mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: case succ.succ.refine_1
R₀ : Type u_1
R : Type u_2
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
this :
  ∀ {M : Type u} [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ≤ Module.annihilator R M → P M
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R₀ M
inst✝¹ : Module R M
inst✝ : IsScalarTower R₀ R M
n : ℕ
ih :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ^ (n + 1) ≤ Module.annihilator R M → P M
hn : Jac ^ (n + 1) ≤ (Jac • ⊤).annihilator
⊢ Jac ^ (n + 1) ≤ Module.annihilator R ↥(Jac • ⊤)

Tactic:
]

Goal state after:
case succ.succ.refine_1
R₀ : Type u_1
R : Type u_2
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
this :
  ∀ {M : Type u} [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ≤ Module.annihilator R M → P M
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R₀ M
inst✝¹ : Module R M
inst✝ : IsScalarTower R₀ R M
n : ℕ
ih :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ^ (n + 1) ≤ Module.annihilator R M → P M
hn : Jac ^ (n + 1) ≤ (Jac • ⊤).annihilator
⊢ Jac ^ (n + 1) ≤ Module.annihilator R ↥(Jac • ⊤)
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
⊢ closure[inst✝] (closure[inst✝] Z ∩ closedPoints X) ⊆ closure[inst✝] Z \ Z

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
⊢ closure[inst✝] (closure[inst✝] Z ∩ closedPoints X) ⊆ closure[inst✝] Z \ Z
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
⊢ (⋃ i ∈ Set.Icc 0 n, {I | absNorm I = i}).Finite

Tactic:
]

Goal state after:
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
inst✝ : CharZero S
n : ℕ
⊢ (⋃ i ∈ Set.Icc 0 n, {I | absNorm I = i}).Finite
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
s : Set (PrimeSpectrum R)
hs : IsOpen s
hs' : s.Nonempty
I : Ideal R
hI' : sᶜ = zeroLocus ↑I
this : ∀ (x : PrimeSpectrum R), x.asIdeal.IsMaximal → ↑I ⊆ ↑x.asIdeal
⊢ I.jacobson = ⊥.jacobson

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
s : Set (PrimeSpectrum R)
hs : IsOpen s
hs' : s.Nonempty
I : Ideal R
hI' : sᶜ = zeroLocus ↑I
this : ∀ (x : PrimeSpectrum R), x.asIdeal.IsMaximal → ↑I ⊆ ↑x.asIdeal
⊢ I.jacobson = ⊥.jacobson
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: case succ.succ.refine_2
R₀ : Type u_1
R : Type u_2
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
this :
  ∀ {M : Type u} [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ≤ Module.annihilator R M → P M
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R₀ M
inst✝¹ : Module R M
inst✝ : IsScalarTower R₀ R M
n : ℕ
ih :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ^ (n + 1) ≤ Module.annihilator R M → P M
hn : Jac ^ (n + 1 + 1) ≤ Module.annihilator R M
⊢ ∀ (x : M), ∀ r ∈ ↑(Jac ^ (n + 1)), r • x ∈ Jac • ⊤

Tactic:
]

Goal state after:
case succ.succ.refine_2
R₀ : Type u_1
R : Type u_2
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
Jac : Ideal R := Ring.jacobson R
h0 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M]
    [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑Jac → P M
h1 :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [inst_3 : IsScalarTower R₀ R M],
    let N := Jac • ⊤;
    P ↥N → P (M ⧸ N) → P M
this :
  ∀ {M : Type u} [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ≤ Module.annihilator R M → P M
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R₀ M
inst✝¹ : Module R M
inst✝ : IsScalarTower R₀ R M
n : ℕ
ih :
  ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : Module R₀ M] [inst_2 : Module R M] [IsScalarTower R₀ R M],
    Jac ^ (n + 1) ≤ Module.annihilator R M → P M
hn : Jac ^ (n + 1 + 1) ≤ Module.annihilator R M
⊢ ∀ (x : M), ∀ r ∈ ↑(Jac ^ (n + 1)), r • x ∈ Jac • ⊤
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
⊢ closure[inst✝] Z ∩ closedPoints X ⊆ closure[inst✝] Z ∧ closure[inst✝] Z ∩ ∅ ⊆ ∅

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
⊢ closure[inst✝] Z ∩ closedPoints X ⊆ closure[inst✝] Z ∧ closure[inst✝] Z ∩ ∅ ⊆ ∅
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
⊢ ∀ (Z : Set (PrimeSpectrum R)), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints (PrimeSpectrum R)).Nonempty

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
⊢ ∀ (Z : Set (PrimeSpectrum R)), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints (PrimeSpectrum R)).Nonempty
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: case mp
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
this : Z ⊆ closure[inst✝] Z ∧ Z = ∅
⊢ False

Tactic:
]

Goal state after:
case mp
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z
Z : Set X
hZ : Z ≠ ∅
hZ' : IsClosed[inst✝] (closure[inst✝] Z \ Z)
e : Z ∩ closedPoints X = ∅
this : Z ⊆ closure[inst✝] Z ∧ Z = ∅
⊢ False
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
n : ℕ
inst✝ : CharZero S
this✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }
this : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }
⊢ Nat.card { I // absNorm I ≤ n } = Nat.card { x // x ∈ (Ideal S)⁰ ∧ absNorm x ≤ n } + 1

Tactic:
]

Goal state after:
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
n : ℕ
inst✝ : CharZero S
this✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }
this : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }
⊢ Nat.card { I // absNorm I ≤ n } = Nat.card { x // x ∈ (Ideal S)⁰ ∧ absNorm x ≤ n } + 1
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
H : JacobsonSpace (PrimeSpectrum R)
I : Ideal R
hI : I.IsRadical
⊢ I.jacobson.radical ≤ I

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
H : JacobsonSpace (PrimeSpectrum R)
I : Ideal R
hI : I.IsRadical
⊢ I.jacobson.radical ≤ I
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
H : JacobsonSpace (PrimeSpectrum R)
I : Ideal R
hI : I.IsRadical
⊢ zeroLocus ↑I ∩ closedPoints (PrimeSpectrum R) ⊆ zeroLocus ↑I.jacobson

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
H : JacobsonSpace (PrimeSpectrum R)
I : Ideal R
hI : I.IsRadical
⊢ zeroLocus ↑I ∩ closedPoints (PrimeSpectrum R) ⊆ zeroLocus ↑I.jacobson
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Z : Set X
hZ : IsClosed[inst✝] Z
⊢ ¬((closure[inst✝] (Z ∩ closedPoints X))ᶜ ∩ Z).Nonempty

Tactic:
]

Goal state after:
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Z : Set X
hZ : IsClosed[inst✝] Z
⊢ ¬((closure[inst✝] (Z ∩ closedPoints X))ᶜ ∩ Z).Nonempty
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
⊢ [IsOpen {x}, IsClopen {x}, IsClosed {x} ∧ StableUnderGeneralization {x}].TFAE

Tactic:
tfae_have 2 → 3
  | h => ⟨h.isClosed, h.isOpen.stableUnderGeneralization⟩

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
⊢ [IsOpen {x}, IsClopen {x}, IsClosed {x} ∧ StableUnderGeneralization {x}].TFAE
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
⊢ IsOpen {x}

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
⊢ IsOpen {x}
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Z : Set X
hZ : IsClosed[inst✝] Z
H' : ((closure[inst✝] (Z ∩ closedPoints X))ᶜ ∩ Z).Nonempty
this : ¬Z ∩ closedPoints X ⊆ closure[inst✝] (Z ∩ closedPoints X)
⊢ False

Tactic:
]

Goal state after:
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Z : Set X
hZ : IsClosed[inst✝] Z
H' : ((closure[inst✝] (Z ∩ closedPoints X))ᶜ ∩ Z).Nonempty
this : ¬Z ∩ closedPoints X ⊆ closure[inst✝] (Z ∩ closedPoints X)
⊢ False
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
S : Set X
hS : IsLocallyClosed S
⊢ S ⊆ closure[inst✝¹] (S ∩ closedPoints X)

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
S : Set X
hS : IsLocallyClosed S
⊢ S ⊆ closure[inst✝¹] (S ∩ closedPoints X)
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
this : {x} = (⋃ p ∈ {p | IsMin p ∧ p ≠ x}, closure {p})ᶜ
⊢ IsClosed (⋃ p ∈ {p | IsMin p ∧ p ≠ x}, closure {p})

Tactic:
]

Goal state after:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
this : {x} = (⋃ p ∈ {p | IsMin p ∧ p ≠ x}, closure {p})ᶜ
⊢ IsClosed (⋃ p ∈ {p | IsMin p ∧ p ≠ x}, closure {p})
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
hf : IsOpenEmbedding f
inst✝ : JacobsonSpace Y
x : X
hx : x ∈ closedPoints X
⊢ IsLocallyClosed (f '' {x})

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
hf : IsOpenEmbedding f
inst✝ : JacobsonSpace Y
x : X
hx : x ∈ closedPoints X
⊢ IsLocallyClosed (f '' {x})
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_2
Y : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace Y
hf : IsOpenEmbedding f
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace Y
hf : IsOpenEmbedding f
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_2
Y : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace Y
hf : IsClosedEmbedding f
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace Y
hf : IsClosedEmbedding f
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty
Module: Mathlib.RingTheory.Spectrum.Prime.Jacobson

Goal state before:
pp: case mp
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
p : PrimeSpectrum R
tfae_1_to_2 : IsOpen {p} → IsClopen {p}
tfae_2_to_3 : IsClopen {p} → IsClosed {p} ∧ StableUnderGeneralization {p}
h₁ : IsMax p
h₂ : IsMin p
i✝ : PrimeSpectrum R
a✝ : IsMin i✝
⊢ i✝ ≤ p → i✝ = p

Tactic:
]

Goal state after:
case mp
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
p : PrimeSpectrum R
tfae_1_to_2 : IsOpen {p} → IsClopen {p}
tfae_2_to_3 : IsClopen {p} → IsClosed {p} ∧ StableUnderGeneralization {p}
h₁ : IsMax p
h₂ : IsMin p
i✝ : PrimeSpectrum R
a✝ : IsMin i✝
⊢ i✝ ≤ p → i✝ = p
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
⊢ IsClosed[inst✝¹] (⋃ x ∈ closedPoints X, {x})

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
⊢ IsClosed[inst✝¹] (⋃ x ∈ closedPoints X, {x})
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
⊢ ∀ (s : Set X), IsOpen[inst✝¹] s

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
⊢ ∀ (s : Set X), IsOpen[inst✝¹] s
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
⊢ IsClosed[inst✝¹] (⋃ x ∈ sᶜ, {x})

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
⊢ IsClosed[inst✝¹] (⋃ x ∈ sᶜ, {x})
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
x : X
x✝ : x ∈ sᶜ
⊢ x ∈ Set.univ

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
x : X
x✝ : x ∈ sᶜ
⊢ x ∈ Set.univ
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty

Tactic:
]

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
Z : Set X
hZ : ∃ i, (Z ∩ ↑(U i)).Nonempty
hZ' : IsLocallyClosed Z
⊢ (Z ∩ closedPoints X).Nonempty

Tactic:
]

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
Z : Set X
hZ : ∃ i, (Z ∩ ↑(U i)).Nonempty
hZ' : IsLocallyClosed Z
⊢ (Z ∩ closedPoints X).Nonempty
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: case e'_3
X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
Z : Set X
hZ' : IsLocallyClosed Z
i : ι
x : X
hx : x ∈ Z
hx' : x ∈ ↑(U i)
y : ↥(U i)
hy : y ∈ Subtype.val ⁻¹' Z
hy' : y ∈ closedPoints ↥(U i)
j : ι
h : ↑y ∉ U j
⊢ ∀ (x : ↥(U j)), x ∉ Subtype.val ⁻¹' {↑y}

Tactic:
]

Goal state after:
case e'_3
X : Type u_2
inst✝ : TopologicalSpace X
ι : Type u_1
U : ι → Opens X
hU : IsOpenCover U
H : ∀ (i : ι), JacobsonSpace ↥(U i)
Z : Set X
hZ' : IsLocallyClosed Z
i : ι
x : X
hx : x ∈ Z
hx' : x ∈ ↑(U i)
y : ↥(U i)
hy : y ∈ Subtype.val ⁻¹' Z
hy' : y ∈ closedPoints ↥(U i)
j : ι
h : ↑y ∉ U j
⊢ ∀ (x : ↥(U j)), x ∉ Subtype.val ⁻¹' {↑y}
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace X
S : Set X
hS : IsLocallyClosed S
hf₁ : Continuous[inst✝², inst✝¹] f
hf₂ : IsClosedMap f
hfS : (f '' S).Finite
hS'' : S.Nonempty
hS' : IsIrreducible (S ∩ closedPoints X)
⊢ IsIrreducible (S ∩ closedPoints X)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : JacobsonSpace X
S : Set X
hS : IsLocallyClosed S
hf₁ : Continuous[inst✝², inst✝¹] f
hf₂ : IsClosedMap f
hfS : (f '' S).Finite
hS'' : S.Nonempty
hS' : IsIrreducible (S ∩ closedPoints X)
⊢ IsIrreducible (S ∩ closedPoints X)
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
n : ℕ
inst✝ : CharZero S
this✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }
this : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }
e : { I // absNorm I ≤ n } ≃ { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n } ⊕ { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n } :=
  (Equiv.subtypeEquivRight ⋯).trans
    (subtypeOrEquiv (fun I ↦ I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n) (fun I ↦ I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n) ⋯)
I : Ideal S
h : absNorm I = 0
⊢ 0 ≤ n

Tactic:
]

Goal state after:
S : Type u_1
inst✝⁴ : CommRing S
inst✝³ : IsDedekindDomain S
inst✝² : Free ℤ S
inst✝¹ : Module.Finite ℤ S
n : ℕ
inst✝ : CharZero S
this✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }
this : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }
e : { I // absNorm I ≤ n } ≃ { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n } ⊕ { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n } :=
  (Equiv.subtypeEquivRight ⋯).trans
    (subtypeOrEquiv (fun I ↦ I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n) (fun I ↦ I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n) ⋯)
I : Ideal S
h : absNorm I = 0
⊢ 0 ≤ n
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: R : Type u_2
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinianRing R
tfae_2_iff_3 : IsNoetherian R M ↔ IsArtinian R M
tfae_2_to_1 : IsNoetherian R M → Module.Finite R M
tfae_1_to_3 : Module.Finite R M → IsArtinian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M].TFAE

Tactic:
]

Goal state after:
R : Type u_2
M : Type u
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsArtinianRing R
tfae_2_iff_3 : IsNoetherian R M ↔ IsArtinian R M
tfae_2_to_1 : IsNoetherian R M → Module.Finite R M
tfae_1_to_3 : Module.Finite R M → IsArtinian R M
⊢ [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsNoetherian R M ∧ IsArtinian R M].TFAE
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : s.Finite
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R M

Tactic:
]

Goal state after:
M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : s.Finite
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R M
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : Finite ↑s
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R M

Tactic:
]

Goal state after:
M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : Finite ↑s
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R M
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : Finite ↑s
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R ↥(⨆ i, R ∙ ↑i)

Tactic:
]

Goal state after:
M : Type u
inst✝⁴ : AddCommGroup M
R : Type u_3
inst✝³ : CommRing R
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : IsArtinian R M
s : Set M
fin : Finite ↑s
span : ⨆ i, R ∙ ↑i = ⊤
⊢ IsNoetherian R ↥(⨆ i, R ∙ ↑i)