mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Localization

Goal state before:
pp: case refine_2
R : Type u_1
inst✝³ : CommSemiring R
A : Type u_2
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
q : Ideal R
hqp : q.IsPrime
inst✝ : IsLocalization.AtPrime A q
I : Ideal R
hIq : q ∈ I.minimalPrimes
this : IsLocalRing A
J : Ideal A
hJ : J ∈ Ideal.under R ⁻¹' I.minimalPrimes
⊢ q ≤ Ideal.under R J

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
inst✝³ : CommSemiring R
A : Type u_2
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
q : Ideal R
hqp : q.IsPrime
inst✝ : IsLocalization.AtPrime A q
I : Ideal R
hIq : q ∈ I.minimalPrimes
this : IsLocalRing A
J : Ideal A
hJ : J ∈ Ideal.under R ⁻¹' I.minimalPrimes
⊢ q ≤ Ideal.under R J
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Localization

Goal state before:
pp: case refine_2
R : Type u_1
inst✝³ : CommSemiring R
A : Type u_2
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
q : Ideal R
hqp : q.IsPrime
inst✝ : IsLocalization.AtPrime A q
I : Ideal R
hIq : q ∈ I.minimalPrimes
this✝ : IsLocalRing A
J : Ideal A
hJ : J ∈ Ideal.under R ⁻¹' I.minimalPrimes
this : Disjoint ↑q.primeCompl ↑(Ideal.under R J)
⊢ Ideal.under R J ≤ q

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
inst✝³ : CommSemiring R
A : Type u_2
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
q : Ideal R
hqp : q.IsPrime
inst✝ : IsLocalization.AtPrime A q
I : Ideal R
hIq : q ∈ I.minimalPrimes
this✝ : IsLocalRing A
J : Ideal A
hJ : J ∈ Ideal.under R ⁻¹' I.minimalPrimes
this : Disjoint ↑q.primeCompl ↑(Ideal.under R J)
⊢ Ideal.under R J ≤ q
Module: Mathlib.Topology.Compactness.Bases

Goal state before:
pp: X : Type u_1
ι : Type u_2
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u_1
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
⊢ ⋃ i, (b ∘ f') i ⊆ ⋃ i, (b ∘ f') i

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Compactness.Bases

Goal state before:
pp: case refine_2
X : Type u_1
ι : Type u_2
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u_1
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i
j : Y
hi : f' j ∈ ↑(Finset.image f' t)
⊢ b (f' j) ⊆ ⋃ i, (b ∘ f') i

Tactic:
]

Goal state after:
case refine_2
X : Type u_1
ι : Type u_2
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u_1
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i
j : Y
hi : f' j ∈ ↑(Finset.image f' t)
⊢ b (f' j) ⊆ ⋃ i, (b ∘ f') i
Module: Mathlib.Topology.Compactness.Bases

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
b : Set (Set X)
hb : IsTopologicalBasis (range fun i ↦ ↑i)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
hb' : b = range fun i ↦ ↑i
⊢ ∃ s, U = ⋃₀ (Subtype.val '' ↑s)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
b : Set (Set X)
hb : IsTopologicalBasis (range fun i ↦ ↑i)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
hb' : b = range fun i ↦ ↑i
⊢ ∃ s, U = ⋃₀ (Subtype.val '' ↑s)
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : Lattice α
le : α → α → Prop
eq_le : le = LE.le
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
a✝ b✝ c✝ : α
hab : a✝ ≤ b✝
hbc : b✝ ≤ c✝
⊢ a✝ ≤ c✝

Tactic:
]

Goal state after:
α : Type u
c : Lattice α
le : α → α → Prop
eq_le : le = LE.le
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
a✝ b✝ c✝ : α
hab : a✝ ≤ b✝
hbc : b✝ ≤ c✝
⊢ a✝ ≤ c✝
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : Lattice α
le : α → α → Prop
eq_le : le = LE.le
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
a✝ b✝ c✝ : α
hab : a✝ ≤ b✝
hbc : b✝ ≤ c✝
⊢ a✝ ≤ c✝

Tactic:
exact le_trans hab hbc

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_4
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K
⊢ (∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) ↔
    x ∈ range ⇑(algebraMap R K)

Tactic:
]

Goal state after:
R : Type u_4
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x : K
⊢ (∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) ↔
    x ∈ range ⇑(algebraMap R K)
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_4
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
y : R
v : Ideal R
hv : v.IsMaximal
⊢ (algebraMap R K) y = (algebraMap R K) y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
r r' : R
eq : (toPiLocalization R) r = (toPiLocalization R) r'
⊢ 1 * r = 1 * r'

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
r r' : R
eq : (toPiLocalization R) r = (toPiLocalization R) r'
⊢ 1 * r = 1 * r'
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : GeneralizedHeytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
himp : α → α → α
eq_himp : himp = HImp.himp
⊢ ∀ (x y : α), x ≤ y ↔ le x y

Tactic:
]

Goal state after:
α : Type u
c : GeneralizedHeytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
himp : α → α → α
eq_himp : himp = HImp.himp
⊢ ∀ (x y : α), x ≤ y ↔ le x y
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
r r' : R
eq : (toPiLocalization R) r = (toPiLocalization R) r'
⊢ 1 * r = 1 * r'

Tactic:
by_contra ne

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
r r' : R
eq : (toPiLocalization R) r = (toPiLocalization R) r'
ne : ¬1 * r = 1 * r'
⊢ False
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ mapPiLocalization ((↑f).comp ↑f.symm) ⋯ = RingHom.id (PiLocalization S)

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ mapPiLocalization ((↑f).comp ↑f.symm) ⋯ = RingHom.id (PiLocalization S)
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : GeneralizedCoheytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
sdiff : α → α → α
eq_sdiff : sdiff = SDiff.sdiff
⊢ ∀ (x y : α), x ≤ y ↔ le x y

Tactic:
]

Goal state after:
α : Type u
c : GeneralizedCoheytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
sdiff : α → α → α
eq_sdiff : sdiff = SDiff.sdiff
⊢ ∀ (x y : α), x ≤ y ↔ le x y
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ mapPiLocalization ((↑f.symm).comp ↑f) ⋯ = RingHom.id (PiLocalization R)

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ mapPiLocalization ((↑f.symm).comp ↑f) ⋯ = RingHom.id (PiLocalization R)
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : HeytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
himp : α → α → α
eq_himp : himp = HImp.himp
compl : α → α
eq_compl : compl = Compl.compl
⊢ ∀ (x y : α), x ≤ y ↔ le x y

Tactic:
]

Goal state after:
α : Type u
c : HeytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
himp : α → α → α
eq_himp : himp = HImp.himp
compl : α → α
eq_compl : compl = Compl.compl
⊢ ∀ (x y : α), x ≤ y ↔ le x y
Module: Mathlib.Order.Copy

Goal state before:
pp: α : Type u
c : CoheytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
sdiff : α → α → α
eq_sdiff : sdiff = SDiff.sdiff
hnot : α → α
eq_hnot : hnot = HNot.hnot
⊢ ∀ (x y : α), x ≤ y ↔ le x y

Tactic:
]

Goal state after:
α : Type u
c : CoheytingAlgebra α
le : α → α → Prop
eq_le : le = LE.le
top : α
eq_top : top = ⊤
bot : α
eq_bot : bot = ⊥
sup : α → α → α
eq_sup : sup = max
inf : α → α → α
eq_inf : inf = min
sdiff : α → α → α
eq_sdiff : sdiff = SDiff.sdiff
hnot : α → α
eq_hnot : hnot = HNot.hnot
⊢ ∀ (x y : α), x ≤ y ↔ le x y
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
J : Ideal ((i : ι) → R i)
max : J.IsMaximal
notMem : { asIdeal := J, isPrime := ⋯ } ∉ Set.range (PrimeSpectrum.sigmaToPi R)
r : (i : ι) → R i
this : r = 0
hr : 0 { asIdeal := J, isMaximal := max } = 1
⊢ False

Tactic:
]

