mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
⊢ φ = ψ

Tactic:
have : Fact (0 ∈ s') := ⟨0, Fact.out, coe_zero⟩

Goal state after:
X : Type u_1
inst✝⁹ : TopologicalSpace X
inst✝⁸ : Zero X
A : Type u_2
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : TopologicalSpace A
inst✝³ : IsSemitopologicalRing A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : T2Space A
s : Set ℝ≥0
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : C(↑s, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
hφ : Continuous[_, inst✝⁴] ⇑φ
hψ : Continuous[_, inst✝⁴] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
s' : Set ℝ := toReal '' s
e : ↑s ≃ₜ ↑s' := isEmbedding_coe.homeomorphImage s
this : Fact (0 ∈ s')
⊢ φ = ψ
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakBilin

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_4
F : Type u_5
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : CommSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommMonoid F
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
l : Filter α
f : α → WeakBilin B
x : WeakBilin B
hB : Function.Injective ⇑B
⊢ Tendsto ((fun x y ↦ (B x) y) ∘ f) l (𝓝 fun y ↦ (B x) y) ↔ Tendsto (fun i ↦ ⇑(B (f i))) l (𝓝 ⇑(B x))

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set 𝕜
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ ⊤ ≤ StarAlgHom.equalizer φ ψ

Tactic:
← polynomialFunctions.starClosure_topologicalClosure s

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : Algebra 𝕜 A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set 𝕜
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
⊢ (polynomialFunctions s).starClosure.topologicalClosure ≤ StarAlgHom.equalizer φ ψ
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakDual

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_4
inst✝⁶ : CommSemiring 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
x : E
⊢ (toWeakSpaceCLM 𝕜 E) x = (toWeakSpace 𝕜 E) x

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case mem
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
hf : restrict s (ContinuousMap.id 𝕜) ∈ {restrict s (ContinuousMap.id 𝕜)}
⊢ restrict s (ContinuousMap.id 𝕜) ∈ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: case inl
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
φ : ↑(characterSpace 𝕜 A)
h₁ : φ 1 * (1 - φ 1) = 0
h₂ : φ 1 = 0
this : ∀ (a : A), φ (a * 1) = 0
⊢ φ 1 = 1

Tactic:
exact False.elim (φ.prop.1 <| ContinuousLinearMap.ext <| by simpa only [mul_one] using! this)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case f
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| fun y ↦ y - φ y • 1

pp: case l₁
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x ⊓ 𝓟 ↑s

pp: case l₂
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x

Tactic:
rfl

Goal state after:
case l₁
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x ⊓ 𝓟 ↑s

case l₂
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case l₁
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x ⊓ 𝓟 ↑s

pp: case l₂
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x

Tactic:
rfl

Goal state after:
case l₂
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
⊢ Dense ↑(adjoin 𝕜 {ContinuousMapZero.id s})

Tactic:
dense_iff_closure_eq,

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
⊢ closure ↑(adjoin 𝕜 {ContinuousMapZero.id s}) = Set.univ
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
⊢ (∀ (b : Set β), (∀ x ∈ b, b ∈ 𝓝 x) → ∀ (a : α), (f a ∩ b).Nonempty → ∀ᶠ (x' : α) in 𝓝 a, (f x' ∩ b).Nonempty) ↔
    ∀ (u : Set β), (∀ x ∈ u, u ∈ 𝓝 x) → ∀ x ∈ {x | (f x ∩ u).Nonempty}, {x | (f x ∩ u).Nonempty} ∈ 𝓝 x

Tactic:
mem_ofPred,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
⊢ (∀ (b : Set β), (∀ x ∈ b, b ∈ 𝓝 x) → ∀ (a : α), (f a ∩ b).Nonempty → ∀ᶠ (x' : α) in 𝓝 a, (f x' ∩ b).Nonempty) ↔
    ∀ (u : Set β), (∀ x ∈ u, u ∈ 𝓝 x) → ∀ (x : α), (f x ∩ u).Nonempty → {x | (f x ∩ u).Nonempty} ∈ 𝓝 x
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
hf : ∀ (u : Set β), IsOpen[inst✝] u → f x ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ u
u : Set β
hu : IsClosed[inst✝] u
t : Set β
ht_open : IsOpen[inst✝] t
ht : f x ∩ u ⊆ t
⊢ f x ⊆ t ∪ uᶜ

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
s : Set α
x : α
u : Set β
hu : IsClosed[inst✝] u
t : Set β
hf : ∀ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ t ∪ uᶜ
ht_open : IsOpen[inst✝] t
ht : f x ∩ u ⊆ t
⊢ ∀ᶠ (x' : α) in 𝓝[s] x, f x' ∩ u ⊆ t

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
x₀ : α
hf : LowerHemicontinuousAt f x₀
x : ℕ → α
hx : Tendsto x atTop (𝓝 x₀)
y₀ : β
hy₀ : y₀ ∈ f x₀
inst✝ : (𝓝 y₀).IsCountablyGenerated
U : ℕ → Set β
hU : ∀ (i : ℕ), y₀ ∈ U i ∧ IsOpen[inst✝¹] (U i)
hUbasis : (𝓝 y₀).HasAntitoneBasis fun i ↦ U i
hev : ∀ (k : ℕ), ∀ᶠ (n : ℕ) in atTop, (f (x n) ∩ U k).Nonempty
g : ℕ → ℕ := fun n ↦ Nat.findGreatest (fun k ↦ (f (x n) ∩ U k).Nonempty) n
key : ∀ (n k : ℕ), k ≤ n → (f (x n) ∩ U k).Nonempty → (f (x n) ∩ U (g n)).Nonempty
y : ℕ → β := fun n ↦ if h : (f (x n) ∩ U (g n)).Nonempty then h.some else y₀
hy : ∀ (n : ℕ), (f (x n) ∩ U (g n)).Nonempty → y n ∈ f (x n) ∩ U (g n)
⊢ ∀ (x_1 : ℕ), (f (x x_1) ∩ U 0).Nonempty → y x_1 ∈ f (x x_1)

Tactic:
grind

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

Goal state before:
pp: case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ insert U ((fun x ↦ U ∩ x) '' v) ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} ∧
    s ∈ {s | s ⊆ U ∧ ∀ U_1 ∈ insert U ((fun x ↦ U ∩ x) '' v), (s ∩ U_1).Nonempty} ∧
      {s | s ⊆ U ∧ ∀ U_1 ∈ insert U ((fun x ↦ U ∩ x) '' v), (s ∩ U_1).Nonempty} ⊆ 𝒫 U ∩ ⋂ a ∈ v, {s | (s ∩ a).Nonempty}

Tactic:
mem_ofPred,

Goal state after:
case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ ((insert U ((fun x ↦ U ∩ x) '' v)).Finite ∧ ∀ U_1 ∈ insert U ((fun x ↦ U ∩ x) '' v), IsOpen[inst✝] U_1) ∧
    (s ⊆ U ∧ ∀ U_1 ∈ insert U ((fun x ↦ U ∩ x) '' v), (s ∩ U_1).Nonempty) ∧
      {s | s ⊆ U ∧ ∀ U_1 ∈ insert U ((fun x ↦ U ∩ x) '' v), (s ∩ U_1).Nonempty} ⊆ 𝒫 U ∩ ⋂ a ∈ v, {s | (s ∩ a).Nonempty}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
⊢ {x | x.1 ⊆ x.2 ∧ x.2 ⊆ x.1} = SetRel.id

Tactic:
← subset_antisymm_iff,

Goal state after:
α : Type u_1
⊢ {x | x.1 = x.2} = SetRel.id
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
h✝ : s.Nonempty
u : Set (Set α)
hu₁ : u ⊆ {U | IsOpen[inst✝] U}
hu₂ : u.Finite
hu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}
v : Set (Set α)
hv₁ : v ⊆ {U | IsOpen[inst✝] U}
hv₂ : v.Finite
hv : (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {U | IsOpen[inst✝] U}
ht₁ : (powerset '' u).Finite ∧ ((fun V ↦ {s | (s ∩ V).Nonempty}) '' v).Finite
ht₂ :
  powerset '' u ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' v ⊆
    powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
U : Set α := ⋂₀ u
hU : IsOpen[inst✝] U
hs : s ⊆ U ∧ ∀ i ∈ v, s ∈ {s | (s ∩ i).Nonempty}
⊢ s ⊆ U ∧ s.Nonempty ∧ ∀ ⦃x : Set α⦄, x ∈ v → (s ∩ x).Nonempty

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
s : Set α
hs₁ : s ⊆ ⋃₀ u
hs₂ : ∀ U ∈ u, (s ∩ U).Nonempty
f : Set α → Set α
hfB : ∀ U ∈ u, f U ∈ B
hfU : ∀ U ∈ u, f U ⊆ U
hfs : ∀ U ∈ u, (s ∩ f U).Nonempty
⊢ (⋃₀ u, f '' u).2 ⊆ B ∧ ∀ U ∈ (⋃₀ u, f '' u).2, U ⊆ (⋃₀ u, f '' u).1

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
s : Set α
hs₁ : s ⊆ ⋃₀ u
hs₂ : ∀ U ∈ u, (s ∩ U).Nonempty
f : Set α → Set α
hfB : ∀ U ∈ u, f U ∈ B
hfU : ∀ U ∈ u, f U ⊆ U
hfs : ∀ U ∈ u, (s ∩ f U).Nonempty
⊢ s ∈ {s | s ⊆ (⋃₀ u, f '' u).1 ∧ ∀ U ∈ (⋃₀ u, f '' u).2, (s ∩ U).Nonempty}

Tactic:
grind

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝¹³ : NormedField 𝕜
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra 𝕜 A
inst✝¹⁰ : CompleteSpace A
B : Type u_3
inst✝⁹ : NonUnitalNormedRing B
inst✝⁸ : NormedSpace 𝕜 B
inst✝⁷ : CompleteSpace B
inst✝⁶ : IsScalarTower 𝕜 B B
inst✝⁵ : SMulCommClass 𝕜 B B
inst✝⁴ : ProperSpace 𝕜
inst✝³ : NormedSpace ℝ B
inst✝² : IsScalarTower ℝ B B
inst✝¹ : SMulCommClass ℝ B B
a : B
inst✝ : CompactSpace ↑(quasispectrum ℝ a)
⊢ CompactSpace ↑(quasispectrum ℝ≥0 a)

Tactic:
by
  rw [← isCompact_iff_compactSpace] at *
  rw [← quasispectrum.preimage_algebraMap ℝ]
  exact isClosed_nonneg.isClosedEmbedding_subtypeVal.isCompact_preimage <| by assumption

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
ε : ℝ≥0∞
hε : ε < 1
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop

Tactic:
by_cases h : ε = 0

Goal state after:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
ε : ℝ≥0∞
hε : ε < 1
h : ε = 0
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop

case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
ε : ℝ≥0∞
hε : ε < 1
h : ¬ε = 0
⊢ ε * spectralRadius 𝕜 a ≤ liminf (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1
z : 𝕜
hz : z ∈ {0}ᶜ
⊢ resolvent a z = z⁻¹ • resolvent (z⁻¹ • a) 1

Tactic:
lift z to 𝕜ˣ using Ne.isUnit hz

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1
z : 𝕜ˣ
hz : ↑z ∈ {0}ᶜ
⊢ resolvent a ↑z = (↑z)⁻¹ • resolvent ((↑z)⁻¹ • a) 1
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {U | IsOpen[inst✝] U}
⊢ IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)

pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] (f ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
· exact h₁ U hU

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
x✝ :
  (∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)) ∧
    ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
h₁ : ∀ (U : Set β), IsOpen[inst✝] U → IsOpen[inst✝¹] (f ⁻¹' 𝒫 U)
h₂ : ∀ (F : Set β), IsClosed[inst✝] F → IsClosed[inst✝¹] (f ⁻¹' 𝒫 F)
U : Set β
hU : U ∈ {V | IsOpen[inst✝] V}
⊢ IsOpen[inst✝¹] (f ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : ∀ i ∈ 𝓤 α, ∃ x ∈ Prod.mk s ⁻¹' (hausdorffEntourage ∘ id) i, x ∈ {s | TotallyBounded s}
U : Set (α × α)
hU : U ∈ 𝓤 α
V : SetRel α α
hV : V ∈ 𝓤 α
hVU : V.comp V ⊆ U
t : Set α
ht : t ∈ {s | TotallyBounded s}
hst : s ⊆ V.preimage t
u : Set α
hu : u.Finite
htu : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ V}
⊢ ∀ (x x_1 x_2 : α), x_2 ∈ u → (x_1, x_2) ∈ V → (x, x_1) ∈ V → ∃ i ∈ u, ∃ b, (x, b) ∈ V ∧ (b, i) ∈ V

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
hf : UniformContinuous f
U : Set (β × β)
hU : U ∈ 𝓤 β
⊢ ∀ᶠ (a : Set α × Set α) in 𝓤 (Set α), ((fun x ↦ f '' x) a.1, (fun x ↦ f '' x) a.2) ∈ hausdorffEntourage U

Tactic:
filter_upwards [Filter.mem_lift' (hf hU)] with ⟨s, t⟩ ⟨h₁, h₂⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
hf : UniformContinuous f
U : Set (β × β)
hU : U ∈ 𝓤 β
s t : Set α
h₁ : (s, t).1 ⊆ SetRel.preimage ((fun x ↦ (f x.1, f x.2)) ⁻¹' U) (s, t).2
h₂ : (s, t).2 ⊆ SetRel.image ((fun x ↦ (f x.1, f x.2)) ⁻¹' U) (s, t).1
⊢ (f '' (s, t).1, f '' (s, t).2) ∈ hausdorffEntourage U
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht✝ : TotallyBounded t
U : SetRel α α
hU : U ∈ 𝓤 α
u : Set α
hu : u.Finite
ht : t ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}
s : Set α
hs : s ⊆ t
⊢ u ∩ U.image s ∈ 𝒫 u

Tactic:
grind

Goal state after:
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}
L : Set α
hL : L ∈ s
hLu : L ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hLT : L ⊆ ⋃ i ∈ T, i
⊢ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ {x | (x ∩ L).Nonempty}

Tactic:
grind

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

Goal state before:
pp: 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).Nonempty
L : Set α
h : L ∈ s
⊢ L \ ⋃₀ u ⊆ K \ ⋃₀ u ∩ L

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
x : α
hx : x ∈ s
x✝ : ℕ+
f : Fin ↑x✝ → α
hf : f ∈ univ.pi fun x ↦ s
⊢ range f ⊆ s

Tactic:
grind

Goal state after:
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

Tactic:
grind

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

Goal state before:
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
u : Set α
hus : u ∈ s
huV : (u ∩ V).Nonempty
this : s ⊆ 𝒫 U ∪ {v | (v ∩ V).Nonempty}
v : Set α
hvs : v ∈ s
hvU : v ∈ 𝒫 U
hvV : v ∈ {s | (s ∩ V).Nonempty}
⊢ v ∩ V ⊆ ⋃₀ s ∩ (U ∩ V)

Tactic:
grind

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

Goal state before:
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
⊢ ¬Disjoint t U

Tactic:
grind

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

Goal state before:
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 : ¬Disjoint t U
⊢ ¬Disjoint t V

Tactic:
grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set α
f : α → Set β
hs : IsPreconnected s
hf : ContinuousOn f s
h : ∃ x ∈ s, IsPreconnected (f x)
⊢ ∃ t ∈ f '' s, IsPreconnected t

Tactic:
grind

Goal state after: