l3lab/ntp-mathlib huggingface dataset:
lemma prod_fac :
    (W₁.Q.prod W₂.Q) ⋙ prodLift F hF = F := by
  

Proof state:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
⊢ Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ prodLift F hF = F

Next tactic:
rw [← Functor.uncurry_obj_curry_obj_flip_flip', prod_fac₂, Functor.flip_flip, prod_fac₁,
    Functor.uncurry_obj_curry_obj]
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  

Proof state:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ F₁ = F₂

Next tactic:
apply Functor.curry_obj_injective
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  

Proof state:
case h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ curry.obj F₁ = curry.obj F₂

Next tactic:
apply Construction.uniq
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  apply Construction.uniq
  

Proof state:
case h.h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ MorphismProperty.Q W₁ ⋙ curry.obj F₁ = MorphismProperty.Q W₁ ⋙ curry.obj F₂

Next tactic:
apply Functor.flip_injective
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  apply Construction.uniq
  apply Functor.flip_injective
  

Proof state:
case h.h.h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₁) = Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₂)

Next tactic:
apply Construction.uniq
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  apply Construction.uniq
  apply Functor.flip_injective
  apply Construction.uniq
  

Proof state:
case h.h.h.h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₁) =
    MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₂)

Next tactic:
apply Functor.flip_injective
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  apply Construction.uniq
  apply Functor.flip_injective
  apply Construction.uniq
  apply Functor.flip_injective
  

Proof state:
case h.h.h.h.h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ Functor.flip (MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₁)) =
    Functor.flip (MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₂))

Next tactic:
apply Functor.uncurry_obj_injective
lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
    (h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
      F₁ = F₂ := by
  apply Functor.curry_obj_injective
  apply Construction.uniq
  apply Functor.flip_injective
  apply Construction.uniq
  apply Functor.flip_injective
  apply Functor.uncurry_obj_injective
  

Proof state:
case h.h.h.h.h.h
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₃, u₃} D₁
inst✝³ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝² : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝¹ : MorphismProperty.ContainsIdentities W₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F : C₁ × C₂ ⥤ E
hF : MorphismProperty.IsInvertedBy (MorphismProperty.prod W₁ W₂) F
F₁ F₂ : MorphismProperty.Localization W₁ × MorphismProperty.Localization W₂ ⥤ E
h :
  Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₁ =
    Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂) ⋙ F₂
⊢ uncurry.obj (Functor.flip (MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₁))) =
    uncurry.obj (Functor.flip (MorphismProperty.Q W₂ ⋙ Functor.flip (MorphismProperty.Q W₁ ⋙ curry.obj F₂)))

Next tactic:
simpa only [Functor.uncurry_obj_curry_obj_flip_flip] using h
/-- If `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` are localization functors
for `W₁ : MorphismProperty C₁` and `W₂ : MorphismProperty C₂` respectively,
and if both `W₁` and `W₂` contain identites, then the product
functor `L₁.prod L₂ : C₁ × C₂ ⥤ D₁ × D₂` is a localization functor for `W₁.prod W₂`. -/
instance prod [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] :
    (L₁.prod L₂).IsLocalization (W₁.prod W₂) := by
  

Proof state:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₃, u₃} D₁
inst✝⁴ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : MorphismProperty.ContainsIdentities W₂
inst✝¹ : IsLocalization L₁ W₁
inst✝ : IsLocalization L₂ W₂
⊢ IsLocalization (Functor.prod L₁ L₂) (MorphismProperty.prod W₁ W₂)

Next tactic:
haveI := Construction.prodIsLocalization W₁ W₂
/-- If `L₁ : C₁ ⥤ D₁` and `L₂ : C₂ ⥤ D₂` are localization functors
for `W₁ : MorphismProperty C₁` and `W₂ : MorphismProperty C₂` respectively,
and if both `W₁` and `W₂` contain identites, then the product
functor `L₁.prod L₂ : C₁ × C₂ ⥤ D₁ × D₂` is a localization functor for `W₁.prod W₂`. -/
instance prod [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] :
    (L₁.prod L₂).IsLocalization (W₁.prod W₂) := by
  haveI := Construction.prodIsLocalization W₁ W₂
  

Proof state:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₃
D₂ : Type u₄
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₃, u₃} D₁
inst✝⁴ : Category.{v₄, u₄} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : MorphismProperty.ContainsIdentities W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : MorphismProperty.ContainsIdentities W₂
inst✝¹ : IsLocalization L₁ W₁
inst✝ : IsLocalization L₂ W₂
this : IsLocalization (Functor.prod (MorphismProperty.Q W₁) (MorphismProperty.Q W₂)) (MorphismProperty.prod W₁ W₂)
⊢ IsLocalization (Functor.prod L₁ L₂) (MorphismProperty.prod W₁ W₂)

Next tactic:
exact of_equivalence_target (W₁.Q.prod W₂.Q) (W₁.prod W₂) (L₁.prod L₂)
    ((uniq W₁.Q L₁ W₁).prod (uniq W₂.Q L₂ W₂))
    (NatIso.prod (compUniqFunctor W₁.Q L₁ W₁) (compUniqFunctor W₂.Q L₂ W₂))
/-- The submonoid fixing a set under a `MulAction`. -/
@[to_additive " The additive submonoid fixing a set under an `AddAction`. "]
def fixingSubmonoid (s : Set α) : Submonoid M
    where
  carrier := { ϕ : M | ∀ x : s, ϕ • (x : α) = x }
  one_mem' _ := one_smul _ _
  mul_mem' {x y} hx hy z := by 

Proof state:
M : Type u_1
α : Type u_2
inst✝¹ : Monoid M
inst✝ : MulAction M α
s : Set α
x y : M
hx : x ∈ {ϕ | ∀ (x : ↑s), ϕ • ↑x = ↑x}
hy : y ∈ {ϕ | ∀ (x : ↑s), ϕ • ↑x = ↑x}
z : ↑s
⊢ (x * y) • ↑z = ↑z

Next tactic:
rw [mul_smul, hy z, hx z]
/-- The subgroup fixing a set under a `MulAction`. -/
@[to_additive " The additive subgroup fixing a set under an `AddAction`. "]
def fixingSubgroup (s : Set α) : Subgroup M :=
  { fixingSubmonoid M s with inv_mem' := fun hx z => by 

Proof state:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
src✝ : Submonoid M := fixingSubmonoid M s
x✝ : M
hx : x✝ ∈ { toSubsemigroup := src✝.toSubsemigroup, one_mem' := (_ : 1 ∈ src✝.carrier) }.toSubsemigroup.carrier
z : ↑s
⊢ x✝⁻¹ • ↑z = ↑z

Next tactic:
rw [inv_smul_eq_iff, hx z]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I : Ideal R
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I

Next tactic:
refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
by_cases h_prM : M.IsPrime
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · 

Proof state:
case pos
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : Ideal.IsPrime M
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
use {⟨M, h_prM⟩}
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    

Proof state:
case h
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : Ideal.IsPrime M
⊢ Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) ≤ M

Next tactic:
rw [Multiset.map_singleton, Multiset.prod_singleton]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  

Proof state:
case neg
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
by_cases htop : M = ⊤
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · 

Proof state:
case pos
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : M = ⊤
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
rw [htop]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    

Proof state:
case pos
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : M = ⊤
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) ⊤

Next tactic:
exact ⟨0, le_top⟩
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  

Proof state:
case neg
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
⊢ ∀ z ∉ M, M < M + span R {z}

Next tactic:
intro z hz
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
z : R
hz : z ∉ M
⊢ M < M + span R {z}

Next tactic:
refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
z : R
hz : z ∉ M
m_eq : M = M + span R {z}
⊢ z ∈ M

Next tactic:
rw [m_eq]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
z : R
hz : z ∉ M
m_eq : M = M + span R {z}
⊢ z ∈ M + span R {z}

Next tactic:
exact Ideal.mem_sup_right (mem_span_singleton_self z)