mathlib-initiative/mathlib-tactics huggingface dataset:
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
⊢ ContinuousOn (fun z ↦ ∑ i ∈ s, f i z) (σₙ R a)

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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
⊢ ContinuousOn (fun z ↦ ∑ i ∈ s, f i z) (σₙ R a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
C : ℝ
hf : MonotoneOn f s
h : ∀ x ∈ s, |f x| ≤ C
⊢ ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Set.Icc p.1 p.2) ≤ ENNReal.ofReal (2 * C)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
C : ℝ
hf : MonotoneOn f s
h : ∀ x ∈ s, |f x| ≤ C
⊢ ⨆ p ∈ {p | p.1 ∈ s ∧ p.2 ∈ s ∧ p.1 ≤ p.2}, eVariationOn f (s ∩ Set.Icc p.1 p.2) ≤ ENNReal.ofReal (2 * C)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ cfcₙ (∑ i, f ↑i) a = ∑ i, cfcₙ (f ↑i) a

Tactic:
]

Goal state after:
case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ cfcₙ (∑ i, f ↑i) a = ∑ i, cfcₙ (f ↑i) a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
C : ℝ
hf : MonotoneOn f s
h : ∀ x ∈ s, |f x| ≤ C
a b : α
as : a ∈ s
bs : b ∈ s
hab : a ≤ b
⊢ ENNReal.ofReal (f b - f a) ≤ ENNReal.ofReal (2 * C)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
f : α → ℝ
s : Set α
C : ℝ
hf : MonotoneOn f s
h : ∀ x ∈ s, |f x| ≤ C
a b : α
as : a ∈ s
bs : b ∈ s
hab : a ≤ b
⊢ ENNReal.ofReal (f b - f a) ≤ ENNReal.ofReal (2 * C)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
F : Type u_3
inst✝ : PseudoEMetricSpace F
f : E → F
C : ℝ≥0
t : Set E
h : LipschitzOnWith C f t
g : α → E
s : Set α
hg : MapsTo g s t
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
⊢ ∑ i ∈ Finset.range n, ↑C * edist (g (u (i + 1))) (g (u i)) =
    ∑ i ∈ Finset.range n, ↑C * edist (g (u (i + 1))) (g (u i))

Tactic:
]

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

Goal state before:
pp: case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict (∑ i, f ↑i), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) (∑ x, { toFun := (σₙ R a).domRestrict (f ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
]

Goal state after:
case pos
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (σₙ R a)
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict (∑ i, f ↑i), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) (∑ x, { toFun := (σₙ R a).domRestrict (f ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
F : Type u_3
inst✝ : PseudoEMetricSpace F
f : E → F
C : ℝ≥0
t : Set E
h : LipschitzOnWith C f t
g : α → E
s : Set α
hg : MapsTo g s t
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
⊢ ↑C * eVariationOn g s ≤ ↑C * eVariationOn g s

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict fun x ↦ s • f x, continuous_toFun := ⋯, map_zero' := ⋯ } =
    s • (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }

Tactic:
]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict fun x ↦ s • f x, continuous_toFun := ⋯, map_zero' := ⋯ } =
    s • (cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict ((s • 1) • f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) ((s • 1) • { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
]

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
ha : p a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict ((s • 1) • f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) ((s • 1) • { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
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
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict fun x ↦ star (f x), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) (star { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
]

Goal state after:
case pos
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
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict fun x ↦ star (f x), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) (star { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inl
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

pp: case neg.inr.inl.hf
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)
⊢ ¬ContinuousOn (fun x ↦ star (f x)) (σₙ R a)

Tactic:
]

Goal state after:
case neg.inr.inl.hf
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)
⊢ ¬ContinuousOn (fun x ↦ star (f x)) (σₙ R a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inr.inr
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
h0 : ¬f 0 = 0
⊢ 0 = 0

pp: case neg.inr.inr.hf
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
h0 : ¬f 0 = 0
⊢ ¬star (f 0) = 0

Tactic:
]

Goal state after:
case neg.inr.inr.hf
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
h0 : ¬f 0 = 0
⊢ ¬star (f 0) = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : Nontrivial R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : StarRing A
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Module R A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMulZeroClass S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
a : A
ha : p a
⊢ s • a = s • 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
a : A
ha : p a
⊢ star a = star a

Tactic:
]

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

Goal state before:
pp: case mp
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
ha : p a
f : R → R
⊢ (fun x ↦ cfcₙ x a) f ∈ ↑(NonUnitalStarAlgHom.range (cfcₙHom ha))

Tactic:
exact cfcₙ_cases _ a f (zero_mem _) fun hf hf₀ ha ↦ ⟨_, rfl⟩

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✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
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
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
⊢ Set.range ⇑{ toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 } = f '' σₙ R a

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
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
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
⊢ Set.range ⇑{ toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 } = f '' σₙ R a
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = 0 then 1 else 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 0 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = 0 then 1 else 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = n then 1 else 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.eval 1 (↑(n.choose ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = n then 1 else 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.derivative (↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * (1 - X) ^ (n + 1 - (ν + 1))) =
    (↑n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1))

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ Polynomial.derivative (↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * (1 - X) ^ (n + 1 - (ν + 1))) =
    (↑n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1))
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ ↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * ↑(n - ν) * (1 - X) ^ (n - (ν + 1)) =
    ↑(n + 1) * (↑(n.choose (ν + 1)) * X ^ (ν + 1)) * (1 - X) ^ (n - (ν + 1))

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ ↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) * ↑(n - ν) * (1 - X) ^ (n - (ν + 1)) =
    ↑(n + 1) * (↑(n.choose (ν + 1)) * X ^ (ν + 1)) * (1 - X) ^ (n - (ν + 1))
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case refine_2.e_a
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ ↑(n - ν) * ↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) = ↑(n + 1) * ↑(n.choose (ν + 1)) * X ^ (ν + 1)

Tactic:
]

Goal state after:
case refine_2.e_a
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ ↑(n - ν) * ↑((n + 1).choose (ν + 1)) * X ^ (ν + 1) = ↑(n + 1) * ↑(n.choose (ν + 1)) * X ^ (ν + 1)
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
ν n✝ : ℕ
⊢ Polynomial.derivative (bernsteinPolynomial R (n✝ + 1) (ν + 1)) =
    (↑n✝ + 1) * (bernsteinPolynomial R (n✝ + 1 - 1) ν - bernsteinPolynomial R (n✝ + 1 - 1) (ν + 1))

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
ν n✝ : ℕ
⊢ Polynomial.derivative (bernsteinPolynomial R (n✝ + 1) (ν + 1)) =
    (↑n✝ + 1) * (bernsteinPolynomial R (n✝ + 1 - 1) ν - bernsteinPolynomial R (n✝ + 1 - 1) (ν + 1))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
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
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
x : ↑(σₙ R a)
⊢ (σₙ R a).domRestrict f x ∈ f '' σₙ R a

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
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
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
x : ↑(σₙ R a)
⊢ (σₙ R a).domRestrict f x ∈ f '' σₙ R a
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
n k ν : ℕ
⊢ k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
n k ν : ℕ
⊢ k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ.succ
R : Type u_1
inst✝ : CommRing R
k : ℕ
ih : ∀ (n ν : ℕ), k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
n ν : ℕ
h : k + 1 ≤ ν
⊢ Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) ν)) = 0

Tactic:
]

Goal state after:
case succ.succ
R : Type u_1
inst✝ : CommRing R
k : ℕ
ih : ∀ (n ν : ℕ), k ≤ ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n (ν + 1))) = 0
n ν : ℕ
h : k + 1 ≤ ν
⊢ Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) ν)) = 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case pos.succ.inr
R : Type u_1
inst✝ : CommRing R
ν : ℕ
ih :
  ∀ (n : ℕ), ν ≤ n → eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)
n : ℕ
h : ν + 1 ≤ n
h' : ν ≤ n - 1
h'' : ν > 0
this : n - 1 - (ν - 1) = n - ν
⊢ ↑n * eval (↑(n - ν)) (ascPochhammer R ν) = (↑(n - ν) + ↑ν) * eval (↑(n - ν)) (ascPochhammer R ν)

Tactic:
]

Goal state after:
case pos.succ.inr
R : Type u_1
inst✝ : CommRing R
ν : ℕ
ih :
  ∀ (n : ℕ), ν ≤ n → eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)
n : ℕ
h : ν + 1 ≤ n
h' : ν ≤ n - 1
h'' : ν > 0
this : n - 1 - (ν - 1) = n - ν
⊢ ↑n * eval (↑(n - ν)) (ascPochhammer R ν) = (↑(n - ν) + ↑ν) * eval (↑(n - ν)) (ascPochhammer R ν)