mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case h₂
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts b ⇑ContinuousMap.realToNNReal

pp: case h₂.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint b

pp: case h₂.ht
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖b‖₊ ≤ ‖b‖₊

Tactic:
]

Goal state after:
case h₂
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts b ⇑ContinuousMap.realToNNReal

case h₂.ha
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint b

case h₂.ht
A : Type u_1
inst✝ : CStarAlgebra A
a b : A
ha₁ : IsSelfAdjoint a
hb₁ : IsSelfAdjoint b
ha₂ : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
hb₂ : SpectrumRestricts b ⇑ContinuousMap.realToNNReal
⊢ ‖b‖₊ ≤ ‖b‖₊
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ ∀ x ∈ (fun x ↦ id x ^ 2) '' spectrum ℝ a, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
⊢ ∀ x ∈ (fun x ↦ id x ^ 2) '' spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
ha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x
ha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x
⊢ a = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
ha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x
ha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x
⊢ a = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
ha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x
ha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x
⊢ ∀ y ∈ spectrum ℝ a, y = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsSelfAdjoint a
ha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x
ha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x
⊢ ∀ y ∈ spectrum ℝ a, y = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
⊢ ∀ x ∈ spectrum ℝ (r • a), 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
r : ℝ
hr : 0 ≤ r
⊢ ∀ x ∈ spectrum ℝ (r • a), 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case neg
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
a✝ : Nontrivial A
x : ℝ
r : ℝˣ
hr : 0 ≤ ↑r
hx : r⁻¹ • x ∈ spectrum ℝ a
hr' : ¬↑r = 0
⊢ 0 ≤ x

Tactic:
]

Goal state after:
case neg
A : Type u_2
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : ∀ x ∈ spectrum ℝ a, 0 ≤ x
a✝ : Nontrivial A
x : ℝ
r : ℝˣ
hr : 0 ≤ ↑r
hx : r⁻¹ • x ∈ spectrum ℝ a
hr' : ¬↑r = 0
⊢ 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
⊢ ∀ x ∈ (fun x ↦ x⁻ ^ 3) '' spectrum ℝ a, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
⊢ ∀ x ∈ (fun x ↦ x⁻ ^ 3) '' spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁺) 1 = 1

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁺) 1 = 1
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁻) 1 = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
⊢ cfc (fun x ↦ x⁻) 1 = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)) ⇑ContinuousMap.realToNNReal

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfc (fun x ↦ x⁻) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁻

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ
⊢ cfc (fun x ↦ x⁻) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ≥0
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) ((algebraMap ℝ≥0 ℝ) r)) = (algebraMap ℝ A) ((algebraMap ℝ≥0 ℝ) r)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
r : ℝ≥0
⊢ cfc (fun x ↦ x⁺) ((algebraMap ℝ A) ((algebraMap ℝ≥0 ℝ) r)) = (algebraMap ℝ A) ((algebraMap ℝ≥0 ℝ) r)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_3
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (↑2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2)) ⇑ContinuousMap.realToNNReal

Tactic:
]

Goal state after:
case refine_3
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : -(star c * c) = a⁻ ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
⊢ SpectrumRestricts (↑2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2)) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁵ : Ring A
inst✝⁴ : Algebra ℝ A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
n : ℕ
⊢ (algebraMap ℝ≥0 A) ↑n = (algebraMap ℝ≥0 A) ↑n

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : SMulCommClass ℂ A A
inst✝⁴ : IsScalarTower ℂ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : StarModule ℂ A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
x : A
⊢ x = x

Tactic:
]

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

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : cfc 0 a = cfc (fun x ↦ x⁻ ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
b : A
a : A := star b * b
a_def : a = star b * b
ha : IsSelfAdjoint a
c : A := b * a⁻
h_eq_negPart_a : cfc 0 a = cfc (fun x ↦ x⁻ ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
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
x : A
x✝ : x ∈ ⊤
⊢ (↑(ℜ x))⁺ - (↑(ℜ x))⁻ + (I • (↑(ℑ x))⁺ - I • (↑(ℑ x))⁻) ∈ span ℂ {a | 0 ≤ a}

Tactic:
]

Goal state after:
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
x : A
x✝ : x ∈ ⊤
⊢ (↑(ℜ x))⁺ - (↑(ℜ x))⁻ + (I • (↑(ℑ x))⁺ - I • (↑(ℑ x))⁻) ∈ span ℂ {a | 0 ≤ a}
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 : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case add
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a x y : A
x_mem : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
y_mem : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : SpectrumRestricts x ⇑ContinuousMap.realToNNReal
hy : SpectrumRestricts y ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts (x + y) ⇑ContinuousMap.realToNNReal

Tactic:
]

Goal state after:
case add
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a x y : A
x_mem : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
y_mem : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : SpectrumRestricts x ⇑ContinuousMap.realToNNReal
hy : SpectrumRestricts y ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts (x + y) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case add
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a x y : A
x_mem : 0 ≤ x
y_mem : 0 ≤ y
hx : SpectrumRestricts x ⇑ContinuousMap.realToNNReal
hy : SpectrumRestricts y ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts (x + y) ⇑ContinuousMap.realToNNReal

Tactic:
]

Goal state after:
case add
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a x y : A
x_mem : 0 ≤ x
y_mem : 0 ≤ y
hx : SpectrumRestricts x ⇑ContinuousMap.realToNNReal
hy : SpectrumRestricts y ⇑ContinuousMap.realToNNReal
⊢ SpectrumRestricts (x + y) ⇑ContinuousMap.realToNNReal
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 = 0 ∨ x ∈ spectrum ℂ a → 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
x : ℂ
⊢ x = 0 ∨ x ∈ spectrum ℂ a → 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
⊢ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
⊢ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x
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
⊢ ‖a ^ (1 / 2)‖₊ = ‖a‖₊ ^ (1 / 2)

Tactic:
]

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)‖₊ = ‖a‖₊ ^ (1 / 2)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ x = y

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ x = y
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ x - y = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ x - y = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ ↑x - ↑y = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x y : A
hxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal
hyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal
⊢ ↑x - ↑y = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
r : ℝ≥0
hr : 0 < ↑r
⊢ ‖a ^ r‖₊ = ‖a ^ r‖₊

pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
r : ℝ≥0
hr : 0 < ↑r
⊢ 0 < r

pp: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
r : ℝ≥0
hr : 0 < ↑r
⊢ 0 < r

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
r : ℝ≥0
hr : 0 < ↑r
⊢ 0 < r

A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : NormedAlgebra ℝ A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : 0 ≤ a
r : ℝ≥0
hr : 0 < ↑r
⊢ 0 < r
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
h : x ≤ y
s : A
hs₁ : IsSelfAdjoint s
left✝ : QuasispectrumRestricts s ⇑ContinuousMap.realToNNReal
hs₂ : y = x + s * s
⊢ y = x + s * s

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
h : x ≤ y
s : A
hs₁ : IsSelfAdjoint s
left✝ : QuasispectrumRestricts s ⇑ContinuousMap.realToNNReal
hs₂ : y = x + s * s
⊢ y = x + s * s
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x y : A
h : x ≤ y
s : A
hs₁ : IsSelfAdjoint s
left✝ : QuasispectrumRestricts s ⇑ContinuousMap.realToNNReal
hs₂ : s * s = y - x
⊢ y = x + s * s

Tactic:
by rwa [eq_sub_iff_add_eq', eq_comm] at hs₂

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

Goal state before:
pp: case mpr.mem
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p s : A
⊢ ∀ x ∈ spectrum ℝ (star ↑s * ↑s), 0 ≤ x

Tactic:
]

Goal state after:
case mpr.mem
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p s : A
⊢ ∀ x ∈ spectrum ℝ (star ↑s * ↑s), 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case mpr.zero
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p : A
⊢ IsSelfAdjoint 0 ∧ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x

Tactic:
]

Goal state after:
case mpr.zero
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝ : PartialOrder A := spectralOrder A
x p : A
⊢ IsSelfAdjoint 0 ∧ ∀ x ∈ spectrum ℝ ↑0, 0 ≤ x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case mpr.add
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝¹ : PartialOrder A := spectralOrder A
x✝ p x y : A
hx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : IsSelfAdjoint ↑x ∧ SpectrumRestricts ↑x ⇑ContinuousMap.realToNNReal
hy : IsSelfAdjoint ↑y ∧ SpectrumRestricts ↑y ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint (↑x + ↑y) ∧ SpectrumRestricts (↑x + ↑y) ⇑ContinuousMap.realToNNReal

Tactic:
]

Goal state after:
case mpr.add
A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
x✝¹ : PartialOrder A := spectralOrder A
x✝ p x y : A
hx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
hx : IsSelfAdjoint ↑x ∧ SpectrumRestricts ↑x ⇑ContinuousMap.realToNNReal
hy : IsSelfAdjoint ↑y ∧ SpectrumRestricts ↑y ⇑ContinuousMap.realToNNReal
⊢ IsSelfAdjoint (↑x + ↑y) ∧ SpectrumRestricts (↑x + ↑y) ⇑ContinuousMap.realToNNReal
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁹ : PartialOrder A
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : CStarRing A
inst✝⁵ : NormedSpace ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
a b : A
ha : 0 ≤ a
⊢ ‖star b * a * b‖ = ‖star b * a * b‖

Tactic:
]

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 ∈ spectrum ℝ ↑a, 0 ≤ x

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
⊢ ∀ x ∈ spectrum ℝ ↑a, 0 ≤ x
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 : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
x✝ : PartialOrder (Unitization ℂ A) := spectralOrder (Unitization ℂ A)
this : StarOrderedRing (Unitization ℂ A)
⊢ ↑a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
x✝ : PartialOrder (Unitization ℂ A) := spectralOrder (Unitization ℂ A)
this : StarOrderedRing (Unitization ℂ A)
⊢ ↑a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
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 : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
x✝ : PartialOrder (Unitization ℂ A) := spectralOrder (Unitization ℂ A)
this✝ : StarOrderedRing (Unitization ℂ A)
this :
  (Unitization.inrNonUnitalStarAlgHom ℂ A) a ∈
    AddSubmonoid.closure (Set.range (⇑(Unitization.inrNonUnitalStarAlgHom ℂ A) ∘ fun s ↦ star s * s))
⊢ ↑a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
x✝ : PartialOrder (Unitization ℂ A) := spectralOrder (Unitization ℂ A)
this✝ : StarOrderedRing (Unitization ℂ A)
this :
  (Unitization.inrNonUnitalStarAlgHom ℂ A) a ∈
    AddSubmonoid.closure (Set.range (⇑(Unitization.inrNonUnitalStarAlgHom ℂ A) ∘ fun s ↦ star s * s))
⊢ ↑a ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)