p ∣ b

Tactic:
simp [ha0]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
eif : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
h : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : a * b = 0
hb0 : b = 0
⊢ p ∣ a ∨ p ∣ b

Tactic:
simp [hb0]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
eif : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
h : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x = 0
⊢ False

Tactic:
simp_all
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
eif : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
h : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
⊢ x ≠ 0

Tactic:
have hx0 : x ≠ 0 := fun hx0 => by simp_all
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
eif : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a
uif :
  ∀ (f g : Multiset α),
    (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
p : α
this : DecidableEq α := Classical.decEq α
hpi : Irreducible p
a b : α
h : p ∣ a * b
x : α
hx : a * b = p * x
hab0 : ¬a * b = 0
hx0 : x ≠ 0
⊢ a ≠ 0

Tactic:
have ha0 : a ≠ 0 := left_ne_zero_of_mul hab0