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

Goal state before:
pp: case left
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
u : Set ((i : ι) × X i)
hu : IsCompact u
s : Finset ι
hs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ
i : ι
⊢ IsCompact (Sigma.mk i ⁻¹' u)

pp: case right
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
u : Set ((i : ι) × X i)
hu : IsCompact u
s : Finset ι
hs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ
⊢ ((↑s).sigma fun i ↦ Sigma.mk i ⁻¹' u) = u

Tactic:
·

Goal state after:
case left
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
u : Set ((i : ι) × X i)
hu : IsCompact u
s : Finset ι
hs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ
i : ι
⊢ IsCompact (Sigma.mk i ⁻¹' u)
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case right
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
u : Set ((i : ι) × X i)
hu : IsCompact u
s : Finset ι
hs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ
⊢ ((↑s).sigma fun i ↦ Sigma.mk i ⁻¹' u) = u

Tactic:
·

Goal state after:
case right
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
u : Set ((i : ι) × X i)
hu : IsCompact u
s : Finset ι
hs : u ⊆ ⋃ i ∈ s, Sigma.mk i '' Sigma.mk i ⁻¹' univ
⊢ ((↑s).sigma fun i ↦ Sigma.mk i ⁻¹' u) = u
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ (cocompact X).coprod (cocompact Y) ≤ cocompact (X × Y)

pp: case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ cocompact (X × Y) ≤ (cocompact X).coprod (cocompact Y)

Tactic:
·

Goal state after:
case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ (cocompact X).coprod (cocompact Y) ≤ cocompact (X × Y)
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ cocompact (X × Y) ≤ (cocompact X).coprod (cocompact Y)

Tactic:
·

Goal state after:
case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ cocompact (X × Y) ≤ (cocompact X).coprod (cocompact Y)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
this :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
    IsClosed s →
      (∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) →
        ∀ x ∈ s, ∀ y ∈ s, ¬x = y → ∀ (U : Set X), IsOpen U → Xor (x ∈ U) (y ∈ U) → x ∈ U ∧ y ∉ U → False
h : ¬(x ∈ U ∧ y ∉ U)
⊢ False

pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
h : x ∈ U ∧ y ∉ U
⊢ False

Tactic:
·

Goal state after:
case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
this :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
    IsClosed s →
      (∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) →
        ∀ x ∈ s, ∀ y ∈ s, ¬x = y → ∀ (U : Set X), IsOpen U → Xor (x ∈ U) (y ∈ U) → x ∈ U ∧ y ∉ U → False
h : ¬(x ∈ U ∧ y ∉ U)
⊢ False
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
h : x ∈ U ∧ y ∉ U
⊢ False

Tactic:
obtain ⟨hxU, hyU⟩ := h

Goal state after:
X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsClosed[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsClosed[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
hxU : x ∈ U
hyU : y ∉ U
⊢ False
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case ht
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ∀ a ∈ t.ite s s' ∩ frontier t, f a = g a

pp: case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure[inst✝²] t)

pp: case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (t.ite s s' ∩ closure[inst✝²] tᶜ)

Tactic:
·

Goal state after:
case ht
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ∀ a ∈ t.ite s s' ∩ frontier t, f a = g a
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure[inst✝²] t)

pp: case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (t.ite s s' ∩ closure[inst✝²] tᶜ)

Tactic:
·

Goal state after:
case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure[inst✝²] t)
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (t.ite s s' ∩ closure[inst✝²] tᶜ)

Tactic:
·

Goal state after:
case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (t.ite s s' ∩ closure[inst✝²] tᶜ)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: Y : Type u_2
inst✝ : TopologicalSpace Y
S : Set Y
⊢ DiscreteTopology ↑S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥

Tactic:
by
  simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]

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

Goal state before:
pp: case mp
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
hs : IsClosed[topologicalSpace] s
H : IsCompact s
⊢ ∀ (i : ι), IsCompact (eval i '' s)

pp: case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
hs : IsClosed[topologicalSpace] s
H : ∀ (i : ι), IsCompact (eval i '' s)
⊢ IsCompact s

Tactic:
·

Goal state after:
case mp
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
hs : IsClosed[topologicalSpace] s
H : IsCompact s
⊢ ∀ (i : ι), IsCompact (eval i '' s)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsOpen[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsOpen[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
this :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
    IsOpen s →
      (∀ t ⊆ s, t.Nonempty → IsOpen t → t = s) →
        ∀ x ∈ s, ∀ y ∈ s, ¬x = y → ∀ (U : Set X), IsOpen U → Xor (x ∈ U) (y ∈ U) → x ∈ U ∧ y ∉ U → False
h : ¬(x ∈ U ∧ y ∉ U)
⊢ False

pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsOpen[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsOpen[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
h : x ∈ U ∧ y ∉ U
⊢ False

Tactic:
·

Goal state after:
case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsOpen[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsOpen[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
this :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
    IsOpen s →
      (∀ t ⊆ s, t.Nonempty → IsOpen t → t = s) →
        ∀ x ∈ s, ∀ y ∈ s, ¬x = y → ∀ (U : Set X), IsOpen U → Xor (x ∈ U) (y ∈ U) → x ∈ U ∧ y ∉ U → False
h : ¬(x ∈ U ∧ y ∉ U)
⊢ False
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case mp
X : Type u_3
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ irreducibleComponents X → s ∈ {s | Maximal (fun x ↦ IsClosed[inst✝] x ∧ IsIrreducible x) s}

pp: case mpr
X : Type u_3
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ {s | Maximal (fun x ↦ IsClosed[inst✝] x ∧ IsIrreducible x) s} → s ∈ irreducibleComponents X

Tactic:
·

Goal state after:
case mp
X : Type u_3
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ irreducibleComponents X → s ∈ {s | Maximal (fun x ↦ IsClosed[inst✝] x ∧ IsIrreducible x) s}
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case mpr
X : Type u_3
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ {s | Maximal (fun x ↦ IsClosed[inst✝] x ∧ IsIrreducible x) s} → s ∈ irreducibleComponents X

Tactic:
·

Goal state after:
case mpr
X : Type u_3
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ {s | Maximal (fun x ↦ IsClosed[inst✝] x ∧ IsIrreducible x) s} → s ∈ irreducibleComponents X
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
hs : IsClosed[topologicalSpace] s
H : ∀ (i : ι), IsCompact (eval i '' s)
⊢ IsCompact s

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
hs : IsClosed[topologicalSpace] s
H : ∀ (i : ι), IsCompact (eval i '' s)
⊢ IsCompact s
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case mp
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t
s : Set X
sS : s ⊆ S
⊢ ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s

pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
s : Set X
sS : s ⊆ S
⊢ ∃ u, IsOpen[inst✝] u ∧ u ∩ S = s

Tactic:
·

Goal state after:
case mp
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t
s : Set X
sS : s ⊆ S
⊢ ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsOpen[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsOpen[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
h : x ∈ U ∧ y ∉ U
⊢ False

Tactic:
obtain ⟨hxU, hyU⟩ := h

Goal state after:
X✝ : Type u_1
inst✝² : TopologicalSpace X✝
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Set X
hs : IsOpen[inst✝¹] s
hmin : ∀ t ⊆ s, t.Nonempty → IsOpen[inst✝¹] t → t = s
x : X
hx : x ∈ s
y : X
hy : y ∈ s
hxy : ¬x = y
U : Set X
hUo : IsOpen[inst✝¹] U
hU : Xor (x ∈ U) (y ∈ U)
hxU : x ∈ U
hyU : y ∉ U
⊢ False
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case mp
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
⊢ (∃ K, IsCompact K ∧ s ⊆ K) → ∀ (i : ι), ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki

pp: case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
⊢ (∀ (i : ι), ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki) → ∃ K, IsCompact K ∧ s ⊆ K

Tactic:
·

Goal state after:
case mp
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
⊢ (∃ K, IsCompact K ∧ s ⊆ K) → ∀ (i : ι), ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
⊢ (∀ (i : ι), ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki) → ∃ K, IsCompact K ∧ s ⊆ K

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
s : Set ((i : ι) → X i)
⊢ (∀ (i : ι), ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki) → ∃ K, IsCompact K ∧ s ⊆ K
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
s : Set X
sS : s ⊆ S
⊢ ∃ u, IsOpen[inst✝] u ∧ u ∩ S = s

Tactic:
·

Goal state after:
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
s : Set X
sS : s ⊆ S
⊢ ∃ u, IsOpen[inst✝] u ∧ u ∩ S = s
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case a.inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
t : Finset X
hts : t ⊂ s
htne : t.Nonempty
hto : IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}

pp: case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}

Tactic:
·

Goal state after:
case a.inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
t : Finset X
hts : t ⊂ s
htne : t.Nonempty
hto : IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case pos
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcne : c.Nonempty
⊢ ∃ ub ∈ opens, ∀ s ∈ c, s ⊆ ub

pp: case neg
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcne : ¬c.Nonempty
⊢ ∃ ub ∈ opens, ∀ s ∈ c, s ⊆ ub

Tactic:
·

Goal state after:
case pos
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
hcne : c.Nonempty
⊢ ∃ ub ∈ opens, ∀ s ∈ c, s ⊆ ub
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}

Tactic:
·

Goal state after:
case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
x : X
hx : ↑s = {x}
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}

pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x, ↑s = {x}

Tactic:
·

Goal state after:
case a.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
x : X
hx : ↑s = {x}
⊢ ∃ x ∈ ↑s, IsOpen[inst✝¹] {x}
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case h.refine_1
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Sᶜ ⊆ ⋃₀ c

pp: case h.refine_2
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ IsOpen[inst✝¹] (⋃₀ c)

pp: case h.refine_3
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ.Nonempty

Tactic:
·

Goal state after:
case h.refine_1
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ Sᶜ ⊆ ⋃₀ c
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
hf' : Injective f
⊢ IsDiscrete (f ⁻¹' s)

Tactic:
refine .of_nhdsWithin fun x hx ↦ ?_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
hf' : Injective f
x : X
hx : x ∈ f ⁻¹' s
⊢ 𝓝[f ⁻¹' s] x ≤ pure x
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
H : ∀ (x : Y), IsDiscrete (f ⁻¹' {x})
⊢ IsDiscrete (f ⁻¹' s)

Tactic:
refine .of_nhdsWithin fun x hx ↦ ?_

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
H : ∀ (x : Y), IsDiscrete (f ⁻¹' {x})
x : X
hx : x ∈ f ⁻¹' s
⊢ 𝓝[f ⁻¹' s] x ≤ pure x
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case h.refine_2
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ IsOpen[inst✝¹] (⋃₀ c)

pp: case h.refine_3
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ.Nonempty

Tactic:
·

Goal state after:
case h.refine_2
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ IsOpen[inst✝¹] (⋃₀ c)
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: case h.refine_3
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ.Nonempty

Tactic:
·

Goal state after:
case h.refine_3
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ.Nonempty
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ = ⋂ U, (↑U)ᶜ

pp: case h.refine_3
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋂ U, (↑U)ᶜ).Nonempty

Tactic:
·

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed[inst✝¹] S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀compl : Sᶜ ⊆ U₀
⊢ (⋃₀ c)ᶜ = ⋂ U, (↑U)ᶜ