mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
⊢ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, ∀ᶠ (x : ℕ → α) in 𝓝 u, x i.1 < x i.2

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
⊢ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, ∀ᶠ (x : ℕ → α) in 𝓝 u, x i.1 < x i.2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict fun x ↦ -f x, continuous_toFun := ⋯ } =
    (cfcHom ha) (-{ toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })

Tactic:
]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict fun x ↦ -f x, continuous_toFun := ⋯ } =
    (cfcHom ha) (-{ toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
hg' : ContinuousOn (fun x ↦ (algebraMap R S) (g (f x))) (σₙ S a)
⊢ ContinuousOn (⇑(algebraMap R S) ∘ g) (σₙ R a)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
hg' : ContinuousOn (fun x ↦ (algebraMap R S) (g (f x))) (σₙ S a)
⊢ ContinuousOn (⇑(algebraMap R S) ∘ g) (σₙ R a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 = 0

pp: case neg.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ ¬ContinuousOn (fun x ↦ -f x) (spectrum R a)

Tactic:
]

Goal state after:
case neg.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ ¬ContinuousOn (fun x ↦ -f x) (spectrum R a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommRing R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
⊢ -a = -a

Tactic:
]

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

Goal state before:
pp: case neg.inl
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
this : ¬ContinuousOn (fun x ↦ (algebraMap R S) (g (f x))) (σₙ S a)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : CommRing R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R A p
f : R → R
a : A
inst✝ : UniqueHom R A
hf : ContinuousOn f ((fun x ↦ -x) '' spectrum R a)
ha : p a
⊢ cfc f (-a) = cfc f (-a)

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : ¬p a
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝¹ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
⊢ c < eVariationOn f s

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝¹ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
⊢ c < eVariationOn f s
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case neg.inr
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬g 0 = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬f 0 = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ g x
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } ≤
    (cfcHom ha) { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ }

Tactic:
]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : PartialOrder R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : ContinuousSqrt R
inst✝⁶ : StarOrderedRing R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ g x
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ (cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } ≤
    (cfcHom ha) { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ }
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
h : ⇑f = (σₙ R a).domRestrict (Function.extend Subtype.val (⇑f) g)
hg : ContinuousOn (Function.extend Subtype.val (⇑f) g) (σₙ R a)
⊢ f 0 = ↑0

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
h : ⇑f = (σₙ R a).domRestrict (Function.extend Subtype.val (⇑f) g)
hg : ContinuousOn (Function.extend Subtype.val (⇑f) g) (σₙ R a)
⊢ f 0 = ↑0
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝¹ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
J : Set ℕ
J_fin : J.Finite
k : ℕ → Set α
k_mem : ∀ (i : ℕ), k i ∈ 𝓝 (u i)
hk : J.pi k ⊆ {x | (fun b ↦ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2) x}
⊢ c < eVariationOn f s

Tactic:
have A i (hi : i ∈ Iic n) : ∃ x, (f x ∈ t i ∧ x ∈ k i) ∧ x ∈ s :=
    ((((mapClusterPt_iff_frequently.1 (hg (u i) (u_mem i hi)) (t i) (t_mem i))).and_eventually
      (mem_nhdsWithin_of_mem_nhds (k_mem i))).and_eventually self_mem_nhdsWithin).exists

Goal state after:
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f g : α → E
s : Set α
hg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f
n : ℕ
u : ℕ → α
u_mono : StrictMonoOn u (Iic n)
u_mem : ∀ i ∈ Iic n, u i ∈ s
this✝¹ : Nonempty α
c : ℝ≥0∞
hc : c < ∑ x ∈ Finset.range n, edist (g (u (x + 1))) (g (u x))
this✝ : ∀ᶠ (b : ℕ → E) in Filter.pi fun i ↦ 𝓝 (g (u i)), c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)
I : Set ℕ
I_fin : I.Finite
t : ℕ → Set E
t_mem : ∀ (i : ℕ), t i ∈ 𝓝 (g (u i))
ht : I.pi t ⊆ {b | c < ∑ i ∈ Finset.range n, edist (b (i + 1)) (b i)}
this : ∀ᶠ (b : ℕ → α) in Filter.pi fun i ↦ 𝓝 (u i), ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2
J : Set ℕ
J_fin : J.Finite
k : ℕ → Set α
k_mem : ∀ (i : ℕ), k i ∈ 𝓝 (u i)
hk : J.pi k ⊆ {x | (fun b ↦ ∀ i ∈ {i ∈ Finset.Iic n ×ˢ Finset.Iic n | i.1 < i.2}, b i.1 < b i.2) x}
A : ∀ i ∈ Iic n, ∃ x, (f x ∈ t i ∧ x ∈ k i) ∧ x ∈ s
⊢ c < eVariationOn f s
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : UniqueHom R A
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
this : NonUnitalContinuousFunctionalCalculus R A p
⊢ IsClosedEmbedding ⇑(nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalClosedEmbeddingContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : UniqueHom R A
inst✝ : CompleteSpace R
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
this : NonUnitalContinuousFunctionalCalculus R A p
⊢ IsClosedEmbedding ⇑(nonUnitalStarAlgHom (cfcₙHom ⋯) ⋯)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
f' : R → R
h : ⇑f = (σₙ R a).domRestrict f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ (cfcₙHom ha) f = (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f', continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
f' : R → R
h : ⇑f = (σₙ R a).domRestrict f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ (cfcₙHom ha) f = (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f', continuous_toFun := ⋯, map_zero' := ⋯ }
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommSemiring R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (∀ (a_1 : ↑(spectrum R a)), 0 a_1 ≤ { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, 0 ≤ f x

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : CommSemiring R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : ContinuousSqrt R
inst✝⁷ : StarOrderedRing R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ (∀ (a_1 : ↑(spectrum R a)), 0 a_1 ≤ { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, 0 ≤ f x