mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁷ : NontriviallyNormedField 𝕜₁
inst✝¹⁶ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : Module 𝕜₂ F
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformSpace F
inst✝⁸ : IsUniformAddGroup E
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : ContinuousSMul 𝕜₁ E
inst✝⁵ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝⁴ : PolynormableSpace 𝕜₂ F
inst✝³ : ContinuousSMul 𝕜₂ F
inst✝² : T2Space F
l : Filter α
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
g : α → E →SL[σ₁₂] F
f : E → F
h : Tendsto (fun n x ↦ (g n) x) l (𝓝 f)
u : ℕ → α
hu : Tendsto u atTop l
x : E
⊢ IsVonNBounded 𝕜₂ (range fun i ↦ (g (u i)) x)

Tactic:
tendsto_pi_nhds

Goal state after:
α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁷ : NontriviallyNormedField 𝕜₁
inst✝¹⁶ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁵ : RingHomIsometric σ₁₂
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : Module 𝕜₂ F
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformSpace F
inst✝⁸ : IsUniformAddGroup E
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : ContinuousSMul 𝕜₁ E
inst✝⁵ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
inst✝⁴ : PolynormableSpace 𝕜₂ F
inst✝³ : ContinuousSMul 𝕜₂ F
inst✝² : T2Space F
l : Filter α
inst✝¹ : l.IsCountablyGenerated
inst✝ : l.NeBot
g : α → E →SL[σ₁₂] F
f : E → F
h✝ : Tendsto (fun n x ↦ (g n) x) l (𝓝 f)
h : ∀ (x : E), Tendsto (fun i ↦ (g i) x) l (𝓝 (f x))
__LinearMap✝ : E →ₛₗ[σ₁₂] F := linearMapOfTendsto f (fun a ↦ ↑(g a)) h✝
u : ℕ → α
hu : Tendsto u atTop l
x : E
⊢ IsVonNBounded 𝕜₂ (range fun i ↦ (g (u i)) x)
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : PseudoMetricSpace P
inst✝² : NormedAddTorsor V P
𝕜 : Type u_5
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 V
p₁ p₂ : P
c : 𝕜
⊢ dist ((homothety p₁ c) p₂) p₂ = ‖1 - c‖ * dist p₁ p₂

Tactic:
rw [homothety_eq_lineMap, dist_lineMap_right]

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

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁸ : Ring k
inst✝⁷ : PartialOrder k
inst✝⁶ : IsOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
a b : E
r r' : k
inst✝ : PosSMulReflectLT k E
h : r < r'
⊢ (1 - r) • a + r • b < (1 - r') • a + r' • b ↔ a < b

Tactic:
← lt_sub_iff_add_lt,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁸ : Ring k
inst✝⁷ : PartialOrder k
inst✝⁶ : IsOrderedRing k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
a b : E
r r' : k
inst✝ : PosSMulReflectLT k E
h : r < r'
⊢ (1 - r) • a < (1 - r') • a + r' • b - r • b ↔ a < b
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≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
this : ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t
⊢ ∃ b ∈ ↑I, nndist f b < ε

Tactic:
obtain ⟨g, hgI, hg, hgt⟩ := this
    refine ⟨f * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, ?_⟩
    rw [nndist_eq_nnnorm]
    refine (nnnorm_lt_iff _ hε).2 fun x => ?_
    simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply]
    by_cases hx : x ∈ t
    · simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapCLM_apply,
        map_one, mul_one, sub_self, nnnorm_zero] using! hε
    · refine lt_of_le_of_lt ?_ (half_lt_self hε)
      have :=
        calc
          ‖((1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
              ‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
            simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
              Pi.one_apply, Function.comp_apply, algebraMapCLM_apply]
          _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
            simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, NNReal.coe_sub (hg x),
              NNReal.coe_one, sub_smul, one_smul]
          _ ≤ 1 := (nnnorm_algebraMap_nnreal 𝕜 (1 - g x)).trans_le tsub_le_self
      calc
        ‖f x - f x * (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
            ‖f x * (1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
          simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
        _ ≤ ε / 2 * ‖(1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
          ((nnnorm_mul_le _ _).trans
            (mul_le_mul_left (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
        _ ≤ ε / 2 := by simpa only [mul_one] using! mul_le_mul_right this _

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∉ (interior k)ᶜ
⊢ x ∈ k

Tactic:
simp only [mem_compl_iff, not_not] at hx

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
x : X
hx : x ∈ interior k
⊢ x ∈ k
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
s : Opens X
⊢ (idealOfSet 𝕜 ↑s).IsMaximal ↔ IsCoatom s

Tactic:
by
  rw [Ideal.isMaximal_def]
  refine (idealOpensGI X 𝕜).isCoatom_iff (fun I hI => ?_) s
  rw [← Ideal.isMaximal_def] at hI
  exact idealOfSet_ofIdeal_isClosed inferInstance

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := ⋯
c : Urysohns.CU P := ⋯
⊢ tsupport ⇑{ toFun := c.lim, continuous_toFun := ⋯ } ⊆ s ∧
    EqOn (⇑{ toFun := c.lim, continuous_toFun := ⋯ }) 1 t ∧
      ∀ (x : X), { toFun := c.lim, continuous_toFun := ⋯ } x ∈ Icc 0 1

Tactic:
simp only [ContinuousMap.coe_mk]

Goal state after:
case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ tsupport c.lim ⊆ s ∧ EqOn c.lim 1 t ∧ ∀ (x : X), c.lim x ∈ Icc 0 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Bijective (characterSpaceToSpectrum a)

Tactic:
refine ⟨fun φ ψ h => starAlgHomClass_ext ℂ ?_ ?_ ?_, ?_⟩
  · exact (map_continuous φ)
  · exact (map_continuous ψ)
  · simpa only [characterSpaceToSpectrum, Subtype.mk_eq_mk,
      ContinuousMap.coe_mk] using h
  · rintro ⟨z, hz⟩
    have hz' := (StarSubalgebra.spectrum_eq (hS := isClosed ℂ a)
      (a := ⟨a, self_mem ℂ a⟩) ▸ hz)
    rw [CharacterSpace.mem_spectrum_iff_exists] at hz'
    obtain ⟨φ, rfl⟩ := hz'
    exact ⟨φ, rfl⟩

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
f : C(↑(spectrum ℂ a), ℂ)
⊢ Commute (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) (star f))
    (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) f)

Tactic:
exact Commute.all (star f) f |>.map _

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : 0 = cfc (fun x ↦ x⁻ ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
← cfc_zero a (R := ℝ)

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : cfc 0 a = cfc (fun x ↦ x⁻ ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

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

Tactic:
rw [StarOrderedRing.nonneg_iff] at ha
    induction ha using AddSubmonoid.closure_induction with
    | mem x hx =>
      obtain ⟨b, rfl⟩ := hx
      exact spectrum_star_mul_self_nonneg
    | zero =>
      nontriviality A
      simp
    | add x y x_mem y_mem hx hy =>
      rw [← SpectrumRestricts.nnreal_iff] at hx hy ⊢
      rw [← StarOrderedRing.nonneg_iff] at x_mem y_mem
      exact hx.nnreal_add (.of_nonneg x_mem) (.of_nonneg y_mem) hy

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
x : A
hx : x ∈ {a | 0 ≤ a}
hx_pos : 0 < ‖x‖
this : (↑r * ↑‖x‖⁻¹)⁻¹ • (↑r * ↑‖x‖⁻¹) • x = x
⊢ (r * ‖x‖⁻¹) • x ∈ {x | 0 ≤ x}

Tactic:
exact smul_nonneg (by positivity) hx

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

Goal state before:
pp: A : Type u_1
inst✝²² : PartialOrder A
inst✝²¹ : Ring A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : Algebra ℝ A
inst✝¹⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁵ : NonnegSpectrumClass ℝ A
inst✝¹⁴ : IsSemitopologicalRing A
inst✝¹³ : T2Space A
B : Type u_2
inst✝¹² : PartialOrder B
inst✝¹¹ : Ring B
inst✝¹⁰ : StarRing B
inst✝⁹ : TopologicalSpace B
inst✝⁸ : StarOrderedRing B
inst✝⁷ : Algebra ℝ B
inst✝⁶ : ContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : ContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : StarOrderedRing (A × B)
inst✝¹ : NonnegSpectrumClass ℝ B
inst✝ : NonnegSpectrumClass ℝ (A × B)
a : A
b : B
x : ℝ
ha : IsUnit a
hb : IsUnit b
ha' : 0 ≤ a
hb' : 0 ≤ b
⊢ rpow (a, b) x = (a ^ x, b ^ x)

Tactic:
have ha'' : 0 ∉ spectrum ℝ≥0 a := spectrum.zero_notMem _ ha

Goal state after:
A : Type u_1
inst✝²² : PartialOrder A
inst✝²¹ : Ring A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : StarOrderedRing A
inst✝¹⁷ : Algebra ℝ A
inst✝¹⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹⁵ : NonnegSpectrumClass ℝ A
inst✝¹⁴ : IsSemitopologicalRing A
inst✝¹³ : T2Space A
B : Type u_2
inst✝¹² : PartialOrder B
inst✝¹¹ : Ring B
inst✝¹⁰ : StarRing B
inst✝⁹ : TopologicalSpace B
inst✝⁸ : StarOrderedRing B
inst✝⁷ : Algebra ℝ B
inst✝⁶ : ContinuousFunctionalCalculus ℝ B IsSelfAdjoint
inst✝⁵ : ContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint
inst✝⁴ : IsSemitopologicalRing B
inst✝³ : T2Space B
inst✝² : StarOrderedRing (A × B)
inst✝¹ : NonnegSpectrumClass ℝ B
inst✝ : NonnegSpectrumClass ℝ (A × B)
a : A
b : B
x : ℝ
ha : IsUnit a
hb : IsUnit b
ha' : 0 ≤ a
hb' : 0 ≤ b
ha'' : 0 ∉ spectrum ℝ≥0 a
⊢ rpow (a, b) x = (a ^ x, b ^ x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case frequently
X : Type u_1
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✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)
hf :
  ∃ᶠ (g : C(↑s, 𝕜)) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcHomSuperset ⋯ ⋯) g
ε : ℝ
hε : 0 < ε
⊢ ∃ F,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
      ∀ (y : X), ((cfcHomSuperset ⋯ ⋯) f, F y) ∈ {p | dist p.1 p.2 ≤ ε}

Tactic:
simp only [Set.mem_ofPred_eq, dist_eq_norm]

Goal state after:
case frequently
X : Type u_1
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✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)
hf :
  ∃ᶠ (g : C(↑s, 𝕜)) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcHomSuperset ⋯ ⋯) g
ε : ℝ
hε : 0 < ε
⊢ ∃ F,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
      ∀ (y : X), ‖(cfcHomSuperset ⋯ ⋯) f - F y‖ ≤ ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
x : ↑{a | p a ∧ spectrum 𝕜 a ⊆ s}
⊢ {a | p a ∧ spectrum 𝕜 a ⊆ s}.domRestrict (cfc f) x =
    (cfcHomSuperset ⋯ ⋯) { toFun := s.domRestrict f, continuous_toFun := ⋯ }

Tactic:
cfcHomSuperset_apply,

Goal state after:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
x : ↑{a | p a ∧ spectrum 𝕜 a ⊆ s}
⊢ {a | p a ∧ spectrum 𝕜 a ⊆ s}.domRestrict (cfc f) x =
    (cfcHom ⋯)
      ({ toFun := s.domRestrict f, continuous_toFun := ⋯ }.comp { toFun := Subtype.map id ⋯, continuous_toFun := ⋯ })
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case inr
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
htriv : 0 ≤ a
h₁ : 0 < x
⊢ sqrt (a ^ ↑x) = a ^ (↑x / 2)

Tactic:
have h₂ : (x : ℝ) / 2 = NNReal.toReal (x / 2) := by simp
      have h₃ : 0 < x / 2 := by positivity
      rw [← nnrpow_eq_rpow h₁, h₂, ← nnrpow_eq_rpow h₃, sqrt_nnrpow (A := A)]

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

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ if hx : x₀ ∈ t then S ⟨x₀, hx⟩ else ∅

pp: case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ x ∈ t, ContinuousOn f (if hx : x ∈ t then S ⟨x, hx⟩ else ∅)

Tactic:
all_goals simp +contextual only [↓reduceDIte]

Goal state after:
case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ (x₀ : X) (a_1 : x₀ ∈ t), ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ S ⟨x₀, ⋯⟩

case refine_2
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra 𝕜 A
inst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ∀ (x : X) (a : x ∈ t), ContinuousOn f (S ⟨x, ⋯⟩)
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → Module ℂ (E i)
inst✝² : (i : ι) → SMul A (E i)
inst✝¹ : (i : ι) → CStarModule A (E i)
inst✝ : StarOrderedRing A
x : C⋆ᵐᵒᵈ(A, (i : ι) → E i)
i : ι
⊢ ‖x i‖ ≤ ‖x‖

Tactic:
let _ : NormedAddCommGroup C⋆ᵐᵒᵈ(A, Π i, E i) := normedAddCommGroup A
  refine abs_le_of_sq_le_sq' ?_ (by positivity) |>.2
  rw [pi_norm_sq, norm_sq_eq A]
  refine CStarAlgebra.norm_le_norm_of_nonneg_of_le inner_self_nonneg ?_
  exact Finset.single_le_sum (fun j _ ↦ inner_self_nonneg (A := A) (x := x j)) (Finset.mem_univ i)

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

Goal state before:
pp: case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) f), continuous_toFun := ⋯,
        map_zero' := hf0 }
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) g), continuous_toFun := ⋯,
        map_zero' := hg0 } ≤
    edist f g

Tactic:
rw [← ContinuousMapZero.isometry_toContinuousMap.edist_eq,
      edist_continuousRestrict_of_singleton hf hg]

Goal state after:
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
A : Matrix m n α
f : α → β
hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊
⊢ ‖A.map f‖₊ = ‖A‖₊

Tactic:
simp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]

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

Goal state before:
pp: case frequently
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)₀
hf :
  ∃ᶠ (g : C(↑s, 𝕜)₀) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
ε : ℝ
hε : 0 < ε
⊢ ∃ F,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
      ∀ (y : X), ((cfcₙHomSuperset ⋯ ⋯) f, F y) ∈ {p | dist p.1 p.2 ≤ ε}

Tactic:
simp only [Set.mem_ofPred_eq, dist_eq_norm]

Goal state after:
case frequently
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)₀
hf :
  ∃ᶠ (g : C(↑s, 𝕜)₀) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
ε : ℝ
hε : 0 < ε
⊢ ∃ F,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
      ∀ (y : X), ‖(cfcₙHomSuperset ⋯ ⋯) f - F y‖ ≤ ε