awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
h : Membership.mem (γ := Submonoid R) (closure {r : R | Prime r}) 0
rcases : ∀ (l : Multiset R) (hl : ∀ y ∈ l, Membership.mem (γ := Set R) {r : R | Prime r} y) (hprod : l.prod = 0), False
⊢ False

Tactic:
rcases exists_multiset_of_mem_closure h with ⟨l, hl, hprod⟩

Premises used: Submonoid.exists_multiset_of_mem_closure
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
h : Membership.mem (γ := Submonoid R) (closure {r : R | Prime r}) 0
l : Multiset R
hl : ∀ y ∈ l, Membership.mem (γ := Set R) {r : R | Prime r} y
hprod : l.prod = 0
⊢ False

Tactic:
not_prime_zero (hl 0 (Multiset.prod_eq_zero_iff.1 hprod))

Premises used: not_prime_zero, Multiset.prod_eq_zero_iff, Iff.mp, OfNat.ofNat
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
h : Membership.mem (γ := Submonoid R) (closure {r : R | Prime r}) 0
l : Multiset R
hl : ∀ y ∈ l, Membership.mem (γ := Set R) {r : R | Prime r} y
hprod : l.prod = 0
⊢ False

Tactic:
exact not_prime_zero (hl 0 (Multiset.prod_eq_zero_iff.1 hprod))

Premises used: not_prime_zero, Multiset.prod_eq_zero_iff, Iff.mp, OfNat.ofNat
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
⊢ 0 ∉ closure {r : R | Prime r}

Tactic:
have hzero : 0 ∉ closure {r : R | Prime r} := fun h ↦ by
    rcases exists_multiset_of_mem_closure h with ⟨l, hl, hprod⟩
    exact not_prime_zero (hl 0 (Multiset.prod_eq_zero_iff.1 hprod))
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
rcases :
  ∀ (T : Ideal R)
    (hT : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) T)
    (hT₂ :
      ∀ (I : Ideal R),
        Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I →
          T ≤ I → I = T),
    False
⊢ False

Tactic:
rcases exists_mem_kaplanskySet_eq_of_le hzero with ⟨T, hT, hT₂⟩

Premises used: _private.0.UniqueFactorizationMonoid.exists_mem_kaplanskySet_eq_of_le
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) T
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
⊢ T ≠ ⊥

Tactic:
have hT₃ : T ≠ ⊥ := fun h₂ ↦ ha (span_singleton_eq_bot.1 ((h₂ ▸ hT₂) _ h zero_le))
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) T
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
hT₃ : T ≠ ⊥
⊢ T.IsPrime

Tactic:
have Tpri := isPrime_of_maximally_disjoint T _ hT (fun J hJ H ↦ hJ.ne (hT₂ J H hJ.le).symm)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) T
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
hT₃ : T ≠ ⊥
Tpri : T.IsPrime
rcases : ∀ (x : R) (H₃ : x ∈ T) (H₄ : Prime x), False
⊢ False

Tactic:
rcases (H T) hT₃ Tpri with ⟨x, H₃, H₄⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) T
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
hT₃ : T ≠ ⊥
Tpri : T.IsPrime
x : R
H₃ : x ∈ T
H₄ : Prime x
rw :
  ∀ (hT : ∀ (x : R), x ∉ (↑T : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R))
    (hT₂ :
      ∀ (I : Ideal R),
        Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I →
          T ≤ I → I = T)
    (hT₃ : T ≠ ⊥) (Tpri : T.IsPrime) (x : R) (H₃ : x ∈ T) (H₄ : Prime x), False
⊢ False

Tactic:
rw [mem_kaplanskySet_iff, eq_empty_iff_forall_notMem] at hT

Premises used: Set.eq_empty_iff_forall_notMem, _private.0.UniqueFactorizationMonoid.mem_kaplanskySet_iff
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : ∀ (x : R), x ∉ (↑T : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
hT₃ : T ≠ ⊥
Tpri : T.IsPrime
x : R
H₃ : x ∈ T
H₄ : Prime x
⊢ False

Tactic:
hT x ⟨H₃, subset_closure H₄⟩

Premises used: And.intro, Submonoid.subset_closure
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
a : R
ha : a ≠ 0
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
hzero : 0 ∉ closure {r : R | Prime r}
h : Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) (span {a})
T : Ideal R
hT : ∀ (x : R), x ∉ (↑T : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
hT₂ :
  ∀ (I : Ideal R),
    Membership.mem (γ := Set (Ideal R)) (kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))) I → T ≤ I → I = T
hT₃ : T ≠ ⊥
Tpri : T.IsPrime
x : R
H₃ : x ∈ T
H₄ : Prime x
⊢ False

Tactic:
exact hT x ⟨H₃, subset_closure H₄⟩

Premises used: And.intro, Submonoid.subset_closure
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
⊢ UniqueFactorizationMonoid R

Tactic:
UniqueFactorizationMonoid.of_exists_prime_factors fun a ha ↦ ?_

Premises used: UniqueFactorizationMonoid.of_exists_prime_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
refine : ∀ (a : R) (ha : a ≠ 0), ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a
⊢ UniqueFactorizationMonoid R

Tactic:
refine UniqueFactorizationMonoid.of_exists_prime_factors fun a ha ↦ ?_

Premises used: UniqueFactorizationMonoid.of_exists_prime_factors
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
⊢ span {a} ∉ kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))

Tactic:
have ha₂ := span_notMem_kaplanskySet ha H
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : span {a} ∉ kaplanskySet (toSubsemigroup (closure {r : R | Prime r}))
⊢ ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅

Tactic:
rw [mem_kaplanskySet_iff] at ha₂

Premises used: _private.0.UniqueFactorizationMonoid.mem_kaplanskySet_iff
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
rcases :
  ∀ (x : R) (hx : x ∈ (↑(span {a}) : Set R))
    (hx₂ : x ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)),
    ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a

Tactic:
rcases nonempty_iff_ne_empty.2 ha₂ with ⟨x, hx, hx₂⟩

Premises used: Iff.mpr, Set.nonempty_iff_ne_empty
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
hx₂ : x ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
obtain : ∀ (b : R) (hb : b * a = x), ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a

Tactic:
obtain ⟨b, hb⟩ := mem_span_singleton'.1 hx

Premises used: Ideal.mem_span_singleton', Iff.mp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
hx₂ : x ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
b : R
hb : b * a = x
⊢ a * b ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)

Tactic:
rw [← hb, mul_comm] at hx₂

Premises used: mul_comm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
hx₂ : x ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
b : R
hb : b * a = x
hx₃ : a * b ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
⊢ b * a = x

Tactic:
rw [← hb, mul_comm] at hx₂

Premises used: mul_comm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
b : R
hx₂ : a * b ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
hb : b * a = x
⊢ Subset.{u_1} (α := Set R) {r : R | Prime r} {r : R | IsUnit r ∨ Prime r}

Tactic:
grind
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
b : R
hx₂ : a * b ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
hb : b * a = x
⊢ LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})

Tactic:
have hsubset : closure {r : R | Prime r} ≤ closure {r : R | IsUnit r ∨ Prime r} :=
    closure_mono (by grind)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a : R
ha : a ≠ 0
ha₂ : ¬(↑(span {a}) : Set R) ∩ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R) = ∅
x : R
hx : x ∈ (↑(span {a}) : Set R)
b : R
hx₂ : a * b ∈ (↑(toSubsemigroup (closure {r : R | Prime r}) : Subsemigroup R) : Set R)
hb : b * a = x
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
⊢ Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) a

Tactic:
have := divisor_closure_eq_closure _ _ (hsubset hx₂)
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z : R
hz : Membership.mem (γ := Set R) {r : R | IsUnit r ∨ Prime r} z
mem : ∀ (h : IsUnit z), ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z
mem₁ : ∀ (h : Prime z), ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z

Tactic:
rcases hz with h | h
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z : R
h : IsUnit z
⊢ (∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Prime b) ∧ Associated (Multiset.prod ∅) z

Tactic:
simpa using (associated_one_iff_isUnit.2 h).symm

Premises used: Iff.mpr, associated_one_iff_isUnit, Associated.symm
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z : R
h : IsUnit z
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z

Tactic:
exact ⟨∅, by simpa using (associated_one_iff_isUnit.2 h).symm⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z : R
h : Prime z
⊢ (∀ (b : R), Membership.mem (γ := Multiset R) {z} b → Prime b) ∧ Associated (Multiset.prod {z}) z

Tactic:
simpa
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z : R
h : Prime z
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z

Tactic:
exact ⟨{z}, by simpa⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
⊢ (∀ (b : R), Membership.mem (γ := Multiset R) ∅ b → Prime b) ∧ Associated.{u_1} (M := R) (Multiset.prod ∅) 1

Tactic:
simp
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod 1

Tactic:
exact ⟨∅, by simp⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
h₁ : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z₁
h₂ : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z₂
mul :
  ∀ (S₁ : Multiset R) (hS₁pri : ∀ b ∈ S₁, Prime b) (hS₁ : Associated S₁.prod z₁),
    ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)

Tactic:
obtain ⟨S₁, hS₁pri, hS₁⟩ := h₁
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
h₂ : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
mul :
  ∀ (S₂ : Multiset R) (hS₂pri : ∀ b ∈ S₂, Prime b) (hS₂ : Associated S₂.prod z₂),
    ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)

Tactic:
obtain ⟨S₂, hS₂pri, hS₂⟩ := h₂
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
S₂ : Multiset R
hS₂pri : ∀ b ∈ S₂, Prime b
hS₂ : Associated S₂.prod z₂
⊢ ∀ b ∈ S₁ + S₂, Prime b

Tactic:
grind
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
S₂ : Multiset R
hS₂pri : ∀ b ∈ S₂, Prime b
hS₂ : Associated S₂.prod z₂
mul : Associated (S₁ + S₂).prod (z₁ * z₂)
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)

Tactic:
refine ⟨S₁ + S₂, by grind, ?_⟩
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
S₂ : Multiset R
hS₂pri : ∀ b ∈ S₂, Prime b
hS₂ : Associated S₂.prod z₂
mul : Associated (S₁.prod * S₂.prod) (z₁ * z₂)
⊢ Associated (S₁ + S₂).prod (z₁ * z₂)

Tactic:
rw [Multiset.prod_add]

Premises used: Multiset.prod_add
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
S₂ : Multiset R
hS₂pri : ∀ b ∈ S₂, Prime b
hS₂ : Associated S₂.prod z₂
⊢ Associated (S₁.prod * S₂.prod) (z₁ * z₂)

Tactic:
hS₁.mul_mul hS₂

Premises used: Associated.mul_mul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
z₁ z₂ : R
hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁
hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂
S₁ : Multiset R
hS₁pri : ∀ b ∈ S₁, Prime b
hS₁ : Associated S₁.prod z₁
S₂ : Multiset R
hS₂pri : ∀ b ∈ S₂, Prime b
hS₂ : Associated S₂.prod z₂
⊢ Associated (S₁.prod * S₂.prod) (z₁ * z₂)

Tactic:
exact hS₁.mul_mul hS₂

Premises used: Associated.mul_mul
-- Module: Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x
a x b : R
hsubset : LE.le.{u_1} (α := Submonoid R) (closure {r : R | Prime r}) (closure {r : R | IsUnit r ∨ Prime r})
this : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) a
mem :
  ∀ (z : R) (hz : Membership.mem (γ := Set R) {r : R | IsUnit r ∨ Prime r} z),
    ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z
one : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod 1
mul :
  ∀ (z₁ z₂ : R) (hz₁ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₁)
    (hz₂ : Membership.mem (γ := Submonoid R) (closure {r : R | IsUnit r ∨ Prime r}) z₂)
    (h₁ : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z₁)
    (h₂ : ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod z₂),
    ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod (z₁ * z₂)
⊢ ∃ (f : Multiset R), (∀ b ∈ f, Prime b) ∧ Associated f.prod a

Tactic:
induction this using closure_induction with
  | mem z hz =>
      rcases hz with h | h
      · exact ⟨∅, by simpa using (associated_one_iff_isUnit.2 h).symm⟩
      · exact ⟨{z}, by simpa⟩
  | one => exact ⟨∅, by simp⟩
  | mul z₁ z₂ hz₁ hz₂ h₁ h₂ =>
      obtain ⟨S₁, hS₁pri, hS₁⟩ := h₁
      obtain ⟨S₂, hS₂pri, hS₂⟩ := h₂
      refine ⟨S₁ + S₂, by grind, ?_⟩
      rw [Multiset.prod_add]
      exact hS₁.mul_mul hS₂

Premises used: Submonoid.mul, Multiset.prod_add, Submonoid.one