mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case inr.inr.hs
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
hf : s.Finite
hne : s.Nonempty
ho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ s, IsCompact t
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (Set α)},
    s.Finite →
      s.Nonempty → (∀ t ∈ s, IsOpen t ∨ IsClosed t) → (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen t) → IsCompact (⋂₀ s)
a : Set (Set α) := ⋯
b : Set (Set α) := ⋯
heq : s = a ∪ b
t : Set α
ht : t ∈ s
hno : ¬IsOpen[inst✝¹] t
ha : a.Nonempty
⊢ IsCompact (⋂₀ a)

pp: case ht
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
hf : s.Finite
hne : s.Nonempty
ho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ s, IsCompact t
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (Set α)},
    s.Finite →
      s.Nonempty → (∀ t ∈ s, IsOpen t ∨ IsClosed t) → (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen t) → IsCompact (⋂₀ s)
a : Set (Set α) := ⋯
b : Set (Set α) := ⋯
heq : s = a ∪ b
t : Set α
ht : t ∈ s
hno : ¬IsOpen[inst✝¹] t
ha : a.Nonempty
⊢ IsClosed[inst✝¹] (⋂₀ b)

Tactic:
·

Goal state after:
case inr.inr.hs
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
hf : s.Finite
hne : s.Nonempty
ho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ s, IsCompact t
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (Set α)},
    s.Finite →
      s.Nonempty → (∀ t ∈ s, IsOpen t ∨ IsClosed t) → (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen t) → IsCompact (⋂₀ s)
a : Set (Set α) := ⋯
b : Set (Set α) := ⋯
heq : s = a ∪ b
t : Set α
ht : t ∈ s
hno : ¬IsOpen[inst✝¹] t
ha : a.Nonempty
⊢ IsCompact (⋂₀ a)
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case ht
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
hf : s.Finite
hne : s.Nonempty
ho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ s, IsCompact t
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (Set α)},
    s.Finite →
      s.Nonempty → (∀ t ∈ s, IsOpen t ∨ IsClosed t) → (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen t) → IsCompact (⋂₀ s)
a : Set (Set α) := ⋯
b : Set (Set α) := ⋯
heq : s = a ∪ b
t : Set α
ht : t ∈ s
hno : ¬IsOpen[inst✝¹] t
ha : a.Nonempty
⊢ IsClosed[inst✝¹] (⋂₀ b)

Tactic:
·

Goal state after:
case ht
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
hf : s.Finite
hne : s.Nonempty
ho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ s, IsCompact t
this :
  ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (Set α)},
    s.Finite →
      s.Nonempty → (∀ t ∈ s, IsOpen t ∨ IsClosed t) → (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen t) → IsCompact (⋂₀ s)
a : Set (Set α) := ⋯
b : Set (Set α) := ⋯
heq : s = a ∪ b
t : Set α
ht : t ∈ s
hno : ¬IsOpen[inst✝¹] t
ha : a.Nonempty
⊢ IsClosed[inst✝¹] (⋂₀ b)
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ IsCompact (⨆ s ∈ t, ↑s).carrier

pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ f '' ↑(⨆ s ∈ t, ↑s) = U

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ IsCompact (⨆ s ∈ t, ↑s).carrier
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ f '' ↑(⨆ s ∈ t, ↑s) = U

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ f '' ↑(⨆ s ∈ t, ↑s) = U
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ ⋃ i, ⋃ (x : i ∈ Us), ⋃ (_ : ⟨i, ⋯⟩ ∈ t), f '' ↑i = ⋃ i ∈ t, f '' ↑↑i

pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ ⋃ i ∈ t, f '' ↑↑i = U

Tactic:
·

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ ⋃ i, ⋃ (x : i ∈ Us), ⋃ (_ : ⟨i, ⋯⟩ ∈ t), f '' ↑i = ⋃ i ∈ t, f '' ↑↑i
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ ⋃ i ∈ t, f '' ↑↑i = U

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PrespectralSpace X
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
h : IsOpenMap f
U : Set Y
hs : U ⊆ Set.range f
hU : IsOpen[inst✝¹] U
hc : IsCompact U
Us : Set (Opens X)
hUs : Us ⊆ {U | IsCompact ↑U}
heq : { carrier := f ⁻¹' U, is_open' := ⋯ } = sSup Us
t : Finset ↑Us
ht : U ⊆ ⋃ i ∈ t, f '' ↑↑i
⊢ ⋃ i ∈ t, f '' ↑↑i = U
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case insert.inl
α✝ : Type u_1
inst✝² : TopologicalSpace α✝
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
a✝ : Set α
ha : a✝ ∉ ∅
hs : ∅.Finite
ih :
  (∀ t ∈ ∅, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →
    (∀ t ∈ ∅, IsCompact t) → (∀ t ∈ ∅, IsOpen[inst✝¹] t) → ∅.Nonempty → IsCompact (⋂₀ ∅)
ho : ∀ t ∈ insert a✝ ∅, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ insert a✝ ∅, IsCompact t
h : ∀ t ∈ insert a✝ ∅, IsOpen[inst✝¹] t
⊢ (insert a✝ ∅).Nonempty → IsCompact (⋂₀ insert a✝ ∅)

pp: case insert.inr
α✝ : Type u_1
inst✝² : TopologicalSpace α✝
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s✝ : Set (Set α)
a✝ : Set α
s : Set (Set α)
ha : a✝ ∉ s
hs : s.Finite
ih :
  (∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →
    (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen[inst✝¹] t) → s.Nonempty → IsCompact (⋂₀ s)
ho : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ insert a✝ s, IsCompact t
h : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t
hne : s.Nonempty
⊢ (insert a✝ s).Nonempty → IsCompact (⋂₀ insert a✝ s)

Tactic:
·

Goal state after:
case insert.inl
α✝ : Type u_1
inst✝² : TopologicalSpace α✝
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s : Set (Set α)
a✝ : Set α
ha : a✝ ∉ ∅
hs : ∅.Finite
ih :
  (∀ t ∈ ∅, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →
    (∀ t ∈ ∅, IsCompact t) → (∀ t ∈ ∅, IsOpen[inst✝¹] t) → ∅.Nonempty → IsCompact (⋂₀ ∅)
ho : ∀ t ∈ insert a✝ ∅, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ insert a✝ ∅, IsCompact t
h : ∀ t ∈ insert a✝ ∅, IsOpen[inst✝¹] t
⊢ (insert a✝ ∅).Nonempty → IsCompact (⋂₀ insert a✝ ∅)
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case insert.inr
α✝ : Type u_1
inst✝² : TopologicalSpace α✝
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s✝ : Set (Set α)
a✝ : Set α
s : Set (Set α)
ha : a✝ ∉ s
hs : s.Finite
ih :
  (∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →
    (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen[inst✝¹] t) → s.Nonempty → IsCompact (⋂₀ s)
ho : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ insert a✝ s, IsCompact t
h : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t
hne : s.Nonempty
⊢ (insert a✝ s).Nonempty → IsCompact (⋂₀ insert a✝ s)

Tactic:
·

Goal state after:
case insert.inr
α✝ : Type u_1
inst✝² : TopologicalSpace α✝
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : QuasiSeparatedSpace α
s✝ : Set (Set α)
a✝ : Set α
s : Set (Set α)
ha : a✝ ∉ s
hs : s.Finite
ih :
  (∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t) →
    (∀ t ∈ s, IsCompact t) → (∀ t ∈ s, IsOpen[inst✝¹] t) → s.Nonempty → IsCompact (⋂₀ s)
ho : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t
hc : ∀ t ∈ insert a✝ s, IsCompact t
h : ∀ t ∈ insert a✝ s, IsOpen[inst✝¹] t
hne : s.Nonempty
⊢ (insert a✝ s).Nonempty → IsCompact (⋂₀ insert a✝ s)
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case inl
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSeparatedSpace α
inst✝ : CompactSpace α
hf : ∅.Finite
ho : ∀ t ∈ ∅, IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
hc : ∀ t ∈ ∅, IsCompact t
⊢ IsCompact (⋂₀ ∅)

pp: case inr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSeparatedSpace α
inst✝ : CompactSpace α
s : Set (Set α)
hf : s.Finite
ho : ∀ t ∈ s, IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
hc : ∀ t ∈ s, IsCompact t
hne : s.Nonempty
⊢ IsCompact (⋂₀ s)

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSeparatedSpace α
inst✝ : CompactSpace α
hf : ∅.Finite
ho : ∀ t ∈ ∅, IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
hc : ∀ t ∈ ∅, IsCompact t
⊢ IsCompact (⋂₀ ∅)
Module: Mathlib.Topology.QuasiSeparated

Goal state before:
pp: case inr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSeparatedSpace α
inst✝ : CompactSpace α
s : Set (Set α)
hf : s.Finite
ho : ∀ t ∈ s, IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
hc : ∀ t ∈ s, IsCompact t
hne : s.Nonempty
⊢ IsCompact (⋂₀ s)

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSeparatedSpace α
inst✝ : CompactSpace α
s : Set (Set α)
hf : s.Finite
ho : ∀ t ∈ s, IsOpen[inst✝²] t ∨ IsClosed[inst✝²] t
hc : ∀ t ∈ s, IsCompact t
hne : s.Nonempty
⊢ IsCompact (⋂₀ s)
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
s t : Set X
hst : s ⊆ t
x✝ : ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U
W : Set X
Wc : IsCompact W
Wo : IsOpen[inst✝¹] W
hKW : t ⊆ W
hWU : W ⊆ U
⊢ ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ⊆ W ∧ W ⊆ U

pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ ⦃s t : Set X⦄,
    (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ⊆ W ∧ W ⊆ U) →
      (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U) → ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ∪ t ⊆ W ∧ W ⊆ U

pp: case refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
s t : Set X
hst : s ⊆ t
x✝ : ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U
W : Set X
Wc : IsCompact W
Wo : IsOpen[inst✝¹] W
hKW : t ⊆ W
hWU : W ⊆ U
⊢ ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ⊆ W ∧ W ⊆ U
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ ⦃s t : Set X⦄,
    (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ⊆ W ∧ W ⊆ U) →
      (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U) → ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ∪ t ⊆ W ∧ W ⊆ U

pp: case refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ ⦃s t : Set X⦄,
    (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ⊆ W ∧ W ⊆ U) →
      (∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U) → ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ s ∪ t ⊆ W ∧ W ⊆ U
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: case refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U

Tactic:
·

Goal state after:
case refine_3
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ W, IsCompact W ∧ IsOpen[inst✝¹] W ∧ t ⊆ W ∧ W ⊆ U
Module: Mathlib.Order.Ideal

Goal state before:
pp: case none
P : Type u_1
inst✝¹ : Preorder P
p : P
ι : Type u_2
inst✝ : Encodable ι
𝒟 : ι → Cofinal P
n : ℕ
⊢ sequenceOfCofinals p 𝒟 n ≤
    match none with
    | none => sequenceOfCofinals p 𝒟 n
    | some i => (𝒟 i).above (sequenceOfCofinals p 𝒟 n)

pp: case some
P : Type u_1
inst✝¹ : Preorder P
p : P
ι : Type u_2
inst✝ : Encodable ι
𝒟 : ι → Cofinal P
n : ℕ
val✝ : ι
⊢ sequenceOfCofinals p 𝒟 n ≤
    match some val✝ with
    | none => sequenceOfCofinals p 𝒟 n
    | some i => (𝒟 i).above (sequenceOfCofinals p 𝒟 n)

Tactic:
·

Goal state after:
case none
P : Type u_1
inst✝¹ : Preorder P
p : P
ι : Type u_2
inst✝ : Encodable ι
𝒟 : ι → Cofinal P
n : ℕ
⊢ sequenceOfCofinals p 𝒟 n ≤
    match none with
    | none => sequenceOfCofinals p 𝒟 n
    | some i => (𝒟 i).above (sequenceOfCofinals p 𝒟 n)
Module: Mathlib.Order.Ideal

Goal state before:
pp: case some
P : Type u_1
inst✝¹ : Preorder P
p : P
ι : Type u_2
inst✝ : Encodable ι
𝒟 : ι → Cofinal P
n : ℕ
val✝ : ι
⊢ sequenceOfCofinals p 𝒟 n ≤
    match some val✝ with
    | none => sequenceOfCofinals p 𝒟 n
    | some i => (𝒟 i).above (sequenceOfCofinals p 𝒟 n)

Tactic:
·

Goal state after:
case some
P : Type u_1
inst✝¹ : Preorder P
p : P
ι : Type u_2
inst✝ : Encodable ι
𝒟 : ι → Cofinal P
n : ℕ
val✝ : ι
⊢ sequenceOfCofinals p 𝒟 n ≤
    match some val✝ with
    | none => sequenceOfCofinals p 𝒟 n
    | some i => (𝒟 i).above (sequenceOfCofinals p 𝒟 n)
Module: Mathlib.Topology.Sets.Compacts

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
K : Compacts α
U : Opens α
h : U ∈ K.openRcNhds
x : ↥K
⊢ ↑{ carrier := closure[inst✝] ↑U, isCompact' := ⋯ } ∈ 𝓝 ↑x

Tactic:
have H : (U : Set α) ∈ 𝓝 (x : α) :=
    U.isOpen.mem_nhds <| Compacts.subset_of_mem_openRcNhds h (by simp)

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
K : Compacts α
U : Opens α
h : U ∈ K.openRcNhds
x : ↥K
H : ↑U ∈ 𝓝 ↑x
⊢ ↑{ carrier := closure[inst✝] ↑U, isCompact' := ⋯ } ∈ 𝓝 ↑x
Module: Mathlib.Order.Ideal

Goal state before:
pp: case refine_1
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ∀ I ∈ SetLike.coe '' S, IsIdeal I

pp: case refine_2
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ∀ (x y : Ideal P), x ≤ y → ↑x ⊆ ↑y

pp: case refine_3
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ≠ ⊤

pp: case refine_4
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ∈ upperBounds S

Tactic:
·

Goal state after:
case refine_1
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ∀ I ∈ SetLike.coe '' S, IsIdeal I
Module: Mathlib.Order.Ideal

Goal state before:
pp: case refine_2
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ∀ (x y : Ideal P), x ≤ y → ↑x ⊆ ↑y

pp: case refine_3
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ≠ ⊤

pp: case refine_4
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ∈ upperBounds S

Tactic:
·

Goal state after:
case refine_2
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ∀ (x y : Ideal P), x ≤ y → ↑x ⊆ ↑y
Module: Mathlib.Order.Ideal

Goal state before:
pp: case refine_3
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ≠ ⊤

pp: case refine_4
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ∈ upperBounds S

Tactic:
·

Goal state after:
case refine_3
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ≠ ⊤
Module: Mathlib.Order.Ideal

Goal state before:
pp: case refine_4
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ∈ upperBounds S

Tactic:
·

Goal state after:
case refine_4
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
S : Set (Ideal P)
hS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S
hS₂ : S.Nonempty
hS₃ : ⊤ ∉ S
⊢ ⋯.toIdeal ∈ upperBounds S
Module: Mathlib.Order.Ideal

Goal state before:
pp: case inl
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
hI : ⊤.IsProper
⊢ ∃ J, ⊤ ≤ J ∧ J.IsMaximal

pp: case inr
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
I : Ideal P
hI : I.IsProper
J : Ideal P
hJ : IsCoatom J
hJ' : I ≤ J
⊢ ∃ J, I ≤ J ∧ J.IsMaximal

Tactic:
·

Goal state after:
case inl
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
hI : ⊤.IsProper
⊢ ∃ J, ⊤ ≤ J ∧ J.IsMaximal
Module: Mathlib.Order.Ideal

Goal state before:
pp: case inr
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
I : Ideal P
hI : I.IsProper
J : Ideal P
hJ : IsCoatom J
hJ' : I ≤ J
⊢ ∃ J, I ≤ J ∧ J.IsMaximal

Tactic:
·

Goal state after:
case inr
P : Type u_1
inst✝¹ : LE P
inst✝ : OrderTop P
I : Ideal P
hI : I.IsProper
J : Ideal P
hJ : IsCoatom J
hJ' : I ≤ J
⊢ ∃ J, I ≤ J ∧ J.IsMaximal
Module: Mathlib.Topology.Constructible

Goal state before:
pp: case refine_1
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsRetrocompact s
t : Set X
htopen : IsOpen[inst✝] t
htcomp : IsCompact t
⊢ IsCompact (Subtype.val ⁻¹' t)

pp: case refine_2
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsSpectralMap Subtype.val
t : Set X
htcomp : IsCompact t
htopen : IsOpen[inst✝] t
⊢ IsCompact (s ∩ t)

Tactic:
·

Goal state after:
case refine_1
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsRetrocompact s
t : Set X
htopen : IsOpen[inst✝] t
htcomp : IsCompact t
⊢ IsCompact (Subtype.val ⁻¹' t)
Module: Mathlib.Topology.Constructible

Goal state before:
pp: case refine_2
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsSpectralMap Subtype.val
t : Set X
htcomp : IsCompact t
htopen : IsOpen[inst✝] t
⊢ IsCompact (s ∩ t)

Tactic:
·

Goal state after:
case refine_2
X : Type u_2
inst✝ : TopologicalSpace X
s : Set X
hs : IsSpectralMap Subtype.val
t : Set X
htcomp : IsCompact t
htopen : IsOpen[inst✝] t
⊢ IsCompact (s ∩ t)
Module: Mathlib.Topology.Sober

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x : α
S : Set α
hS : IsClosed S
hxS : x ∈ S
this : closure {x} ⊆ S
⊢ closure {x} = S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z

Tactic:
subset_antisymm_iff,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
x : α
S : Set α
hS : IsClosed S
hxS : x ∈ S
this : closure {x} ⊆ S
⊢ closure {x} ⊆ S ∧ S ⊆ closure {x} ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z
Module: Mathlib.Topology.Spectral.Prespectral

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
Z U₁ U₂ : Set X
hU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁
hU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂
hU₁Z : (Z ∩ U₁).Nonempty
hU₂Z : (Z ∩ U₂).Nonempty
hU₁₂ : Z ∩ (U₁ ∩ U₂) ⊆ ∅
x₁ : X
hx₁ : x₁ ∈ U₁ ∧ x₁ ∈ Z ∧ x₁ ∉ U₂
x₂ : X
hx₂ : x₂ ∈ U₂ ∧ x₂ ∈ Z ∧ x₂ ∉ U₁
W₁ : Set X
hW₁ : W₁ ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U} ∧ x₁ ∈ W₁ ∧ W₁ ⊆ U₁
W₂ : Set X
hW₂ : W₂ ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U} ∧ x₂ ∈ W₂ ∧ W₂ ⊆ U₂
z : X
hz : z ∈ Z
⊢ z ∈ W₁ᶜ ∪ W₂ᶜ

Tactic:
·

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PrespectralSpace X
Z U₁ U₂ : Set X
hU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁
hU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂
hU₁Z : (Z ∩ U₁).Nonempty
hU₂Z : (Z ∩ U₂).Nonempty
hU₁₂ : Z ∩ (U₁ ∩ U₂) ⊆ ∅
x₁ : X
hx₁ : x₁ ∈ U₁ ∧ x₁ ∈ Z ∧ x₁ ∉ U₂
x₂ : X
hx₂ : x₂ ∈ U₂ ∧ x₂ ∈ Z ∧ x₂ ∉ U₁
W₁ : Set X
hW₁ : W₁ ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U} ∧ x₁ ∈ W₁ ∧ W₁ ⊆ U₁
W₂ : Set X
hW₂ : W₂ ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U} ∧ x₂ ∈ W₂ ∧ W₂ ⊆ U₂
z : X
hz : z ∈ Z
⊢ z ∈ W₁ᶜ ∪ W₂ᶜ
Module: Mathlib.Topology.Sober

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S

Tactic:
·

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
ι : Type u_3
U : ι → Opens α
hU : IsOpenCover U
hU' : ∀ (i : ι), QuasiSober ↥(U i)
⊢ ∀ {S : Set α}, IsIrreducible S → IsClosed S → ∃ x, IsGenericPoint x S