mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.Spaces.WeakBilin

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕝 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : CommSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
inst✝ : ContinuousAdd 𝕜
x✝¹ : WeakBilin B × WeakBilin B
x✝ : F
⊢ ((fun x y ↦ (B x) y) ∘ Prod.fst + (fun x y ↦ (B x) y) ∘ Prod.snd) x✝¹ x✝ =
    ((fun x y ↦ (B x) y) ∘ fun p ↦ p.1 + p.2) x✝¹ x✝

Tactic:
simp only [Function.comp_apply, Pi.add_apply, map_add, LinearMap.add_apply]

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s : Set α
x : α
inst✝ : TopologicalSpace β
⊢ (∀ (i : Set β), IsOpen[inst✝] i ∧ f x ⊆ i → ∀ᶠ (x' : α) in 𝓝[s] x, i ∈ 𝓝ˢ (f x')) ↔
    ∀ (i : Set β), IsOpen[inst✝] i ∧ f x ⊆ i → f ⁻¹' Iic i ∈ 𝓝[s] x

Tactic:
refine forall₂_congr fun u ⟨hu, hfu⟩ ↦ ?_

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
s : Set α
x : α
inst✝ : TopologicalSpace β
u : Set β
x✝ : IsOpen[inst✝] u ∧ f x ⊆ u
hu : IsOpen[inst✝] u
hfu : f x ⊆ u
⊢ (∀ᶠ (x' : α) in 𝓝[s] x, u ∈ 𝓝ˢ (f x')) ↔ f ⁻¹' Iic u ∈ 𝓝[s] x
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
hf : UpperHemicontinuous f
x : α
⊢ x ∈ {a | f a = ∅} → {a | f a = ∅} ∈ 𝓝 x

Tactic:
(hx : f x = ∅)

Goal state after:
α : Type u_3
β : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → Set β
hf : UpperHemicontinuous f
x : α
hx : f x = ∅
⊢ {a | f a = ∅} ∈ 𝓝 x
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s K : Set α
hKs : K ∈ 𝒫 closure[inst✝] s
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
ht₁ : K ⊆ ⋃₀ u
ht₂ : ∀ U ∈ u, (K ∩ U).Nonempty
U : ↑u
x : α
hxK : x ∈ K
hxV : x ∈ ↑U
⊢ (↑U ∩ s).Nonempty

Tactic:
exact mem_closure_iff.mp (hKs hxK) _ (hu₂ _ U.prop) hxV

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

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
⊢ (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1

Tactic:
simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)

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

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✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y‖₊ < ‖a‖₊⁻¹

Tactic:
simpa only [← coe_inv h, mem_ball_zero_iff, Metric.eball_coe] using! hy

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

Goal state before:
pp: 𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
h_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x \ σ 𝕜 ↑x)
⊢ frontier (σ 𝕜 ↑x) ∪ frontier (σ 𝕜 x) ⊆ frontier (σ 𝕜 ↑x)

Tactic:
frontier_subset_frontier,

Goal state after:
𝕜 : Type u_3
A : Type u_4
SA : Type u_5
inst✝⁵ : NormedRing A
inst✝⁴ : CompleteSpace A
inst✝³ : SetLike SA A
inst✝² : SubringClass SA A
inst✝¹ : NormedField 𝕜
inst✝ : NormedAlgebra 𝕜 A
instSMulMem : SMulMemClass SA 𝕜 A
S : SA
hS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑S
x : ↥S
this : CompleteSpace ↥S
h_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (σ 𝕜 x \ σ 𝕜 ↑x)
⊢ frontier (σ 𝕜 ↑x) ∪ frontier (σ 𝕜 ↑x) ⊆ frontier (σ 𝕜 ↑x)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a : A
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
ha : 0 ≤ a
⊢ x ≤ ‖a‖₊

Tactic:
by
  simpa [cfc_id ℝ≥0 a] using apply_le_nnnorm_cfc_nnreal id a hx

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z✝ : ℂ
hf : Differentiable ℂ (expMulMulExp a b x)
z : ℂ
⊢ ‖↑(expUnitary (2 • ℑ (z • star b))) * x * ↑(expUnitary (2 • ℑ (star z • a)))‖ = ‖x‖

Tactic:
CStarRing.norm_mul_coe_unitary,

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
a b x : A
inst✝¹ : IsStarNormal a
inst✝ : IsStarNormal b
h : SemiconjBy x a b
z✝ : ℂ
hf : Differentiable ℂ (expMulMulExp a b x)
z : ℂ
⊢ ‖↑(expUnitary (2 • ℑ (z • star b))) * x‖ = ‖x‖
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty
this : Tendsto (⇑f) (cocompact α) (𝓝 0)
⊢ f ∈ range toBCF

Tactic:
exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩

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

Goal state before:
pp: R : Type u_4
M : Type u_5
inst✝³ : SeminormedCommRing R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Function.Injective ⇑(topDualPairing R M)

Tactic:
exact ContinuousLinearMap.coe_injective

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
x✝¹ : E
x✝ : x✝¹ ∈ -s
⊢ x✝¹ ∈ balancedHull ℝ s

Tactic:
rw [mem_balancedHull_iff]

Goal state after:
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x✝¹ : E
x✝ : x✝¹ ∈ -s
⊢ ∃ r, ‖r‖ ≤ 1 ∧ x✝¹ ∈ r • s
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
h_deriv₂ :
  ∀ (a b : A),
    deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2
h₄ : a ^ 2 + 2 • (-a * star a) + star a ^ 2 = star a ^ 2 + 2 • (star a * -a) + a ^ 2
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-(x • a)) * NormedSpace.exp (x • star a) =
      NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))
h₃ :
  deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))) y) 0 =
    a ^ 2 + 2 • (-a * star a) + star a ^ 2
⊢ IsStarNormal a

Tactic:
rw [isStarNormal_iff, commute_iff_eq]

Goal state after:
case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A
h_deriv :
  ∀ (a b c : A) (y : ℝ),
    deriv (fun x ↦ NormedSpace.exp (x • a) * c * NormedSpace.exp (x • b)) y =
      NormedSpace.exp (y • a) * (a * c + c * b) * NormedSpace.exp (y • b)
h_deriv₂ :
  ∀ (a b : A),
    deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • a) * NormedSpace.exp (x • b)) y) 0 = a ^ 2 + 2 • (a * b) + b ^ 2
h₄ : a ^ 2 + 2 • (-a * star a) + star a ^ 2 = star a ^ 2 + 2 • (star a * -a) + a ^ 2
key :
  ∀ (x : ℝ),
    NormedSpace.exp (-(x • a)) * NormedSpace.exp (x • star a) =
      NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))
h₃ :
  deriv (fun y ↦ deriv (fun x ↦ NormedSpace.exp (x • star a) * NormedSpace.exp (-(x • a))) y) 0 =
    a ^ 2 + 2 • (-a * star a) + star a ^ 2
⊢ star a * a = a * star a
Module: Mathlib.Analysis.Normed.Algebra.Basic

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ φ ∈ ⇑toStrongDual ⁻¹' Metric.closedBall 0 ‖1‖

Tactic:
rw [Set.mem_preimage, mem_closedBall_zero_iff]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedRing A
inst✝² : NormedAlgebra 𝕜 A
inst✝¹ : CompleteSpace A
inst✝ : ProperSpace 𝕜
φ : WeakDual 𝕜 A
hφ : φ ∈ characterSpace 𝕜 A
⊢ ‖toStrongDual φ‖ ≤ ‖1‖
Module: Mathlib.LinearAlgebra.AffineSpace.Ordered

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
h : 0 < r * (b - a)
⊢ f (r * (b - a) + a) ≤ r • (f b - f a) + f a ↔
    (b - a)⁻¹ • r⁻¹ • f (r * (b - a) + a) ≤ (b - a)⁻¹ • (f b - f a) + (b - a)⁻¹ • r⁻¹ • f a

Tactic:
← smul_add,

Goal state after:
k : Type u_1
E : Type u_2
inst✝⁷ : Field k
inst✝⁶ : LinearOrder k
inst✝⁵ : AddCommGroup E
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : Module k E
inst✝¹ : IsStrictOrderedModule k E
inst✝ : PosSMulReflectLE k E
f : k → E
a b r : k
h : 0 < r * (b - a)
⊢ f (r * (b - a) + a) ≤ r • (f b - f a) + f a ↔
    (b - a)⁻¹ • r⁻¹ • f (r * (b - a) + a) ≤ (b - a)⁻¹ • (f b - f a + r⁻¹ • f a)
Module: Mathlib.Analysis.CStarAlgebra.GelfandDuality

Goal state before:
pp: A : Type u_1
inst✝ : CommCStarAlgebra A
rng : StarSubalgebra ℂ C(↑(characterSpace ℂ A), ℂ) :=
  let __Subalgebra := (gelfandTransform ℂ A).range;
  { toSubalgebra := __Subalgebra, star_mem' := ⋯ }
h : rng.topologicalClosure = rng
x✝¹ x✝ : ↑(characterSpace ℂ A)
⊢ (∀ f ∈ (fun f ↦ ⇑f) '' ↑rng.toSubalgebra, f x✝¹ = f x✝) → x✝¹ = x✝

Tactic:
exact fun h => Subtype.ext (ContinuousLinearMap.ext fun a =>
    h (gelfandTransform ℂ A a) ⟨gelfandTransform ℂ A a, ⟨a, rfl⟩, rfl⟩)

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

Goal state before:
pp: case pos
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
r : ℝ≥0
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) (↑r • a) = ↑r • cfcₙ (fun x ↦ x⁺) a

Tactic:
← cfcₙ_comp_smul ..,

Goal state after:
case pos
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
r : ℝ≥0
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ (↑r • x)⁺) a = ↑r • cfcₙ (fun x ↦ x⁺) a
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ tsupport c.lim ⊆ s

Tactic:
apply Subset.trans _ (compl_subset_comm.mp hscompl_subset_u)
    rw [← IsClosed.closure_eq (isClosed_compl_iff.mpr huIsOpen)]
    apply closure_mono
    exact Disjoint.subset_compl_right (disjoint_of_subset_right subset_closure
      (Disjoint.symm (Urysohns.CU.disjoint_C_support_lim c)))

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
⊢ 1⁺ = 1

Tactic:
by
  rw [CFC.posPart_def, cfcₙ_eq_cfc]
  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
⊢ ((algebraMap ℝ A) r)⁺ = (algebraMap ℝ A) r⁺

Tactic:
by
  rw [CFC.posPart_def, cfcₙ_eq_cfc]
  simp

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«0».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)

pp: case h.left.«0».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖

Tactic:
· simp
        cfc_tac

Goal state after:
case h.left.«0».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨0, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«1».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)

pp: case h.left.«1».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)‖ ≤ ‖a‖

Tactic:
· simp
        cfc_tac

Goal state after:
case h.left.«1».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨1, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«2».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)

pp: case h.left.«2».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖

Tactic:
· simp
        cfc_tac

Goal state after:
case h.left.«2».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨2, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart

Goal state before:
pp: case h.left.«3».left
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ 0 ≤ ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)

pp: case h.left.«3».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)‖ ≤ ‖a‖

Tactic:
· simp
        cfc_tac

Goal state after:
case h.left.«3».right
A : Type u_2
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ ‖![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] ((fun i ↦ i) ⟨3, ⋯⟩)‖ ≤ ‖a‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case inr
X : Type u_1
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 : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
hf : ContinuousOn f (spectrum R a)
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

Tactic:
by_cases ha : p a

Goal state after:
case pos
X : Type u_1
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 : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))

case neg
X : Type u_1
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 : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (spectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (spectrum R a)
hl : l.NeBot
hf : ContinuousOn f (spectrum R a)
ha : ¬p a
⊢ Tendsto (fun x ↦ cfc (F x) a) l (𝓝 (cfc f a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 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
⊢ LipschitzOnWith 1 (fun f ↦ cfc ((toFun {spectrum R a}) f) a)
    {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}

Tactic:
by_cases ha : p a

Goal state after:
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
⊢ LipschitzOnWith 1 (fun f ↦ cfc ((toFun {spectrum R a}) f) a)
    {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}

case neg
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
⊢ LipschitzOnWith 1 (fun f ↦ cfc ((toFun {spectrum R a}) f) a)
    {f | ContinuousOn ((toFun {spectrum R a}) f) (spectrum R a)}
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 : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) ≤ a • f x + b • f y

Tactic:
← Unitization.inr_le_iff _ _

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConvexOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ↑(f (a • x + b • y)) ≤ ↑(a • f x + b • f y)
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 : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x + b • f y ≤ f (a • x + b • y)

Tactic:
← Unitization.inr_le_iff _ _

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : A → A
s : Set A
hf : ∀ (x : A), IsSelfAdjoint (f x)
hf₂ : ConcaveOn ℝ s (inr ∘ f)
x : A
hx : x ∈ s
y : A
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ ↑(a • f x + b • f y) ≤ ↑(f (a • x + b • y))
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) ^ (-1)
⊢ (a ^ x)⁻¹ʳ = a ^ (-x)

Tactic:
← inverse_eq_rpow_neg_one (by grind)

Goal state after:
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)