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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
this : (ofReal ⁻¹' (σ ℂ a)ᶜ).Nonempty
⊢ ofReal '' ofReal ⁻¹' (σ ℂ a)ᶜ ⊆ (σ ℂ a)ᶜ ∩ {z | 0 ≤ z.im} ∩ ((σ ℂ a)ᶜ ∩ {z | z.im ≤ 0})

Tactic:
simp

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

Goal state before:
pp: case inl
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedRing A
inst✝² : StarRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p
a : A
ha : p a
this : CompactSpace ↑(σₙ 𝕜 a)
f : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜
ι : C(↑(σ 𝕜 a), ↑(σₙ 𝕜 a)) := { toFun := Set.inclusion ⋯, continuous_toFun := ⋯ }
hx : 0 ∈ σₙ 𝕜 a
⊢ ‖f 0‖ ≤ ‖(↑f).comp ι‖

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → Compacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
x : α
x✝ : γ
⊢ x✝ ∈ ↑(Compacts.map (g x) ⋯ (f x)) ↔ x✝ ∈ ↑(Compacts.map (Function.uncurry g) hg ({x} ×ˢ f x))

Tactic:
simp

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

Goal state before:
pp: F : Type u_1
A : Type u_2
inst✝¹ : CStarAlgebra A
inst✝ : FunLike F A ℂ
hF : AlgHomClass F ℂ A ℂ
φ : F
a : A
s : ↥(selfAdjoint A)
⊢ (↑(φ ↑s).re)⋆ = ↑(φ ↑s).re

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
⊢ ↑(expUnitary 0) = ↑1

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (Compacts α)
⊢ ⋃ x, ↑x = univ

Tactic:
eq_univ_iff_forall,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (Compacts α)
⊢ ∀ (x : α), x ∈ ⋃ x, ↑x
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : SeparableSpace (Compacts α)
h✝ : Nonempty α
⊢ SeparableSpace α

Tactic:
obtain ⟨s, hs₁, hs₂⟩ := exists_countable_dense (Compacts α)

Goal state after:
case inr
α : Type u_1
inst✝ : TopologicalSpace α
x✝ : SeparableSpace (Compacts α)
h✝ : Nonempty α
s : Set (Compacts α)
hs₁ : s.Countable
hs₂ : Dense s
⊢ SeparableSpace α
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
⊢ LocallyConnectedSpace (Compacts α)

Tactic:
locallyConnectedSpace_iff_isTopologicalBasis_isOpen_isPreconnected

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
⊢ IsTopologicalBasis {s | IsOpen s ∧ IsPreconnected s}
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Zero β
inst✝² : FunLike F α β
inst✝¹ : ZeroAtInftyContinuousMapClass F α β
inst✝ : CompactSpace α
f : C(α, β)
⊢ Tendsto (⇑f) (cocompact α) (𝓝 0)

Tactic:
simp

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

Goal state before:
pp: α : Type u
β : Type v
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Zero β
G : Type u_2
inst✝² : FunLike G α β
inst✝¹ : ContinuousMapClass G α β
inst✝ : CompactSpace α
⊢ ∀ (f : G), Tendsto (⇑f) (cocompact α) (𝓝 0)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
h : B.SeparatingRight
⊢ 0 ∈ B.polar Set.univ

Tactic:
simp only [zero_mem_polar]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
hcomm : Commute (x • a) (-x • star a)
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) = ↑(expUnitary (x • 2 • ℑ a))

Tactic:
← exp_add_of_commute hcomm

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
hcomm : Commute (x • a) (-x • star a)
⊢ NormedSpace.exp (x • a + -x • star a) = ↑(expUnitary (x • 2 • ℑ a))
Module: Mathlib.Analysis.LocallyConvex.AbsConvexOpen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommMonoid E
inst✝² : SeminormedRing 𝕜
inst✝¹ : SMul 𝕜 E
inst✝ : PartialOrder 𝕜
⊢ ∃ x, True

Tactic:
Subtype.exists

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommMonoid E
inst✝² : SeminormedRing 𝕜
inst✝¹ : SMul 𝕜 E
inst✝ : PartialOrder 𝕜
⊢ ∃ a, ∃ (_ : 0 ∈ a ∧ IsOpen[inst✝⁴] a ∧ AbsConvex 𝕜 a), True
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_2.inl
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
K : NonemptyCompacts α
hu : ∅.Finite
huB : ∅ ⊆ B
hK₁ : ↑K.toCompacts ⊆ ⋃₀ ∅
hK₂ : ∀ U ∈ ∅, (↑K.toCompacts ∩ U).Nonempty
⊢ ¬(⋃₀ ∅).Nonempty

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 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
hv_basis : (𝓝 0).HasAntitoneBasis v
φ : ℕ → ℕ
hφ_add : ∀ ⦃m n : ℕ⦄, m < n → v (φ n) + v (φ n) ⊆ v (φ m)
hφ_basis : (𝓝 0).HasAntitoneBasis (v ∘ φ)
n : ℕ
⊢ n < n + 1

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (NonemptyCompacts α)
⊢ ⋃ x, ↑x = univ

Tactic:
eq_univ_iff_forall,

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
h : CompactSpace (NonemptyCompacts α)
⊢ ∀ (x : α), x ∈ ⋃ x, ↑x
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: 𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
⊢ x ∈ 𝕜 ∙ x

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
hval : g x = ↑‖x‖
⊢ ∃ g, ‖g‖ = 1 ∧ g x = ↑‖x‖

Tactic:
refine ⟨g, le_antisymm ?_ ?_, hval⟩

Goal state after:
case refine_1
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
hval : g x = ↑‖x‖
⊢ ‖g‖ ≤ 1

case refine_2
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : ‖x‖ ≠ 0
hhomothety : ∀ (c : 𝕜), ‖(LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) c‖ = ‖x‖ * ‖c‖
coord : ↥(𝕜 ∙ x) →L[𝕜] 𝕜 := ↑(ofHomothety (LinearEquiv.toSpanNonzeroSingleton 𝕜 E x ⋯) ‖x‖ ⋯ hhomothety).symm
g : StrongDual 𝕜 E
hg : (∀ (x_1 : ↥(𝕜 ∙ x)), g ↑x_1 = (↑‖x‖ • coord) x_1) ∧ ‖g‖ = ‖↑‖x‖ • coord‖
hval : g x = ↑‖x‖
⊢ 1 ≤ ‖g‖
Module: Mathlib.Analysis.Normed.Module.HahnBanach

Goal state before:
pp: 𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
hx : ‖x‖ = 0
⊢ ‖0‖ ≤ 1

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Extend

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ F
inst✝ : IsScalarTower ℝ 𝕜 F
fr : StrongDual ℝ F
x : F
⊢ ‖fr x‖ = ‖re (fr.extendRCLike x)‖

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
⊢ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) 0 = 0

Tactic:
simp

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

Goal state before:
pp: V : Type u_1
P : Type u_2
W : Type u_3
Q : Type u_4
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : PseudoMetricSpace P
inst✝⁵ : NormedAddTorsor V P
inst✝⁴ : NormedAddCommGroup W
inst✝³ : MetricSpace Q
inst✝² : NormedAddTorsor W Q
inst✝¹ : NormedSpace ℝ V
inst✝ : NormedSpace ℝ W
f : P → Q
h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
hfc :
  Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : P := Classical.arbitrary P
p : P
⊢ f p =
    ↑((AddMonoidHom.ofMapMidpoint ℝ ℝ (⇑(AffineEquiv.vaddConst ℝ (f c)).symm ∘ f ∘ ⇑(AffineEquiv.vaddConst ℝ c)) ⋯
                ⋯).toRealLinearMap
            ⋯)
        (p -ᵥ c) +ᵥ
      f c

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_4
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
f : 𝕜 → 𝕜
x₀ b : 𝕜
hb : b < x₀
hbf : f b < f x₀
hf : f x₀ = 0
⊢ f b < 0

pp: 𝕜 : Type u_4
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
f : 𝕜 → 𝕜
x₀ b : 𝕜
hb : b < x₀
hbf : 0 < slope f x₀ b
hf : f x₀ = 0
⊢ b < x₀

Tactic:
hf

Goal state after:
𝕜 : Type u_4
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
f : 𝕜 → 𝕜
x₀ b : 𝕜
hb : b < x₀
hbf : f b < 0
hf : f x₀ = 0
⊢ f b < 0

𝕜 : Type u_4
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
f : 𝕜 → 𝕜
x₀ b : 𝕜
hb : b < x₀
hbf : 0 < slope f x₀ b
hf : f x₀ = 0
⊢ b < x₀
Module: Mathlib.Topology.ContinuousMap.ContinuousSqrt

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
x✝¹ : 𝕜 × 𝕜
x✝ : x✝¹.1 ≤ x✝¹.2
⊢ 0 ≤ (ofReal ∘ (fun x ↦ √x) ∘ ⇑re ∘ fun z ↦ z.2 - z.1) x✝¹

Tactic:
simp

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

Goal state before:
pp: ⊢ ∀ (x : ℝ≥0 × ℝ≥0), x.1 ≤ x.2 → 0 ≤ (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) x

Tactic:
simp

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
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
y : X
⊢ ((↑(algebraMapCLM ℝ≥0 𝕜)).comp (g₁ + g₂)) y = ((↑(algebraMapCLM ℝ≥0 𝕜)).comp g₁ + (↑(algebraMapCLM ℝ≥0 𝕜)).comp g₂) y

Tactic:
simp

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)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t

Tactic:
obtain ⟨c, hc, hgc'⟩ : ∃ c > 0, t ⊆ {y | c ≤ g' y} :=
    t.eq_empty_or_nonempty.elim
      (fun ht' => ⟨1, zero_lt_one, fun y hy => False.elim (by rwa [ht'] at hy)⟩) fun ht' =>
      let ⟨x, hx, hx'⟩ := ht.isCompact.exists_isMinOn ht' (map_continuous g').continuousOn
      ⟨g' x, hgt' x hx, hx'⟩

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
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)
hI' : (↑(algebraMapCLM ℝ≥0 𝕜)).comp g' ∈ I
hgt' : ∀ x ∈ t, 0 < g' x
c : ℝ≥0
hc : c > 0
hgc' : t ⊆ {y | c ≤ g' y}
⊢ ∃ g, (↑(algebraMapCLM ℝ≥0 𝕜)).comp g ∈ I ∧ (∀ (x : X), g x ≤ 1) ∧ Set.EqOn (⇑g) 1 t
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: case empty
A : Type u_1
inst✝ : NonUnitalCommCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
⊢ ‖∑ i ∈ ∅, f i‖₊ = ∅.sup fun x ↦ ‖f x‖₊

Tactic:
simp

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

Goal state before:
pp: case empty
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
ι : Type u_2
f : ι → A
h0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
h : ∀ i ∈ ∅, IsSelfAdjoint (f i)
⊢ ‖∑ i ∈ ∅, f i‖₊ = ∅.sup fun x ↦ ‖f x‖₊

Tactic:
simp

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

Goal state before:
pp: 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)
this : f j * ∑ i ∈ s, f i = 0
⊢ j ∈ insert j s

Tactic:
simp

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

Goal state before:
pp: case neg.inl
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
a : A
ha : ¬IsSelfAdjoint a
⊢ (0 • a)⁺ = 0 • a⁺

Tactic:
simp

Goal state after:
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 ↔ 0 ≤ a

Tactic:
refine ⟨fun ha ↦ ha ▸ posPart_nonneg a, fun ha ↦ ?_⟩

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 : 0 ≤ a
⊢ a⁺ = a
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⁺ - a⁻ = a⁺

Tactic:
simp

Goal state after:
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⁺ - a⁻) = a⁻

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : α → γ → β
K : ℝ≥0
h : ∀ (c : α), LipschitzWith K (f c)
⊢ UniformEquicontinuous f

Tactic:
uniformEquicontinuous_iff_uniformContinuous

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PseudoEMetricSpace γ
inst✝ : PseudoEMetricSpace β
f : α → γ → β
K : ℝ≥0
h : ∀ (c : α), LipschitzWith K (f c)
⊢ UniformContinuous (⇑ofFun ∘ Function.swap f)
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f : α →ᵤ[𝔖] β
⊢ (ofFun 𝔖) (UniformFun.toFun (UniformFun.ofFun ((toFun 𝔖) f))) = f

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
this : Fact (0 ∈ s)
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
key : cfcₙHomSuperset ha has = ψ
⊢ { toFun := fun x ↦ (↑x)⁺, continuous_toFun := ⋯ } 0 = 0

Tactic:
simp

Goal state after:
A : Type u_1
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : Module ℝ A
inst✝⁹ : SMulCommClass ℝ A A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a b c : A
habc : a = b - c
hbc : b * c = 0
hb : 0 ≤ b
hc : 0 ≤ c
hb' : IsSelfAdjoint b
hc' : IsSelfAdjoint (-c)
ha : IsSelfAdjoint a
s : Set ℝ := σₙ ℝ a ∪ σₙ ℝ b ∪ σₙ ℝ (-c)
hs : CompactSpace ↑s
has : σₙ ℝ a ⊆ s
hbs : σₙ ℝ b ⊆ s
hcs : σₙ ℝ (-c) ⊆ s
this : Fact (0 ∈ s)
mul₁ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hb' hbs) f * (cfcₙHomSuperset hc' hcs) g = 0
mul₂ : ∀ (f g : C(↑s, ℝ)₀), (cfcₙHomSuperset hc' hcs) f * (cfcₙHomSuperset hb' hbs) g = 0
ψ : C(↑s, ℝ)₀ →⋆ₙₐ[ℝ] A :=
  let __src := ↑(cfcₙHomSuperset hb' hbs) + ↑(cfcₙHomSuperset hc' hcs);
  { toFun := ⇑(cfcₙHomSuperset hb' hbs) + ⇑(cfcₙHomSuperset hc' hcs), map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯,
    map_mul' := ⋯, map_star' := ⋯ }
key : cfcₙHomSuperset ha has = ψ
⊢ ↑0 ≤ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case hom_id
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
⊢ ((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
      (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) =
    a

pp: case hom_map_spectrum
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
⊢ ∀ (f : C(↑(spectrum ℂ a), ℂ)),
    spectrum ℂ (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) f) = Set.range ⇑f

pp: case predicate_hom
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
⊢ ∀ (f : C(↑(spectrum ℂ a), ℂ)), IsStarNormal (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) f)

Tactic:
case hom_id => exact congr_arg Subtype.val <| continuousFunctionalCalculus_map_id a

Goal state after:
case hom_map_spectrum
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
⊢ ∀ (f : C(↑(spectrum ℂ a), ℂ)),
    spectrum ℂ (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) f) = Set.range ⇑f

case predicate_hom
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
this : Isometry ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
⊢ ∀ (f : C(↑(spectrum ℂ a), ℂ)), IsStarNormal (((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)) f)
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:
SpectrumRestricts.nnreal_iff,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ ∀ x ∈ spectrum ℝ (a ^ 2), 0 ≤ x
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
⊢ 0⁻ = 0

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁺) 1 = 1

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁻) 1 = 0

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺

Tactic:
simp

Goal state after: