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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
L : Set α
hL : L ∈ s
hLu : L ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hLT : L ⊆ ⋃ i ∈ T, i
⊢ ∃ Q ⊆ S, Q.Finite ∧ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ Q

Tactic:
refine ⟨(fun U => {s | (s ∩ U).Nonempty}) '' T, by grind [image_subset_iff], hT.image _, ?_⟩

Goal state after:
case pos
α : Type u_1
inst✝ : TopologicalSpace α
K : Set α
hK : IsCompact K
s : Set (Set α)
hsK : s ⊆ 𝒫 K
hs : ∀ L ∈ s, IsCompact L
S : Set (Set (Set α))
hS : S ⊆ powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V}
hKS : {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ S
u : Set (Set α) := {U | IsOpen[inst✝] U ∧ {s | (s ∩ U).Nonempty} ∈ S}
L : Set α
hL : L ∈ s
hLu : L ⊆ ⋃ i ∈ u, i
T : Set (Set α)
hTS : T ⊆ u
hT : T.Finite
hLT : L ⊆ ⋃ i ∈ T, i
⊢ {t | t ⊆ K ∧ ∀ L ∈ s, (t ∩ L).Nonempty} ⊆ ⋃₀ ((fun U ↦ {s | (s ∩ U).Nonempty}) '' T)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
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 : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
⊢ IsClosedEmbedding (Unitization.inr ∘ ⇑(cfcₙHom ha))

Tactic:
convert! isClosedEmbedding_cfcₙAux hp₁ a ha

Goal state after:
𝕜 : Type u_1
A : Type u_2
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 : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
⊢ Unitization.inr ∘ ⇑(cfcₙHom ha) = ⇑(cfcₙAux ⋯ a ha)
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: case refine_1
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
y : ℝ≥0
hy : ↑y ∈ spectrum ℝ a
hx : ↑t - ↑y ∈ {↑t} - spectrum ℝ a
hty : -↑t ≤ ↑y
hyt : ↑y ≤ ↑t
⊢ ↑‖↑t - ↑y‖₊ ≤ ↑t

Tactic:
← NNReal.coe_sub (by exact_mod_cast hyt)

Goal state after:
case refine_1
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
y : ℝ≥0
hy : ↑y ∈ spectrum ℝ a
hx : ↑t - ↑y ∈ {↑t} - spectrum ℝ a
hty : -↑t ≤ ↑y
hyt : ↑y ≤ ↑t
⊢ ↑‖↑(t - y)‖₊ ≤ ↑t
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case refine_1
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖0‖ ≤ c

pp: case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ ≤ c

Tactic:
· simpa using (norm_nonneg _).trans <| h 0 (quasispectrum.zero_mem 𝕜 a)

Goal state after:
case refine_2
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 : { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ ‖(cfcₙHom ha) { toFun := (σₙ 𝕜 a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }‖ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝¹¹ : TopologicalSpace A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : PartialOrder A
inst✝⁸ : StarRing A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : IsSemitopologicalRing A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : SMulCommClass ℝ A A
inst✝² : T2Space A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
ha : 0 ≤ a
⊢ cfcₙ f a = cfcₙ (fun x ↦ ↑(f x.toNNReal)) a

Tactic:
apply (QuasispectrumRestricts.nnreal_of_nonneg ha).cfcₙ_eq_restrict _
    NNReal.isClosedEmbedding_coe ha (.of_nonneg ha)

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (spectralRadius 𝕜 a)⁻¹
z : 𝕜
z_mem : z ∈ Metric.closedBall 0 ↑r
hu : IsUnit (1 - z • a)
H₁ : Differentiable 𝕜 fun w ↦ 1 - w • a
⊢ DifferentiableAt 𝕜 (fun z ↦ (1 - z • a)⁻¹ʳ) z

Tactic:
exact DifferentiableAt.comp z (differentiableAt_inverse hu) H₁.differentiableAt

