mathlib-initiative/mathlib-tactics huggingface dataset:
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
K : Set (WeakDual 𝕜 E)
K_cpt : IsCompact K
this : CompactSpace ↑K
⊢ MetrizableSpace ↑K

Tactic:
obtain ⟨gs, gs_cont, gs_sep⟩ := exists_countable_separating 𝕜 E

Goal state after:
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
K : Set (WeakDual 𝕜 E)
K_cpt : IsCompact K
this : CompactSpace ↑K
gs : ℕ → WeakDual 𝕜 E → 𝕜
gs_cont : ∀ (n : ℕ), Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (gs n)
gs_sep : ∀ ⦃x y : WeakDual 𝕜 E⦄, x ≠ y → ∃ n, gs n x ≠ gs n y
⊢ MetrizableSpace ↑K
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
s : Set X
f : C(X, R)
⊢ f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0

Tactic:
mem_idealOfSet

Goal state after:
X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
s : Set X
f : C(X, R)
⊢ (¬∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0) ↔ ∃ x ∈ sᶜ, f x ≠ 0
Module: Mathlib.Topology.ContinuousMap.Ideals

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
f : C(X, R)
⊢ f ∈ idealOfSet R ∅ ↔ f ∈ ⊥

Tactic:
by
    simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
      DFunLike.ext_iff, zero_apply]

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

Goal state before:
pp: case refine_3.refine_2
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
s₁ s₂ : Set X
g₁ : C(X, ℝ≥0)
hI₁ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ ∈ I
hgt₁ : ∀ x ∈ s₁, 0 < g₁ x
g₂ : C(X, ℝ≥0)
hI₂ : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂ ∈ I
hgt₂ : ∀ x ∈ s₂, 0 < g₂ x
x : X
hx : x ∈ s₁ ∪ s₂
⊢ 0 < (g₁ + g₂) x

Tactic:
· rcases hx with (hx | hx)
        · simpa using add_lt_add_of_lt_of_le (hgt₁ x hx) zero_le
        · simpa using add_lt_add_of_le_of_lt zero_le (hgt₂ x hx)

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
s : Set X
x : X
hx : x ∈ (closure sᶜ)ᶜ
⊢ ∃ f ∈ idealOfSet 𝕜 s, f x ≠ 0

Tactic:
mem_idealOfSet

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
s : Set X
x : X
hx : x ∈ (closure sᶜ)ᶜ
⊢ ∃ f, (∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0) ∧ f x ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ cfcₙ (fun x ↦ x⁺) a * a⁻ = 0

Tactic:
negPart_def

Goal state after:
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ a⁻ * cfcₙ (fun x ↦ x⁺) a = 0

Tactic:
negPart_def

Goal state after:
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a - a⁻ = a

Tactic:
negPart_def

Goal state after:
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a - cfcₙ (fun x ↦ x⁻) a = a
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 : -(star c * c) = a⁻ ^ 3
⊢ ∀ x ∈ spectrum ℝ (cfc (fun x ↦ x⁻) a ^ 3), 0 ≤ x

Tactic:
← cfc_pow (ha := ha) ..,

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 : -(star c * c) = a⁻ ^ 3
⊢ ∀ x ∈ spectrum ℝ (cfc (fun x ↦ x⁻ ^ 3) a), 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h₂
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
r : ℝ
hr : 0 < r
⊢ Metric.closedBall 0 (r / 2) ⊆ Metric.ball 0 r

Tactic:
exact Metric.closedBall_subset_ball <| half_lt_self hr

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
ha'' : 0 ∉ spectrum ℝ≥0 a
hb'' : 0 ∉ spectrum ℝ≥0 b
⊢ ContinuousOn (fun x_1 ↦ x_1 ^ x) (spectrum ℝ≥0 a ∪ spectrum ℝ≥0 b)

Tactic:
cfc_cont_tac

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)}
⊢ ContinuousOn (fun fa ↦ cfc ((toFun {k}) ((ofFun {k}) ((toFun {t | IsCompact t ∧ t ⊆ s}) fa.1))) fa.2)
    ({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)})

Tactic:
refine continuousOn_cfc_setProd hk |>.comp'
    (uniformContinuous_ofFun_toFun_of_mem 𝕜 {t | IsCompact t ∧ t ⊆ s} _ ⟨hk, hks⟩ |>.prodMap
      uniformContinuous_id).continuous.continuousOn ?_

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)}
⊢ 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})
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: case neg.inl
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
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
H : ¬0 ≤ a
⊢ sqrt a⁻¹ʳ = (sqrt a)⁻¹ʳ

pp: case neg.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
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
H : ¬IsUnit a
⊢ sqrt a⁻¹ʳ = (sqrt a)⁻¹ʳ

Tactic:
· rw [sqrt_of_not_nonneg H, inverse_zero]
      by_cases hunit : IsUnit a
      · have h₂ : ¬0 ≤ inverse a := by grind [CFC.ringInverse_nonneg_iff_nonneg_of_isUnit]
        rw [sqrt_of_not_nonneg h₂]
      · simp [inverse_non_unit _ hunit]

Goal state after:
case neg.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
ha : ¬IsStrictlyPositive a
ha' : ¬IsUnit (sqrt a)
H : ¬IsUnit a
⊢ sqrt a⁻¹ʳ = (sqrt a)⁻¹ʳ
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
⊢ ‖x + y‖ ^ 2 ≤ (‖x‖ + ‖y‖) ^ 2

Tactic:
calc _ ≤ ‖⟪x, x⟫ + ⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ := by
          simp only [norm_eq_sqrt_norm_inner_self (A := A), inner_add_right, inner_add_left,
            ← add_assoc, norm_nonneg, Real.sq_sqrt]
          exact norm_add₃_le
      _ ≤ ‖⟪x, x⟫‖ + ‖⟪y, x⟫‖ + ‖⟪x, y⟫‖ + ‖⟪y, y⟫‖ := by gcongr; exact norm_add_le _ _
      _ ≤ ‖⟪x, x⟫‖ + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖⟪y, y⟫‖ := by gcongr <;> exact norm_inner_le E
      _ = ‖x‖ ^ 2 + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖y‖ ^ 2 := by
          simp [norm_eq_sqrt_norm_inner_self (A := A)]
      _ = (‖x‖ + ‖y‖) ^ 2 := by simp only [add_pow_two, add_left_inj]; ring

Goal state after:
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
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
⊢ ‖v‖ = sSup {x | ∃ w, ∃ (_ : ‖w‖ ≤ 1), ‖⟪w, v⟫_A‖ = x}

Tactic:
refine Eq.symm <| IsGreatest.csSup_eq ⟨⟨‖v‖⁻¹ • v, ?_, ?_⟩, ?_⟩

Goal state after:
case refine_1
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
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
⊢ ‖‖v‖⁻¹ •> v‖ ≤ 1

case refine_2
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
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
⊢ ‖⟪‖v‖⁻¹ •> v, v⟫_A‖ = ‖v‖

case refine_3
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
v : E
instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯
instNS : NormedSpace ℂ E := NormedSpace.ofCore ⋯
⊢ ‖v‖ ∈ upperBounds {x | ∃ w, ∃ (_ : ‖w‖ ≤ 1), ‖⟪w, v⟫_A‖ = 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✝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
g : C(↑s, 𝕜)₀
g_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
hg : ‖f - g‖ ≤ ε
x : X
⊢ ‖(f - g).comp { toFun := Subtype.map id ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }‖ ≤ ε

Tactic:
rw [ContinuousMapZero.norm_def, ContinuousMap.norm_le _ hε.le] at hg ⊢

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
g : C(↑s, 𝕜)₀
g_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
hg : ∀ (x : ↑s), ‖↑(f - g) x‖ ≤ ε
x : X
⊢ ∀ (x_1 : ↑(quasispectrum 𝕜 (a x))),
    ‖↑((f - g).comp { toFun := Subtype.map id ⋯, continuous_toFun := ⋯, map_zero' := ⋯ }) x_1‖ ≤ ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 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)
f : C(↑s, 𝕜)₀
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)
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) f

Tactic:
by
  have : CompactSpace s := by rwa [isCompact_iff_compactSpace] at hs
  induction f using ContinuousMapZero.induction_on_of_compact with
  | zero => simpa [map_zero] using! continuous_const
  | id => simpa only [cfcₙHomSuperset_id]
  | star_id => simp only [map_star, cfcₙHomSuperset_id]; fun_prop
  | add f g hf hg => simpa only [map_add] using! hf.add hg
  | mul f g hf hg => simpa only [map_mul] using! hf.mul hg
  | smul r f hf => simpa only [map_smul] using! hf.const_smul r
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_ofPred_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcₙHomSuperset_apply]
    rw [isometry_cfcₙHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcₙHom (ha' x)))]
    rw [ContinuousMapZero.norm_def, ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop

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

Goal state before:
pp: A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √‖M * star M‖ * √(‖v‖ ^ 2)

Tactic:
← Real.sqrt_mul

Goal state after:
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ √‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ √(‖M * star M‖ * ‖v‖ ^ 2)

case hx
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2
h₂ : ‖v‖ = √(‖v‖ ^ 2)
⊢ 0 ≤ ‖M * star M‖
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
ℓ : StrongDual 𝕜 E
Y : Submodule 𝕜 E := (↑ℓ).ker
z : E
hz : z ∈ Yᗮ
z_ne_0 : z ≠ 0
x : E
h₁ : ℓ z • x - ℓ x • z ∈ Y
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [(Y.mem_orthogonal' z).mp hz]
                exact h₁

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Dual

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ⟪(toDual 𝕜 E).symm y, x⟫ = y x

Tactic:
rw [← toDual_apply_apply]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
x : E
y : StrongDual 𝕜 E
⊢ ((toDual 𝕜 E) ((toDual 𝕜 E).symm y)) x = y x
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
F : Set (Set E)
hF : F.Nonempty
hA : ∀ B ∈ F, IsExtreme 𝕜 A B
⊢ IsExtreme 𝕜 A (⋂ B ∈ F, B)

Tactic:
by
  have := hF.to_subtype
  simpa only [iInter_subtype] using isExtreme_iInter fun i : F ↦ hA _ i.2

Goal state after:
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: case refine_1.a
𝕜 : Type u_1
ι : Type u_4
M : ι → Type u_5
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : (i : ι) → Set (M i)
x : (i : ι) → M i
hx : ∀ (i : ι), x i ∈ s i
h :
  ∀ (x₁ : (i : ι) → M i),
    (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → M i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x
i : ι
x₁ : M i
hx₁ : x₁ ∈ s i
x₂ : M i
hx₂ : x₂ ∈ s i
hi : x i ∈ openSegment 𝕜 x₁ x₂
⊢ x ∈ openSegment 𝕜 (update x i x₁) (update x i x₂)

Tactic:
← Pi.image_update_openSegment

Goal state after:
case refine_1.a
𝕜 : Type u_1
ι : Type u_4
M : ι → Type u_5
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : (i : ι) → AddCommGroup (M i)
inst✝ : (i : ι) → Module 𝕜 (M i)
s : (i : ι) → Set (M i)
x : (i : ι) → M i
hx : ∀ (i : ι), x i ∈ s i
h :
  ∀ (x₁ : (i : ι) → M i),
    (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → M i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x
i : ι
x₁ : M i
hx₁ : x₁ ∈ s i
x₂ : M i
hx₂ : x₂ ∈ s i
hi : x i ∈ openSegment 𝕜 x₁ x₂
⊢ x ∈ update x i '' openSegment 𝕜 x₁ x₂
Module: Mathlib.Analysis.CStarAlgebra.Projection

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p q : A
hp : IsStarProjection p
hq : IsStarProjection q
tfae_1_to_2 : p ≤ q → q * p = p
⊢ [p ≤ q, q * p = p, p * q = p, IsStarProjection (q - p), IsIdempotentElem (q - p)].TFAE

Tactic:
tfae_have 2 → 3 := fun h ↦ by
    simpa [hp.isSelfAdjoint.star_eq, hq.isSelfAdjoint.star_eq] using congr(star $h)

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p q : A
hp : IsStarProjection p
hq : IsStarProjection q
tfae_1_to_2 : p ≤ q → q * p = p
tfae_2_to_3 : q * p = p → p * q = p
⊢ [p ≤ q, q * p = p, p * q = p, IsStarProjection (q - p), IsIdempotentElem (q - p)].TFAE
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: x y : Circle
z : ℂ
hz : z ≠ 0
⊢ ‖(starRingEnd ℂ) z / z‖ = 1

Tactic:
by
    rw [norm_div, RCLike.norm_conj, div_self]; exact norm_ne_zero_iff.mpr hz

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i✝ : ι
b : B
a : F
i : ι
⊢ ContinuousOn (Z.localTrivAsPartialEquiv i).invFun (Z.localTrivAsPartialEquiv i).target

Tactic:
by
    refine continuousOn_isOpen_of_generateFrom fun t ht ↦ ?_
    simp only [exists_prop, mem_iUnion, mem_singleton_iff] at ht
    obtain ⟨j, s, s_open, ts⟩ : ∃ j s, IsOpen s ∧
      t = (localTrivAsPartialEquiv Z j).source ∩ localTrivAsPartialEquiv Z j ⁻¹' s := ht
    rw [ts]
    simp only [preimage_inter]
    let e := Z.localTrivAsPartialEquiv i
    let e' := Z.localTrivAsPartialEquiv j
    let f := e.symm.trans e'
    have : IsOpen (f.source ∩ f ⁻¹' s) := by
      rw [PartialEquiv.EqOnSource.source_inter_preimage_eq (Z.localTrivAsPartialEquiv_trans i j)]
      exact (continuousOn_open_iff (Z.trivChange i j).open_source).1
        (Z.trivChange i j).continuousOn _ s_open
    convert! this using 1
    dsimp [f, PartialEquiv.trans_source]
    rw [← preimage_comp, inter_assoc]

Goal state after: