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

Goal state before:
pp: case smul
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f✝ : C(↑s, 𝕜)
r : 𝕜
f : C(↑s, 𝕜)
hx✝ : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf : (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩) f = 0
⊢ (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩) (r • f) = 0

Tactic:
map_smul,

Goal state after:
case smul
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f✝ : C(↑s, 𝕜)
r : 𝕜
f : C(↑s, 𝕜)
hx✝ : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf : (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩) f = 0
⊢ r • (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩) f = 0
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 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
map_one' : ∀ (φ : ↑(characterSpace 𝕜 A)), φ 1 = 1
φ : ↑(characterSpace 𝕜 A)
r : 𝕜
⊢ φ (r • 1) = r

Tactic:
map_smul,

Goal state after:
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
map_one' : ∀ (φ : ↑(characterSpace 𝕜 A)), φ 1 = 1
φ : ↑(characterSpace 𝕜 A)
r : 𝕜
⊢ r • φ 1 = r
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case smul
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
r✝ : 𝕜
f✝ : C(↑s, 𝕜)₀
h : φ f✝ * a = 0
⊢ φ (r✝ • f✝) * a = 0

Tactic:
map_smul,

Goal state after:
case smul
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
r✝ : 𝕜
f✝ : C(↑s, 𝕜)₀
h : φ f✝ * a = 0
⊢ r✝ • φ f✝ * a = 0
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case smul
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : a * φ (ContinuousMapZero.id s) = 0
hmul_star_id : a * φ (star (ContinuousMapZero.id s)) = 0
hφ : Continuous ⇑φ
r✝ : 𝕜
f✝ : C(↑s, 𝕜)₀
h : a * φ f✝ = 0
⊢ a * φ (r✝ • f✝) = 0

Tactic:
map_smul,

Goal state after:
case smul
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : a * φ (ContinuousMapZero.id s) = 0
hmul_star_id : a * φ (star (ContinuousMapZero.id s)) = 0
hφ : Continuous ⇑φ
r✝ : 𝕜
f✝ : C(↑s, 𝕜)₀
h : a * φ f✝ = 0
⊢ a * r✝ • φ f✝ = 0
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
⊢ IsClosedMap f ↔ UpperHemicontinuous fun x ↦ f ⁻¹' {x}

Tactic:
by
  rw [isClosedMap_iff_kernImage, upperHemicontinuous_iff_isOpen_preimage_Iic]
  aesop

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
⊢ UniformContinuous fun x ↦ x.1 ∪ x.2

Tactic:
refine Filter.tendsto_lift'.mpr fun U hU => ?_

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∀ᶠ (a : (Set α × Set α) × Set α × Set α) in 𝓤 (Set α × Set α),
    ((fun x ↦ x.1 ∪ x.2) a.1, (fun x ↦ x.1 ∪ x.2) a.2) ∈ hausdorffEntourage U
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A✝ : Type u_2
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜 A✝
inst✝³ : CompleteSpace A✝
A : Type u_3
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : A
inst✝ : IsCompact (spectrum ℝ a)
⊢ IsCompact (spectrum ℝ≥0 a)

Tactic:
← preimage_algebraMap ℝ

Goal state after:
𝕜 : Type u_1
A✝ : Type u_2
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedRing A✝
inst✝⁴ : NormedAlgebra 𝕜 A✝
inst✝³ : CompleteSpace A✝
A : Type u_3
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : A
inst✝ : IsCompact (spectrum ℝ a)
⊢ IsCompact (⇑(algebraMap ℝ≥0 ℝ) ⁻¹' spectrum ℝ a)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: 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✝ : DiscreteUniformity α
⊢ DiscreteUniformity (Set α)

Tactic:
rw [discreteUniformity_iff_setRelId_mem_uniformity]

Goal state after:
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✝ : DiscreteUniformity α
⊢ SetRel.id ∈ 𝓤 (Set α)
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
⊢ UniformContinuous fun x ↦ f '' x

Tactic:
refine Filter.tendsto_lift'.mpr fun U hU => ?_

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
⊢ ∀ᶠ (a : Set α × Set α) in 𝓤 (Set α), ((fun x ↦ f '' x) a.1, (fun x ↦ f '' x) a.2) ∈ hausdorffEntourage U
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case e'_3
α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : T2Space α
inst✝ : CompleteSpace α
⊢ Set.range toCloseds = {s | TotallyBounded ↑s}

Tactic:
exact subset_antisymm
      (Set.range_subset_iff.mpr fun K => K.isCompact.totallyBounded)
      (fun K hK => ⟨⟨K, hK.isCompact_of_isClosed K.isClosed⟩, rfl⟩)

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} ⊆ ⋃ a ∈ T, {s | (s ∩ a).Nonempty}

Tactic:
← ofPred_exists,

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}
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 | ∃ i, ∃ (_ : i ∈ T), (x ∩ i).Nonempty}
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
this : {t | t.Nonempty ∧ t.Finite ∧ t ⊆ s} = ⋃ n, range '' univ.pi fun x ↦ s
n : ℕ+
⊢ ((fun x_1 ↦ x) ∈ univ.pi fun x ↦ s) ∧ (range fun x_1 ↦ x) = {x}

Tactic:
by simpa

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

Goal state before:
pp: 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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
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
x : α
hx : x ∈ ↑K
lx : Filter α := l ⊓ 𝓟 (UniformSpace.ball x U)
le_def : lx = l ⊓ 𝓟 (UniformSpace.ball x U)
hlx : lx.TotallyBounded
⊢ lx.NeBot

Tactic:
le_def,

Goal state after:
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 := f.lift' fun s ↦ ⋃ K ∈ s, ↑K
hl : l.TotallyBounded
L : Compacts α := { carrier := {x | ClusterPt x l}, isCompact' := ⋯ }
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
x : α
hx : x ∈ ↑K
lx : Filter α := l ⊓ 𝓟 (UniformSpace.ball x U)
le_def : lx = l ⊓ 𝓟 (UniformSpace.ball x U)
hlx : lx.TotallyBounded
⊢ (l ⊓ 𝓟 (UniformSpace.ball x U)).NeBot
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 ≤ c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c
h✝ : Nontrivial A
⊢ ‖0‖ ≤ c

Tactic:
by simpa

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
hc : 0 < c
h : ∀ x ∈ σ 𝕜 a, ‖f x‖ < c
h✝ : Nontrivial A
⊢ ‖0‖ < c

Tactic:
by simpa

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

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ S a)
⊢ dist
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        x)
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        x) ≤
    dist g₁ g₂

Tactic:
simpa [halg.dist_eq] using ContinuousMap.dist_apply_le_dist _

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

Goal state before:
pp: case refine_2.inr
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
⊢ ∃ 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}

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
hwK : ∀ 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}

Tactic:
· refine this {W ∈ w | (↑K ∩ W).Nonempty} (fun x hx => ?_) (by grind) (by grind) (by grind)
      obtain ⟨W, hWw, hxW⟩ := mem_sUnion.mp (hKV hx)
      exact mem_sUnion_of_mem hxW ⟨hWw, x, hx, hxW⟩

Goal state after:
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
hwK : ∀ 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}
Module: Mathlib.Analysis.Complex.Convex

Goal state before:
pp: r : ℝ
⊢ IsPathConnected {0}ᶜ

Tactic:
convert!
        (((convex_halfSpace_im_gt 0).isPathConnected ⟨.I, by simp⟩).union
              ((convex_halfSpace_re_gt 0).isPathConnected ⟨1, by simp⟩) ⟨1 + .I, by simp⟩).union
          (((convex_halfSpace_im_lt 0).isPathConnected ⟨-.I, by simp⟩).union
            ((convex_halfSpace_re_lt 0).isPathConnected ⟨-1, by simp⟩) ⟨-1 - .I, by simp⟩)
          ⟨1 - .I, by simp⟩ using 1
      ext x
      refine ⟨?_, by aesop⟩
      simp +contextual [Complex.ext_iff, -not_and, not_and_or, or_imp, ← ne_eq, ← lt_or_lt_iff_ne]

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

Goal state before:
pp: case inl
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h✝ : Subsingleton A
⊢ ‖cfc f a‖₊ ≤ c

Tactic:
rw [Subsingleton.elim (cfc f a) 0]
    simp

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

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
hc : 0 < c
h : ∀ x ∈ σ ℝ≥0 a, f x < c
h✝ : Nontrivial A
⊢ ‖0‖₊ < c

Tactic:
by simpa

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ ↑(f (Real.toNNReal 0)) = 0

Tactic:
by simpa

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✝ : RegularSpace α
U : Set α
hU : U ∈ {U | IsOpen[inst✝³] U}
K : Compacts α
x : α
hx₁ : x ∈ ↑K
hx₂ : x ∈ U
⊢ Disjoint {x} Uᶜ

Tactic:
by simpa

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
x✝¹ : SeparableSpace (Compacts α)
h✝ : Nonempty α
s : Set (Compacts α)
hs₁ : s.Countable
hs₂ : Dense s
U : Set α
hU : IsOpen[inst✝] U
x✝ : U.Nonempty
x : α
hx : x ∈ U
⊢ {x} ∈ {K | ↑K ⊆ U} ∩ {K | (↑K ∩ U).Nonempty}

Tactic:
by simpa

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
⊢ B.polar ↑m = ↑(⨅ x ∈ m, (B x).ker)

Tactic:
ext; simp [polar_subMulAction]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
⊢ ∃ x,
    (𝓝 0).HasAntitoneBasis x ∧
      ∀ (n : ℕ),
        IsOpen[inst✝⁵] (x n) ∧ AbsConvex 𝕜 (x n) ∧ x (n + 1) + x (n + 1) ⊆ x n ∧ closure[inst✝⁵] (x (n + 1)) ⊆ x n

Tactic:
have hv_open (n : ℕ) : IsOpen (v n) := isOpen_interior.absConvexHull 𝕜 (hu_zero n)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
⊢ ∃ x,
    (𝓝 0).HasAntitoneBasis x ∧
      ∀ (n : ℕ),
        IsOpen[inst✝⁵] (x n) ∧ AbsConvex 𝕜 (x n) ∧ x (n + 1) + x (n + 1) ⊆ x n ∧ closure[inst✝⁵] (x (n + 1)) ⊆ x n
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: case refine_1
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
hval : g x = ↑‖x‖
⊢ ‖g‖ ≤ 1

Tactic:
simp only [hg.right, norm_smul, norm_algebraMap', norm_norm]

Goal state after:
case refine_1
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
hval : g x = ↑‖x‖
⊢ ‖x‖ * ‖coord‖ ≤ 1
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module ℝ E
inst✝ : PolynormableSpace ℝ E
S : Subspace ℝ E
f : Dual ℝ ↥S
p : Seminorm ℝ E
hp_cont : Continuous[inst✝², _] ⇑p
hp : ∀ (x : ↥S), f x ≤ p ↑x
⊢ ∃ g, (∀ (x : ↥S), g ↑x = f x) ∧ ∀ (x : E), |g x| ≤ p x

Tactic:
obtain ⟨g, hg, hl⟩ := f.exists_extension_of_le_seminorm_real S hp
  exact ⟨⟨g, (PolynormableSpace.withSeminorms ℝ E).continuous_real_rng g
    ⟨{⟨p, hp_cont⟩}, 1, fun x ↦ by simpa using (le_abs_self _).trans (hl x)⟩⟩, hg, hl⟩

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.HahnBanach

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : NormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
inst✝ : PolynormableSpace 𝕜 E
S : Submodule 𝕜 E
f : Dual 𝕜 ↥S
p : Seminorm 𝕜 E
hp_cont : Continuous[inst✝², _] ⇑p
hp : ∀ (x : ↥S), ‖f x‖ ≤ p ↑x
g : Dual 𝕜 E
hg : ∀ (x : ↥S), g ↑x = f x
hle : ∀ (x : E), ‖g x‖ ≤ p x
⊢ (normSeminorm 𝕜 𝕜).comp g ≤ 1 • {⟨p, hp_cont⟩}.sup fun p ↦ ↑p

Tactic:
by simpa

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
h : 0 < r * (b - a)
⊢ f (r * (b - a) + a) ≤ r • f b - r • f a + f a ↔
    (r * (b - a))⁻¹ • f (r * (b - a) + a) ≤ (b - a)⁻¹ • f b - (b - a)⁻¹ • f a + (r * (b - a))⁻¹ • f a

Tactic:
mul_inv_rev,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
h : 0 < r * (b - a)
⊢ f (r * (b - a) + a) ≤ r • f b - r • f a + f a ↔
    ((b - a)⁻¹ * r⁻¹) • f (r * (b - a) + a) ≤ (b - a)⁻¹ • f b - (b - a)⁻¹ • f a + ((b - a)⁻¹ * r⁻¹) • f a
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: α : Type u_4
𝕜 : Type u_5
E : Type u_6
F : Type u_7
inst✝⁵ : TopologicalSpace α
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
g : α → WeakBilin B
h : ∀ (y : F), Continuous[inst✝⁵, _] fun a ↦ re ((B (g a)) y)
x : F
this :
  Continuous[inst✝⁵, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦
    ↑(re ((B (g a)) x)) - ↑(re ((B (g a)) (I • x))) * I
⊢ Continuous[inst✝⁵, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦ (B (g a)) x

Tactic:
by simpa

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
a b : A
h : a * b = 0
⊢ a * -b = 0

Tactic:
by simpa

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
⊢ a * -b = 0

Tactic:
by simpa

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

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a b : A
ha : IsSelfAdjoint a
hb : IsSelfAdjoint b
hab : a * b = 0
⊢ a * -b = 0

Tactic:
by simpa

Goal state after: