awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M : Type u_4
N : Type u_5
inst : Monoid M
inst_1 : Monoid N
x z : M × N
⊢ x ~ᵤ z ↔ x.1 ~ᵤ z.1 ∧ x.2 ~ᵤ z.2

Tactic:
⟨fun ⟨u, hu⟩ => ⟨⟨(MulEquiv.prodUnits.toFun u).1, (Prod.eq_iff_fst_eq_snd_eq.1 hu).1⟩,
    ⟨(MulEquiv.prodUnits.toFun u).2, (Prod.eq_iff_fst_eq_snd_eq.1 hu).2⟩⟩,
  fun ⟨⟨u₁, h₁⟩, ⟨u₂, h₂⟩⟩ =>
    ⟨MulEquiv.prodUnits.invFun (u₁, u₂), Prod.eq_iff_fst_eq_snd_eq.2 ⟨h₁, h₂⟩⟩⟩

Premises used: Iff.mpr, _private.0.Prod.associated_iff.match_1, MulEquiv.toEquiv, Prod.mk, Equiv.toFun, Prod.fst, _private.0.Prod.associated_iff.match_2, And, Associated, And.right, And.left, And.intro, Iff.intro, Exists.intro, Iff.mp, Prod.eq_iff_fst_eq_snd_eq, MulEquiv.prodUnits, Prod.snd, Equiv.invFun
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M : Type u_4
inst : CommMonoid M
ι : Type u_5
f g : ι → M
h : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → f i ~ᵤ g i
empty : Associated.{u_4} (M := M) 1 1
⊢ Associated.{u_4} (M := M) (∏ i ∈ ∅, f i) (∏ i ∈ ∅, g i)

Tactic:
simp only [Finset.prod_empty]

Premises used: Finset.prod_empty
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M : Type u_4
inst : CommMonoid M
ι : Type u_5
f g : ι → M
h : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → f i ~ᵤ g i
⊢ Associated.{u_4} (M := M) 1 1

Tactic:
rfl
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M : Type u_4
inst : CommMonoid M
ι : Type u_5
s_1 : Finset ι
f g : ι → M
j : ι
s : Finset ι
hjs : j ∉ s
IH : ∀ (h : ∀ i ∈ s, f i ~ᵤ g i), Associated.{u_4} (M := M) (∏ i ∈ s, f i) (∏ i ∈ s, g i)
h : ∀ i ∈ insert j s, f i ~ᵤ g i
⊢ Associated.{u_4} (M := M) (∏ i ∈ insert j s, f i) (∏ i ∈ insert j s, g i)

Tactic:
grind [Associated.mul_mul]

Premises used: Associated.mul_mul
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M : Type u_4
inst : CommMonoid M
ι : Type u_5
s : Finset ι
f g : ι → M
h : ∀ i ∈ s, f i ~ᵤ g i
empty :
  ∀ (h : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → f i ~ᵤ g i),
    Associated.{u_4} (M := M) (∏ i ∈ ∅, f i) (∏ i ∈ ∅, g i)
insert :
  ∀ (j : ι) (s : Finset ι) (hjs : j ∉ s)
    (IH : ∀ (h : ∀ i ∈ s, f i ~ᵤ g i), Associated.{u_4} (M := M) (∏ i ∈ s, f i) (∏ i ∈ s, g i))
    (h : ∀ i ∈ Insert.insert j s, f i ~ᵤ g i),
    Associated.{u_4} (M := M) (∏ i ∈ Insert.insert j s, f i) (∏ i ∈ Insert.insert j s, g i)
⊢ Associated.{u_4} (M := M) (∏ i ∈ s, f i) (∏ i ∈ s, g i)

Tactic:
induction s using Finset.induction with
  | empty =>
    simp only [Finset.prod_empty]
    rfl
  | insert j s hjs IH =>
    classical
    convert_to (∏ i ∈ insert j s, f i) ~ᵤ (∏ i ∈ insert j s, g i)
    grind [Associated.mul_mul]

Premises used: Finset.prod_empty, Insert.insert, Associated.mul_mul
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s : Multiset M₀
⊢ (∀ (r : M₀), Membership.mem (γ := Multiset M₀) 0 r → Prime r) →
    p ∣ Multiset.prod 0 → ∃ (q : M₀), Membership.mem (γ := Multiset M₀) 0 q ∧ p ~ᵤ q

Tactic:
simp [mt isUnit_iff_dvd_one.2 hp.not_unit]

Premises used: Iff.mpr, mt, Prime.not_unit, isUnit_iff_dvd_one
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ (a ::ₘ s).prod
⊢ p ∣ a * s.prod

Tactic:
rw [Multiset.prod_cons] at hps

Premises used: Multiset.prod_cons
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
inl : ∀ (h : p ∣ a), ∃ q ∈ a ::ₘ s, p ~ᵤ q
inr : ∀ (h : p ∣ s.prod), ∃ q ∈ a ::ₘ s, p ~ᵤ q
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
rcases hp.dvd_or_dvd hps with h | h

Premises used: Prime.dvd_or_dvd
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ a
⊢ Prime a

Tactic:
have hap := hs a (Multiset.mem_cons.2 (Or.inl rfl))
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ a
hap : Prime a
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
⟨a, Multiset.mem_cons_self a _, hp.associated_of_dvd hap h⟩

Premises used: Prime.associated_of_dvd, And.intro, Exists.intro, Multiset.mem_cons_self
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ a
hap : Prime a
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
exact ⟨a, Multiset.mem_cons_self a _, hp.associated_of_dvd hap h⟩
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ s.prod
inr : ∀ (q : M₀) (hq₁ : q ∈ s) (hq₂ : p ~ᵤ q), ∃ q ∈ a ::ₘ s, p ~ᵤ q
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
rcases ih (fun r hr => hs _ (Multiset.mem_cons.2 (Or.inr hr))) h with ⟨q, hq₁, hq₂⟩

Premises used: Iff.mpr, Multiset.mem_cons, Or.inr
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ s.prod
q : M₀
hq₁ : q ∈ s
hq₂ : p ~ᵤ q
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
⟨q, Multiset.mem_cons.2 (Or.inr hq₁), hq₂⟩

Premises used: Iff.mpr, Multiset.mem_cons, And.intro, Exists.intro, Or.inr
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
p : M₀
hp : Prime p
s_1 : Multiset M₀
a : M₀
s : Multiset M₀
ih : (∀ r ∈ s, Prime r) → p ∣ s.prod → ∃ q ∈ s, p ~ᵤ q
hs : ∀ r ∈ a ::ₘ s, Prime r
hps : p ∣ a * s.prod
h : p ∣ s.prod
q : M₀
hq₁ : q ∈ s
hq₂ : p ~ᵤ q
⊢ ∃ q ∈ a ::ₘ s, p ~ᵤ q

Tactic:
exact ⟨q, Multiset.mem_cons.2 (Or.inr hq₁), hq₂⟩
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
obtain :
  ∀ (m : Multiset M₀) (hm : ∀ y ∈ m, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y)
    (hprod : m.prod = x * y), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
obtain ⟨m, hm, hprod⟩ := exists_multiset_of_mem_closure hxy

Premises used: Submonoid.exists_multiset_of_mem_closure
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : ∀ (y : M₀), Membership.mem (γ := Multiset M₀) 0 y → Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : Multiset.prod 0 = x * y
empty : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} x
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
apply subset_closure

Premises used: Submonoid.subset_closure
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : ∀ (y : M₀), Membership.mem (γ := Multiset M₀) 0 y → Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : 1 = x * y
empty : IsUnit x
⊢ IsUnit x ∨ Prime x

Tactic:
left
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : ∀ (y : M₀), Membership.mem (γ := Multiset M₀) 0 y → Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : 1 = x * y
⊢ IsUnit x

Tactic:
.of_mul_eq_one _ hprod.symm

Premises used: IsUnit.of_mul_eq_one, Eq.symm
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : ∀ (y : M₀), Membership.mem (γ := Multiset M₀) 0 y → Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : 1 = x * y
⊢ IsUnit x

Tactic:
exact .of_mul_eq_one _ hprod.symm
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : ∀ y ∈ c ::ₘ s, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : (c ::ₘ s).prod = x * y
⊢ (IsUnit c ∨ Prime c) ∧ ∀ a ∈ s, IsUnit a ∨ Prime a

Tactic:
simp only [Multiset.mem_cons, forall_eq_or_imp, Set.mem_setOf] at hm

Premises used: forall_eq_or_imp, Set.mem_setOf, Multiset.mem_cons
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : (c ::ₘ s).prod = x * y
hm : (IsUnit c ∨ Prime c) ∧ ∀ a ∈ s, IsUnit a ∨ Prime a
⊢ c * s.prod = x * y

Tactic:
simp only [Multiset.prod_cons] at hprod

Premises used: Multiset.prod_cons
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hm : (IsUnit c ∨ Prime c) ∧ ∀ a ∈ s, IsUnit a ∨ Prime a
hprod : c * s.prod = x * y
cons :
  ∀ (hs : ∀ a ∈ s, IsUnit a ∨ Prime a) (ha₁ : IsUnit c),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
cons₁ :
  ∀ (hs : ∀ a ∈ s, IsUnit a ∨ Prime a) (ha₂ : Prime c),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
obtain ⟨ha₁ | ha₂, hs⟩ := hm
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
cons : ∀ (k : M₀) (hk : c * k = 1), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
rcases ha₁.exists_right_inv with ⟨k, hk⟩

Premises used: IsUnit.exists_right_inv
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
hind x (y * k) ?_ hs ?_

Premises used: HMul.hMul
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * (y * k))
cons₁ : s.prod = x * (y * k)
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x

Tactic:
refine hind x (y * k) ?_ hs ?_

Premises used: HMul.hMul
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (k ::ₘ c ::ₘ s).prod
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * (y * k))

Tactic:
simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]

Premises used: Multiset.prod_cons, mul_assoc, mul_comm
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (k ::ₘ c ::ₘ s).prod

Tactic:
multiset_prod_mem _ _ (Multiset.forall_mem_cons.2 ⟨subset_closure ?_,
          Multiset.forall_mem_cons.2 ⟨subset_closure ?_, fun t ht => subset_closure (hs t ht)⟩⟩)

Premises used: Iff.mpr, Set.Mem, Submonoid.multiset_prod_mem, And.intro, Submonoid.subset_closure, Multiset.forall_mem_cons
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} k
cons₁ : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (k ::ₘ c ::ₘ s).prod

Tactic:
refine multiset_prod_mem _ _ (Multiset.forall_mem_cons.2 ⟨subset_closure ?_,
          Multiset.forall_mem_cons.2 ⟨subset_closure ?_, fun t ht => subset_closure (hs t ht)⟩⟩)

Premises used: Iff.mpr, Submonoid.multiset_prod_mem, And.intro, Submonoid.subset_closure, Multiset.forall_mem_cons
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : IsUnit k
⊢ Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} k

Tactic:
left
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ IsUnit k

Tactic:
.of_mul_eq_one_right _ hk

Premises used: IsUnit.of_mul_eq_one_right
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ IsUnit k

Tactic:
exact .of_mul_eq_one_right _ hk
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : IsUnit c
⊢ Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} c

Tactic:
left
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ IsUnit c

Tactic:
ha₁
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ IsUnit c

Tactic:
exact ha₁
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : c * s.prod = x * y ∨ k = 0
⊢ s.prod = x * (y * k)

Tactic:
rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]

Premises used: Multiset.prod, mul_assoc, mul_eq_mul_right_iff, mul_comm, mul_one
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
cons : c * s.prod = x * y
⊢ c * s.prod = x * y ∨ k = 0

Tactic:
left
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ c * s.prod = x * y

Tactic:
hprod
-- Module: Mathlib.Algebra.BigOperators.Associated
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀) (hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y))
    (hm : ∀ y ∈ s, IsUnit y ∨ Prime y) (hprod : s.prod = x * y),
    Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
x y : M₀
hxy : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * y)
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ c * s.prod = x * y

Tactic:
exact hprod