awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
refine_1 : ∀ (h : x ∈ p), Irreducible x
refine_1₁ : ∀ (h : x ∈ factors c), Irreducible x
⊢ Irreducible x

Tactic:
obtain h | h := Multiset.mem_add.1 hx

Premises used: Multiset.mem_add, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ p
⊢ Irreducible x

Tactic:
hp x h
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ p
⊢ Irreducible x

Tactic:
exact hp x h
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ factors c
⊢ Irreducible x

Tactic:
irreducible_of_factor _ h

Premises used: UniqueFactorizationMonoid.irreducible_of_factor
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
x : Associates α
hx : x ∈ p + factors c
h : x ∈ factors c
⊢ Irreducible x

Tactic:
exact irreducible_of_factor _ h

Premises used: UniqueFactorizationMonoid.irreducible_of_factor
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
refine_2 : p.prod * c = p.prod * (factors c).prod
⊢ q.prod = (p + factors c).prod

Tactic:
rw [eqc, Multiset.prod_add]

Premises used: Multiset.prod_add
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
refine_2 : c = (factors c).prod
⊢ p.prod * c = p.prod * (factors c).prod

Tactic:
congr
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
⊢ c = (factors c).prod

Tactic:
associated_iff_eq.mp (factors_prod fun hc => ?_).symm

Premises used: associated_iff_eq, Associated.symm, UniqueFactorizationMonoid.factors_prod, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
refine_2 : ∀ (hc : c = 0), False
⊢ c = (factors c).prod

Tactic:
refine associated_iff_eq.mp (factors_prod fun hc => ?_).symm

Premises used: associated_iff_eq, Associated.symm, UniqueFactorizationMonoid.factors_prod, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ False

Tactic:
not_irreducible_zero (hq _ ?_)

Premises used: not_irreducible_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
refine_2 : 0 ∈ q
⊢ False

Tactic:
refine not_irreducible_zero (hq _ ?_)

Premises used: not_irreducible_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : UniqueFactorizationMonoid α
inst_2 : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
refine_2 : q.prod = 0
refine_2₁ : p.prod * c = 0
refine_2₂ : p.prod * 0 = 0
refine_2₃ refine_2₄ : Eq.{u_1 + 1} (α := Associates α) 0 0
⊢ 0 ∈ q

Tactic:
rw [← prod_eq_zero_iff, eqc, hc, mul_zero]

Premises used: Multiset.prod_eq_zero_iff, MulZeroClass.mul_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit

Tactic:
RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt

Premises used: RelHomClass.wellFounded, wellFounded_lt, RelHom.mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
refine_1 : α → ℕ∞
refine_2 :
  ∀ {a b : α},
    DvdNotUnit a b →
      LT.lt (α := ℕ∞)
        (if h : a = 0 then ⊤
        else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞))
        (if h : b = 0 then ⊤
        else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
⊢ WellFounded.{u_1 + 1} (α := α) DvdNotUnit

Tactic:
refine RelHomClass.wellFounded
        (RelHom.mk ?_ ?_ : (DvdNotUnit : α → α → Prop) →r ((· < ·) : ℕ∞ → ℕ∞ → Prop)) wellFounded_lt

Premises used: RelHomClass.wellFounded, wellFounded_lt, RelHom.mk
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a : α
pos : (h : a = 0) → ℕ∞
neg : (h : ¬a = 0) → ℕ∞
⊢ ℕ∞

Tactic:
by_cases h : a = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a : α
h : a = 0
⊢ ℕ∞

Tactic:
exact ⊤
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a : α
h : ¬a = 0
⊢ ℕ∞

Tactic:
exact ↑(Multiset.card (Classical.choose (pf a h)))

Premises used: Classical.choose, Multiset.card
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
refine_2 :
  LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
⊢ LT.lt (α := ℕ∞)
    (if h : a = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞))
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))

Tactic:
rw [dif_neg ane0]

Premises used: dif_neg
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
pos :
  ∀ (h : b = 0),
    LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
      (if h : b = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
neg :
  ∀ (h : ¬b = 0),
    LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
      (if h : b = 0 then ⊤
      else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))

Tactic:
by_cases h : b = 0
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : b = 0
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))

Tactic:
simp [h, lt_top_iff_ne_top]

Premises used: lt_top_iff_ne_top
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
neg :
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card
⊢ LT.lt (α := ℕ∞) (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card : ℕ∞)
    (if h : b = 0 then ⊤
    else (↑(Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card : ℕ∞))

Tactic:
rw [dif_neg h, Nat.cast_lt]

Premises used: dif_neg, Nat.cast_lt
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ c ≠ 0