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

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

Tactic:
Set.compl_empty_iff

Goal state after:
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : resolventSet ℂ a = Set.univ
⊢ resolventSet ℂ a = Set.univ
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : LocallyConvexSpace 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0 ∧ Convex 𝕜 s
⊢ ∃ i', (i' ∈ 𝓝 0 ∧ AbsConvex 𝕜 i') ∧ id i' ⊆ id s

Tactic:
refine ⟨convexHull 𝕜 (balancedCore 𝕜 s), ?_, convexHull_min (balancedCore_subset s) hs.2⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : LocallyConvexSpace 𝕜 E
inst✝ : ContinuousSMul 𝕜 E
s : Set E
hs : s ∈ 𝓝 0 ∧ Convex 𝕜 s
⊢ (convexHull 𝕜) (balancedCore 𝕜 s) ∈ 𝓝 0 ∧ AbsConvex 𝕜 ((convexHull 𝕜) (balancedCore 𝕜 s))
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

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

Tactic:
ring

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsor

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ p₃ p₄ : P
⊢ ‖midpoint 𝕜 p₁ p₂ -ᵥ midpoint 𝕜 p₃ p₄‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖

Tactic:
midpoint_vsub_midpoint

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
𝕜 : Type u_5
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : Invertible 2
p₁ p₂ p₃ p₄ : P
⊢ ‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), approx n c x ≤ 1
c : CU P
⊢ approx (n + 1) c x ≤ 1

Tactic:
simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, ← div_eq_inv_mul]
    have := add_le_add (ihn (left c)) (ihn (right c))
    norm_num at this
    exact Iff.mpr (div_le_one zero_lt_two) this

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
n : ℕ
c : CU P
y : X
hxl : x ∉ c.left.left.U
hyl : y ∉ c.left.left.U
r : ℝ := (3 / 4) ^ n
ihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ r
hydl : dist (c.left.right.lim y) (c.left.right.lim x) ≤ r
hydr : dist (c.right.lim y) (c.right.lim x) ≤ r
⊢ (r / 2 + r) / 2 = r * (3 / 4)

Tactic:
ring

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

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

Tactic:
rw [← ofReal_mul, Real.mul_self_sqrt, hx', add_sub_cancel]

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

Tactic:
rw [← neg_inj, neg_neg, eq_comm]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ SpectrumRestricts (a ^ 2) ⇑ContinuousMap.realToNNReal

Tactic:
rw [SpectrumRestricts.nnreal_iff, ← cfc_id (R := ℝ) a, ← cfc_pow .., cfc_map_spectrum ..]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ ∀ x ∈ (fun x ↦ id x ^ 2) '' spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ SpectrumRestricts (a ^ 2) ⇑ContinuousMap.realToNNReal

Tactic:
rw [SpectrumRestricts.nnreal_iff, ← cfc_id (R := ℝ) a, ← cfc_pow .., cfc_map_spectrum ..]
  rintro - ⟨x, -, rfl⟩
  exact sq_nonneg x

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : SMulCommClass ℂ A A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
x : A
⊢ ↑(ℜ x) + I • ↑(ℑ x) = x

Tactic:
realPart_add_I_smul_imaginaryPart x

Goal state after:
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : SMulCommClass ℂ A A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
x : A
⊢ x = x
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
ha : 0 ≤ a
⊢ a ^ 1 = a

Tactic:
rw [nnrpow_one a]

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℂ E
inst✝² : SMul A E
inst✝¹ : Norm E
inst✝ : CStarModule A E
x : E
⊢ ‖x‖ ^ 2 = ‖⟪x, x⟫‖

Tactic:
simp [norm_eq_sqrt_norm_inner_self (A := A)]

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℂ E
inst✝² : SMul A E
inst✝¹ : Norm E
inst✝ : CStarModule A E
x : E
⊢ 0 ≤ ‖x‖

Tactic:
simp [norm_eq_sqrt_norm_inner_self (A := A)]

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℂ E
inst✝² : SMul A E
inst✝¹ : Norm E
inst✝ : CStarModule A E
⊢ ‖0‖ = 0

Tactic:
simp [norm_eq_sqrt_norm_inner_self (A := A)]

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

Goal state before:
pp: A : Type u_1
E : Type u_2
inst✝⁶ : NonUnitalCStarAlgebra A
inst✝⁵ : PartialOrder A
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℂ E
inst✝² : SMul A E
inst✝¹ : Norm E
inst✝ : CStarModule A E
x : E
h : x = 0
⊢ ‖x‖ = 0

Tactic:
simp [h, norm_eq_sqrt_norm_inner_self (A := A)]

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
x y : E
⊢ ‖⟪x, x⟫‖ + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖⟪y, y⟫‖ = ‖x‖ ^ 2 + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ + ‖y‖ ^ 2

Tactic:
simp [norm_eq_sqrt_norm_inner_self (A := A)]

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
x y : E
⊢ ‖x‖ ^ 2 + ‖y‖ * ‖x‖ + ‖x‖ * ‖y‖ = ‖x‖ ^ 2 + 2 * ‖x‖ * ‖y‖

Tactic:
ring

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalCStarAlgebra A
inst✝⁸ : PartialOrder A
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : Module ℂ E
inst✝⁵ : SMul A E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : Module ℂ F
inst✝² : SMul A F
inst✝¹ : CStarModule A E
inst✝ : CStarModule A F
x : C⋆ᵐᵒᵈ(A, E × F)
⊢ ‖x.1‖ ^ 2 + 2 * ‖x.1‖ * ‖x.2‖ + ‖x.2‖ ^ 2 = (‖x.1‖ + ‖x.2‖) ^ 2

Tactic:
ring

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
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
this : ∀ (a b : A), 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b)
⊢ DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ({x | 0 ≤ x} ∩ Metric.ball 0 1)

Tactic:
rintro a ⟨(ha₁ : 0 ≤ a), ha₂⟩ b ⟨(hb₁ : 0 ≤ b), hb₂⟩

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
this : ∀ (a b : A), 0 ≤ a → 0 ≤ b → ‖a‖ < 1 → ‖b‖ < 1 → a ≤ cfcₙ f (cfcₙ g a + cfcₙ g b)
a : A
ha₁ : 0 ≤ a
ha₂ : a ∈ Metric.ball 0 1
b : A
hb₁ : 0 ≤ b
hb₂ : b ∈ Metric.ball 0 1
⊢ ∃ z ∈ {x | 0 ≤ x} ∩ Metric.ball 0 1, (fun x1 x2 ↦ x1 ≤ x2) a z ∧ (fun x1 x2 ↦ x1 ≤ x2) b z
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✝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 : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
⊢ Tendsto (fun x ↦ cfcₙ (F x) a) l (𝓝 (cfcₙ f a))

Tactic:
let s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
    have hs : s ∈ l := hF.and hF0
    rw [← tendsto_comap'_iff (i := ((↑) : s → X)) (by simpa)]
    conv =>
      enter [1, x]
      rw [Function.comp_apply, cfcₙ_apply (hf := x.2.1) (hf0 := x.2.2)]
    rw [cfcₙ_apply ..]
    apply cfcₙHom_continuous _ |>.tendsto _ |>.comp
    rw [ContinuousMapZero.isEmbedding_toContinuousMap.isInducing.tendsto_nhds_iff]
    change Tendsto (fun x : s ↦ (⟨_, x.2.1.domRestrict⟩ : C(quasispectrum R a, R))) _
      (𝓝 ⟨_, hf.domRestrict⟩)
    rw [hf.tendsto_domRestrict_iff_tendstoUniformlyOn (fun x ↦ x.2.1)]
    intro t
    simp only [eventually_comap, Subtype.forall]
    peel h_tendsto t with ht x _
    simp_all

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
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn g (σₙ ℝ≥0 a)

Tactic:
have (x) (hx : x ∈ σₙ ℝ≥0 a) : 1 - x ≠ 0 := by
          refine tsub_pos_of_lt ?_ |>.ne'
          exact lt_of_le_of_lt (le_nnnorm_of_mem_quasispectrum hx) ha₂
        fun_prop

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

Goal state before:
pp: case inr
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
i : m
x✝ : i ∈ Finset.univ
h✝ : Nonempty n
x : n → α := fun j ↦ unitOf (A i j)
hxn : ‖x‖₊ = 1
hN : ∑ i_1, ‖A i i_1‖₊ ≤ N
⊢ ∑ j, ‖A i j‖₊ ≤ N

Tactic:
exact hN

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

Goal state before:
pp: case neg
A : Type u_2
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra ℝ A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : IsTopologicalRing A
inst✝³ : T2Space A
inst✝² : PartialOrder A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : StarOrderedRing A
a b : A
hb : Commute a b
f : ℝ≥0 → ℝ≥0
ha : ¬0 ≤ a
⊢ Commute (cfc f a) b

Tactic:
simp [cfc_apply_of_not_predicate a ha]

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

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra 𝕜 A
inst✝⁴ : ContinuousFunctionalCalculus 𝕜 A p
inst✝³ : CompleteSpace A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (univ ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
⊢ AEStronglyMeasurable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ

Tactic:
exact aeStronglyMeasurable_mkD_restrict_of_uncurry _ _ hf

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

Goal state before:
pp: case refine_1
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : MeasurableSpace X
μ : Measure X
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra 𝕜 A
inst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p
inst✝⁴ : CompleteSpace A
inst✝³ : NormedSpace ℝ A
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
f : X → 𝕜 → 𝕜
bound : X → ℝ
a : A
inst✝ : SecondCountableTopologyEither X C(↑(spectrum 𝕜 a), 𝕜)
hf : ContinuousOn (Function.uncurry f) (univ ×ˢ spectrum 𝕜 a)
bound_ge : ∀ᵐ (x : X) ∂μ, ∀ z ∈ spectrum 𝕜 a, ‖f x z‖ ≤ bound x
bound_int : HasFiniteIntegral bound μ
ha : p a
this : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
⊢ AEStronglyMeasurable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ

Tactic:
exact aeStronglyMeasurable_mkD_restrict_of_uncurry _ _ hf

Goal state after:
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
x : X
| cfcₙ (f x) a

Tactic:
rw [cfcₙ_eq_cfcₙL_mkD _ a]

Goal state after:
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
x : X
| (cfcₙL ha) (mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0)
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)
⊢ ‖v‖ * ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ = ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖ ^ 2

Tactic:
ring

Goal state after:
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✝⁹ : RCLike 𝕜
inst✝⁸ : MeasurableSpace X
μ : Measure X
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 : NonUnitalContinuousFunctionalCalculus 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) (quasispectrum 𝕜 a)
hf₂ : ∀ᵐ (x : X) ∂μ, f x 0 = 0
hf₃ : Integrable (fun x ↦ mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
⊢ cfcₙ (fun z ↦ ∫ (x : X), f x z ∂μ) a = ∫ (x : X), cfcₙ (f x) a ∂μ

Tactic:
have key₁ (z : quasispectrum 𝕜 a) :
      ∫ x, f x z ∂μ = (∫ x, mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z := by
    rw [integral_apply hf₃]
    refine integral_congr_ae ?_
    filter_upwards [hf₁, hf₂] with x cont_x zero_x
    rw [mkD_apply_of_continuousOn cont_x zero_x]
  have key₂ (z : quasispectrum 𝕜 a) :
      ∫ x, f x z ∂μ = mkD ((quasispectrum 𝕜 a).domRestrict (fun z ↦ ∫ x, f x z ∂μ)) 0 z := by
    rw [mkD_apply_of_continuousOn]
    · rw [continuousOn_iff_continuous_domRestrict]
      refine continuous_congr key₁ |>.mpr ?_
      exact map_continuous (∫ x, mkD ((quasispectrum 𝕜 a).domRestrict (f x)) 0 ∂μ)
    · exact integral_eq_zero_of_ae hf₂
  simp_rw [cfcₙ_eq_cfcₙL_mkD _ a, cfcₙL_integral a _ hf₃ ha]
  congr
  ext z
  rw [← key₁, key₂]

Goal state after: