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

Goal state before:
pp: case mem
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝¹ f✝ f : C(↑s, 𝕜)
hf : f ∈ {(toContinuousMapOnAlgHom s) X} ∪ star {(toContinuousMapOnAlgHom s) X}
⊢ p f

Tactic:
push _ ∈ _ at hf
    rw [star_eq_iff_star_eq, eq_comm (b := f)] at hf
    obtain (rfl | rfl) := hf
    all_goals simpa only [toContinuousMapOnAlgHom_apply, toContinuousMapOn_X_eq_restrict_id]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
f : C(↑s, 𝕜)
hf₂ : f ∈ RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)
hf₁ : ∃ x, (∃ a, a • 1 = x) ∧ ∃ y ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}, x + y = f
⊢ f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}

Tactic:
obtain ⟨-, ⟨r, rfl⟩, f, hf, rfl⟩ := hf₁

Goal state after:
case mp
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
h0 : 0 ∈ s
r : 𝕜
f : C(↑s, 𝕜)
hf : f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
hf₂ : r • 1 + f ∈ RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)
⊢ r • 1 + f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
U : SetRel β β
hUsub : id U ⊆ V
hUmem : U ∈ uniformity β
hUopen : IsOpen[instTopologicalSpaceProd] U
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
hFN_image : F N x₀ ⊆ U.image (f x₀)
hF : ∀ (n : ι), ∀ x ∈ s, UpperHemicontinuousWithinAt (F n) s x
hFN_uhc : ∀ᶠ (x : α) in 𝓝[s] x₀, F N x ⊆ U.image (f x₀)
x : α
hFNx : F N x ⊆ U.image (f x₀)
hx_s : x ∈ s
y : β
hy : y ∈ f x
⊢ y ∈ u

Tactic:
obtain ⟨z, hzFN, hyz⟩ := ((mem_hausdorffEntourage U (f x) (F N x)).mp (hN x hx_s)).1 hy

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hf_compact : ∀ x ∈ s, IsCompact (f x)
x₀ : α
hx₀s : x₀ ∈ s
u : Set β
hu : IsOpen[inst✝.toTopologicalSpace] u
hx₀u : f x₀ ⊆ u
W : Set (β × β)
hW : W ∈ uniformity β
left✝ : IsOpen[instTopologicalSpaceProd] W
hWu : ∀ x ∈ f x₀, ball x W ⊆ u
V : Set (β × β)
hV : V ∈ uniformity β
hVsym : SetRel.IsSymm V
hVcomp : SetRel.comp V V ⊆ W
U : SetRel β β
hUsub : id U ⊆ V
hUmem : U ∈ uniformity β
hUopen : IsOpen[instTopologicalSpaceProd] U
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
hFN_image : F N x₀ ⊆ U.image (f x₀)
hF : ∀ (n : ι), ∀ x ∈ s, UpperHemicontinuousWithinAt (F n) s x
hFN_uhc : ∀ᶠ (x : α) in 𝓝[s] x₀, F N x ⊆ U.image (f x₀)
x : α
hFNx : F N x ⊆ U.image (f x₀)
hx_s : x ∈ s
y : β
hy : y ∈ f x
z : β
hzFN : z ∈ F N x
hyz : (y, z) ∈ U
⊢ y ∈ 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}
⊢ ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ U}

Tactic:
refine ⟨u, hu, ?_⟩

Goal state after:
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}
⊢ s ⊆ ⋃ y ∈ u, {x | (x, y) ∈ U}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : TotallyBounded s
⊢ ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsOpen[instTopologicalSpaceProd] i' → {a | s ⊆ i'.preimage a} ∩ {a | a ⊆ i'.image s} ∈ 𝓝 s

Tactic:
intro U ⟨hU₁, hU₂⟩

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU₁ : U ∈ 𝓤 α
hU₂ : IsOpen[instTopologicalSpaceProd] U
⊢ {a | s ⊆ U.preimage a} ∩ {a | a ⊆ U.image s} ∈ 𝓝 s
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 \ U ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
⊢ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} \ 𝒫 U ⊆ ⋃ a ∈ T, {s | (s ∩ a).Nonempty}

Tactic:
rintro t ⟨⟨htK, -⟩, htU⟩

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
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hKT : K \ U ⊆ ⋃ i ∈ T, i
t : Set α
htU : t ∉ 𝒫 U
htK : t ⊆ K
⊢ t ∈ ⋃ a ∈ T, {s | (s ∩ a).Nonempty}
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' := ⋯ }
⊢ ∀ (i' : SetRel α α),
    i' ∈ 𝓤 α ∧ IsClosed[instTopologicalSpaceProd] i' →
      (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id i') ∈ f

Tactic:
intro U ⟨hU₁, hU₂⟩

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
⊢ (fun y ↦ (y, L)) ⁻¹' Prod.map SetLike.coe SetLike.coe ⁻¹' hausdorffEntourage (id U) ∈ f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case inl
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✝ : Subsingleton A
⊢ ‖cfc f a‖ ≤ c

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

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

Goal state before:
pp: case inl
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✝ : Subsingleton A
⊢ ‖cfc f a‖ < c

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

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

Goal state before:
pp: case right.left
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ QuasispectrumRestricts (cfcₙ (fun x ↦ √x) a) ⇑ContinuousMap.realToNNReal

pp: case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a

Tactic:
· simpa only [QuasispectrumRestricts.nnreal_iff, cfcₙ_map_quasispectrum (√·) a,
      Set.mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
        using fun x _ ↦ Real.sqrt_nonneg x

Goal state after:
case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: A : Type u_4
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
a : A
ha : (spectrum ℝ a).Nonempty
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx₁ : x ∈ spectrum ℝ a
hx₂ : ↑‖x‖₊ = spectralRadius ℝ a
⊢ (spectralRadius ℝ a).toNNReal ∈ spectrum ℝ≥0 a

Tactic:
rw [← hx₂, ENNReal.toNNReal_coe, ← spectrum.algebraMap_mem_iff ℝ, NNReal.algebraMap_eq_coe]

Goal state after:
A : Type u_4
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
inst✝ : CompleteSpace A
a : A
ha : (spectrum ℝ a).Nonempty
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
hx₁ : x ∈ spectrum ℝ a
hx₂ : ↑‖x‖₊ = spectralRadius ℝ a
⊢ ↑‖x‖₊ ∈ spectrum ℝ a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsSemitopologicalRing A
inst✝² : T2Space A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ → ℝ
a : A
hf_nonneg : ∀ x ∈ σ ℝ a, 0 ≤ f x
ha : 0 ≤ a
x : ℝ
hx : x ∈ σ ℝ a
⊢ f x = ↑(f ↑x.toNNReal).toNNReal

Tactic:
x.coe_toNNReal (spectrum_nonneg_of_nonneg ha hx),

Goal state after:
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsSemitopologicalRing A
inst✝² : T2Space A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ → ℝ
a : A
hf_nonneg : ∀ x ∈ σ ℝ a, 0 ≤ f x
ha : 0 ≤ a
x : ℝ
hx : x ∈ σ ℝ a
⊢ f x = ↑(f x).toNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
⊢ dist (↑g₁ x) (↑g₂ x) ≤
    dist
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))

Tactic:
let x' : σₙ S a := Subtype.map (algebraMap R S) (fun _ ↦ quasispectrum.algebraMap_mem S) x

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ dist (↑g₁ x) (↑g₂ x) ≤
    dist
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₁.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
      ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        (g₂.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }))
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k ∈ σ 𝕜 ↑u
⊢ 1 ≤ ‖k‖

Tactic:
← Unitary.val_toUnits_apply u

Goal state after:
case refine_2
𝕜 : Type u_1
inst✝⁵ : NormedField 𝕜
E : Type u_2
inst✝⁴ : NormedRing E
inst✝³ : StarRing E
inst✝² : CStarRing E
inst✝¹ : NormedAlgebra 𝕜 E
inst✝ : CompleteSpace E
u : ↥(unitary E)
a✝ : Nontrivial E
k : 𝕜
hk : k ∈ σ 𝕜 ↑(toUnits u)
⊢ 1 ≤ ‖k‖
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
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✝ : T2Space α
K₁ K₂ : Compacts α
h : K₁ ≠ K₂
this :
  ∀ (K₁ K₂ : Compacts α),
    K₁ ≠ K₂ → ¬K₁ ≤ K₂ → (fun x y ↦ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) K₁ K₂
h' : ¬¬K₁ ≤ K₂
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ K₁ ∈ u ∧ K₂ ∈ v ∧ Disjoint u v

Tactic:
grind [Disjoint.symm, le_antisymm]

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✝ : CompactSpace α
⊢ IsCompact univ

Tactic:
simpa using isCompact_subsets_of_isCompact isCompact_univ

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
⊢ {a | (↑a).Nonempty} ∩ {a | ↑a ⊆ s} ⊆ closure ({a | (↑a).Nonempty} ∩ {a | (↑a).Finite ∧ ↑a ⊆ s})

Tactic:
Compacts.coe_nonempty,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
⊢ {a | a ≠ ⊥} ∩ {a | ↑a ⊆ s} ⊆ closure ({a | a ≠ ⊥} ∩ {a | (↑a).Finite ∧ ↑a ⊆ s})
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f g : α →C₀ β
h : f.toBCF = g.toBCF
⊢ f = g

Tactic:
ext x
  simpa only using! DFunLike.congr_fun h x

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

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ s = ∅ → (absConvexHull 𝕜) s = ∅

Tactic:
· rintro rfl
    exact absConvexHull_empty

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ (↑(nndist p₁ p₂))⁻¹ * (dist c₁ c₂ * dist p₁ p₂)

Tactic:
← dist_nndist,

Goal state after:
W : Type u_3
Q : Type u_4
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : Q
h : p₁ ≠ p₂
c₁ c₂ : 𝕜
⊢ dist c₁ c₂ ≤ (dist p₁ p₂)⁻¹ * (dist c₁ c₂ * dist p₁ p₂)
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: case ha
k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a a' b : E
r : k
ha : a < a'
hr : r < 1
⊢ 0 < 1 - r

Tactic:
exact sub_pos.2 hr

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

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∃ g', (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I ∧ ∀ x ∈ ∅, 0 < g' x

Tactic:
refine ⟨0, ?_, fun x hx => False.elim hx⟩

Goal state after:
case refine_1
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ (↑(algebraMapCLM ℝ≥0 𝕜)).comp 0 ∈ I
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
⊢ ‖a + b‖ = max ‖a‖ ‖b‖

Tactic:
have hcomm₁ := ha.commute_star_left hcomm

Goal state after:
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
⊢ ‖a + b‖ = max ‖a‖ ‖b‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsSelfAdjoint a
⊢ a⁻ = 0 ↔ 0 ≤ a

Tactic:
rw [← posPart_eq_self, eq_comm (b := a)]
  nth_rw 2 [← posPart_sub_negPart a]
  simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : γ → α →ᵤ β
K : ℝ≥0
s : Set γ
⊢ LipschitzOnWith K f s ↔ ∀ (c : α), LipschitzOnWith K (fun x ↦ toFun (f x) c) s

Tactic:
simp [lipschitzOnWith_iff_restrict, lipschitzWith_iff]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : γ → α →ᵤ β
K : ℝ≥0
s : Set γ
⊢ (∀ (c : α), LipschitzWith K fun x ↦ toFun (f ↑x) c) ↔ ∀ (c : α), LipschitzWith K (s.domRestrict fun x ↦ toFun (f x) c)
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
f : γ → α →ᵤ[𝔖] β
K : ℝ≥0
s : Set γ
⊢ LipschitzOnWith K f s ↔ ∀ c ∈ ⋃₀ 𝔖, LipschitzOnWith K (fun x ↦ (toFun 𝔖) (f x) c) s

Tactic:
simp [lipschitzOnWith_iff_restrict, lipschitzWith_iff]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
𝔖 : Set (Set α)
inst✝¹ : PseudoEMetricSpace β
inst✝ : Finite ↑𝔖
f : γ → α →ᵤ[𝔖] β
K : ℝ≥0
s : Set γ
⊢ (∀ (c : α), ∀ x ∈ 𝔖, c ∈ x → LipschitzWith K fun x ↦ (toFun 𝔖) (f ↑x) c) ↔
    ∀ (c : α), ∀ x ∈ 𝔖, c ∈ x → LipschitzWith K (s.domRestrict fun x ↦ (toFun 𝔖) (f x) c)
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

Tactic:
have : BddAbove (.range fun x : ⋃₀ 𝔖 ↦ dist (toFun 𝔖 f x) (toFun 𝔖 g x)) := by
        use (Metric.ediam (.univ : Set β)).toReal
        simp +contextual [mem_upperBounds, eq_comm (a := dist _ _), ← edist_dist,
          ← ENNReal.ofReal_le_iff_le_toReal BoundedSpace.bounded_univ.ediam_ne_top,
          Metric.edist_le_ediam_of_mem]

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
this : BddAbove (Set.range fun x ↦ dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖(cfcHom ⋯) ((Homeomorph.compStarAlgEquiv' ℂ ℂ (Homeomorph.setCongr ⋯)) (toContinuousMapHom f))‖ = ‖f‖

Tactic:
rw [norm_cfcHom (a : Unitization ℂ A), StarAlgEquiv.norm_map]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
f : C(↑(σₙ ℂ a), ℂ)₀
⊢ ‖toContinuousMapHom f‖ = ‖f‖