mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
f : C(↑s, 𝕜)₀
h : ∃ᶠ (g : C(↑s, 𝕜)₀) in nhds f, φ g * a = 0
⊢ Continuous fun g ↦ φ g * a

Tactic:
by fun_prop

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

Goal state before:
pp: case frequently
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : φ (ContinuousMapZero.id s) * a = 0
hmul_star_id : φ (star (ContinuousMapZero.id s)) * a = 0
hφ : Continuous ⇑φ
f : C(↑s, 𝕜)₀
h : ∃ᶠ (g : C(↑s, 𝕜)₀) in nhds f, φ g * a = 0
⊢ φ f * a = 0

Tactic:
| frequently f h =>

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

Goal state before:
pp: 𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : a * φ (ContinuousMapZero.id s) = 0
hmul_star_id : a * φ (star (ContinuousMapZero.id s)) = 0
hφ : Continuous ⇑φ
f : C(↑s, 𝕜)₀
h : ∃ᶠ (g : C(↑s, 𝕜)₀) in nhds f, a * φ g = 0
⊢ Continuous fun g ↦ a * φ g

Tactic:
by fun_prop

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

Goal state before:
pp: case frequently
𝕜 : Type u_2
A : Type u_3
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalSemiring A
inst✝⁷ : Star A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : SeparatelyContinuousMul A
inst✝⁴ : T2Space A
inst✝³ : DistribMulAction 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
s : Set 𝕜
inst✝¹ : Fact (0 ∈ s)
inst✝ : CompactSpace ↑s
φ : C(↑s, 𝕜)₀ →⋆ₙₐ[𝕜] A
a : A
hmul_id : a * φ (ContinuousMapZero.id s) = 0
hmul_star_id : a * φ (star (ContinuousMapZero.id s)) = 0
hφ : Continuous ⇑φ
f : C(↑s, 𝕜)₀
h : ∃ᶠ (g : C(↑s, 𝕜)₀) in nhds f, a * φ g = 0
⊢ a * φ f = 0

Tactic:
| frequently f h =>

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
inst✝ : TopologicalSpace β
⊢ UpperHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ (u : Set β), IsOpen[inst✝] u → f x ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ u

Tactic:
simp [upperHemicontinuousOn_iff, upperHemicontinuousWithinAt_iff_forall_isOpen]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), LowerHemicontinuousOn (F n) s
x₀ : α
hx₀s : x₀ ∈ s
V : Set β
hV : IsOpen[inst✝.toTopologicalSpace] V
y₀ : β
hy₀f : y₀ ∈ f x₀
hy₀V : y₀ ∈ V
W : Set (β × β)
hW : W ∈ uniformity β
hWsub : ball y₀ W ⊆ V
U₁ : Set (β × β)
hU₁ : U₁ ∈ uniformity β
hU₁sym : SetRel.IsSymm U₁
hU₁comp : SetRel.comp U₁ U₁ ⊆ W
U : Set (β × β)
hU : U ∈ uniformity β
hUsym : SetRel.IsSymm U
hUcomp : SetRel.comp U U ⊆ U₁
hU_le_U₁ : U ⊆ U₁
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
z₀ : β
hz₀FN : z₀ ∈ F N x₀
hz₀y₀ : (y₀, z₀) ∈ U
U' : SetRel β β
hU'sub : id U' ⊆ U
hU'mem : U' ∈ uniformity β
hU'open : IsOpen[instTopologicalSpaceProd] U'
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, (f x' ∩ V).Nonempty

Tactic:
have hmeet₀ : (F N x₀ ∩ ball z₀ U').Nonempty := ⟨z₀, hz₀FN, mem_ball_self z₀ hU'mem⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), LowerHemicontinuousOn (F n) s
x₀ : α
hx₀s : x₀ ∈ s
V : Set β
hV : IsOpen[inst✝.toTopologicalSpace] V
y₀ : β
hy₀f : y₀ ∈ f x₀
hy₀V : y₀ ∈ V
W : Set (β × β)
hW : W ∈ uniformity β
hWsub : ball y₀ W ⊆ V
U₁ : Set (β × β)
hU₁ : U₁ ∈ uniformity β
hU₁sym : SetRel.IsSymm U₁
hU₁comp : SetRel.comp U₁ U₁ ⊆ W
U : Set (β × β)
hU : U ∈ uniformity β
hUsym : SetRel.IsSymm U
hUcomp : SetRel.comp U U ⊆ U₁
hU_le_U₁ : U ⊆ U₁
hHU : hausdorffEntourage U ∈ uniformity (Set β)
N : ι
hN : ∀ x ∈ s, (f x, F N x) ∈ hausdorffEntourage U
z₀ : β
hz₀FN : z₀ ∈ F N x₀
hz₀y₀ : (y₀, z₀) ∈ U
U' : SetRel β β
hU'sub : id U' ⊆ U
hU'mem : U' ∈ uniformity β
hU'open : IsOpen[instTopologicalSpaceProd] U'
hmeet₀ : (F N x₀ ∩ ball z₀ U').Nonempty
⊢ ∀ᶠ (x' : α) in 𝓝[s] x₀, (f x' ∩ V).Nonempty
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    fun r ↦ ↑ₐ r - a

Tactic:
by fun_prop

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

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
s t : Set α
inst✝ : T1Space α
⊢ (∀ x ∈ s, ∃ y ∈ t, x = y) ∧ t ⊆ closure[inst✝¹] s ↔ s ⊆ t ∧ t ⊆ closure[inst✝¹] s

Tactic:
existsAndEq,

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
s t : Set α
inst✝ : T1Space α
⊢ (∀ x ∈ s, x ∈ t ∧ True) ∧ t ⊆ closure[inst✝¹] s ↔ s ⊆ t ∧ t ⊆ closure[inst✝¹] s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
hs : IsPreconnected s
x : α
hx : x ∈ s
this : {t | t.Nonempty ∧ t.Finite ∧ t ⊆ s} = ⋃ n, range '' univ.pi fun x ↦ s
n : ℕ+
⊢ ContinuousOn range (univ.pi fun x ↦ s)

Tactic:
by fun_prop

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

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

Tactic:
by
  apply (SpectrumRestricts.nnreal_of_nonneg ha).cfc_eq_restrict _
    NNReal.isClosedEmbedding_coe ha (.of_nonneg ha)

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

Goal state before:
pp: s t : Set ℝ
⊢ ⇑equivRealProdLm ⁻¹' (convexHull ℝ) (s ×ˢ t) = (convexHull ℝ) s ×ℂ (convexHull ℝ) t

Tactic:
rw [convexHull_prod]; rfl

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

Goal state before:
pp: A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
⊢ (spectrum ℂ a).Nonempty

Tactic:
by_contra! h
  have H₀ : resolventSet ℂ a = Set.univ := by rwa [spectrum, Set.compl_empty_iff] at h
  have H₁ : Differentiable ℂ fun z : ℂ => resolvent a z := fun z =>
    hasDerivAt_resolvent_const_left (H₀.symm ▸ Set.mem_univ z : z ∈ resolventSet ℂ a)
      |>.differentiableAt
  /- Since `resolvent a` tends to zero at infinity, by Liouville's theorem `resolvent a = 0`,
  which contradicts that `resolvent a z` is invertible. -/
  have H₃ := H₁.apply_eq_of_tendsto_cocompact 0 <| by
    simpa [Metric.cobounded_eq_cocompact] using resolvent_tendsto_cobounded a (𝕜 := ℂ)
  exact not_isUnit_zero <| H₃ ▸ (isUnit_resolvent.mp <| H₀.symm ▸ Set.mem_univ 0)

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
x : ℝ≥0
hx : x ∈ σₙ ℝ≥0 a
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
⊢ f x ≤ ‖cfcₙ f a‖₊

Tactic:
revert hx
  exact (IsGreatest.nnnorm_cfcₙ_nnreal f a hf hf0 ha |>.2 ⟨x, ·, rfl⟩)

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 → β
f : α → Compacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
x : α
⊢ Continuous[inst✝¹, inst✝] (g x)

Tactic:
by fun_prop

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

Goal state before:
pp: α✝ : Type u_1
inst✝¹ : TopologicalSpace α✝
α : Type u_1
inst✝ : TopologicalSpace α
K L : Compacts α
hK : K ≠ ⊥
hL : IsPreconnected ↑L
hKL : K ≤ L
⊢ ContinuousOn (fun x ↦ K ⊔ x) {K | (↑K).Nonempty ∧ ↑K ⊆ ↑L}

Tactic:
by fun_prop

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

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
inst✝¹ : FunLike F α β
inst✝ : ZeroAtInftyContinuousMapClass F α β
f : α →C₀ β
f' : α → β
h : f' = ⇑f
⊢ Continuous[inst✝⁴, inst✝³] ⇑f

Tactic:
exact f.continuous_toFun

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✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f : α → β
inst✝ : LocallyConnectedSpace α
basis :
  IsTopologicalBasis
    ((fun u ↦ {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}) ''
      {u | u.Finite ∧ u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}})
u : Set (Set α)
hu' : u.Finite
hu : u ⊆ {s | IsOpen[inst✝³] s ∧ IsPreconnected s}
this✝ : Fintype ↑u := hu'.fintype
this :
  {K | ↑K ⊆ ⋃₀ u ∧ (∀ U ∈ u, (↑K ∩ U).Nonempty) ∧ (↑K).Finite} =
    Finset.univ.sup '' univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U}
⊢ ContinuousOn Finset.univ.sup (univ.pi fun U ↦ {K | (↑K).Nonempty ∧ (↑K).Finite ∧ ↑K ⊆ ↑U})

Tactic:
by fun_prop

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
⊢ ‖expMulMulExp a b x z - 0‖ = ‖x‖

Tactic:
sub_zero,

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
⊢ ‖expMulMulExp a b x z‖ = ‖x‖
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 → β
f : α → NonemptyCompacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
x : α
⊢ Continuous[inst✝¹, inst✝] (g x)

Tactic:
by fun_prop

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

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
h : LocallyConnectedSpace (NonemptyCompacts α)
x : α
U : Set α
x✝ : x ∈ U ∧ IsOpen[inst✝] U
hx : x ∈ U
hU : IsOpen[inst✝] U
V : Set (NonemptyCompacts α)
hV₁ : IsOpen V
hV₂ : IsPreconnected V
hxV : {x} ∈ V
hKV : V ⊆ {K | ↑K ⊆ U}
⊢ ContinuousOn SetLike.coe V

Tactic:
by fun_prop

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

Goal state before:
pp: 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✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
inst✝⁹ : AddCommGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₁ E
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : UniformSpace E
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup E
inst✝² : IsUniformAddGroup F
inst✝¹ : ContinuousSMul 𝕜₁ E
inst✝ : BarrelledSpace 𝕜₁ E
𝓕 : ι → E →SL[σ₁₂] F
q : SeminormFamily 𝕜₂ F κ
hq : WithSeminorms q
H : ∀ (k : κ) (x : E), BddAbove (range fun i ↦ (q k) ((𝓕 i) x))
⊢ UniformEquicontinuous (DFunLike.coe ∘ 𝓕)

Tactic:
refine (hq.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup (toLinearMap ∘ 𝓕)).mpr ?_
  intro k
  -- By assumption the supremum `⊔ i, q k (𝓕 i x)` is well-defined for all `x`, hence the
  -- supremum `⊔ i, (q k) ∘ (𝓕 i)` is well defined in the lattice of seminorms.
  have : BddAbove (range fun i ↦ (q k).comp (𝓕 i).toLinearMap) := by
    rw [Seminorm.bddAbove_range_iff]
    exact H k
  -- By definition of the lattice structure on seminorms, `⊔ i, (q k) ∘ (𝓕 i)` is the *pointwise*
  -- supremum of the continuous seminorms `(q k) ∘ (𝓕 i)`. Since `E` is barrelled, this supremum
  -- is continuous.
  exact ⟨this, Seminorm.continuous_iSup _
    (fun i ↦ (hq.continuous_seminorm k).comp (𝓕 i).continuous) this⟩

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

Goal state before:
pp: X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
I : Ideal C(X, R)
x : X
⊢ x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0

Tactic:
by
  rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_ofPred]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
⊢ ContinuousOn (ofReal ∘ (fun x ↦ √x) ∘ ⇑re ∘ fun z ↦ z.2 - z.1) {x | x.1 ≤ x.2}

Tactic:
by fun_prop

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

Goal state before:
pp: ⊢ ContinuousOn (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) {x | x.1 ≤ x.2}

Tactic:
by fun_prop

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
⊢ ‖(1 - (↑(algebraMapCLM ℝ≥0 𝕜)).comp g) x‖₊ = ‖1 - (algebraMap ℝ≥0 𝕜) (g x)‖₊

Tactic:
simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
              Pi.one_apply, Function.comp_apply, algebraMapCLM_apply]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed[inst✝²] k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous[inst✝², _] f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 0 s
hft : EqOn (⇑{ toFun := f, continuous_toFun := hf }) 1 (interior k)ᶜ
h'f : ∀ (x : X), { toFun := f, continuous_toFun := hf } x ∈ Icc 0 1
A : t ⊆ (interior k)ᶜ
⊢ Continuous[inst✝², _] fun x ↦ 1 - f x

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
h's : IsGδ s
ht : IsClosed[inst✝²] t
hd : Disjoint s t
U : ℕ → Set X
U_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)
hU : s = ⋂ n, U n
m : Set X
m_comp : IsCompact m
sm : s ⊆ interior m
mt : m ⊆ tᶜ
f : ℕ → C(X, ℝ)
fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s
fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ
_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)
f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1
u : ℕ → ℝ
u_pos : ∀ (i : ℕ), 0 < u i
u_sum : Summable u
hu : ∑' (i : ℕ), u i = 1
g : X → ℝ := fun x ↦ ∑' (n : ℕ), u n * (f n) x
hgmc : EqOn g 0 mᶜ
I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n
S : ∀ (x : X), Summable fun n ↦ u n * (f n) x
n : ℕ
⊢ Continuous[inst✝², _] fun x ↦ u n * (f n) x

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: 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 : Set X
⊢ ∀ {u : Set X},
    IsClosed[inst✝¹] c →
      P c u →
        IsOpen[inst✝¹] u →
          c ⊆ u → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u ∧ P c v ∧ P (closure[inst✝¹] v) u

Tactic:
u0

Goal state after:
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 u0 : Set X
⊢ IsClosed[inst✝¹] c →
    P c u0 →
      IsOpen[inst✝¹] u0 →
        c ⊆ u0 → ∃ v, IsOpen[inst✝¹] v ∧ c ⊆ v ∧ closure[inst✝¹] v ⊆ u0 ∧ P c v ∧ P (closure[inst✝¹] v) u0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
hr : r ≤ 0
a : A
⊢ (- -r • a)⁺ = -r • a⁻

Tactic:
rw [neg_smul, ← smul_neg, posPart_smul_of_nonneg (neg_nonneg.mpr hr), posPart_neg]

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

Goal state before:
pp: 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
hr : 0 ≤ r
a : A
| (r • a)⁻

Tactic:
← neg_neg r,

Goal state after:
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
hr : 0 ≤ r
a : A
| (- -r • a)⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
hr : r ≤ 0
a : A
| (r • a)⁻

Tactic:
← neg_neg r,

Goal state after:
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
hr : r ≤ 0
a : A
| (- -r • a)⁻
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
⊢ Continuous fun x ↦ (↑x)⁺

Tactic:
by fun_prop

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

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
x : ℂ
⊢ x ∈ σₙ ℂ a → 0 ≤ x

Tactic:
by
    rw [mem_quasispectrum_iff]
    refine (Or.elim · ge_of_eq fun hx ↦ ?_)
    obtain ⟨y, hy, rfl⟩ := (IsSelfAdjoint.of_nonneg ha).spectrumRestricts.algebraMap_image ▸ hx
    simpa using spectrum_nonneg_of_nonneg ha hy

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

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ ‖sqrt a‖₊ = NNReal.sqrt ‖a‖₊

Tactic:
sqrt_eq_nnrpow,

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
⊢ ‖a ^ (1 / 2)‖₊ = NNReal.sqrt ‖a‖₊
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric

Goal state before:
pp: case inl
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
hx' : 0 ≤ x
⊢ ‖x‖ ≤ max ‖a⁺‖ ‖a⁻‖

pp: case inr
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
hx' : x ≤ 0
⊢ ‖x‖ ≤ max ‖a⁺‖ ‖a⁻‖

Tactic:
· apply le_max_of_le_left
    refine le_of_eq_of_le ?_ <| norm_apply_le_norm_cfcₙ _ a hx
    rw [posPart_eq_self.mpr hx']

Goal state after:
case inr
A : Type u_1
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace ℝ A
inst✝³ : SMulCommClass ℝ A A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : StarRing A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
x : ℝ
hx : x ∈ quasispectrum ℝ a
hx' : x ≤ 0
⊢ ‖x‖ ≤ max ‖a⁺‖ ‖a⁻‖