Tactic:
mt (fun con => ?_) h

Premises used: mt
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
refine : ∀ (con : c = 0), b = 0
⊢ c ≠ 0

Tactic:
refine mt (fun con => ?_) h

Premises used: mt
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
con : c = 0
rw : a * c = 0
rw₁ : a * 0 = 0
rw₂ rw₃ : Eq.{u_1 + 1} (α := α) 0 0
⊢ b = 0

Tactic:
rw [b_eq, con, mul_zero]

Premises used: MulZeroClass.mul_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
⊢ c ≠ 0

Tactic:
have cne0 : c ≠ 0 := by
          refine mt (fun con => ?_) h
          rw [b_eq, con, mul_zero]
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card +
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).card

Tactic:
lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))

Premises used: Iff.mpr, lt_add_of_pos_right, associated_one_iff_isUnit, Multiset.card_pos, Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card +
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).card =
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).card

Tactic:
(Multiset.card_add _ _).symm

Premises used: Multiset.card_add, Eq.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).card =
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card

Tactic:
Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)

Premises used: Classical.choose_spec, prime_factors_unique, And.left, Multiset.card_eq_card_of_rel
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
neg : ∀ (con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0), c ~ᵤ 1
neg₁ :
  ∀
    x ∈
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯,
    Prime x
neg₂ :
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).card <
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).card

Tactic:
calc
          Multiset.card (Classical.choose (pf a ane0)) <
              _ + Multiset.card (Classical.choose (pf c cne0)) :=
            lt_add_of_pos_right _
              (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))
          _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) :=
            (Multiset.card_add _ _).symm
          _ = Multiset.card (Classical.choose (pf b h)) :=
            Multiset.card_eq_card_of_rel
            (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)

Premises used: Iff.mpr, lt_add_of_pos_right, Multiset.card_add, associated_one_iff_isUnit, Classical.choose_spec, prime_factors_unique, And.left, HAdd.hAdd, Multiset.card_pos, Iff.mp, Classical.choose, Multiset.card, Eq.symm, Multiset.card_eq_card_of_rel
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
⊢ c ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod

Tactic:
(Classical.choose_spec (pf c cne0)).2.symm

Premises used: Classical.choose_spec, Associated.symm, And.right
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
h₁ : 1 = (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod
⊢ c ~ᵤ 1

Tactic:
convert! (Classical.choose_spec (pf c cne0)).2.symm

Premises used: Classical.choose_spec, Associated.symm, And.right
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
con : Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯ = 0
h₁ : Eq.{u_1 + 1} (α := α) 1 (Multiset.prod 0)
h₂ h₃ : Eq.{u_1 + 1} (α := α) 1 1
⊢ 1 = (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod

Tactic:
rw [con, Multiset.prod_zero]

Premises used: Multiset.prod_zero
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
x : α
hadd :
  x ∈
    Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
      Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
⊢ x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ ∨
    x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯

Tactic:
rw [Multiset.mem_add] at hadd

Premises used: Multiset.mem_add
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
x : α
hadd :
  x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ ∨
    x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
neg : ∀ (h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯), Prime x
neg₁ : ∀ (h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯), Prime x
⊢ Prime x

Tactic:
rcases hadd with h | h
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h_1 : ¬b = 0
cne0 : c ≠ 0
x : α
h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯
apply : a ≠ 0
⊢ Prime x

Tactic:
apply (Classical.choose_spec (pf _ _)).1 _ h

Premises used: Classical.choose_spec, And.left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h_1 : ¬b = 0
cne0 : c ≠ 0
x : α
h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
apply : c ≠ 0
⊢ Prime x

Tactic:
apply (Classical.choose_spec (pf _ _)).1 _ h

Premises used: Classical.choose_spec, And.left
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h_1 : ¬b = 0
cne0 : c ≠ 0
x : α
h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯
⊢ a ≠ 0

Tactic:
assumption
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h_1 : ¬b = 0
cne0 : c ≠ 0
x : α
h : x ∈ Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯
⊢ c ≠ 0

Tactic:
assumption
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
neg :
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯ +
        Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod

Tactic:
rw [Multiset.prod_add]

Premises used: Multiset.prod_add
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic
Goal:
α : Type u_1
inst : CommMonoidWithZero α
inst_1 : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ (f : Multiset α), (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
trans :
  (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    a * c
trans₁ : a * c ~ᵤ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod
⊢ (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ a) ⋯).prod *
      (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ c) ⋯).prod ~ᵤ
    (Classical.choose (p := fun (x : Multiset α) ↦ (∀ b ∈ x, Prime b) ∧ x.prod ~ᵤ b) ⋯).prod

Tactic:
trans a * c