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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
x : X
⊢ ∀ (y : X), U x y ∈ 𝓝 y

Tactic:
y

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := fun x y ↦ {z | f z - ε < (g x y) z}
x y : X
⊢ U x y ∈ 𝓝 y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case inr
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
r : ℝ
hr : r ≤ 0
⊢ φ ((algebraMap ℝ C(X, ℝ)) (- -r)).toNNReal - φ ((algebraMap ℝ C(X, ℝ)) (-r)).toNNReal = (algebraMap ℝ A) (- -r)

Tactic:
← neg_nonneg

Goal state after:
case inr
X : Type u_1
inst✝⁵ : TopologicalSpace X
A : Type u_2
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
r : ℝ
hr : 0 ≤ -r
⊢ φ ((algebraMap ℝ C(X, ℝ)) (- -r)).toNNReal - φ ((algebraMap ℝ C(X, ℝ)) (-r)).toNNReal = (algebraMap ℝ A) (- -r)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case e'_5
X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ f

Tactic:
congr

Goal state after:
case e'_5.e_6
X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ ({ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case e'_6
X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = φ 0

Tactic:
congr

Goal state after:
case e'_6.e_6
X : Type u_1
inst✝³ : TopologicalSpace X
A : Type u_2
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra ℝ A
φ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A
f : C(X, ℝ≥0)
⊢ (-{ toFun := toReal, continuous_toFun := continuous_coe }.comp f).toNNReal = 0
Module: Mathlib.Analysis.Real.Spectrum

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
⊢ SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
by
  refine ⟨fun h x hx ↦ ?_, fun h ↦ ?_⟩
  · obtain ⟨x, -, rfl⟩ := h.algebraMap_image.symm ▸ hx
    exact coe_nonneg x
  · exact .of_subset_range_algebraMap (fun _ ↦ Real.toNNReal_coe) fun x hx ↦ ⟨⟨x, h x hx⟩, rfl⟩

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

Goal state before:
pp: X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := compStarAlgHom' ℝ ℝ { toFun := Subtype.val, continuous_toFun := ⋯ }
AK : Subalgebra ℝ C(↑K, ℝ) := Subalgebra.map (↑restrict_on_K) A
x : ↑K
⊢ ∀ ⦃y : ↑K⦄, x ≠ y → ∃ f ∈ (fun f ↦ ⇑f) '' ↑AK, f x ≠ f y

Tactic:
y

Goal state after:
X : Type u_2
inst✝ : TopologicalSpace X
A : Subalgebra ℝ C(X, ℝ)
hA : A.SeparatesPoints
f : C(X, ℝ)
K : Set X
hK : IsCompact K
ε : ℝ
pos : 0 < ε
restrict_on_K : C(X, ℝ) →⋆ₐ[ℝ] C(↑K, ℝ) := compStarAlgHom' ℝ ℝ { toFun := Subtype.val, continuous_toFun := ⋯ }
AK : Subalgebra ℝ C(↑K, ℝ) := Subalgebra.map (↑restrict_on_K) A
x y : ↑K
⊢ x ≠ y → ∃ f ∈ (fun f ↦ ⇑f) '' ↑AK, f x ≠ f y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case e'_5
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ f

Tactic:
congr

Goal state after:
case e'_5.e_6
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ ({ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case e'_6
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ φ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = φ 0

Tactic:
congr

Goal state after:
case e'_6.e_6
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : Zero X
A : Type u_2
inst✝² : NonUnitalRing A
inst✝¹ : StarRing A
inst✝ : Module ℝ A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f : C(X, ℝ≥0)₀
⊢ (-{ toFun := toReal, continuous_toFun := continuous_coe, map_zero' := ⋯ }.comp f).toNNReal = 0
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case calc_1
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ BddAbove
    ((fun x_1 ↦ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) x_1‖) ''
      Metric.closedBall 0 1)

pp: case calc_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) b‖ =
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖

Tactic:
· refine ⟨‖(star x * x).fst‖ + ‖(star x * x).snd‖, ?_⟩
    rintro _ ⟨y, hy, rfl⟩
    refine (norm_add_le _ _).trans ?_
    gcongr
    · rw [Algebra.algebraMap_eq_smul_one]
      refine (norm_smul _ _).trans_le ?_
      simpa only [mul_one] using!
        mul_le_mul_of_nonneg_left (mem_closedBall_zero_iff.1 hy) (norm_nonneg (star x * x).fst)
    · exact (unit_le_opNorm _ y <| mem_closedBall_zero_iff.1 hy).trans (opNorm_mul_apply_le _ _ _)

Goal state after:
case calc_2
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖((algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2) b‖ =
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : CompactSpace ↑s
⊢ (adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}).topologicalClosure =
    (adjoin 𝕜 {(toContinuousMapOnAlgHom s) X}).topologicalClosure

Tactic:
congr

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : CompactSpace ↑s
⊢ ContinuousMap.id 𝕜 = X.toContinuousMap
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ characterSpace 𝕜 A ↔ φ ∈ {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y}

Tactic:
refine ⟨?_, ?_⟩

Goal state after:
case refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ characterSpace 𝕜 A → φ ∈ {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y}

case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : CommRing 𝕜
inst✝⁷ : NoZeroDivisors 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra 𝕜 A
inst✝ : Nontrivial 𝕜
φ : WeakDual 𝕜 A
⊢ φ ∈ {φ | φ 1 = 1 ∧ ∀ (x y : A), φ (x * y) = φ x * φ y} → φ ∈ characterSpace 𝕜 A
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁹ : CommSemiring R
inst✝²⁸ : Nontrivial R
inst✝²⁷ : StarRing R
inst✝²⁶ : MetricSpace R
inst✝²⁵ : IsTopologicalSemiring R
inst✝²⁴ : ContinuousStar R
inst✝²³ : CommRing S
inst✝²² : Algebra R S
inst✝²¹ : NonUnitalRing A
inst✝²⁰ : StarRing A
inst✝¹⁹ : TopologicalSpace A
inst✝¹⁸ : Module R A
inst✝¹⁷ : IsScalarTower R A A
inst✝¹⁶ : SMulCommClass R A A
inst✝¹⁵ : NonUnitalRing B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Module R B
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : SMulCommClass R B B
inst✝⁹ : Module S A
inst✝⁸ : Module S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : NonUnitalContinuousFunctionalCalculus R A p
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMapZero.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (quasispectrum R a)
hf₀ : f 0 = 0
hφ : Continuous[inst✝¹⁹, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₙₐ[R] B := ⋯
this : Continuous[inst✝¹⁹, inst✝¹³] ⇑ψ
h_spec : quasispectrum R (φ a) ⊆ quasispectrum R a
hψa : q (ψ a)
ι : C(↑(quasispectrum R (ψ a)), ↑(quasispectrum R a))₀ := ⋯
⊢ (cfcₙHom hψa) (ContinuousMapZero.id (quasispectrum R (ψ a))) =
    ((cfcₙHom hψa).comp (ContinuousMapZero.nonUnitalStarAlgHom_precomp R ι)) (ContinuousMapZero.id (quasispectrum R a))

Tactic:
congr

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

Goal state before:
pp: F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| Tendsto (fun y ↦ y - φ y • 1) (𝓝 x ⊓ 𝓟 ↑s) (𝓝 x)

Tactic:
congr

Goal state after:
case f
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| fun y ↦ y - φ y • 1

case l₁
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x ⊓ 𝓟 ↑s

case l₂
F : Type u_2
S : Type u_3
K : Type u_4
A : Type u_5
inst✝¹³ : CommRing K
inst✝¹² : Ring A
inst✝¹¹ : Algebra K A
inst✝¹⁰ : TopologicalSpace K
inst✝⁹ : T1Space K
inst✝⁸ : TopologicalSpace A
inst✝⁷ : ContinuousSub A
inst✝⁶ : ContinuousSMul K A
inst✝⁵ : FunLike F A K
inst✝⁴ : AlgHomClass F K A K
inst✝³ : SetLike S A
inst✝² : OneMemClass S A
inst✝¹ : AddSubgroupClass S A
inst✝ : SMulMemClass S K A
φ : F
hφ : Continuous[inst✝⁸, inst✝¹⁰] ⇑φ
s : S
x : A
hxs : (𝓝 x ⊓ 𝓟 ↑s).NeBot
hxφ : φ x = 0
| 𝓝 x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := ⋯
this : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := ⋯
⊢ (cfcHom hψa) (ContinuousMap.restrict (spectrum R (ψ a)) (ContinuousMap.id R)) =
    ((cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι))
      (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R))

Tactic:
congr

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ hausdorffEntourage U ∈ ((𝓤 α).lift' hausdorffEntourage).lift' fun s ↦ SetRel.comp s s

Tactic:
obtain ⟨V, hV, hVU⟩ := comp_mem_uniformity_sets hU

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : UniformSpace α
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
U : Set (α × α)
hU : U ∈ 𝓤 α
V : Set (α × α)
hV : V ∈ 𝓤 α
hVU : SetRel.comp V V ⊆ U
⊢ hausdorffEntourage U ∈ ((𝓤 α).lift' hausdorffEntourage).lift' fun s ↦ SetRel.comp s s
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
s : Set α
hs : IsCompact s
⊢ 𝓝 s = 𝓝 s

Tactic:
refine le_antisymm ?_ hs.totallyBounded.nhds_vietoris_le_nhds_hausdorff
  simp_rw [TopologicalSpace.nhds_generateFrom, le_iInf₂_iff, Filter.le_principal_iff]
  rintro _ ⟨hs', (⟨U, hU, rfl⟩ | ⟨U, hU, rfl⟩)⟩
  · obtain ⟨V : SetRel α α, hV₁, hV₂⟩ :=
      hs.nhdsSet_basis_uniformity (𝓤 α).basis_sets |>.mem_iff.mp (hU.mem_nhdsSet.mpr hs')
    filter_upwards [UniformSpace.ball_mem_nhds _ (Filter.mem_lift' hV₁)]
      with t ⟨_, ht⟩
    exact ht.trans fun x ⟨y, hy, hxy⟩ => hV₂ <| Set.mem_biUnion hy hxy
  · exact (UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen hU).mem_nhds hs'

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
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ f a‖

Tactic:
obtain ⟨x, hx⟩ := NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrum a
      |>.image_of_continuousOn hf.norm |>.exists_isGreatest <|
      (quasispectrum.nonempty 𝕜 a).image _

Goal state after:
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
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : ℝ
hx : IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) x
⊢ IsGreatest ((fun x ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ f a‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : ∀ x ∈ σₙ ℝ a, 0 ≤ x
x : ℝ
hx : 0 ≤ x
⊢ √x * √x = id x

Tactic:
simp [← sq, Real.sq_sqrt hx]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : ProperSpace 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : CompleteSpace A
a : A
⊢ spectrum 𝕜 ((unitizationAlgEquiv 𝕜).symm ↑a) = (spectrum 𝕜 ∘ fun x ↦ toLp 1 ↑x) a

Tactic:
congr

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
hconst : Tendsto (fun _n ↦ ↑‖a‖₊) atTop (𝓝 ↑‖a‖₊)
n : ℕ
⊢ ↑‖a‖₊ = ((fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) ∘ fun n ↦ 2 ^ n) n

Tactic:
Function.comp_apply,

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
hconst : Tendsto (fun _n ↦ ↑‖a‖₊) atTop (𝓝 ↑‖a‖₊)
n : ℕ
⊢ ↑‖a‖₊ = ↑‖a ^ 2 ^ n‖₊ ^ (1 / ↑(2 ^ n))
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

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
r : ℝ≥0
hr : ↑r < (spectralRadius 𝕜 a)⁻¹
z : 𝕜
z_mem : z ∈ Metric.closedBall 0 ↑r
hu : IsUnit (1 - z • a)
⊢ DifferentiableAt 𝕜 (fun z ↦ (1 - z • a)⁻¹ʳ) z

Tactic:
have H₁ : Differentiable 𝕜 fun w : 𝕜 => 1 - w • a := (differentiable_id.smul_const a).const_sub 1

Goal state after:
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
r : ℝ≥0
hr : ↑r < (spectralRadius 𝕜 a)⁻¹
z : 𝕜
z_mem : z ∈ Metric.closedBall 0 ↑r
hu : IsUnit (1 - z • a)
H₁ : Differentiable 𝕜 fun w ↦ 1 - w • a
⊢ DifferentiableAt 𝕜 (fun z ↦ (1 - z • a)⁻¹ʳ) z
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
n : ℕ
⊢ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n) = ((fun x ↦ x ^ 2) ∘ fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) n

Tactic:
Function.comp_apply,

Goal state after:
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
n : ℕ
⊢ ↑‖(a⋆ * a) ^ n‖₊ ^ (1 / ↑n) = (↑‖a ^ n‖₊ ^ (1 / ↑n)) ^ 2
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ liminf (fun i ↦ 1 / ↑‖a ^ i‖.toNNReal ^ (1 / ↑i)) atTop = liminf (fun n ↦ 1 / ↑(‖a ^ n‖.toNNReal ^ (1 / ↑n))) atTop

Tactic:
congr

Goal state after:
case e_u
A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ (fun i ↦ 1 / ↑‖a ^ i‖.toNNReal ^ (1 / ↑i)) = fun n ↦ 1 / ↑(‖a ^ n‖.toNNReal ^ (1 / ↑n))
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
K : Compacts α
L : Compacts β
x : α
y : β
hx : x ∈ K
hy : y ∈ L
hxy : (x, y) ∈ U
V : Set α
W : Set β
hV : IsOpen[inst✝¹] V
hW : IsOpen[inst✝] W
hxV : x ∈ V
hyW : y ∈ W
hVW : V ×ˢ W ⊆ U
⊢ ∃ t ⊆ {a | ((SetLike.coe ∘ fun p ↦ p.1 ×ˢ p.2) a ∩ V ×ˢ W).Nonempty}, IsOpen t ∧ (K, L) ∈ t

Tactic:
Function.comp_apply,

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
U : Set (α × β)
hU : U ∈ {V | IsOpen[instTopologicalSpaceProd] V}
K : Compacts α
L : Compacts β
x : α
y : β
hx : x ∈ K
hy : y ∈ L
hxy : (x, y) ∈ U
V : Set α
W : Set β
hV : IsOpen[inst✝¹] V
hW : IsOpen[inst✝] W
hxV : x ∈ V
hyW : y ∈ W
hVW : V ×ˢ W ⊆ U
⊢ ∃ t ⊆ {a | (↑(a.1 ×ˢ a.2) ∩ V ×ˢ W).Nonempty}, IsOpen t ∧ (K, L) ∈ t
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
toFun✝¹ : α → β
continuous_toFun✝¹ : Continuous[inst✝⁴, inst✝³] toFun✝¹
zero_at_infty'✝¹ : Tendsto { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹ }.toFun (cocompact α) (𝓝 0)
toFun✝ : α → β
continuous_toFun✝ : Continuous[inst✝⁴, inst✝³] toFun✝
zero_at_infty'✝ : Tendsto { toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun (cocompact α) (𝓝 0)
h :
  (fun f ↦ f.toFun) { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, zero_at_infty' := zero_at_infty'✝¹ } =
    (fun f ↦ f.toFun) { toFun := toFun✝, continuous_toFun := continuous_toFun✝, zero_at_infty' := zero_at_infty'✝ }
⊢ { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, zero_at_infty' := zero_at_infty'✝¹ } =
    { toFun := toFun✝, continuous_toFun := continuous_toFun✝, zero_at_infty' := zero_at_infty'✝ }

Tactic:
congr

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
⊢ (topDualPairing R M).ker = ⊥

Tactic:
LinearMap.ker_eq_bot

Goal state after:
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)
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
c : CU P
y : X
⊢ dist (c.lim y) (c.lim x) ≤ (3 / 4) ^ 0

Tactic:
rw [pow_zero]

Goal state after:
case zero
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Set X → Prop
h0 : 0 < 2⁻¹
h1234 : 2⁻¹ < 3 / 4
h1 : 3 / 4 < 1
x : X
x✝ : True
c : CU P
y : X
⊢ dist (c.lim y) (c.lim x) ≤ 1
Module: Mathlib.Analysis.Normed.Module.WeakDual

Goal state before:
pp: case h.right
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
w : WeakDual 𝕜 E
⊢ ∀ ⦃y : WeakDual 𝕜 E⦄, w ≠ y → ∃ n, w (denseSeq E n) ≠ y (denseSeq E n)

Tactic:
y

Goal state after:
case h.right
𝕜 : Type u_1
E : Type u_3
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : SeparableSpace E
w y : WeakDual 𝕜 E
⊢ w ≠ y → ∃ n, w (denseSeq E n) ≠ y (denseSeq E n)
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 : x₀ < b
⊢ 0 < slope f x₀ b ↔ f x₀ < f b

Tactic:
by
  simp [slope, hb]

Goal state after: