mathlib-initiative/mathlib-tactics huggingface dataset:
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 g : α → Set β
x : α
hf : UpperHemicontinuousWithinAt f univ x
hg : UpperHemicontinuousWithinAt g univ x
⊢ UpperHemicontinuousWithinAt (fun x ↦ f x ∪ g x) univ x

Tactic:
exact hf.union hg

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
⊢ IsClopen {∅}

Tactic:
rw [← powerset_empty]
  exact ⟨isClosed_empty.powerset_vietoris, isOpen_empty.powerset_vietoris⟩

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a.refine_1
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
x : α
hx : x ∈ s
z : α
hz : z ∈ t
y : α
hxy : (x, y) ∈ U
hyz : (y, z) ∈ V
⊢ x ∈ U.preimage (U.image s ∩ V.preimage t)

Tactic:
exact ⟨y, ⟨⟨x, hx, hxy⟩, ⟨z, hz, hyz⟩⟩, hxy⟩

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
μ : 𝕜
h : IsUnit ((algebraMap 𝕜 A) μ - ↑x)
hμ₁ : ClusterPt μ (𝓟 (σ 𝕜 x)ᶜ)
hμ₂ : μ ∉ (σ 𝕜 x)ᶜ
⊢ map (fun x_1 ↦ (algebraMap 𝕜 ↥S) x_1 - x) (𝓝 μ) ≤ 𝓝 ((algebraMap 𝕜 ↥S) μ - x)

Tactic:
by rw [← Filter.Tendsto, ← ContinuousAt]; fun_prop

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝ : CompleteSpace A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ (cfcₙAux ⋯ a ha) f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A)

Tactic:
have h₁ := (continuous_cfcₙAux hp₁ a ha).range_subset_closure_image_dense
    (ContinuousMapZero.adjoin_id_dense (σₙ 𝕜 a)) ⟨f, rfl⟩
  rw [← SetLike.mem_coe]
  refine closure_minimal ?_ ?_ h₁
  · rw [← NonUnitalStarSubalgebra.coe_map, SetLike.coe_subset_coe, NonUnitalStarSubalgebra.map_le]
    apply NonUnitalStarAlgebra.adjoin_le
    apply Set.singleton_subset_iff.mpr
    rw [SetLike.mem_coe, NonUnitalStarSubalgebra.mem_comap, cfcₙAux_id hp₁ a ha]
    exact ⟨a, rfl⟩
  · simp only [NonUnitalStarAlgHom.coe_range]
    convert! IsClosed.preimage (Unitization.continuous_fst (𝕜 := 𝕜)) isClosed_singleton
    aesop

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}

Tactic:
change closure (SetLike.coe ⁻¹' {K : Set α | K.Finite ∧ K ⊆ s}) =
    SetLike.coe ⁻¹' (closure s).powerset
  rw [isEmbedding_coe.closure_eq_preimage_closure_image, image_preimage_eq_of_subset ?_,
    vietoris.closure_finite_subsets]
  exact fun K ⟨hK, _⟩ => ⟨⟨K, hK.isCompact⟩, rfl⟩

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 \ σ 𝕜 ↑x) ⊆ frontier (σ 𝕜 ↑x)

Tactic:
grw [sdiff_eq_compl_inter, frontier_inter_subset, inter_subset_left, inter_subset_right,
    frontier_compl, frontier_subset_frontier, union_self]

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

Goal state before:
pp: 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
a : A
f : C(↑(σₙ 𝕜 a), 𝕜)₀
ha : p a
⊢ ‖(cfcₙHom ⋯) f‖ = ‖f‖

Tactic:
by
  refine isometry_cfcₙHom a |>.norm_map_of_map_zero (map_zero _) f

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

Goal state before:
pp: 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
a : A
ha : p a
⊢ IsGreatest ((fun x ↦ ‖x‖₊) '' σₙ 𝕜 a) ‖a‖₊

Tactic:
by
  simpa only [cfcₙ_id 𝕜 a] using! IsGreatest.nnnorm_cfcₙ (id : 𝕜 → 𝕜) a

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] V
hu : u.Finite
huB : u ⊆ B
huV : ∀ U ∈ u, U ⊆ V
K : Compacts α
hKu : ∀ U ∈ u, (↑K ∩ U).Nonempty
w : Set (Set α)
hKV : ↑K ⊆ ⋃₀ w
hwB : w ⊆ B
hwV : ⋃₀ w ⊆ V
hwK : ∀ W ∈ w, (↑K ∩ W).Nonempty
hw : w.Finite
huw : u ⊆ w
⊢ w ∈ {u | u.Finite ∧ u ⊆ B} ∧
    K ∈ {K | ↑K ⊆ ⋃₀ w ∧ ∀ U ∈ w, (↑K ∩ U).Nonempty} ∧
      {K | ↑K ⊆ ⋃₀ w ∧ ∀ U ∈ w, (↑K ∩ U).Nonempty} ⊆ SetLike.coe ⁻¹' {s | s ⊆ V ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
grind (splits := 12)

Goal state after:
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
hU : U ∈ nhds 0
W : Set E
hW₁ : W ∈ nhds 0
hW₂ : ∀ v ∈ W, ∀ w ∈ W, v + w ∈ U
⊢ ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U

Tactic:
obtain ⟨V, hV₁,hV₂, hV₃⟩ := (locallyConvexSpace_iff_exists_convex_subset_zero ℝ E).mp ‹_› W hW₁

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
W : Set E
hW₁ : W ∈ nhds 0
hW₂ : ∀ v ∈ W, ∀ w ∈ W, v + w ∈ U
V : Set E
hV₁ : V ∈ nhds 0
hV₂ : Convex ℝ V
hV₃ : V ⊆ W
⊢ ∃ t, t.Finite ∧ (convexHull ℝ) s ⊆ ⋃ y ∈ t, y +ᵥ U
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) • y = (fun x ↦ r • x) y

Tactic:
ring_nf

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Completeness

Goal state before:
pp: case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
⊢ ∃ C, ∀ (x : E'), ‖(linearMapOfMemClosureRangeCoe f ⋯) x‖ ≤ C * ‖x‖

Tactic:
rcases isBounded_iff_forall_norm_le.1 hs with ⟨C, hC⟩

Goal state after:
case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f✝ g : E →SL[σ₁₂] F
E' : Type u_6
inst✝² : SeminormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
inst✝ : RingHomIsometric σ₁₂
f : E' → F
s : Set (E' →SL[σ₁₂] F)
hs : Bornology.IsBounded s
hf : f ∈ closure[Pi.topologicalSpace] (DFunLike.coe '' s)
C : ℝ
hC : ∀ x ∈ s, ‖x‖ ≤ C
⊢ ∃ C, ∀ (x : E'), ‖(linearMapOfMemClosureRangeCoe f ⋯) x‖ ≤ C * ‖x‖
Module: Mathlib.Analysis.LocallyConvex.Barrelled

Goal state before:
pp: α : Type u_1
ι : Type u_2
κ : Type u_3
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_6
F : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜₁
inst✝⁹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝⁸ : RingHomIsometric σ₁₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜₁ E
inst✝ : BaireSpace E
p : Seminorm 𝕜₁ E
hp : LowerSemicontinuous ⇑p
h₁ : ∀ (n : ℕ), IsClosed[inst✝³] (p.closedBall 0 ↑n)
h₂ : ⋃ n, p.closedBall 0 ↑n = univ
n : ℕ
x : E
hxn : x ∈ interior {y | p y ≤ ↑n}
⊢ p x ≤ ↑n

Tactic:
convert! interior_subset hxn

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

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ a < (lineMap a b) r ↔ 0 < r

Tactic:
rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero]

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

Goal state before:
pp: k : Type u_1
E : Type u_2
inst✝⁷ : Ring k
inst✝⁶ : LinearOrder k
inst✝⁵ : IsStrictOrderedRing k
inst✝⁴ : AddCommGroup E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : Module k E
inst✝ : IsStrictOrderedModule k E
a b : E
r : k
h : a < b
⊢ (lineMap a b) r < a ↔ r < 0

Tactic:
rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed[inst✝¹] s
ht : IsClosed[inst✝¹] t
hd : Disjoint s t
P : Set X → Set X → Prop := fun x x_1 ↦ True
c u : Set X
c_closed : IsClosed[inst✝¹] c
u_open : IsOpen[inst✝¹] u
cu : c ⊆ u
v : Set X
v_open : IsOpen[inst✝¹] v
cv : c ⊆ v
hv : closure[inst✝¹] v ⊆ u
⊢ ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u ∧ P c v ∧ P (closure[inst✝¹] v) u

Tactic:
exact ⟨v, v_open, cv, hv, trivial, trivial⟩

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✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f : α →ᵤ[𝔖] β
⊢ ∃ y, (ofFun 𝔖) (UniformFun.toFun y) = f

Tactic:
exact ⟨UniformFun.ofFun (toFun 𝔖 f), by 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 ℂ A
inst✝⁷ : SMulCommClass ℂ A A
inst✝⁶ : IsScalarTower ℂ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
⊢ span ℂ {a | 0 ≤ a} = ⊤

Tactic:
by
  refine eq_top_iff.mpr fun x _ => ?_
  rw [← CStarAlgebra.linear_combination_nonneg x]
  apply_rules [sub_mem, Submodule.smul_mem, add_mem]
  all_goals
    refine subset_span ?_
    first | apply CFC.negPart_nonneg | apply CFC.posPart_nonneg

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedAlgebra ℝ A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : ContinuousStar A
inst✝ : CompleteSpace A
r : ℝ
⊢ ContinuousOn (fun x ↦ x ^ r) {a | IsStrictlyPositive a}

Tactic:
by
  refine continuousOn_id.cfc_nnreal_of_mem_nhdsSet _ (s := {0}ᶜ) ?_
  simp_rw [nhdsSet_iUnion, Filter.mem_iSup, isOpen_compl_singleton.mem_nhdsSet]
  exact fun a ha ↦ by simpa using spectrum.zero_notMem _ ha.isUnit

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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
⊢ ∀ x ∈ σₙ ℝ a, 0 ≤ x

Tactic:
by
    rw [Unitization.quasispectrum_eq_spectrum_inr' _ ℂ]
    -- should this actually be an instance on the `Unitization`? (probably scoped)
    let _ := CStarAlgebra.spectralOrder A⁺¹
    have := CStarAlgebra.spectralOrderedRing A⁺¹
    apply spectrum_nonneg_of_nonneg
    rw [StarOrderedRing.nonneg_iff] at ha ⊢
    have := AddSubmonoid.mem_map_of_mem (Unitization.inrNonUnitalStarAlgHom ℂ A) ha
    rw [AddMonoidHom.map_mclosure, ← Set.range_comp] at this
    apply AddSubmonoid.closure_mono ?_ this
    rintro _ ⟨s, rfl⟩
    exact ⟨s, by simp⟩

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
y : ℝ
ha : 0 ≤ a
⊢ a ^ y = cfc (fun x ↦ x ^ y) a

Tactic:
by
  rw [CFC.rpow_def, cfc_nnreal_eq_real ..]
  refine cfc_congr ?_
  intro x hx
  simp only [NNReal.coe_rpow, Real.coe_toNNReal']
  grind

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ
hx : x ≠ 0
ha : IsStrictlyPositive a
⊢ (a ^ x⁻¹) ^ x = a

Tactic:
by
  simpa using rpow_rpow_inv a x⁻¹ (inv_ne_zero hx)

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

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : Ring A
inst✝⁷ : StarRing A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : IsSemitopologicalRing A
inst✝ : T2Space A
a : A
x : ℝ≥0
ha : 0 ≤ a
hx : ¬x = 0
h₁ : 0 ≤ ↑x
⊢ sqrt a ^ ↑x = a ^ (↑x / 2)

Tactic:
rw [sqrt_eq_rpow, rpow_rpow_of_exponent_nonneg _ _ _ (by simp) h₁, one_div_mul_eq_div]

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
⊢ a ≤ (algebraMap ℝ A) ‖a‖

Tactic:
by
  by_cases! nontriv : Nontrivial A
  · refine le_algebraMap_of_spectrum_le fun r hr => ?_
    calc r ≤ ‖r‖ := Real.le_norm_self r
      _ ≤ ‖a‖ := spectrum.norm_le_norm_of_mem hr
  · simp

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

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

Tactic:
simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr]

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

Goal state before:
pp: l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ (Finset.univ.sup fun i ↦ ∑ k, ∑ j, ‖A i j‖₊ * ‖B j k‖₊) = Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊ * ∑ k, ‖B j k‖₊

Tactic:
@Finset.sum_comm m,

Goal state after:
l : Type u_2
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : Fintype l
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSeminormedRing α
A : Matrix l m α
B : Matrix m n α
⊢ (Finset.univ.sup fun i ↦ ∑ y, ∑ x, ‖A i y‖₊ * ‖B y x‖₊) = Finset.univ.sup fun i ↦ ∑ j, ‖A i j‖₊ * ∑ k, ‖B j k‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : CompleteSpace A
s : Set 𝕜
x✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A
f✝ : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a✝ : A
x✝ :
  (f✝, a✝) ∈
    {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ {a | p a ∧ s ∈ 𝓝ˢ (quasispectrum 𝕜 a)}
hf✝ : (f✝, a✝).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha✝ : p (f✝, a✝).2
has : s ∈ 𝓝ˢ (quasispectrum 𝕜 (f✝, a✝).2)
hs : IsCompact (quasispectrum 𝕜 a✝)
k : Set 𝕜
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum 𝕜 a✝)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum 𝕜 x)}
f : 𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜
a : A
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : (f, a).2 ∈ {a | p a ∧ s ∈ 𝓝ˢ (quasispectrum 𝕜 a)}
left✝ : (f, a).1 ∈ Set.univ
ha' : (f, a).2 ∈ {x | k ∈ 𝓝ˢ (quasispectrum 𝕜 x)}
⊢ (fun fa ↦ fa) (f, a) ∈ {f | ContinuousOn ((toFun {k}) f) k ∧ f 0 = 0} ×ˢ {a | p a ∧ quasispectrum 𝕜 a ⊆ k}

Tactic:
exact ⟨⟨hf.1.mono hks, hf.2⟩, ha.1, subset_of_mem_nhdsSet ha'⟩

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

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

Tactic:
rwa [map_star, cfcₙHom_id]

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

Goal state before:
pp: A : Type u_2
inst✝¹¹ : NonUnitalNormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedSpace ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : ContinuousStar A
inst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ f x) {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}

Tactic:
have : {a : A | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}.EqOn (cfcₙ f)
      (cfcₙ (fun x : ℝ ↦ f x.toNNReal)) :=
    fun a ha ↦ cfcₙ_nnreal_eq_real _ _ ha.1

Goal state after:
A : Type u_2
inst✝¹¹ : NonUnitalNormedRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : NormedSpace ℝ A
inst✝⁸ : IsScalarTower ℝ A A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : ContinuousStar A
inst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : T2Space A
inst✝ : IsSemitopologicalRing A
s : Set ℝ≥0
hs : IsCompact s
f : ℝ≥0 → ℝ≥0
hf : ContinuousOn f s
hf0 : f 0 = 0
this : Set.EqOn (cfcₙ f) (cfcₙ fun x ↦ ↑(f x.toNNReal)) {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
⊢ ContinuousOn (fun x ↦ cfcₙ f x) {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ s}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: A : Type u_2
inst✝¹² : NonUnitalNormedRing A
inst✝¹¹ : StarRing A
inst✝¹⁰ : NormedSpace ℝ A
inst✝⁹ : IsScalarTower ℝ A A
inst✝⁸ : SMulCommClass ℝ A A
inst✝⁷ : ContinuousStar A
inst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : NonnegSpectrumClass ℝ A
inst✝² : T2Space A
inst✝¹ : IsSemitopologicalRing A
inst✝ : CompleteSpace A
s : Set ℝ≥0
x✝¹ : (ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0) × A
f✝ : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a✝ : A
x✝ :
  (f✝, a✝) ∈
    {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0} ×ˢ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)}
hf✝ : (f✝, a✝).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha✝ : 0 ≤ (f✝, a✝).2
has : s ∈ 𝓝ˢ (quasispectrum ℝ≥0 (f✝, a✝).2)
hs : IsCompact (quasispectrum ℝ≥0 a✝)
k : Set ℝ≥0
hks : id k ⊆ s
hka : k ∈ 𝓝ˢ (quasispectrum ℝ≥0 a✝)
hk : IsCompact k
this : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}
f : ℝ≥0 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] ℝ≥0
a : A
hf : (f, a).1 ∈ {f | ContinuousOn ((toFun {t | IsCompact t ∧ t ⊆ s}) f) s ∧ f 0 = 0}
ha : (f, a).2 ∈ {a | 0 ≤ a ∧ s ∈ 𝓝ˢ (quasispectrum ℝ≥0 a)}
left✝ : (f, a).1 ∈ Set.univ
ha' : (f, a).2 ∈ {x | k ∈ 𝓝ˢ (quasispectrum ℝ≥0 x)}
⊢ (fun fa ↦ fa) (f, a) ∈ {f | ContinuousOn ((toFun {k}) f) k ∧ f 0 = 0} ×ˢ {a | 0 ≤ a ∧ quasispectrum ℝ≥0 a ⊆ k}

Tactic:
exact ⟨⟨hf.1.mono hks, hf.2⟩, ha.1, subset_of_mem_nhdsSet ha'⟩

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

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
S : Type u_4
A : Type u_5
B : Type u_6
inst✝⁵ : Unique n
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid A
inst✝² : Mul A
inst✝¹ : Star A
inst✝ : Module R A
x✝¹ x✝ : A
⊢ (fun x y ↦ x✝¹ * x✝) = (fun x y ↦ x✝¹) * fun x y ↦ x✝

Tactic:
by ext; simp [mul_apply]

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✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = (mkD ((spectrum 𝕜 a).domRestrict fun z ↦ ∫ (x : X), f x z ∂μ) 0) z

Tactic:
mkD_apply_of_continuousOn

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ∫ (x : X), f x ↑z ∂μ = ∫ (x : X), f x ↑z ∂μ

X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace X
μ : Measure X
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : ContinuousFunctionalCalculus 𝕜 A p
inst✝¹ : CompleteSpace A
inst✝ : NormedSpace ℝ A
f : X → 𝕜 → 𝕜
a : A
hf₁ : ∀ᵐ (x : X) ∂μ, ContinuousOn (f x) (spectrum 𝕜 a)
hf₂ : Integrable (fun x ↦ mkD ((spectrum 𝕜 a).domRestrict (f x)) 0) μ
ha : p a
key₁ : ∀ (z : ↑(spectrum 𝕜 a)), ∫ (x : X), f x ↑z ∂μ = (∫ (x : X), mkD ((spectrum 𝕜 a).domRestrict (f x)) 0 ∂μ) z
z : ↑(spectrum 𝕜 a)
⊢ ContinuousOn (fun z ↦ ∫ (x : X), f x z ∂μ) (spectrum 𝕜 a)