mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Algebra.Unitization

Goal state before:
pp: case refine_2
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
x : Unitization 𝕜 A
⊢ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.toProd.1 + (mul 𝕜 A) x.toProd.2‖ ≤ 2 * max ‖((addEquiv 𝕜 A) x).1‖ ‖((addEquiv 𝕜 A) x).2‖

Tactic:
· nontriviality A
    calc
      ‖algebraMap 𝕜 _ x.fst + mul 𝕜 A x.snd‖ ≤ ‖algebraMap 𝕜 _ x.fst‖ + ‖mul 𝕜 A x.snd‖ :=
        norm_add_le _ _
      _ = ‖x.fst‖ + ‖x.snd‖ := by
        rw [norm_algebraMap', (AddMonoidHomClass.isometry_iff_norm (mul 𝕜 A)).mp (isometry_mul 𝕜 A)]
      _ ≤ _ := (add_le_add (le_max_left _ _) (le_max_right _ _)).trans_eq (two_mul _).symm

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
key : IsUniformInducing ⇑(addEquiv 𝕜 A)
⊢ 𝓤 (Unitization 𝕜 A) = comap (fun x ↦ ((addEquiv 𝕜 A) x.1, (addEquiv 𝕜 A) x.2)) (𝓤 (𝕜 × A))

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
key : IsUniformInducing ⇑(addEquiv 𝕜 A)
⊢ 𝓤 (Unitization 𝕜 A) = comap (fun x ↦ ((addEquiv 𝕜 A) x.1, (addEquiv 𝕜 A) x.2)) (𝓤 (𝕜 × A))
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
⊢ sSup ((fun x ↦ ‖((mul 𝕜 E) a) x‖₊) '' Metric.closedBall 0 1) = ‖a‖₊

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
⊢ sSup ((fun x ↦ ‖((mul 𝕜 E) a) x‖₊) '' Metric.closedBall 0 1) = ‖a‖₊
Module: Mathlib.Analysis.Normed.Algebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NonUnitalNormedRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : RegularNormedAlgebra 𝕜 A
k : 𝕜
x : Unitization 𝕜 A
⊢ ‖k‖ * ‖x‖ ≤ ‖k‖ * ‖x‖

Tactic:
]

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

Goal state before:
pp: case pos
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}
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := fun x ↦ ⋯.choose
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ :=
  fun x ↦ Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhds_y x))
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := fun x ↦ ⟨(ys x).sup' ⋯ fun y ↦ g x y, ⋯⟩
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := fun x ↦ {z | ↑(h x) z < f z + ε}
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯.choose
xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhds)
xs_nonempty : xs.Nonempty
k : ↑L := ⟨xs.inf' xs_nonempty fun x ↦ ↑(h x), ⋯⟩
z : X
⊢ ↑k z < f z + ε ∧ f z - ε < ↑k z

Tactic:
]

Goal state after:
case pos
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}
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := fun x ↦ ⋯.choose
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ :=
  fun x ↦ Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhds_y x))
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := fun x ↦ ⟨(ys x).sup' ⋯ fun y ↦ g x y, ⋯⟩
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := fun x ↦ {z | ↑(h x) z < f z + ε}
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯.choose
xs_w : ⋃ x ∈ xs, W x = ⊤ := Exists.choose_spec (CompactSpace.elim_nhds_subcover W W_nhds)
xs_nonempty : xs.Nonempty
k : ↑L := ⟨xs.inf' xs_nonempty fun x ↦ ↑(h x), ⋯⟩
z : X
⊢ ↑k z < f z + ε ∧ f z - ε < ↑k z
Module: Mathlib.Analysis.Real.Spectrum

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalRing A
inst✝² : Module ℝ A
inst✝¹ : IsScalarTower ℝ A A
inst✝ : SMulCommClass ℝ A A
a : A
⊢ (∀ x ∈ spectrum ℝ ↑a, 0 ≤ x) ↔ ∀ x ∈ spectrum ℝ ↑a, 0 ≤ x

Tactic:
]

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

Goal state before:
pp: case refine_3
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
⊢ ∃ a_1 ∈ (fun x ↦ ‖((mul 𝕜 E) a) x‖₊) '' Metric.closedBall 0 1, r < a_1

Tactic:
]

Goal state after:
case refine_3
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
⊢ ∃ a_1 ∈ (fun x ↦ ‖((mul 𝕜 E) a) x‖₊) '' Metric.closedBall 0 1, r < a_1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
⊢ Set.EqOn (⇑φ) (⇑ψ) (closure ↑(adjoin 𝕜 {ContinuousMapZero.id s}))

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
⊢ Set.EqOn (⇑φ) (⇑ψ) (closure ↑(adjoin 𝕜 {ContinuousMapZero.id s}))
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: case refine_3.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
k : 𝕜
hk₁ : r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < 1 * ‖a‖₊⁻¹
⊢ r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊

Tactic:
]

Goal state after:
case refine_3.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DenselyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing E
inst✝⁴ : StarRing E
inst✝³ : CStarRing E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
a : E
r : NNReal
hr : r * ‖a‖₊⁻¹ < 1
ha : 0 < ‖a‖₊
k : 𝕜
hk₁ : r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < 1 * ‖a‖₊⁻¹
⊢ r * ‖a‖₊⁻¹ * ‖a‖₊⁻¹ < ‖k‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
f : ContinuousMapZero (↑s) 𝕜
hf : f ∈ ↑(adjoin 𝕜 {ContinuousMapZero.id s})
⊢ f ∈ NonUnitalStarAlgHom.equalizer φ ψ

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
f : ContinuousMapZero (↑s) 𝕜
hf : f ∈ ↑(adjoin 𝕜 {ContinuousMapZero.id s})
⊢ f ∈ NonUnitalStarAlgHom.equalizer φ ψ
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 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
⊢ sSup ((fun x_1 ↦ ‖((algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2) x_1‖) '' Metric.closedBall 0 1) ≤
    √‖(algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2‖

Tactic:
]

Goal state after:
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
⊢ sSup ((fun x_1 ↦ ‖((algebraMap 𝕜 (E →L[𝕜] E)) x.toProd.1 + (mul 𝕜 E) x.toProd.2) x_1‖) '' Metric.closedBall 0 1) ≤
    √‖(algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).toProd.1 + (mul 𝕜 E) (x⋆ * x).toProd.2‖
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
f : ContinuousMapZero (↑s) 𝕜
hf : f ∈ ↑(adjoin 𝕜 {ContinuousMapZero.id s})
⊢ ContinuousMapZero.id s ∈ ↑(NonUnitalStarAlgHom.equalizer φ ψ)

Tactic:
]

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
f : ContinuousMapZero (↑s) 𝕜
hf : f ∈ ↑(adjoin 𝕜 {ContinuousMapZero.id s})
⊢ ContinuousMapZero.id s ∈ ↑(NonUnitalStarAlgHom.equalizer φ ψ)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
⊢ Continuous (_root_.toContinuousMap ∘ toNNReal)

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
⊢ Continuous (_root_.toContinuousMap ∘ toNNReal)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Zero X
r : ℝ≥0
f : C(X, ℝ)₀
⊢ r • (-f).toNNReal = r • (-f).toNNReal

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : C(X, ℝ)
⊢ f ∈ ⊤

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : C(X, ℝ)
⊢ f ∈ ⊤
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
g : C(X, ℝ)
H : ‖g - f‖ < ε
m : g ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
g : C(X, ℝ)
H : ‖g - f‖ < ε
m : g ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (r • f).toNNReal - φ (-r • f).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (r • f).toNNReal - φ (-r • f).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous[inst✝¹, _] f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ∀ (x : X), ‖(↑g - { toFun := f, continuous_toFun := c }) x‖ < ε
⊢ ∀ (x : X), ‖↑g x - f x‖ < ε

Tactic:
]

Goal state after:
case h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w : A.SeparatesPoints
f : X → ℝ
c : Continuous[inst✝¹, _] f
ε : ℝ
pos : 0 < ε
g : ↥A
b : ∀ (x : X), ‖(↑g - { toFun := f, continuous_toFun := c }) x‖ < ε
⊢ ∀ (x : X), ‖↑g x - f x‖ < ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (- -r • f).toNNReal - φ (-r • f).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (- -r • f).toNNReal - φ (-r • f).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (- -r • f).toNNReal - φ (-r • f).toNNReal = - -r • (φ f.toNNReal - φ (-f).toNNReal)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r : ℝ
f : C(X, ℝ)₀
hr : 0 < -r
⊢ φ (- -r • f).toNNReal - φ (-r • f).toNNReal = - -r • (φ f.toNNReal - φ (-f).toNNReal)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: case neg
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
r✝ : ℝ
f : C(X, ℝ)₀
r : ℝ≥0
hr : 0 < ↑r
⊢ -(r • φ f.toNNReal) + r • φ (-f).toNNReal = -(r • φ f.toNNReal) + r • φ (-f).toNNReal

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) = 0

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) =
    φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal)

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal) = 0
⊢ φ (f + g).toNNReal - φ (-(f + g)).toNNReal - (φ f.toNNReal - φ (-f).toNNReal + (φ g.toNNReal - φ (-g).toNNReal)) =
    φ (f + g).toNNReal + φ (-f).toNNReal + φ (-g).toNNReal - (φ (-(f + g)).toNNReal + φ f.toNNReal + φ g.toNNReal)
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
hsep : AK.SeparatesPoints
gK : C(↑K, ℝ)
hgKAK✝ : gK ∈ AK
hgapprox : ∀ (x : ↑K), ‖↑⟨gK, hgKAK✝⟩ x - K.domRestrict (⇑f) x‖ < ε
g : C(X, ℝ)
hgA : g ∈ A
hgKAK : ↑restrict_on_K g = gK
x : X
hxK : x ∈ K
⊢ g x = (↑restrict_on_K g) ⟨x, hxK⟩

Tactic:
]

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
hsep : AK.SeparatesPoints
gK : C(↑K, ℝ)
hgKAK✝ : gK ∈ AK
hgapprox : ∀ (x : ↑K), ‖↑⟨gK, hgKAK✝⟩ x - K.domRestrict (⇑f) x‖ < ε
g : C(X, ℝ)
hgA : g ∈ A
hgKAK : ↑restrict_on_K g = gK
x : X
hxK : x ∈ K
⊢ g x = (↑restrict_on_K g) ⟨x, hxK⟩
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    0

Tactic:
]

Goal state after:
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : Zero X
A : Type u_2
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module ℝ A
inst✝¹ : TopologicalSpace A
inst✝ : IsSemitopologicalRing A
φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A
f g : C(X, ℝ)₀
this :
  φ (f * g).toNNReal + φ (-f).toNNReal * φ g.toNNReal + φ f.toNNReal * φ (-g).toNNReal -
      (φ (-(f * g)).toNNReal + φ f.toNNReal * φ g.toNNReal + φ (-f).toNNReal * φ (-g).toNNReal) =
    0
⊢ φ (f * g).toNNReal - φ (-(f * g)).toNNReal -
      (φ f.toNNReal * φ g.toNNReal - φ (-f).toNNReal * φ g.toNNReal -
        (φ f.toNNReal * φ (-g).toNNReal - φ (-f).toNNReal * φ (-g).toNNReal)) =
    0