Goal state after:
ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
J : Ideal ((i : ι) → R i)
max : J.IsMaximal
notMem : { asIdeal := J, isPrime := ⋯ } ∉ Set.range (PrimeSpectrum.sigmaToPi R)
r : (i : ι) → R i
this : r = 0
hr : 0 { asIdeal := J, isMaximal := max } = 1
⊢ False
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: ι : Type u_5
R : ι → Type u_4
inst✝¹ : (i : ι) → CommSemiring (R i)
inst✝ : ∀ (i : ι), Nontrivial (R i)
h : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
⊢ Finite ι

Tactic:
contrapose! h
  exact toPiLocalization_not_surjective_of_infinite _

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
surj✝ : Function.Surjective ⇑(toPiLocalization R)
surj : Function.Surjective (⇑(toPiLocalization (PiLocalization R)).toRingHom ∘ ⇑(toPiLocalization R).toRingHom)
⊢ Finite (MaximalSpectrum R)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
surj✝ : Function.Surjective ⇑(toPiLocalization R)
surj : Function.Surjective (⇑(toPiLocalization (PiLocalization R)).toRingHom ∘ ⇑(toPiLocalization R).toRingHom)
⊢ Finite (MaximalSpectrum R)
Module: Mathlib.Topology.Sets.OpenCover

Goal state before:
pp: ι : Type u_1
X : Type u_3
inst✝¹ : TopologicalSpace X
u : ι → Opens X
hu : ↑⊤ ⊆ ↑(iSup u)
inst✝ : CompactSpace X
⊢ ∃ s, IsOpenCover fun i ↦ u ↑i

Tactic:
]

Goal state after:
ι : Type u_1
X : Type u_3
inst✝¹ : TopologicalSpace X
u : ι → Opens X
hu : ↑⊤ ⊆ ↑(iSup u)
inst✝ : CompactSpace X
⊢ ∃ s, IsOpenCover fun i ↦ u ↑i
Module: Mathlib.Topology.Sets.OpenCover

Goal state before:
pp: case inl
X : Type u_1
ι : Type u_2
inst✝ : TopologicalSpace X
U : ι → Opens X
hn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)
h : ∀ (i : ι), IsPreirreducible ↑(U i)
s : Set X
hs : IsOpen[inst✝] s
hsU : s ⊆ ⋃ i, ↑(U i)
he : s = ∅
⊢ IsPreirreducible ∅

Tactic:
]

Goal state after:
case inl
X : Type u_1
ι : Type u_2
inst✝ : TopologicalSpace X
U : ι → Opens X
hn : Pairwise ((fun x1 x2 ↦ ¬Disjoint x1 x2) on U)
h : ∀ (i : ι), IsPreirreducible ↑(U i)
s : Set X
hs : IsOpen[inst✝] s
hsU : s ⊆ ⋃ i, ↑(U i)
he : s = ∅
⊢ IsPreirreducible ∅
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: case of_equiv
α : Type u_2
inst✝¹ : TopologicalSpace α
ι : Type u_5
inst✝ : Finite ι
α✝ β✝ : Type u_5
e : α✝ ≃ β✝
ih : ∀ (U : α✝ → Opens α), ↑(⨅ i, U i) = ⋂ i, ↑(U i)
U : β✝ → Opens α
⊢ ⋂ i, ↑(U (e i)) = ⋂ x, ↑(U (e x))

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: case h_option
α : Type u_2
inst✝² : TopologicalSpace α
ι : Type u_5
inst✝¹ : Finite ι
α✝ : Type u_5
inst✝ : Fintype α✝
ih : ∀ (U : α✝ → Opens α), ↑(⨅ i, U i) = ⋂ i, ↑(U i)
U : Option α✝ → Opens α
⊢ ↑(U none) ∩ ⋂ i, ↑(U (some i)) = ↑(U none) ∩ ⋂ i, ↑(U (some i))

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
ι : Sort u_5
x : α
s : ι → Opens α
⊢ x ∈ ↑(iSup s) ↔ ∃ i, x ∈ s i

