awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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))
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
have cons₁ : s.prod = x * (y * k)
Tactic: refine hind x (y * k) ?_ hs ?_
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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 * (y * k))
have cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (k ::ₘ c ::ₘ s).prod
Tactic: simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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
have cons : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} k
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)⟩⟩)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (k ::ₘ c ::ₘ s).prod
have cons₁ : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} c
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)⟩⟩)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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 := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} k
have cons : IsUnit k
Tactic: left
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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
⊢ s.prod = x * (y * k)
have cons : c * s.prod = x * y ∨ k = 0
Tactic: rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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
ha₂ : Prime c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
have cons : ∀ (c_1 : M₀) (hc : x = c * c_1), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
Tactic: rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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
ha₂ : Prime c
cons : ∀ (c_1 : M₀) (hc : x = c * c_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
have cons₁ : ∀ (c_1 : M₀) (hc : y = c * c_1), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
Tactic: rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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_1 * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hc : x = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
have cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c_1 * c)
Tactic: rw [hc]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (hypothesis)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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_1 * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hc : x = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c_1 * c)
have hprod₁ : c_1 * s.prod = c_1 * (c * y)
Tactic: rw [hc, mul_assoc] at hprod
Cleared hypotheses: hprod, hc
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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)
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hprod : c_1 * s.prod = c_1 * (c * y)
hc : x = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c_1 * c)
have cons : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} c_1
Tactic: refine Submonoid.mul_mem _ (subset_closure ?_)
          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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)
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hprod : c_1 * s.prod = c_1 * (c * y)
hc : x = c_1 * c
cons : Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} c_1
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c_1 * c)
have cons₁ : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c * y)
Tactic: refine Submonoid.mul_mem _ (subset_closure ?_)
          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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)
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hprod : c_1 * s.prod = c_1 * (c * y)
hc : x = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (c * y)
have cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) s.prod
Tactic: rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (hypothesis)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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_1 * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hc : y = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
have hprod₁ : c_1 * s.prod = c_1 * (x * c)
Tactic: rw [hc, mul_comm x _, mul_assoc, mul_comm c _] at hprod
Cleared hypotheses: hprod, hc
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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)
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hprod : c_1 * s.prod = c_1 * (x * c)
hc : y = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
have cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * c)
Tactic: refine hind x c ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
c_1 : 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)
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c_1
c : M₀
hprod : c_1 * s.prod = c_1 * (x * c)
hc : y = c_1 * c
⊢ Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) (x * c)
have cons : Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) s.prod
Tactic: rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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)
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
have empty : ∀ (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), Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
Tactic: induction m using Multiset.induction generalizing x y with
  | empty =>
    apply subset_closure
    push _ ∈ _
    simp only [Multiset.prod_zero] at hprod
    left; exact .of_mul_eq_one _ hprod.symm
  | cons c s hind =>
    simp only [Multiset.mem_cons, forall_eq_or_imp, Set.mem_setOf] at hm
    simp only [Multiset.prod_cons] at hprod
    simp only [Set.mem_setOf_eq] at hind
    obtain ⟨ha₁ | ha₂, hs⟩ := hm
    · rcases ha₁.exists_right_inv with ⟨k, hk⟩
      refine hind x (y * k) ?_ hs ?_
      · simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]
        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)⟩⟩)
        · left; exact .of_mul_eq_one_right _ hk
        · left; exact ha₁
      · rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
        left; exact hprod
    · rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
      · rw [hc]; rw [hc, mul_assoc] at hprod
        refine Submonoid.mul_mem _ (subset_closure ?_)
          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
        · right; exact ha₂
        rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
        exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
      rw [hc, mul_comm x _, mul_assoc, mul_comm c _] at hprod
      refine hind x c ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod)
      rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
      exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
Cleared hypotheses: x, y, hxy, m, hm, hprod
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
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)
m : Multiset M₀
hm : ∀ y ∈ m, Membership.mem (γ := Set M₀) {r : M₀ | IsUnit r ∨ Prime r} y
hprod : m.prod = x * y
empty :
  ∀ (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), 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
have cons : ∀ (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),
  Membership.mem (γ := Submonoid M₀) (closure {r : M₀ | IsUnit r ∨ Prime r}) x
Tactic: induction m using Multiset.induction generalizing x y with
  | empty =>
    apply subset_closure
    push _ ∈ _
    simp only [Multiset.prod_zero] at hprod
    left; exact .of_mul_eq_one _ hprod.symm
  | cons c s hind =>
    simp only [Multiset.mem_cons, forall_eq_or_imp, Set.mem_setOf] at hm
    simp only [Multiset.prod_cons] at hprod
    simp only [Set.mem_setOf_eq] at hind
    obtain ⟨ha₁ | ha₂, hs⟩ := hm
    · rcases ha₁.exists_right_inv with ⟨k, hk⟩
      refine hind x (y * k) ?_ hs ?_
      · simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]
        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)⟩⟩)
        · left; exact .of_mul_eq_one_right _ hk
        · left; exact ha₁
      · rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
        left; exact hprod
    · rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
      · rw [hc]; rw [hc, mul_assoc] at hprod
        refine Submonoid.mul_mem _ (subset_closure ?_)
          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
        · right; exact ha₂
        rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
        exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
      rw [hc, mul_comm x _, mul_assoc, mul_comm c _] at hprod
      refine hind x c ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod)
      rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
      exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
Cleared hypotheses: x, y, hxy, m, hm, hprod
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
n : M₀
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ n
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
⊢ (a ::ₘ s).prod ∣ n
have cons : a * s.prod ∣ n
Tactic: rw [Multiset.prod_cons]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
n : M₀
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ n
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
⊢ a * s.prod ∣ n
have cons : ∀ (k : M₀) (div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k), a * s.prod ∣ a * k
Tactic: obtain ⟨k, rfl⟩ : a ∣ n := div a (Multiset.mem_cons_self a s)
Cleared hypotheses: n, div
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
⊢ a * s.prod ∣ a * k
have cons : s.prod ∣ k
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (goal)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
⊢ s.prod ∣ k
have cons : ∀ (b : M₀) (b_in_s : b ∈ s), b ∣ k
Tactic: refine induct _ (fun a ha => h a (Multiset.mem_cons_of_mem ha)) (fun b b_in_s => ?_)
      fun a => (Multiset.countP_le_of_le _ (Multiset.le_cons_self _ _)).trans (uniq a)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (hypothesis)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
b : M₀
b_in_s : b ∈ s
⊢ b ∣ k
have b_div_n : b ∣ a * k
Tactic: have b_div_n := div b (Multiset.mem_cons_of_mem b_in_s)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (hypothesis)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
b : M₀
b_in_s : b ∈ s
b_div_n : b ∣ a * k
⊢ b ∣ k
have a_prime : Prime a
Tactic: have a_prime := h a (Multiset.mem_cons_self a s)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Associated (hypothesis)
Goal:
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : IsCancelMulZero M₀
inst_2 : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct :
  ∀ (n : M₀) (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ (a : M₀), countP (Associated a) s ≤ 1), s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
b : M₀
b_in_s : b ∈ s
b_div_n : b ∣ a * k
a_prime : Prime a
⊢ b ∣ k
have b_prime : Prime b
Tactic: have b_prime := h b (Multiset.mem_cons_of_mem b_in_s)