mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: case insert
A : Type u_1
inst✝ : NonUnitalCStarAlgebra 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, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
h : ∀ i ∈ insert j s, IsSelfAdjoint (f i)
⊢ ‖∑ 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 [(h j (by simp)).nnnorm_add_eq_max (by cfc_tac) this]

Goal state after:
case insert
A : Type u_1
inst✝ : NonUnitalCStarAlgebra 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, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊
h : ∀ i ∈ insert j s, IsSelfAdjoint (f i)
⊢ f j * ∑ i ∈ s, f i = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

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

Tactic:
rw [← negPart_eq_neg, eq_comm (b := -a)]

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 : IsSelfAdjoint a
⊢ a⁺ = 0 ↔ -a = a⁻
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts (a + b) ⇑ContinuousMap.realToNNReal

Tactic:
SpectrumRestricts.nnreal_iff_nnnorm (ha₁.add hb₁) (nnnorm_add_le a b),

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖(algebraMap ℝ A) ↑(‖a‖₊ + ‖b‖₊) - (a + b)‖₊ ≤ ‖a‖₊ + ‖b‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖(algebraMap ℝ A) ↑‖a‖₊ - a + ((algebraMap ℝ A) ↑‖b‖₊ - b)‖₊ ≤ ‖a‖₊ + ‖b‖₊

Tactic:
refine nnnorm_add_le _ _ |>.trans ?_

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖(algebraMap ℝ A) ↑‖a‖₊ - a‖₊ + ‖(algebraMap ℝ A) ↑‖b‖₊ - b‖₊ ≤ ‖a‖₊ + ‖b‖₊
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
x : ℝ≥0
hx : x ≠ 0
ha : 0 ≤ a
⊢ (a ^ x) ^ x⁻¹ = a

Tactic:
by
  simp [mul_inv_cancel₀ hx, nnrpow_one _ ha]

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

Goal state before:
pp: case star_id
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
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦
    (cfcHomSuperset ⋯ ⋯) (star (ContinuousMap.restrict s (ContinuousMap.id 𝕜)))

Tactic:
| star_id =>

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
⊢ sqrt a = a ^ (1 / 2)

Tactic:
by
  have : a ^ (1 / 2 : ℝ) = a ^ ((1 / 2 : ℝ≥0) : ℝ) := rfl
  rw [this, ← nnrpow_eq_rpow (by simp), sqrt_eq_nnrpow a]

Goal state after:
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✝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))
this : ∀ (x : ↑t), ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
choose S hS₁ hS₂ hS₃ using this

Goal state after:
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
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t
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
hx : x ≠ 0
ha : IsStrictlyPositive a
this : a ^ (-x) = (a ^ x)⁻¹ʳ
⊢ (a ^ x)⁻¹ʳ = a ^ (-x)

Tactic:
rw [this]

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 : Aˣ
ha : 0 ≤ ↑a
⊢ 1 ≤ ↑a⁻¹ ↔ a ≤ 1

Tactic:
by
  simpa using! CStarAlgebra.le_inv_iff (a := 1) (by simp) ha

Goal state after:
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 : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : ¬f 0 = 0
this : cfcₙ f = fun x ↦ 0
⊢ ConvexOn ℝ s (cfcₙ f)

Tactic:
rw [this]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ → ℝ
s : Set A
hf : ConvexOn ℝ (inr '' s) (cfc f)
inrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A
hf₀ : ¬f 0 = 0
this : cfcₙ f = fun x ↦ 0
⊢ ConvexOn ℝ s fun x ↦ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
A : Type u_2
inst✝¹¹ : NormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedAlgebra ℝ A
inst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁷ : ContinuousStar A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
this : ∀ (x : ↑t), ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum ℝ≥0 x' ⊆ S) ∧ S ⊆ s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t

Tactic:
choose S hS₁ hS₂ hS₃ using this

Goal state after:
X : Type u_1
A : Type u_2
inst✝¹¹ : NormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedAlgebra ℝ A
inst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁷ : ContinuousStar A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, spectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (spectrum ℝ≥0 (a i))
S : ↑t → Set ℝ≥0
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), spectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ContinuousOn (fun x ↦ cfc f (a x)) t
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)
⊢ ‖x‖ ≤ ↑(Fintype.card ι) * ‖(equiv A ((i : ι) → E i)) x‖

Tactic:
calc ‖x‖ ≤ ∑ i, ‖x i‖ := pi_norm_le_sum_norm x
      _ ≤ ∑ _, ‖⇑x‖ := Finset.sum_le_sum fun _ _ ↦ norm_le_pi_norm ..
      _ ≤ Fintype.card ι * ‖⇑x‖ := by simp

Goal state after:
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)), ‖(linearEquiv ℂ A ((i : ι) → E i)) x‖₊ ≤ 1 * ‖x‖₊

Tactic:
simpa using! norm_equiv_le_norm_pi

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

Goal state before:
pp: case star_id
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
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦
    (cfcₙHomSuperset ⋯ ⋯) (star (ContinuousMapZero.id s))

Tactic:
| star_id =>

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

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : SeminormedAddCommGroup α
r : ℝ≥0
hr : 0 < r
A : Matrix m n α
⊢ ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r

Tactic:
simp_rw [nnnorm_def, pi_nnnorm_lt_iff hr]

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

Goal state before:
pp: m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : NontriviallyNormedField α
inst✝ : NormedAlgebra ℝ α
A : Matrix m n α
N : ℝ≥0
hN : ∀ (x : n → α), ‖{ toLinearMap := A.mulVecLin, cont := ⋯ } x‖₊ ≤ N * ‖x‖₊
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
⊢ ‖x‖₊ = 1

Tactic:
simp_rw [x, Pi.nnnorm_def, norm_unitOf, Finset.sup_const Finset.univ_nonempty]

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
this : star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m = cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m
⊢ ‖star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m‖₊ ≤ ε ^ 2

Tactic:
rw [this]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
m✝¹ m✝ : A
ε : ℝ≥0
hε : 0 < ε
x✝ : A
x m : Unitization ℂ A
hm₁ : 0 ≤ m
hm₂ : ‖m‖ < 1
hx₂ : ‖x‖ ≤ 1
hx₀ : 0 ≤ x
g : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹
hx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x
hg : Continuous g
hg' : ContinuousOn (fun y ↦ (1 + ε⁻¹ ^ 2 • y)⁻¹) (σ ℝ≥0 m)
hx : x ∈ Set.Icc 0 1
hx' : x ∈ Set.Icc (cfc g (ε⁻¹ ^ 2 • m)) 1
this : star m * (1 - cfc g (ε⁻¹ ^ 2 • m)) * m = cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m
⊢ ‖cfc (fun y ↦ y * (1 + ε⁻¹ ^ 2 • y)⁻¹ * y) m‖₊ ≤ ε ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 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✝ : 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 ∧ f 0 = 0} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (quasispectrum 𝕜 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : p (f, a).2
has : s ∈ 𝓝ˢ (quasispectrum 𝕜 (f, a).2)
hs : IsCompact (quasispectrum 𝕜 a)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum 𝕜 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum 𝕜 x)}
fa : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
| cfcₙ ((toFun {s}) fa.1)

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

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

Goal state before:
pp: case star_id
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : Algebra 𝕜 A
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
⊢ Commute ((cfcHom ha) (star (ContinuousMap.restrict (spectrum 𝕜 a) (ContinuousMap.id 𝕜)))) b

Tactic:
| star_id =>

Goal state after:
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✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : ContinuousStar A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
this : ∀ (x : ↑t), ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum 𝕜 x' ⊆ S) ∧ S ⊆ s
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t

Tactic:
choose S hS₁ hS₂ hS₃ using this

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : ContinuousStar A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum 𝕜 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum 𝕜 (a i))
S : ↑t → Set 𝕜
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum 𝕜 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Goal state before:
pp: case star_id
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : IsTopologicalRing A
inst✝ : T2Space A
a b : A
ha : p a
hb₁ : Commute a b
hb₂ : Commute (star a) b
⊢ Commute ((cfcₙHom ha) (star (ContinuousMapZero.id (quasispectrum 𝕜 a)))) b

Tactic:
| star_id =>

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

Goal state before:
pp: A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
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 ∧ f 0 = 0} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)}
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : 0 ≤ (f, a).2
has : s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (f, a).2)
hs : IsCompact (quasispectrum ℝ≥0 a)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}
fa : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
| cfcₙ ((toFun {s}) fa.1)

Tactic:
equals cfcₙ (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: X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (a i))
this : ∀ (x : ↑t), ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum ℝ≥0 x' ⊆ S) ∧ S ⊆ s
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t

Tactic:
choose S hS₁ hS₂ hS₃ using this

Goal state after:
X : Type u_1
A : Type u_2
inst✝¹³ : NonUnitalNormedRing A
inst✝¹² : StarRing A
inst✝¹¹ : NormedSpace ℝ A
inst✝¹⁰ : IsScalarTower ℝ A A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : ContinuousStar A
inst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : NonnegSpectrumClass ℝ A
inst✝³ : T2Space A
inst✝² : IsSemitopologicalRing A
inst✝¹ : CompleteSpace A
inst✝ : TopologicalSpace X
s : Set ℝ≥0
f : ℝ≥0 → ℝ≥0
a : X → A
t : Set X
hs : s ∈ 𝓝ˢ (⋃ x ∈ t, quasispectrum ℝ≥0 (a x))
ha_cont : ContinuousOn a t
ha' : ∀ x ∈ t, 0 ≤ a x
hf : ContinuousOn f s
hf0 : f 0 = 0
hs' : ∀ i ∈ t, s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (a i))
S : ↑t → Set ℝ≥0
hS₁ : ∀ (x : ↑t), IsCompact (S x)
hS₂ : ∀ (x : ↑t), ∀ᶠ (x' : A) in 𝓝 (a ↑x), quasispectrum ℝ≥0 x' ⊆ S x
hS₃ : ∀ (x : ↑t), S x ⊆ s
⊢ ContinuousOn (fun x ↦ cfcₙ f (a x)) t
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z

Tactic:
rw [integral_apply hf₂]

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), (mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) z ∂μ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral

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 : MeasurableSpace X
μ : Measure X
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : IsScalarTower 𝕜 A A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus 𝕜 A p
inst✝ : CompleteSpace A
f : X → 𝕜 → 𝕜
a : A
hf : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
⊢ Integrable (fun x ↦ (cfcₙL ha) (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0)) μ

Tactic:
exact cfcₙL_integrable _ _ hf ha

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

Goal state before:
pp: case mp
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : Ring 𝕜
inst✝⁵ : LinearOrder 𝕜
inst✝⁴ : IsStrictOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : IsTorsionFree 𝕜 E
A : Set E
x : E
hxA : x ∈ A
x₁ : E
h₁ : x₁ ∈ A
x₂ : E
h₂ : x₂ ∈ A
⊢ (x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x) → x ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x

Tactic:
rw [← insert_endpoints_openSegment]
    rintro H (rfl | rfl | hx)
    exacts [Or.inl rfl, Or.inr rfl, Or.inl <| (H hx).1]

Goal state after:
Module: Mathlib.Analysis.Complex.Circle

Goal state before:
pp: case refine_2
e : AddChar ℝ Circle
x : ℝ
h : ↑⟨↑(e x), ?refine_1⟩⁻¹ = (starRingEnd ℂ) ↑⟨↑(e x), ?refine_1⟩
⊢ star ↑(e x) = ↑(e (-x))

Tactic:
simp [← h, e.map_neg_eq_inv]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
⊢ EqOn (↑(e.subtypeRestr hV).symm) (inclusion hUV ∘ ↑(e.subtypeRestr hU).symm) (e.subtypeRestr hU).target

Tactic:
set i := Set.inclusion hUV
  intro y hy
  dsimp [OpenPartialHomeomorph.subtypeRestr_def] at hy ⊢
  have hyV : e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target := by
    rw [Opens.openPartialHomeomorphSubtypeCoe_target] at hy ⊢
    exact hUV hy.2
  refine (V.openPartialHomeomorphSubtypeCoe hV).injOn ?_ trivial ?_
  · simp
  · rw [(V.openPartialHomeomorphSubtypeCoe hV).right_inv hyV]
    change _ = U.openPartialHomeomorphSubtypeCoe hU _
    rw [(U.openPartialHomeomorphSubtypeCoe hU).right_inv hy.2]

Goal state after: