mathlib-initiative/mathlib-tactics huggingface dataset:
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

Tactic:
rw [mem_powerset_iff, sdiff_subset_comm, sUnion_eq_biUnion] at hUu

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 \ U ⊆ ⋃ i ∈ u, i
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q
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:
by 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:
by grind

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T2Space α
K L : Compacts α
hKL : (K, L) ∈ {p | Disjoint ↑p.1 ↑p.2}
U V : Set α
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
hKU : ↑K ⊆ U
hLV : ↑L ⊆ V
hUV : Disjoint U V
⊢ {K' | ↑K' ⊆ U} ×ˢ {L' | ↑L' ⊆ V} ⊆ {p | Disjoint ↑p.1 ↑p.2}

Tactic:
by grind

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

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
⊢ ∀ ⦃x : Set α × Set (Set α)⦄,
    x ∈ {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} →
      ∀ x_1 ∈ SetLike.coe ⁻¹' {s | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty},
        ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
          x_1 ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ x.1 ∧ ∀ U ∈ x.2, (s ∩ U).Nonempty}

Tactic:
intro ⟨V, u⟩

Goal state after:
case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
⊢ (V, u) ∈ {p | IsOpen[inst✝] p.1 ∧ p.2.Finite ∧ p.2 ⊆ B ∧ ∀ U ∈ p.2, U ⊆ p.1} →
    ∀ x ∈ SetLike.coe ⁻¹' {s | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty},
      ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
        x ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
⊢ {W | W ∈ w ∧ (↑K ∩ W).Nonempty} ⊆ B

Tactic:
by 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
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
⊢ ⋃₀ {W | W ∈ w ∧ (↑K ∩ W).Nonempty} ⊆ V

Tactic:
by 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
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
this :
  ∀ (w : Set (Set α)),
    ↑K ⊆ ⋃₀ w →
      w ⊆ B →
        ⋃₀ w ⊆ V →
          (∀ W ∈ w, (↑K ∩ W).Nonempty) →
            ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u ⊆ B},
              K ∈ v ∧ v ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}
hwK : ¬∀ W ∈ w, (↑K ∩ W).Nonempty
⊢ ∀ W ∈ {W | W ∈ w ∧ (↑K ∩ W).Nonempty}, (↑K ∩ W).Nonempty

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.Complex.Convex

Goal state before:
pp: z w : ℂ
⊢ (convexHull ℝ)
      (insert (equivRealProd.symm (z.re, z.im))
        (insert (equivRealProd.symm (z.re, w.im))
          (insert (equivRealProd.symm (w.re, z.im)) (⇑equivRealProd.symm '' {(w.re, w.im)})))) =
    (convexHull ℝ) {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w}

Tactic:
image_singleton,

Goal state after:
z w : ℂ
⊢ (convexHull ℝ)
      {equivRealProd.symm (z.re, z.im), equivRealProd.symm (z.re, w.im), equivRealProd.symm (w.re, z.im),
        equivRealProd.symm (w.re, w.im)} =
    (convexHull ℝ) {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w}
Module: Mathlib.Analysis.Complex.Convex

Goal state before:
pp: U : Set ℂ
U_convex : Convex ℝ U
z w : ℂ
hz : z ∈ U
hw : w ∈ U
hzw : ↑z.re + ↑w.im * I ∈ U
hwz : ↑w.re + ↑z.im * I ∈ U
⊢ {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w} ⊆ U

Tactic:
by 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
⊢ SetLike.coe '' {L | ↑L ⊆ K} ⊆ 𝒫 K

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ CompactSpace (Compacts α) ↔ CompactSpace α

Tactic:
by
  refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩
  convert! isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ
  symm
  simp_rw [biUnion_univ, eq_univ_iff_forall, mem_iUnion]
  exact fun x => ⟨{x}, Set.mem_singleton x⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
U : Set α
hU : U ∈ u
x : α
hxK : x ∈ ↑K
hxU : x ∈ U
M : Set α
hM : IsCompact M
right✝ : x ∈ interior M ∧ M ⊆ U ∩ interior L
⊢ M ⊆ U

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
U : Set α
hU : U ∈ u
x : α
hxK : x ∈ ↑K
hxU : x ∈ U
M : Set α
hM : IsCompact M
right✝ : x ∈ interior M ∧ M ⊆ U ∩ interior L
⊢ x ∈ ↑K ∩ interior M

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ M '' u ⊆ 𝒫 L

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ ∀ L ∈ M '' u, IsCompact L

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyCompactSpace α
K : Compacts α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝³] U
hKu₁ : ↑K ⊆ ⋃₀ u
hKu₂ : ∀ U ∈ u, (↑K ∩ U).Nonempty
L : Set α
hL : IsCompact L
hLK : ↑K ⊆ interior L
hLu : L ⊆ ⋃₀ u
M : Set α → Set α
hM : ∀ U ∈ u, IsCompact (M U)
hML : ∀ U ∈ u, M U ⊆ L
hMU : ∀ U ∈ u, M U ⊆ U
hMK : ∀ U ∈ u, (↑K ∩ interior (M U)).Nonempty
⊢ SetLike.coe '' {K' | ↑K' ⊆ L ∧ ∀ U ∈ u, (↑K' ∩ M U).Nonempty} ⊆ {t | t ⊆ L ∧ ∀ L ∈ M '' u, (t ∩ L).Nonempty}

Tactic:
by 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
⊢ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ s} ⊆ {K | (↑K).Nonempty ∧ ↑K ⊆ s}

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
⊢ ∀ u ∈ {s | IsOpen s ∧ IsPreconnected s}, IsOpen u

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f✝ : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
u : Set (Set α)
hu' : u.Finite
hu : u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}
this : Fintype ↑u := hu'.fintype
f : ↑u → Compacts α
hf : f ∈ univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}
⊢ ∀ (i : ↑u), ↑(f i) ⊆ ⋃₀ u

Tactic:
by grind

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f✝ : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
u : Set (Set α)
hu' : u.Finite
hu : u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}
this : Fintype ↑u := hu'.fintype
f : ↑u → Compacts α
hf : f ∈ univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}
⊢ ∀ (i : ↑u), (↑(f i)).Finite

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
⊢ deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2

Tactic:
conv => enter [1, 1, y, 1, x, 1]; rw [← mul_one (exp (x • a))]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a✝ : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a✝) * NormedSpace.exp (-x • star a✝) ∈ unitary A
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-x • a✝) * NormedSpace.exp (x • star a✝) =
      NormedSpace.exp (x • star a✝) * NormedSpace.exp (-x • a✝)
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
a b : A
⊢ deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * 1 * NormedSpace.exp (x • b)) y) 0 =
    a ^ 2 + 2 • (a * b) + b ^ 2
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
K : NonemptyCompacts α
hK₁ : ↑K.toCompacts ⊆ ⋃₀ u
hK₂ : ∀ U ∈ u, (↑K.toCompacts ∩ U).Nonempty
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}

Tactic:
rcases eq_empty_or_nonempty u with rfl | hu'

Goal state after:
case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
K : NonemptyCompacts α
hu : ∅.Finite
huB : ∅ ⊆ B
hK₁ : ↑K.toCompacts ⊆ ⋃₀ ∅
hK₂ : ∀ U ∈ ∅, (↑K.toCompacts ∩ U).Nonempty
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ ∅ ∧ ∀ U ∈ ∅, (↑K ∩ U).Nonempty}

case refine_2.inr
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set (Set α)
hu : u.Finite
huB : u ⊆ B
K : NonemptyCompacts α
hK₁ : ↑K.toCompacts ⊆ ⋃₀ u
hK₂ : ∀ U ∈ u, (↑K.toCompacts ∩ U).Nonempty
hu' : u.Nonempty
⊢ ∃ v ∈ (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) '' {u | u.Finite ∧ u.Nonempty ∧ u ⊆ B},
    K ∈ v ∧ v ⊆ toCompacts ⁻¹' {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ CompactSpace (NonemptyCompacts α) ↔ CompactSpace α

Tactic:
by
  refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩
  convert! isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ
  symm
  simp_rw [biUnion_univ, eq_univ_iff_forall, mem_iUnion]
  exact fun x => ⟨{x}, Set.mem_singleton x⟩

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
⊢ ∀ (i' : Set α), i' ∈ 𝓝 x ∧ IsPreconnected i' → id i' ∈ 𝓝 x

Tactic:
by grind

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
⊢ CompactSpace ↑(characterSpace 𝕜 A)

Tactic:
by
  rw [← isCompact_iff_compactSpace]
  have h : characterSpace 𝕜 A ⊆ toStrongDual ⁻¹' Metric.closedBall 0 ‖(1 : A)‖ := by
    intro φ hφ
    rw [Set.mem_preimage, mem_closedBall_zero_iff]
    exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ :)
  exact (isCompact_closedBall 0 _).of_isClosed_subset CharacterSpace.isClosed h

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsStarNormal a
hb : IsStarNormal b
hcomm : Commute a b
hab : a * b = 0
S : NonUnitalStarSubalgebra ℂ A := (adjoin ℂ {a, b}).topologicalClosure
hS : IsClosed ↑S
hcomm₁ : Commute (star a) b
hcomm₂ : Commute a (star b)
⊢ ∀ x ∈ {a, b}, ∀ y ∈ {a, b}, x * y = y * x

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
j : ι
s : Finset ι
hj : j ∉ s
ih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
h : ∀ i ∈ insert j s, IsSelfAdjoint (f i)
i : ι
hi : i ∈ s
⊢ j ≠ i

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg.inl
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
a : A
ha : ¬IsSelfAdjoint a
⊢ (0 • a)⁺ = 0 • a⁺

pp: case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺

Tactic:
· simp

Goal state after:
case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
⊢ σₙ ℝ a ⊆ s ∧ σₙ ℝ b ⊆ s ∧ σₙ ℝ (-c) ⊆ s

Tactic:
by grind

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
StarOrderedRing.nonneg_iff

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : CStarRing A
inst✝⁵ : NormedSpace ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a b : A
ha : 0 ≤ a
| ‖star (sqrt a * star b)‖ ^ 2

Tactic:
norm_star,

Goal state after:
A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : CStarRing A
inst✝⁵ : NormedSpace ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a b : A
ha : 0 ≤ a
| ‖sqrt a * star b‖ ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
x✝ : PartialOrder (Unitization ℂ A) := ⋯
this : StarOrderedRing (Unitization ℂ A)
⊢ 0 ≤ ↑a

Tactic:
StarOrderedRing.nonneg_iff

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
x✝ : PartialOrder (Unitization ℂ A) := spectralOrder (Unitization ℂ A)
this : StarOrderedRing (Unitization ℂ A)
⊢ ↑a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case inl
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f ⊥ (spectrum R a)
hF : ∀ᶠ (x : X) in ⊥, ContinuousOn (F x) (spectrum R a)
⊢ Tendsto (fun x ↦ cfc (F x) a) ⊥ (𝓝 (cfc f a))

pp: case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

Tactic:
· simp

Goal state after:
case inr
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
y : ℝ
f : ℝ → ℝ
hf₁ : ∀ x ∈ spectrum ℝ a, 0 < f x
hf₂ : ContinuousOn f (spectrum ℝ a)
ha : IsSelfAdjoint a
⊢ ∀ x ∈ f '' spectrum ℝ a, id x ≠ 0 ∨ 0 ≤ y

Tactic:
by grind

Goal state after: