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

Goal state before:
pp: A : Type u_1
inst✝⁴ : Semiring A
inst✝³ : Algebra ℝ A
inst✝² : TopologicalSpace A
inst✝¹ : T2Space A
s : Set ℝ
inst✝ : CompactSpace ↑s
φ ψ : C(↑s, ℝ) →ₐ[ℝ] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ((toContinuousMapOnAlgHom s) X) = ψ ((toContinuousMapOnAlgHom s) X)
this : ⊤ ≤ φ.equalizer ψ
x : C(↑s, ℝ)
⊢ x ∈ ⊤

Tactic:
by trivial

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this✝ : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
this : ∃ᶠ (n : ι) in l, y n ∈ s
⊢ False

Tactic:
by
    apply this
    filter_upwards [hy₀ ht] with n hn hyn
    exact hst.notMem_of_mem_left hyn hn

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

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ : t ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : ∅ ∈ ⋂₀ t
⊢ ∃ x ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U},
    ∅ ∈ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ∧ {s | s ⊆ ⋃₀ x ∧ ∀ U ∈ x, (s ∩ U).Nonempty} ⊆ ⋂₀ t

Tactic:
exists ∅

Goal state after:
case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
t : Set (Set (Set α))
ht₁ : t.Finite
ht₂ : t ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hs : ∅ ∈ ⋂₀ t
⊢ ∅ ∈ {u | u.Finite ∧ ∀ U ∈ u, IsOpen[inst✝] U} ∧
    ∅ ∈ {s | s ⊆ ⋃₀ ∅ ∧ ∀ U ∈ ∅, (s ∩ U).Nonempty} ∧ {s | s ⊆ ⋃₀ ∅ ∧ ∀ U ∈ ∅, (s ∩ U).Nonempty} ⊆ ⋂₀ t
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : TotallyBounded s
U : SetRel α α
hU : U ∈ 𝓤 α
⊢ ∃ t, (↑t, s) ∈ hausdorffEntourage U

Tactic:
obtain ⟨t, ht₁, ht₂⟩ := hs _ (symm_le_uniformity hU)
  lift t to Finset α using ht₁
  classical
  refine ⟨{x ∈ t | ∃ y ∈ s, (x, y) ∈ U}, ?_⟩
  rw [Finset.coe_filter]
  refine ⟨fun _ h => h.2, fun x hx => ?_⟩
  obtain ⟨y, hy, hxy⟩ := Set.mem_iUnion₂.mp (ht₂ hx)
  exact ⟨y, ⟨hy, x, hx, hxy⟩, hxy⟩

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✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ ∀ s ∈ 𝓤 α, (hausdorffEntourage s).IsRefl

Tactic:
intro (U : SetRel α α)

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
U : SetRel α α
⊢ U ∈ 𝓤 α → (hausdorffEntourage U).IsRefl
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✝ : CompactSpace ↑(spectrum ℝ a)
⊢ CompactSpace ↑(spectrum ℝ≥0 a)

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

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

Goal state before:
pp: case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1

Tactic:
have nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹ := by
            simpa only [← coe_inv h, mem_ball_zero_iff, Metric.eball_coe] using! hy
          rwa [← coe_nnnorm, ← Real.lt_toNNReal_iff_coe_lt, Real.toNNReal_one, nnnorm_smul,
            ← NNReal.lt_inv_iff_mul_lt h]

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

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
⊢ ∀ i ∈ 𝓤 α, ∃ t_1, t_1.Finite ∧ ∀ x ⊆ t, ∃ i_1, ∃ (_ : i_1 ∈ t_1), x ∈ {x | (x, i_1) ∈ hausdorffEntourage i}

Tactic:
intro (U : SetRel α α)

Goal state after:
α : Type u_1
inst✝ : UniformSpace α
t : Set α
ht : TotallyBounded t
U : SetRel α α
⊢ U ∈ 𝓤 α → ∃ t_1, t_1.Finite ∧ ∀ x ⊆ t, ∃ i, ∃ (_ : i ∈ t_1), x ∈ {x | (x, i) ∈ hausdorffEntourage 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
h : Inseparable F₁ F₂
x : α
hx₁ : x ∈ F₁
⊢ ∀ {U : Set (α × α)}, U ∈ 𝓤 α → ∃ x_1 ∈ U, ∃ a, (x, a) = x_1 ∧ a ∈ ↑F₂

Tactic:
intro (U : SetRel α α)

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
U : SetRel α α
⊢ U ∈ 𝓤 α → ∃ x_1 ∈ U, ∃ a, (x, a) = x_1 ∧ a ∈ ↑F₂
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
hf : IsInducing f
⊢ TopologicalSpace.vietoris α = induced (fun x ↦ f '' x) (TopologicalSpace.vietoris β)

Tactic:
have : {U : Set α | IsOpen U} = (f ⁻¹' ·) '' {V : Set β | IsOpen V} :=
    Set.ext fun _ => hf.isOpen_iff

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
hf : IsInducing f
this : {U | IsOpen[inst✝¹] U} = (fun x ↦ f ⁻¹' x) '' {V | IsOpen[inst✝] V}
⊢ TopologicalSpace.vietoris α = induced (fun x ↦ f '' x) (TopologicalSpace.vietoris β)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℂ A
inst✝¹ : ContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝ : T2Space A
a : A
f : ℝ → ℝ
ha : IsSelfAdjoint a
⊢ cfc f a = cfc (fun x ↦ ↑(f x.re)) a

Tactic:
by
  exact ha.spectrumRestricts.cfc_eq_restrict (f := Complex.reCLM)
    Complex.isometry_ofReal.isClosedEmbedding ha ha.isStarNormal f

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

Goal state before:
pp: 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
⊢ Isometry ⇑(cfcₙHom ha)

Tactic:
cfcₙHom_eq_restrict f ha ha' haf

Goal state after:
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
⊢ Isometry ⇑(nonUnitalStarAlgHom (cfcₙHom ha') haf)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : ProperSpace 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
⊢ quasispectrum 𝕜 = spectrum 𝕜 ∘ fun x ↦ toLp 1 ↑x

Tactic:
ext1 a

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : ProperSpace 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
a : A
⊢ quasispectrum 𝕜 a = (spectrum 𝕜 ∘ fun x ↦ toLp 1 ↑x) a
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:
rw [← Unitary.val_toUnits_apply u] at hk

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 inl
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {U | IsOpen[instTopologicalSpaceProd] U}
⊢ IsOpen ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' 𝒫 U)

pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
⊢ IsOpen ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)

Tactic:
· rw [isOpen_iff_forall_mem_open]
    intro ⟨K, L⟩ (hKL : (K : Set α) ×ˢ (L : Set β) ⊆ U)
    obtain ⟨V, W, hV, hW, hK, hL, hVW⟩ := generalized_tube_lemma K.isCompact L.isCompact hU hKL
    exact ⟨_, fun ⟨K', L'⟩ ⟨hK', hL'⟩ => (Set.prod_mono hK' hL').trans hVW,
      (isOpen_subsets_of_isOpen hV).prod (isOpen_subsets_of_isOpen hW), hK, hL⟩

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
⊢ IsOpen ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) ⁻¹' (fun V ↦ {s | (s ∩ V).Nonempty}) U)
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)⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsUnit a

Tactic:
rw [← StarMemClass.coe_star, ← MulMemClass.coe_mul, ← spectrum.zero_notMem_iff ℂ, ← spec_eq,
    spectrum.zero_notMem_iff] at ha₁ ha₂

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

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ (a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a * a⋆)

A : Type u_1
inst✝ : CStarAlgebra A
S : StarSubalgebra ℂ A
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
a : ↥S
ha : IsUnit ↑a
ha₁ : 0 ∉ σ ℂ ↑(a⋆ * a)
ha₂ : 0 ∉ σ ℂ ↑(a * a⋆)
spec_eq : ∀ {x : ↥S}, IsSelfAdjoint x → σ ℂ x = σ ℂ ↑x
⊢ IsSelfAdjoint (a⋆ * a)
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 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
⊢ (absConvexHull 𝕜) s = ⋂ t, ⋂ (_ : s ⊆ t), ⋂ (_ : AbsConvex 𝕜 t), t

Tactic:
simp [absConvexHull, iInter_subtype, iInter_and]

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✝)
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 * NormedSpace.exp (y • b) +
      NormedSpace.exp (y • a) * (c * b) * NormedSpace.exp (y • b)

Tactic:
← mul_assoc _ c b,

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✝)
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 * NormedSpace.exp (y • b) +
      NormedSpace.exp (y • a) * c * b * NormedSpace.exp (y • b)
Module: Mathlib.Topology.Sets.VietorisTopology

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

Tactic:
by
  simp [connectedSpace_iff]

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' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔ r' • (f b - f a) ≤ f b - f (r' * (a - b) + b)

pp: case h
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)
⊢ b - a ≠ 0

Tactic:
le_sub_comm,

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' * (a - b) + b) ≤ r' • (f a - f b) + f b ↔ f (r' * (a - b) + b) ≤ f b - r' • (f b - f a)

case h
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)
⊢ b - a ≠ 0
Module: Mathlib.Topology.ContinuousMap.ContinuousSqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
x : 𝕜 × 𝕜
hx : x.1 ≤ x.2
⊢ x.2 = x.1 + (ofReal ∘ (fun x ↦ √x) ∘ ⇑re ∘ fun z ↦ z.2 - z.1) x * (ofReal ∘ (fun x ↦ √x) ∘ ⇑re ∘ fun z ↦ z.2 - z.1) x

Tactic:
simp only [Function.comp_apply]

Goal state after:
𝕜 : Type u_1
inst✝ : RCLike 𝕜
x : 𝕜 × 𝕜
hx : x.1 ≤ x.2
⊢ x.2 = x.1 + ↑√(re (x.2 - x.1)) * ↑√(re (x.2 - x.1))
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 : a = -a⁻
⊢ a ≤ 0

Tactic:
ha,

Goal state after:
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 : a = -a⁻
⊢ -a⁻ ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric

Goal state before:
pp: case pos
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ ‖a⁺‖ ≤ ‖a‖

pp: case neg
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ ‖a⁺‖ ≤ ‖a‖

Tactic:
case neg => simp [CFC.posPart_def, cfcₙ_apply_of_not_predicate a ha]

Goal state after:
case pos
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ ‖a⁺‖ ≤ ‖a‖
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
tfae_1_iff_9 : 0 ≤ a ↔ IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
tfae_1_iff_7 : 0 ≤ a ↔ a = a⁺
tfae_1_iff_8 : 0 ≤ a ↔ IsSelfAdjoint a ∧ a⁻ = 0
tfae_1_to_2 : 0 ≤ a → a = sqrt a * sqrt a
tfae_2_to_3 : a = sqrt a * sqrt a → ∃ b, 0 ≤ b ∧ a = b * b
tfae_3_to_4 : (∃ b, 0 ≤ b ∧ a = b * b) → ∃ b, IsSelfAdjoint b ∧ a = b * b
tfae_4_to_5 : (∃ b, IsSelfAdjoint b ∧ a = b * b) → ∃ b, a = star b * b
tfae_5_to_6 : (∃ b, a = star b * b) → ∃ b, a = b * star b
⊢ [0 ≤ a, a = sqrt a * sqrt a, ∃ b, 0 ≤ b ∧ a = b * b, ∃ b, IsSelfAdjoint b ∧ a = b * b, ∃ b, a = star b * b,
      ∃ b, a = b * star b, a = a⁺, IsSelfAdjoint a ∧ a⁻ = 0,
      IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal].TFAE

Tactic:
tfae_have 6 → 1 := fun ⟨b, hb⟩ => hb ▸ mul_star_self_nonneg _

Goal state after:
A : Type u_1
inst✝¹¹ : PartialOrder A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : TopologicalSpace A
inst✝⁸ : StarRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
tfae_1_iff_9 : 0 ≤ a ↔ IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
tfae_1_iff_7 : 0 ≤ a ↔ a = a⁺
tfae_1_iff_8 : 0 ≤ a ↔ IsSelfAdjoint a ∧ a⁻ = 0
tfae_1_to_2 : 0 ≤ a → a = sqrt a * sqrt a
tfae_2_to_3 : a = sqrt a * sqrt a → ∃ b, 0 ≤ b ∧ a = b * b
tfae_3_to_4 : (∃ b, 0 ≤ b ∧ a = b * b) → ∃ b, IsSelfAdjoint b ∧ a = b * b
tfae_4_to_5 : (∃ b, IsSelfAdjoint b ∧ a = b * b) → ∃ b, a = star b * b
tfae_5_to_6 : (∃ b, a = star b * b) → ∃ b, a = b * star b
tfae_6_to_1 : (∃ b, a = b * star b) → 0 ≤ a
⊢ [0 ≤ a, a = sqrt a * sqrt a, ∃ b, 0 ≤ b ∧ a = b * b, ∃ b, IsSelfAdjoint b ∧ a = b * b, ∃ b, a = star b * b,
      ∃ b, a = b * star b, a = a⁺, IsSelfAdjoint a ∧ a⁻ = 0,
      IsSelfAdjoint a ∧ QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal].TFAE
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
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
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

Tactic:
let s : Set X := {x | ContinuousOn (F x) (spectrum R a)}
    rw [← tendsto_comap'_iff (i := ((↑) : s → X)) (by simpa)]
    conv =>
      enter [1, x]
      rw [Function.comp_apply, cfc_apply (hf := x.2)]
    rw [cfc_apply ..]
    apply cfcHom_continuous _ |>.tendsto _ |>.comp
    rw [hf.tendsto_domRestrict_iff_tendstoUniformlyOn Subtype.property]
    intro t
    simp only [eventually_comap, Subtype.forall]
    peel h_tendsto t with ht x _
    simp_all

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
x : ℝ≥0
ha : 0 ≤ a
hx : ¬x = 0
⊢ sqrt a ^ ↑x = a ^ (↑x / 2)

Tactic:
have h₁ : 0 ≤ (x : ℝ) := NNReal.zero_le_coe

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
x : ℝ≥0
ha : 0 ≤ a
hx : ¬x = 0
h₁ : 0 ≤ ↑x
⊢ sqrt a ^ ↑x = a ^ (↑x / 2)
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
x : A
r : ℝ
hr : 0 ≤ r
⊢ x ∈ Icc 0 ((algebraMap ℝ A) r) ↔ 0 ≤ x ∧ ‖x‖ ≤ r

Tactic:
rw [mem_Icc, and_congr_right_iff, iff_comm]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
r : ℝ
hr : 0 ≤ r
⊢ 0 ≤ x → (‖x‖ ≤ r ↔ x ≤ (algebraMap ℝ A) r)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: A : Type u_2
inst✝⁹ : NormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedAlgebra ℝ A
inst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : ContinuousStar A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
this : Set.EqOn (cfc f) (cfc fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (NNReal.toReal '' s)
a : A
ha : (IsSelfAdjoint a ∧ SpectrumRestricts a ⇑ContinuousMap.realToNNReal) ∧ spectrum ℝ≥0 a ⊆ s
⊢ spectrum ℝ a ⊆ NNReal.toReal '' s

Tactic:
rw [← ha.1.2.algebraMap_image]

Goal state after:
A : Type u_2
inst✝⁹ : NormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedAlgebra ℝ A
inst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : ContinuousStar A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
this : Set.EqOn (cfc f) (cfc fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ spectrum ℝ≥0 a ⊆ s}
hf : ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (NNReal.toReal '' s)
a : A
ha : (IsSelfAdjoint a ∧ SpectrumRestricts a ⇑ContinuousMap.realToNNReal) ∧ spectrum ℝ≥0 a ⊆ s
⊢ ⇑(algebraMap ℝ≥0 ℝ) '' spectrum ℝ≥0 a ⊆ NNReal.toReal '' s