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
ι : Sort u_5
x : α
s : ι → Opens α
⊢ x ∈ ↑(iSup s) ↔ ∃ i, x ∈ s i
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ RingHom.id (PiLocalization S) = RingHom.id (PiLocalization S)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f✝ : R →+* S
hf : Function.Bijective ⇑f✝
f : R ≃+* S := RingEquiv.ofBijective f✝ hf
⊢ RingHom.id (PiLocalization R) = RingHom.id (PiLocalization R)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U V : Opens α
x : α
⊢ x ∈ U ⇨ V ↔ ∃ W, W ⊓ U ≤ V ∧ x ∈ W

Tactic:
by
  simp [himp_eq_sSup]

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
⊢ Function.Surjective ⇑↑((piLocalizationToMaximal ((i : ι) → R i)).comp (toPiLocalization ((i : ι) → R i)))

Tactic:
]

Goal state after:
ι : Type u_5
R : ι → Type u_4
inst✝² : (i : ι) → CommSemiring (R i)
inst✝¹ : ∀ (i : ι), Nontrivial (R i)
inst✝ : Infinite ι
surj : Function.Surjective ⇑(toPiLocalization ((i : ι) → R i))
⊢ Function.Surjective ⇑↑((piLocalizationToMaximal ((i : ι) → R i)).comp (toPiLocalization ((i : ι) → R i)))
Module: Mathlib.RingTheory.Spectrum.Maximal.Localization

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
surj : Function.Surjective (⇑(toPiLocalization (PiLocalization R)).toRingHom ∘ ⇑(toPiLocalization R).toRingHom)
⊢ Finite (PrimeSpectrum R)

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
surj : Function.Surjective (⇑(toPiLocalization (PiLocalization R)).toRingHom ∘ ⇑(toPiLocalization R).toRingHom)
⊢ Finite (PrimeSpectrum R)
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U V : Opens α
⊢ interior (↑U ⇨ ↑V) = interior (↑U ⇨ ↑V)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U : Opens α
⊢ interior (↑U)ᶜ = interior (↑U)ᶜ

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U V : Opens α
i : U ≤ V
⊢ IsOpen[instTopologicalSpaceSubtype] {x | ↑x ∈ ↑U}

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
U V : Opens α
i : U ≤ V
⊢ IsOpen[instTopologicalSpaceSubtype] {x | ↑x ∈ ↑U}
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ Disjoint s (closure[inst✝] s \ s)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ Disjoint s (closure[inst✝] s \ s)
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ s ⊆ closure[inst✝] s

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ s ⊆ closure[inst✝] s
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ closure[inst✝] s \ s = closure[inst✝] s \ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ closure[inst✝] s ∩ sᶜ = closure[inst✝] s \ s

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ closure[inst✝] s ∩ sᶜ = closure[inst✝] s \ s
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U : Opens α
⊢ ¬(↑U).Nonempty ↔ ¬(↑U).Nonempty

Tactic:
]

Goal state after:
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ univ ⊆ closure[inst✝] s ∪ closure[inst✝] (closure[inst✝] s)ᶜ

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ univ ⊆ closure[inst✝] s ∪ closure[inst✝] (closure[inst✝] s)ᶜ
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
U : Opens α
⊢ (↑U).Nonempty ↔ (↑U).Nonempty

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: α : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : IndiscreteTopology α
U : Opens α
⊢ IsOpen[inst✝¹] ↑U

Tactic:
]

Goal state after:
α : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : IndiscreteTopology α
U : Opens α
⊢ IsOpen[inst✝¹] ↑U
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsOpenMap f
s : Set Y
⊢ f ⁻¹' closure[inst✝] s \ f ⁻¹' s ⊆ closure[inst✝¹] (f ⁻¹' s) \ f ⁻¹' s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsOpenMap f
s : Set Y
⊢ f ⁻¹' closure[inst✝] s \ f ⁻¹' s ⊆ closure[inst✝¹] (f ⁻¹' s) \ f ⁻¹' s
Module: Mathlib.Topology.LocalAtTarget

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set β
h : ∀ (x : α), 𝓝 x = comap f (𝓝 (f x))
a : ↑(f ⁻¹' s)
⊢ 𝓝 a = 𝓝 a

Tactic:
]

Goal state after:
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
s : Set Y
⊢ closure[inst✝¹] (f ⁻¹' s) \ f ⁻¹' s ⊆ f ⁻¹' closure[inst✝] s \ f ⁻¹' s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
s : Set Y
⊢ closure[inst✝¹] (f ⁻¹' s) \ f ⁻¹' s ⊆ f ⁻¹' closure[inst✝] s \ f ⁻¹' s
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ s ⊆ s

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: case mp
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
hB : IsBasis B
U : Opens α
⊢ ⋃₀ {s | s ∈ SetLike.coe '' B ∧ s ⊆ ↑U} = ⋃ i ∈ {V | V ∈ B ∧ V ≤ U}, ↑i

Tactic:
]

Goal state after:
case mp
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
hB : IsBasis B
U : Opens α
⊢ ⋃₀ {s | s ∈ SetLike.coe '' B ∧ s ⊆ ↑U} = ⋃ i ∈ {V | V ∈ B ∧ V ≤ U}, ↑i
Module: Mathlib.Topology.LocalAtTarget

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
H : IsProperMap f
s : Set β
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceSubtype] (s.restrictPreimage f) ∧
    IsClosedMap (s.restrictPreimage f) ∧ ∀ (y : ↑s), IsCompact (s.restrictPreimage f ⁻¹' {y})

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
H : IsProperMap f
s : Set β
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceSubtype] (s.restrictPreimage f) ∧
    IsClosedMap (s.restrictPreimage f) ∧ ∀ (y : ↑s), IsCompact (s.restrictPreimage f ⁻¹' {y})
Module: Mathlib.Topology.LocalAtTarget

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
H : IsProperMap f
s : Set β
y : ↑s
⊢ IsCompact (f ⁻¹' {↑y})

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
H : IsProperMap f
s : Set β
y : ↑s
⊢ IsCompact (f ⁻¹' {↑y})
Module: Mathlib.Topology.Sets.Opens

Goal state before:
pp: case mpr
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
h : ∀ (U : Opens α), ∃ Us ⊆ B, U = sSup Us
⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U

Tactic:
]

Goal state after:
case mpr
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
h : ∀ (U : Opens α), ∃ Us ⊆ B, U = sSup Us
⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
Module: Mathlib.Topology.LocalAtTarget

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
H : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
s : Set α
hs : IsOpen[inst✝¹] s
⊢ ∀ (i : ι), IsOpen[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' f '' s)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
H : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
s : Set α
hs : IsOpen[inst✝¹] s
⊢ ∀ (i : ι), IsOpen[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' f '' s)
Module: Mathlib.Order.BooleanSubalgebra

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ a ⊓ bᶜ ∈ L

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ a ⊓ bᶜ ∈ L
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
hf' : IsLocallyClosed (range f)
t : Set Y
ht : IsLocallyClosed t
hs : IsLocallyClosed (f ⁻¹' t)
⊢ IsLocallyClosed (t ∩ range f)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
hf' : IsLocallyClosed (range f)
t : Set Y
ht : IsLocallyClosed t
hs : IsLocallyClosed (f ⁻¹' t)
⊢ IsLocallyClosed (t ∩ range f)
Module: Mathlib.Order.BooleanSubalgebra

Goal state before:
pp: α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ b ⊔ aᶜ ∈ L

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : BooleanAlgebra α
L : BooleanSubalgebra α
a b : α
ha : a ∈ L
hb : b ∈ L
⊢ b ⊔ aᶜ ∈ L
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U Z : Set X
hU : IsOpen[inst✝] U
hZ : IsClosed[inst✝] Z
⊢ closure[inst✝] Z ∪ (frontier (U ∩ Z))ᶜ = univ

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
U Z : Set X
hU : IsOpen[inst✝] U
hZ : IsClosed[inst✝] Z
⊢ closure[inst✝] Z ∪ (frontier (U ∩ Z))ᶜ = univ
Module: Mathlib.Topology.LocallyClosed

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U Z : Set X
hU : IsOpen[inst✝] U
hZ : IsClosed[inst✝] Z
⊢ frontier (U ∩ Z) ⊆ closure[inst✝] Z

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
U Z : Set X
hU : IsOpen[inst✝] U
hZ : IsClosed[inst✝] Z
⊢ frontier (U ∩ Z) ⊆ closure[inst✝] Z