mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ inst✝ = generateFrom ((fun a ↦ a) '' {U | IsOpen[inst✝] U})

Tactic:
image_id',

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
⊢ inst✝ = generateFrom {U | IsOpen[inst✝] U}
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ ∀ (F₁ F₂ : Closeds α), Inseparable F₁ F₂ → F₁ ≤ F₂

Tactic:
refine fun F₁ F₂ h x hx₁ => isClosed_iff_frequently.mp F₂.isClosed _ ?_

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
F₁ F₂ : Closeds α
h : Inseparable F₁ F₂
x : α
hx₁ : x ∈ F₁
⊢ ∃ᶠ (y : α) in 𝓝 x, y ∈ ↑F₂
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁷ : TopologicalSpace A
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : Module ℂ A
inst✝³ : IsScalarTower ℂ A A
inst✝² : SMulCommClass ℂ A A
inst✝¹ : T2Space A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
f : ℂ → ℂ
a : A
hf_real : ∀ x ∈ σₙ ℂ a, star (f x) = f x
ha : IsSelfAdjoint a
⊢ cfcₙ f a = cfcₙ (fun x ↦ (f ↑x).re) a

Tactic:
rw [cfcₙ_real_eq_complex ..]
  refine cfcₙ_congr fun x hx ↦ ?_
  simp_rw [RCLike.star_def, RCLike.conj_eq_iff_re, RCLike.re_eq_complex_re,
    RCLike.ofReal_eq_complex_ofReal] at hf_real
  rw [← QuasispectrumRestricts.real_iff.mp ha.quasispectrumRestricts _ hx, hf_real _ hx]

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

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
k : 𝕜
hk : k ∈ resolventSet 𝕜 a
H₁ :
  HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)
    ((algebraMap 𝕜 A) k - a)
⊢ HasFDerivAt (fun a ↦ (algebraMap 𝕜 A) k - a) (-ContinuousLinearMap.id 𝕜 A) a

Tactic:
simpa using! (hasFDerivAt_const _ a).sub (hasFDerivAt_id a)

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

Goal state before:
pp: A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
H₁ : HasFPowerSeriesOnBall (fun z ↦ (1 - z • a)⁻¹ʳ) (cauchyPowerSeries (fun z ↦ (1 - z • a)⁻¹ʳ) 0 ↑r) 0 ↑r
⊢ ↑r ≤ p.radius

Tactic:
exact ((hasFPowerSeriesOnBall_inverse_one_sub_smul ℂ a).exchange_radius H₁).r_le

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

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
⊢ False

Tactic:
have H₁ : Differentiable ℂ fun z : ℂ => resolvent a z := fun z =>
    hasDerivAt_resolvent_const_left (H₀.symm ▸ Set.mem_univ z : z ∈ resolventSet ℂ a)
      |>.differentiableAt

Goal state after:
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z ↦ resolvent a z
⊢ False
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit ((↑a)⋆ * ↑a)
ha₂ : IsUnit (↑a * (↑a)⋆)
⊢ IsUnit a

Tactic:
have spec_eq {x : S} (hx : IsSelfAdjoint x) : spectrum ℂ x = spectrum ℂ (x : A) :=
    Subalgebra.spectrum_eq_of_isPreconnected_compl S _ <|
      (hx.map S.subtype).isConnected_spectrum_compl.isPreconnected

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : IsUnit ((↑a)⋆ * ↑a)
ha₂ : IsUnit (↑a * (↑a)⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a
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 (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ U ⊆ U ∩ closure {K | (↑K).Finite}

Tactic:
dense_setOfPred_finite.closure_eq,

Goal state after:
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 (Compacts α)
hU :
  U ∈
    (fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}}
this : IsPreconnected (U ∩ {K | (↑K).Finite})
⊢ U ⊆ U ∩ univ
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x ↦ r • x) y ∈ balancedHull ℝ s
this✝ : 0 ≤ 1 + r
this : 0 ≤ 1 - r
⊢ ((1 + r) / 2) • y + ((1 - r) / 2) • -y = (fun x ↦ r • x) y

Tactic:
rw [smul_neg, ← sub_eq_add_neg, ← sub_smul]

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
r : ℝ
hr : ‖r‖ ≤ 1
y : E
hy : y ∈ s
ha : (fun x ↦ r • x) y ∈ balancedHull ℝ s
this✝ : 0 ≤ 1 + r
this : 0 ≤ 1 - r
⊢ ((1 + r) / 2 - (1 - r) / 2) • y = (fun x ↦ r • x) y
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
⊢ (convexHull ℝ) (s ∪ -s) = (convexHull ℝ) (balancedHull ℝ s)

Tactic:
exact le_antisymm (convexHull_mono (union_subset (subset_balancedHull ℝ)
    (fun _ _ => by rw [mem_balancedHull_iff]; use -1; simp_all)))
    (by
      rw [← Convex.convexHull_eq (convex_convexHull ℝ (s ∪ -s))]
      exact convexHull_mono balancedHull_subset_convexHull_union_neg)

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

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
⊢ φ ∈ Submodule.span 𝕜 (Set.range f) ↔ Continuous[⨅ i, induced (⇑(f i)) inferInstance, inferInstance] ⇑φ

Tactic:
suffices
      Continuous[t₁, t𝕜] φ ↔ ∃ s : Finset ι, Continuous[t₂ s, t𝕜] φ by
    simp_rw [this, ← mem_span_iff_continuous_of_finite, Submodule.span_range_eq_iSup,
      iSup_subtype]
    rw [Submodule.mem_iSup_iff_exists_finset]

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
⊢ Continuous[t₁, t𝕜] ⇑φ ↔ ∃ s, Continuous[t₂ s, t𝕜] ⇑φ
Module: Mathlib.Analysis.Normed.Module.RCLike.Extend

Goal state before:
pp: 𝕜✝ : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁷ : RCLike 𝕜✝
𝕜 : Type u_4
F : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
⊢ Continuous (↑extendRCLikeₗ).toFun

Tactic:
(ContinuousLinearMap.isEmbedding_restrictScalars ℝ).continuous_iff

Goal state after:
𝕜✝ : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁷ : RCLike 𝕜✝
𝕜 : Type u_4
F : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : ContinuousSMul 𝕜 F
inst✝¹ : Module ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
⊢ Continuous (ContinuousLinearMap.restrictScalars ℝ ∘ (↑extendRCLikeₗ).toFun)
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
⊢ ∃ gs,
    (∀ (n : ℕ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (gs n)) ∧
      ∀ ⦃x y : WeakDual 𝕜 E⦄, x ≠ y → ∃ n, gs n x ≠ gs n y

Tactic:
use (fun n φ ↦ φ (denseSeq E n))
  constructor
  · exact fun _ ↦ eval_continuous _
  · intro w y w_ne_y
    contrapose! w_ne_y
    exact DFunLike.ext'_iff.mpr <| (map_continuous w).ext_on
      (denseRange_denseSeq E) (map_continuous y) (Set.eqOn_range.mpr (funext w_ne_y))

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

Goal state before:
pp: 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
hε : ε > 0
⊢ ∃ b ∈ ↑I, dist f b < ε

Tactic:
lift ε to ℝ≥0 using hε.lt.le

Goal state after:
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 : ↑ε > 0
⊢ ∃ b ∈ ↑I, dist f b < ↑ε
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: case insert
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 ∈ insert j s, f i‖₊ = (insert j s).sup fun x ↦ ‖f x‖₊

Tactic:
suffices f j * ∑ i ∈ s, f i = 0 by simp_all [nnnorm_add_eq_max this]

Goal state after:
case insert
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‖₊
⊢ f j * ∑ i ∈ s, f i = 0
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
⊢ a⁺ = b ∧ a⁻ = c

Tactic:
and_iff_left_of_imp ?of_b_eq

Goal state after:
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
⊢ a⁺ = b

case of_b_eq
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
⊢ a⁺ = b → a⁻ = c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ Isometry ⇑(cfcHom ha)

Tactic:
rw [cfcHom_eq_of_isStarNormal]
    exact isometry_subtype_coe.comp <| StarAlgEquiv.isometry (continuousFunctionalCalculus a)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.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 : NonnegSpectrumClass ℝ A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ ‖sqrt a‖₊ = NNReal.sqrt ‖a‖₊

Tactic:
by
  rw [sqrt_eq_nnrpow, NNReal.sqrt_eq_rpow]
  exact nnnorm_nnrpow a (by simp) ha

Goal state after:
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≥0
hy : y ≠ 0
ha : 0 ≤ a
⊢ IsUnit (a ^ ↑y) ↔ IsUnit a

Tactic:
refine isUnit_rpow_iff a y ?_ ha

Goal state after:
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≥0
hy : y ≠ 0
ha : 0 ≤ a
⊢ ↑y ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : CompleteSpace A
s : Set 𝕜
x✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
f : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : p (f, a).2
has : s ∈ 𝓝ˢ (spectrum 𝕜 (f, a).2)
hs : IsCompact (spectrum 𝕜 a)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum 𝕜 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)}
fa : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
| cfc ((toFun {s}) fa.1)

Tactic:
equals cfc (toFun {k} (ofFun {k} (toFun {t | IsCompact t ∧ t ⊆ s} fa.1))) => rfl

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

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : CompleteSpace A
s : Set 𝕜
x✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
f : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : p (f, a).2
has : s ∈ 𝓝ˢ (spectrum 𝕜 (f, a).2)
hs : IsCompact (spectrum 𝕜 a)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum 𝕜 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)}
⊢ Set.MapsTo (fun fa ↦ fa)
    ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)} ∩
      Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)})
    ({f | ContinuousOn ((toFun {k}) f) k} ×ˢ {a | p a ∧ spectrum 𝕜 a ⊆ k})

Tactic:
intro (f, a)

Goal state after:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : ContinuousStar A
inst✝ : CompleteSpace A
s : Set 𝕜
x✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
f✝ : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a✝ : A
x✝ : (f✝, a✝) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)}
hf : (f✝, a✝).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : p (f✝, a✝).2
has : s ∈ 𝓝ˢ (spectrum 𝕜 (f✝, a✝).2)
hs : IsCompact (spectrum 𝕜 a✝)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum 𝕜 a✝)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)}
f : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a : A
⊢ (f, a) ∈
      {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (spectrum 𝕜 a)} ∩
        Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum 𝕜 x)} →
    (fun fa ↦ fa) (f, a) ∈ {f | ContinuousOn ((toFun {k}) f) k} ×ˢ {a | p a ∧ spectrum 𝕜 a ⊆ k}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
⊢ a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y

Tactic:
hab

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ a • f x + b • f y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
⊢ a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)

Tactic:
hab

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x + b • f y ≤ f (a • x + b • y)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
fa : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
| cfc ((toFun {s}) fa.1)

Tactic:
equals cfc (toFun {k} (ofFun {k} (toFun {t | IsCompact t ∧ t ⊆ s} fa.1))) => rfl

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
apos : IsStrictlyPositive a
b : A
bpos : IsStrictlyPositive b
⊢ a ≤ b → Ring.inverse b ≤ Ring.inverse a

Tactic:
hab

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
apos : IsStrictlyPositive a
b : A
bpos : IsStrictlyPositive b
hab : a ≤ b
⊢ Ring.inverse b ≤ Ring.inverse a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
x✝ : (f, a) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f, a).2)
hs : IsCompact (spectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
⊢ Set.MapsTo (fun fa ↦ fa)
    ({f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩
      Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)})
    ({f | ContinuousOn ((toFun {k}) f) k} ×ˢ {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ k})

Tactic:
intro (f, a)

Goal state after:
A : Type u_2
inst✝¹⁰ : NormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : NormedAlgebra ℝ A
inst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : ContinuousStar A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f✝ : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a✝ : A
x✝ : (f✝, a✝) ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)}
hf : (f✝, a✝).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s}
ha : 0 ≤ (f✝, a✝).2
has : s ∈ 𝓝ˢ (spectrum ℝ≥0 (f✝, a✝).2)
hs : IsCompact (spectrum ℝ≥0 a✝)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (spectrum ℝ≥0 a✝)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)}
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
⊢ (f, a) ∈
      {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (spectrum ℝ≥0 a)} ∩
        Set.univ ×ˢ {x | k ∈ 𝓝ˢ (spectrum ℝ≥0 x)} →
    (fun fa ↦ fa) (f, a) ∈ {f | ContinuousOn ((toFun {k}) f) k} ×ˢ {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ k}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ b✝ : A
ha✝ : 0 ≤ a✝
hab : a✝ ≤ b✝
a b : Unitization ℂ A
ha : 0 ≤ a
⊢ a ≤ b → ‖a‖ ≤ ‖b‖

Tactic:
hab

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ b✝ : A
ha✝ : 0 ≤ a✝
hab✝ : a✝ ≤ b✝
a b : Unitization ℂ A
ha : 0 ≤ a
hab : a ≤ b
⊢ ‖a‖ ≤ ‖b‖
Module: Mathlib.Analysis.CStarAlgebra.Module.Defs

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ 0 ≤ ⟪x, y⟫_A * ⟪y, x⟫_A

Tactic:
rw [← star_inner x]

Goal state after:
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul A E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
⊢ 0 ≤ ⟪x, y⟫_A * star ⟪x, y⟫_A