mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case pos
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : z = 0
⊢ IsUnit (1 - z • a)

pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
⊢ IsUnit (1 - z • a)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : z = 0
⊢ IsUnit (1 - z • a)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
⊢ IsUnit (1 - z • a)

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
⊢ IsUnit (1 - z • a)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case inl
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
U : Set α
hU : U ∈ {U | IsOpen[inst✝.toTopologicalSpace] U}
hs' : s ∈ 𝒫 U
⊢ 𝒫 U ∈ 𝓝 s

pp: case inr
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
U : Set α
hU : U ∈ {U | IsOpen[inst✝.toTopologicalSpace] U}
hs' : s ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ 𝓝 s

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
U : Set α
hU : U ∈ {U | IsOpen[inst✝.toTopologicalSpace] U}
hs' : s ∈ 𝒫 U
⊢ 𝒫 U ∈ 𝓝 s
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case inr
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
U : Set α
hU : U ∈ {U | IsOpen[inst✝.toTopologicalSpace] U}
hs' : s ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ 𝓝 s

Tactic:
·

Goal state after:
case inr
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
U : Set α
hU : U ∈ {U | IsOpen[inst✝.toTopologicalSpace] U}
hs' : s ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ 𝓝 s
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case hac
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ ‖(a - ↑ₐ z) ^ n‖ ≤ ‖a - ↑ₐ z‖ ^ n

pp: case h
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ n ≤ n + 1

Tactic:
·

Goal state after:
case hac
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ ‖(a - ↑ₐ z) ^ n‖ ≤ ‖a - ↑ₐ z‖ ^ n
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case h
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ n ≤ n + 1

Tactic:
·

Goal state after:
case h
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ⋯
n : ℕ
hn : n ≠ 0
⊢ n ≤ n + 1
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case pos
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∃ L ∈ s, L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, ¬L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∃ L ∈ s, L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, ¬L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, ¬L ⊆ ⋃₀ u
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case e'_5
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ 1 = (↑(Nat.factorial 0))⁻¹ • (a - ↑ₐ z) ^ 0

pp: case e'_6
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ (a - ↑ₐ z) * b = ∑' (b : ℕ), (↑(b + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (b + 1)

Tactic:
·

Goal state after:
case e'_5
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ 1 = (↑(Nat.factorial 0))⁻¹ • (a - ↑ₐ z) ^ 0
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ ⋃₀ u ∈ 𝒫 U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

pp: case neg.inr
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {V | IsOpen[inst✝] V}
hUS : (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ S
hUu : K \ ⋃₀ u ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
·

Goal state after:
case neg.inl
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hUS : 𝒫 U ∈ S
hUu : K \ ⋃₀ u ∈ 𝒫 U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case e'_6
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ (a - ↑ₐ z) * b = ∑' (b : ℕ), (↑(b + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (b + 1)

Tactic:
·

Goal state after:
case e'_6
𝕜 : Type u_1
A : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ spectrum 𝕜 a
this : NormedAlgebra ℚ A
hexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)
b : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
hb : Summable fun n ↦ (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n
h₀ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = (a - ↑ₐ z) * b
h₁ : ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (n + 1) = b * (a - ↑ₐ z)
⊢ (a - ↑ₐ z) * b = ∑' (b : ℕ), (↑(b + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ (b + 1)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case neg.inr
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {V | IsOpen[inst✝] V}
hUS : (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ S
hUu : K \ ⋃₀ u ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
·

Goal state after:
case neg.inr
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
hsu : ∀ L ∈ s, (K \ ⋃₀ u ∩ L).Nonempty
U : Set α
hU : U ∈ {V | IsOpen[inst✝] V}
hUS : (fun V ↦ {s | (s ∩ V).Nonempty}) U ∈ S
hUu : K \ ⋃₀ u ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
h : s ⤳ t
x : α
hx : x ∈ s
⊢ ∃ y ∈ t, x ⤳ y

pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
⊢ s ⤳ t

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
h : s ⤳ t
x : α
hx : x ∈ s
⊢ ∃ y ∈ t, x ⤳ y
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
⊢ s ⤳ t

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
⊢ s ⤳ t
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ 𝒫 U
⊢ ⨅
      s_1 ∈
        {s_2 |
          s ∈ s_2 ∧
            s_2 ∈ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}},
      Filter.principal s_1 ≤
    Filter.principal (𝒫 U)

pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ⨅
      s_1 ∈
        {s_2 |
          s ∈ s_2 ∧
            s_2 ∈ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}},
      Filter.principal s_1 ≤
    Filter.principal ((fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
·

Goal state after:
case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ 𝒫 U
⊢ ⨅
      s_1 ∈
        {s_2 |
          s ∈ s_2 ∧
            s_2 ∈ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}},
      Filter.principal s_1 ≤
    Filter.principal (𝒫 U)
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ⨅
      s_1 ∈
        {s_2 |
          s ∈ s_2 ∧
            s_2 ∈ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}},
      Filter.principal s_1 ≤
    Filter.principal ((fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
·

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s t : Set α
x✝ : (∀ x ∈ s, ∃ y ∈ t, x ⤳ y) ∧ t ⊆ closure[inst✝] s
hst : ∀ x ∈ s, ∃ y ∈ t, x ⤳ y
hts : t ⊆ closure[inst✝] s
U : Set α
hU : U ∈ {U | IsOpen[inst✝] U}
hs : t ∈ (fun V ↦ {s | (s ∩ V).Nonempty}) U
⊢ ⨅
      s_1 ∈
        {s_2 |
          s ∈ s_2 ∧
            s_2 ∈ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}},
      Filter.principal s_1 ≤
    Filter.principal ((fun V ↦ {s | (s ∩ V).Nonempty}) U)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := ⋯
hl : l.TotallyBounded
L : Compacts α := ⋯
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
⊢ ↑K ⊆ U.preimage ↑L

pp: case right
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := ⋯
hl : l.TotallyBounded
L : Compacts α := ⋯
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
⊢ ↑L ⊆ U.image ↑K

Tactic:
·

Goal state after:
case left
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace α
f : Filter (Compacts α)
x✝ : Cauchy f
left✝ : f.NeBot
hf : ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∀ᶠ (K : Compacts α) (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
l : Filter α := ⋯
hl : l.TotallyBounded
L : Compacts α := ⋯
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsClosed[instTopologicalSpaceProd] U
K : Compacts α
hK : ∀ᶠ (K' : Compacts α) in f, (↑K, ↑K') ∈ hausdorffEntourage U
⊢ ↑K ⊆ U.preimage ↑L
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inl
α : Type u_1
inst✝ : TopologicalSpace α
hs : IsPreconnected ∅
⊢ IsPreconnected {t | t.Nonempty ∧ t.Finite ∧ t ⊆ ∅}

pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
x : α
hx : x ∈ s
⊢ IsPreconnected {t | t.Nonempty ∧ t.Finite ∧ t ⊆ s}

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝ : TopologicalSpace α
hs : IsPreconnected ∅
⊢ IsPreconnected {t | t.Nonempty ∧ t.Finite ∧ t ⊆ ∅}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}) ⊆
    ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ IsClosed[Unitization.instUniformSpace.toTopologicalSpace]
    ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))

Tactic:
·

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
h₁ :
  (cfcₙAux ⋯ a ha) f ∈
    closure[Unitization.instUniformSpace.toTopologicalSpace]
      (⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}))
⊢ ⇑(cfcₙAux ⋯ a ha) '' ↑(NonUnitalStarAlgebra.adjoin 𝕜 {ContinuousMapZero.id (σₙ 𝕜 a)}) ⊆
    ↑(NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A))
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case pos
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : t ⊆ U ∨ t ⊆ V
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty

pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : ¬t ⊆ U ∧ ¬t ⊆ V
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : t ⊆ U ∨ t ⊆ V
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case pos.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : t ⊆ U ∨ t ⊆ V
this :
  ∀ (U V : Set α),
    IsOpen[inst✝] U →
      IsOpen[inst✝] V →
        ⋃₀ s ⊆ U ∪ V → t ⊆ U ∨ t ⊆ V → t ⊆ U → (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty
htU : ¬t ⊆ U
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty

pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : t ⊆ U ∨ t ⊆ V
htU : t ⊆ U
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty

Tactic:
·

Goal state after:
case pos.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set (Set α)
hs : IsPreconnected s
t : Set α
hts : t ∈ s
ht : IsPreconnected t
hts' : t ⊆ ⋃₀ s
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
hUV : ⋃₀ s ⊆ U ∪ V
ht' : t ⊆ U ∨ t ⊆ V
this :
  ∀ (U V : Set α),
    IsOpen[inst✝] U →
      IsOpen[inst✝] V →
        ⋃₀ s ⊆ U ∪ V → t ⊆ U ∨ t ⊆ V → t ⊆ U → (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty
htU : ¬t ⊆ U
⊢ (⋃₀ s ∩ U).Nonempty → (⋃₀ s ∩ V).Nonempty → (⋃₀ s ∩ (U ∩ V)).Nonempty