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

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
x : Unitization ℂ A
⊢ ∀ {s : Unitization ℂ A}, IsSelfAdjoint s → ‖ψ s‖₊ ≤ ‖s‖₊

Tactic:
intro s hs

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
a : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
x s : Unitization ℂ A
hs : IsSelfAdjoint s
⊢ ‖ψ s‖₊ ≤ ‖s‖₊
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
⊢ (∀ (i : ι), IsOpen[inst✝] (U i)) → ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i → ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i

Tactic:
hU

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
S : Set (Compacts α)
hS : IsCompact S
ι : Type u_1
U : ι → Set α
hU : ∀ (i : ι), IsOpen[inst✝] (U i)
⊢ ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i → ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i
Module: Mathlib.Analysis.Convex.TotallyBounded

Goal state before:
pp: E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
⊢ U ∈ nhds 0 → ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U

Tactic:
hU

Goal state after:
E : Type u_1
s : Set E
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : UniformSpace E
inst✝² : IsUniformAddGroup E
inst✝¹ : LocallyConvexSpace ℝ E
inst✝ : ContinuousSMul ℝ E
hs : ∀ U ∈ nhds 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, y +ᵥ U
U : Set E
hU : U ∈ nhds 0
⊢ ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : PartialOrder 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : LocallyConvexSpace 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ZeroLEOneClass 𝕜
inst✝ : FirstCountableTopology E
u : ℕ → Set E
hu_basis : (𝓝 0).HasAntitoneBasis u
hu_zero : ∀ (n : ℕ), 0 ∈ interior (u n)
v : ℕ → Set E := fun n ↦ (absConvexHull 𝕜) (interior (u n))
hv_open : ∀ (n : ℕ), IsOpen[inst✝⁵] (v n)
hv_nhds : ∀ (n : ℕ), v n ∈ 𝓝 0
n : ℕ
a✝ : True
m : ℕ
hm : u m ⊆ interior (u n)
⊢ ∃ i, True ∧ u i ⊆ v n

Tactic:
exact ⟨m, trivial, hm.trans subset_absConvexHull⟩

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
S : Set (NonemptyCompacts α)
hs : IsCompact S
⊢ ⋃ K ∈ S, ↑K = ⋃ y ∈ S, ↑y.toCompacts

Tactic:
coe_toCompacts

Goal state after:
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
⊢ (fun x ↦ r • x) y ∈ (convexHull ℝ) (s ∪ -s)

Tactic:
apply segment_subset_convexHull (mem_union_left (-s) hy) (mem_union_right _ (neg_mem_neg.mpr hy))

Goal state after:
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
⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)
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
⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)

Tactic:
refine ⟨(1 + r)/2, (1 - r)/2, by positivity, by positivity, by ring, ?_⟩

Goal state after:
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) • y + ((1 - r) / 2) • -y = (fun x ↦ r • x) y
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : ∀ (s : Set (NonemptyCompacts α)), IsClopen s → s = ∅ ∨ s = univ
⊢ ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ

Tactic:
intro s hs

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : ∀ (s : Set (NonemptyCompacts α)), IsClopen s → s = ∅ ∨ s = univ
s : Set α
hs : IsClopen s
⊢ s = ∅ ∨ s = univ
Module: Mathlib.Analysis.Normed.Module.Dual

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ IsClosed[ContinuousLinearMap.topologicalSpace] (⋂ x ∈ s, {y | ‖((topDualPairing 𝕜 E) y) x‖ ≤ 1})

Tactic:
refine isClosed_biInter fun z _ => ?_

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
z : E
x✝ : z ∈ s
⊢ IsClosed[ContinuousLinearMap.topologicalSpace] {y | ‖((topDualPairing 𝕜 E) y) z‖ ≤ 1}
Module: Mathlib.Analysis.LocallyConvex.WeakDual

Goal state before:
pp: ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
s : Finset ι
| ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, 𝓝 0

Tactic:
iInf_subtype

Goal state after:
ι : Type u_4
𝕜 : Type u_5
E : Type u_6
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : ι → E →ₗ[𝕜] 𝕜
φ : E →ₗ[𝕜] 𝕜
t𝕜 : TopologicalSpace 𝕜 := inferInstance
t₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜
t₂ : Finset ι → TopologicalSpace E := fun s ↦ ⨅ i, induced (⇑(f ↑i)) t𝕜
t₁_group : IsTopologicalAddGroup E
t₂_group : ∀ (s : Finset ι), IsTopologicalAddGroup E
t₁_smul : ContinuousSMul 𝕜 E
t₂_smul : ∀ (s : Finset ι), ContinuousSMul 𝕜 E
s : Finset ι
| ((normSeminorm 𝕜 𝕜).comp φ).ball 0 1 ∈ ⨅ i, ⨅ (h : i ∈ s), 𝓝 0
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: ι : Sort u_1
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : BarrelledSpace 𝕜 E
p : ι → Seminorm 𝕜 E
hp : ∀ (i : ι), Continuous[inst✝¹, _] ⇑(p i)
bdd : BddAbove (range p)
⊢ Continuous[inst✝¹, _] (⨆ i, ⇑(p i))

Tactic:
by
  rw [← Seminorm.coe_iSup_eq bdd]
  refine Seminorm.continuous_of_lowerSemicontinuous _ ?_
  rw [Seminorm.coe_iSup_eq bdd]
  rw [Seminorm.bddAbove_range_iff] at bdd
  convert! lowerSemicontinuous_ciSup (f := fun i x ↦ p i x) bdd (fun i ↦ (hp i).lowerSemicontinuous)
  exact iSup_apply

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
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
g : C(X, ℝ≥0)
hgI : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I
hg : ∀ (x : X), g x ≤ 1
hgt : Set.EqOn (⇑g) 1 t
x : X
hx : x ∉ t
this : ‖(1 - (↑(algebraMapCLM ℝ≥0 𝕜)).comp g) x‖₊ ≤ 1
⊢ ε / 2 * ‖(1 - (↑(algebraMapCLM ℝ≥0 𝕜)).comp g) x‖₊ ≤ ε / 2

Tactic:
simpa only [mul_one] using! mul_le_mul_right this _

Goal state after:
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⁺ * a⁻ = 0

Tactic:
posPart_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 * 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⁻ * a⁺ = 0

Tactic:
posPart_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
⊢ 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
⊢ a⁺ - a⁻ = a

Tactic:
posPart_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 - a⁻ = a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ (-a)⁺ = a⁻

Tactic:
posPart_def,

Goal state after:
case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) (-a) = a⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
ha' : ¬IsSelfAdjoint (-a)
⊢ (-a)⁺ = a⁻

Tactic:
posPart_def,

Goal state after:
case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
ha' : ¬IsSelfAdjoint (-a)
⊢ cfcₙ (fun x ↦ x⁺) (-a) = a⁻
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : (i : ι) → Ring (A i)
inst✝⁷ : (i : ι) → Algebra S (A i)
inst✝⁶ : (i : ι) → Algebra R (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R S (A i)
hinst : IsScalarTower R S ((i : ι) → A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : ContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), ContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMap.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, spectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
i : ι
φ : ((i : ι) → A i) →⋆ₐ[S] A i := Pi.evalStarAlgHom S A i
⊢ cfc f a i = cfc f (a i)

Tactic:
exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i)

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 : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c)) + -(star c * c)) ⇑ContinuousMap.realToNNReal

Tactic:
← sq,

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
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) * ↑(ℑ c)) + -(star c * c)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Goal state before:
pp: ι : Type u_2
C : ι → Type u_3
inst✝¹⁴ : (i : ι) → PartialOrder (C i)
inst✝¹³ : (i : ι) → NonUnitalRing (C i)
inst✝¹² : (i : ι) → TopologicalSpace (C i)
inst✝¹¹ : (i : ι) → StarRing (C i)
inst✝¹⁰ : ∀ (i : ι), StarOrderedRing (C i)
inst✝⁹ : StarOrderedRing ((i : ι) → C i)
inst✝⁸ : (i : ι) → Module ℝ (C i)
inst✝⁷ : ∀ (i : ι), SMulCommClass ℝ (C i) (C i)
inst✝⁶ : ∀ (i : ι), IsScalarTower ℝ (C i) (C i)
inst✝⁵ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus ℝ (C i) IsSelfAdjoint
inst✝⁴ : NonUnitalContinuousFunctionalCalculus ℝ ((i : ι) → C i) IsSelfAdjoint
inst✝³ : ∀ (i : ι), IsSemitopologicalRing (C i)
inst✝² : ∀ (i : ι), T2Space (C i)
inst✝¹ : NonnegSpectrumClass ℝ ((i : ι) → C i)
inst✝ : ∀ (i : ι), NonnegSpectrumClass ℝ (C i)
c : (i : ι) → C i
hc : ∀ (i : ι), 0 ≤ c i
⊢ sqrt c = fun i ↦ sqrt (c i)

Tactic:
simp only [sqrt_eq_nnrpow]
  exact nnrpow_map_pi

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✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : MetricSpace A
inst✝¹ : Algebra R A
inst✝ : IsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{spectrum R a}] R
hf : f ∈ {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}
g : R →ᵤ[{spectrum R a}] R
hg : g ∈ {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}
⊢ edist { toFun := (spectrum R a).domRestrict ((toFun {spectrum R a}) f), continuous_toFun := ⋯ }
      { toFun := (spectrum R a).domRestrict ((toFun {spectrum R a}) g), continuous_toFun := ⋯ } ≤
    edist f g

Tactic:
rw [edist_continuousRestrict_of_singleton hf hg]

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : PartialOrder A
inst✝⁶ : Ring A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarOrderedRing A
inst✝² : Algebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsUnit a
⊢ 0 ≤ a⁻¹ʳ ↔ 0 ≤ a

Tactic:
by
  grind [isStrictlyPositive_ringInverse_iff]

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
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ ‖sqrt a * (1 - e)‖ = 0

Tactic:
← sq_eq_zero_iff,

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a✝ e✝ : A
he✝ : IsStarProjection e✝
ha✝ : 0 ≤ a✝
hae✝ : a✝ ≤ e✝
a e : Unitization ℂ A
he : IsStarProjection e
ha : 0 ≤ a
hae : a ≤ e
⊢ ‖sqrt a * (1 - e)‖ ^ 2 = 0
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 : A
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
⊢ Antitone fun x ↦ a ^ x

Tactic:
by
  intro n m hnm
  simp only
  rw [← cfc_pow_id (R := ℝ) a, ← cfc_pow_id (R := ℝ) a, cfc_le_iff ..]
  rw [CFC.le_one_iff (R := ℝ) a] at ha₁
  peel ha₁ with x hx _
  exact pow_le_pow_of_le_one (spectrum_nonneg_of_nonneg ha₀ hx) (ha₁ x hx) hnm

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
v : C⋆ᵐᵒᵈ(A, m → A)
⊢ (toCLM M) v = (WithCStarModule.equiv A (n → A)).symm fun j ↦ ∑ i, v i * M i j

Tactic:
by
  ext i
  simp [toCLM_apply, Matrix.vecMul, dotProduct]

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

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖M i j‖ = 0

Tactic:
← sq_eq_zero_iff,

Goal state after:
m : Type u_1
n : Type u_2
A : Type u_5
inst✝¹ : Fintype m
inst✝ : NonUnitalCStarAlgebra A
M : CStarMatrix m n A
h : toCLM M = 0
i : m
j : n
⊢ ‖M i j‖ ^ 2 = 0
Module: Mathlib.Analysis.Convex.Extreme

Goal state before:
pp: case mpr
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 ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x

Tactic:
intro H hx

Goal state after:
case mpr
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
H : x ∈ segment 𝕜 x₁ x₂ → x₁ = x ∨ x₂ = x
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ = x ∧ x₂ = x
Module: Mathlib.Analysis.CStarAlgebra.Extreme

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
x✝ :
  (0 ≤ e ∧ ‖e‖ ≤ 1) ∧ ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
h1 : 0 ≤ e
h2 : ‖e‖ ≤ 1
h3 : ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
this : 0 ≤ ↑(2 • e - e * e)
⊢ star (1 - ↑e) * (1 - ↑e) = 1 - ↑(2 • e - e * e)

Tactic:
by simp [LE.le.star_eq, h1, mul_sub, sub_mul, two_smul, sub_sub, add_sub]

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

Goal state before:
pp: x : ℝ
n : ℕ
⊢ exp (n • x) = exp x ^ n

Tactic:
exp_nsmul

Goal state after:
x : ℝ
n : ℕ
⊢ exp x ^ n = exp x ^ n
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace X
f : E → X
G : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G E
hf : IsQuotientMap f
inst✝² : ContinuousConstSMul G E
hfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂
inst✝¹ : TopologicalSpace G
inst✝ : DiscreteTopology G
U : Set E
open_U : IsOpen[inst✝⁶] U
disjoint : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1
pGE : ∀ (g : G) (e : E), f (g • e) = f e
e : E
x✝ : e ∈ f ⁻¹' f '' U
g : G
heU : (fun m ↦ m • e) g ∈ U
he : f ((fun m ↦ m • e) g) = f e
⊢ e ∈ ⋃ g, (fun x ↦ g • x) ⁻¹' U

Tactic:
exact ⟨_, ⟨g, rfl⟩, heU⟩

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

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
r : ℝ
a : A
hr : 0 < r
ha : IsStrictlyPositive a
⊢ log (r • a) = (algebraMap ℝ A) (Real.log r) + log a

Tactic:
grind [log_smul]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case mpr
A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
a✝ : Nontrivial A
x : ℝ
hx : IsLeast (re '' spectrum ℂ u) x
⊢ 2 ^ 2 ≤ ‖u - 1‖ ^ 2 → -1 ∈ spectrum ℂ u

Tactic:
norm_sub_one_sq_eq hu hx

Goal state after:
case mpr
A : Type u_1
inst✝ : CStarAlgebra A
u : A
hu : u ∈ unitary A
a✝ : Nontrivial A
x : ℝ
hx : IsLeast (re '' spectrum ℂ u) x
⊢ 2 ^ 2 ≤ 2 * (1 - x) → -1 ∈ spectrum ℂ u
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: ⊢ Filter.Tendsto (fun i ↦ π * 2⁻¹ * (2 ^ i)⁻¹) Filter.atTop (nhds 0)

Tactic:
rw [← mul_zero (π * 2⁻¹)]

Goal state after:
⊢ Filter.Tendsto (fun i ↦ π * 2⁻¹ * (2 ^ i)⁻¹) Filter.atTop (nhds (π * 2⁻¹ * 0))
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
⊢ ContinuousOn (Subtype.val ∘ argSelfAdjoint) (ball 1 2)

Tactic:
simp only [Function.comp_def, argSelfAdjoint_coe]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
⊢ ContinuousOn (fun x ↦ cfc (fun x ↦ ↑x.arg) ↑x) (ball 1 2)
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: case h.left
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝¹ : (A : Type u_1) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
x✝ : ∀ (A : Type u_1) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing
hx : x ≠ 0
u₁ : ↥(unitary A) := ⋯
u₂ : ↥(unitary A) := ⋯
this : x = ‖x‖ • 2⁻¹ • (↑u₁ + ↑(star u₁) + I • (↑u₂ + ↑(star u₂)))
⊢ x = ∑ i, ![↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹, ↑‖x‖ * 2⁻¹ * I, ↑‖x‖ * 2⁻¹ * I] i • ↑(![u₁, star u₁, u₂, star u₂] i)

Tactic:
conv_lhs => rw [this]
      simp [Fin.sum_univ_four, ← Complex.coe_smul]
      module

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: case mp
A : Type u_1
inst✝ : CStarAlgebra A
⊢ connectedComponent 1 ⊆ Set.range fun y ↦ (List.map expUnitary y).prod

Tactic:
refine IsClopen.connectedComponent_subset ?_ ⟨[], by simp⟩

Goal state after:
case mp
A : Type u_1
inst✝ : CStarAlgebra A
⊢ IsClopen (Set.range fun y ↦ (List.map expUnitary y).prod)
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖A‖ * ‖A‖ ≤ ‖adjoint A ∘SL A‖

Tactic:
← sq,

Goal state after:
case refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
A : E →L[𝕜] F
⊢ ‖A‖ ^ 2 ≤ ‖adjoint A ∘SL A‖
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
u : H →L[𝕜] K
⊢ (∀ (x y : H), ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜) ↔ adjoint u ∘SL u = 1

Tactic:
by
  refine ⟨fun h ↦ ext fun x ↦ ?_, fun h ↦ ?_⟩
  · refine ext_inner_right 𝕜 fun y ↦ ?_
    simpa [star_eq_adjoint, adjoint_inner_left] using h x y
  · simp [← adjoint_inner_left, ← comp_apply, h]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.centroid -ᵥ s.points i = ↑n • (s.faceOppositeCentroid i -ᵥ s.centroid)

Tactic:
by
  rw [← neg_vsub_eq_vsub_rev, point_vsub_centroid_eq_smul_vsub, ← neg_smul_neg,
    neg_vsub_eq_vsub_rev, ← neg_smul, neg_neg]

Goal state after: