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

Goal state before:
pp: case pos.e_6
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
ha : p a
hsum : s.sum f = fun z ↦ ∑ i ∈ s, f i z
hf' : ContinuousOn (∑ i, f ↑i) (spectrum R a)
a✝ : ↑(spectrum R a)
⊢ { toFun := (spectrum R a).domRestrict (∑ i, f ↑i), continuous_toFun := ⋯ } a✝ =
    (∑ x, { toFun := (spectrum R a).domRestrict (f ↑x), continuous_toFun := ⋯ }) a✝

Tactic:
simp

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✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a)
⊢ cfc (∑ i ∈ s, f i) a = ∑ i ∈ s, cfc (f i) a

Tactic:
by
  by_cases ha : p a
  · have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
    have hf' : ContinuousOn (∑ i : s, f i) (spectrum R a) := by
      rw [sum_coe_sort s, hsum]
      exact continuousOn_finsetSum s fun i hi => hf i hi
    rw [← sum_coe_sort s, ← sum_coe_sort s]
    rw [cfc_apply_pi _ a ha (fun ⟨i, hi⟩ => hf i hi), ← map_sum, cfc_apply _ a ha hf']
    congr 1
    ext
    simp
  · simp [cfc_apply_of_not_predicate a ha]

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 : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s] x) (𝓝 0)

Tactic:
have A : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0) := by
    simp_rw [← Iio_inter_Ici, ← inter_assoc]
    exact (hf.mono inter_subset_left).tendsto_eVariationOn_Ici_zero_of_filter (𝓝[s ∩ Iio x] x)
      (fun y hy ↦ inter_mem_nhdsWithin _ (Ici_mem_nhds hy.2))

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 : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
A : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s] x) (𝓝 0)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : CompleteSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
hf : BoundedVariationOn f univ
x : α
x✝ : E
⊢ Iio x = univ ∩ Iio x

Tactic:
simp

Goal state after:
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✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
rw [cfc_apply_of_not_continuousOn a hf, cfc_apply_of_not_continuousOn, star_zero]
      exact fun hf_star ↦ hf <| by simpa using hf_star.star

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 : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
ha : p a
hg : ContinuousOn g (f '' spectrum R a)
hf : ContinuousOn f (spectrum R a)
this : ContinuousOn (g ∘ f) (spectrum R a)
x✝ : R
⊢ x✝ ∈ Set.range ⇑{ toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } ↔ x✝ ∈ f '' spectrum R a

Tactic:
simp

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

Goal state before:
pp: case h
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
ha : p a
hg : ContinuousOn g (f '' spectrum R a)
hf : ContinuousOn f (spectrum R a)
this : ContinuousOn (g ∘ f) (spectrum R a)
sp_eq : spectrum R ((cfcHom ⋯) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) = f '' spectrum R a
x : ↑(spectrum R a)
⊢ ↑x ∈ spectrum R a ∧ f ↑x = (spectrum R a).domRestrict f x

Tactic:
simp

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✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
r x : R
⊢ (fun x ↦ r) x ∈ {r}

Tactic:
simp

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 : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : Nontrivial A
this : 0 = (algebraMap R A) 0
⊢ spectrum R ((algebraMap R A) 0) = {0}

Tactic:
spectrum_algebraMap_eq

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : Nontrivial A
this : 0 = (algebraMap R A) 0
⊢ {0} = {0}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : Nontrivial A
this : 1 = (algebraMap R A) 1
⊢ spectrum R ((algebraMap R A) 1) = {1}

Tactic:
spectrum_algebraMap_eq

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : Nontrivial A
this : 1 = (algebraMap R A) 1
⊢ {1} = {1}
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : Semifield R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring 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
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ 0 ∉ f '' spectrum R a ↔ ∀ x ∈ spectrum R a, f x ≠ 0

Tactic:
simp

Goal state after:
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)
hg0 : g 0 = 0
x : { x // x ∈ σₙ S a }
⊢ Function.extend Subtype.val
      (⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯, map_zero' := ⋯ }.comp
          ({ toFun := (σₙ R a).domRestrict g, continuous_toFun := ⋯, map_zero' := ⋯ }.comp
            { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯, map_zero' := ⋯ })))
      0 ↑x =
    (algebraMap R S) (g (f ↑x))

Tactic:
simp

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
g : R → R
ha : p a
f : C(↑(σₙ R a), R)₀
x✝ : ↑(σₙ R a)
⊢ f x✝ = (σₙ R a).domRestrict (Function.extend Subtype.val (⇑f) g) x✝

Tactic:
simp

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✝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:
simp

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
f : R → R
ha : p a
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
⊢ cfcₙ f a = (cfcₙL ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }

Tactic:
rw [cfcₙ_def, dif_pos ⟨ha, hf, hf0⟩, cfcₙL_apply]

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✝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
⊢ (∀ (x : ↑(spectrum R a)), 0 < { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } x) ↔
    ∀ x ∈ spectrum R a, 0 < f x

Tactic:
simp

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✝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 : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 1
⊢ 0 ≤ 1

Tactic:
simp

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 g : R → R
a : A
⊢ Commute 0 (cfcₙ g a)

Tactic:
simp

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 g : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
ha : p a
⊢ Commute ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) 0

Tactic:
simp

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✝¹⁴ : CommRing R
inst✝¹³ : PartialOrder R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalRing 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 g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hg : ContinuousOn g (spectrum R a)
ha : p a
⊢ (∀ (a_1 : ↑(spectrum R a)),
      { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ } a_1 ≤
        { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, f x ≤ g x

Tactic:
simp

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

Goal state before:
pp: case inl
α : Type u_1
inst✝² : LinearOrder α
E : Type u_2
inst✝¹ : PseudoEMetricSpace E
inst✝ : CompleteSpace E
hE : Nonempty E
f : α → E
hf : BoundedVariationOn f ∅
⊢ ∃ l, Tendsto f (𝓟 ∅ ⊓ atTop) (𝓝 l)

Tactic:
simp

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
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
x✝ : R
⊢ s.sum f x✝ = ∑ i ∈ s, f i x✝

Tactic:
simp

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

Goal state before:
pp: a b : ℝ
⊢ a ∈ Set.univ

Tactic:
simp

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

Goal state before:
pp: a b : ℝ
⊢ b ∈ Set.univ

Tactic:
simp

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

Goal state before:
pp: case pos.e_6
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)
x✝ : ↑(σₙ R a)
⊢ { toFun := (σₙ R a).domRestrict (∑ i, f ↑i), continuous_toFun := ⋯, map_zero' := ⋯ } x✝ =
    (∑ x, { toFun := (σₙ R a).domRestrict (f ↑x), continuous_toFun := ⋯, map_zero' := ⋯ }) x✝

Tactic:
simp

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)
x✝ : R
⊢ x✝ ∈ Set.range ⇑{ toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 } ↔ x✝ ∈ f '' σₙ R a

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

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

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

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

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case e'_3
R : Type u_1
inst✝ : CommRing R
n ν : ℕ
⊢ n + 1 - (ν + 1) = n - ν

Tactic:
simp

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

Goal state before:
pp: case h
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)
⊢ ↑x ∈ σₙ R a ∧ f ↑x = (σₙ R a).domRestrict f x

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

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

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
a✝ : R
⊢ X.toContinuousMap a✝ = (ContinuousMap.id R) a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
s : Set R
a✝ : ↑s
⊢ (X.toContinuousMapOn s) a✝ = (restrict s (ContinuousMap.id R)) a✝

Tactic:
simp

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

Goal state before:
pp: case refine_2
R : Type u_1
α : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
g : R[X]
f : C(α, R)
x : α
n : ℕ
a : R
⊢ ((aeval f) ((monomial n) a)) x = eval (f x) ((monomial n) a)

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
a✝ : R
⊢ (toContinuousMap 1) a✝ = 1 a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
x✝¹ x✝ : R[X]
a✝ : R
⊢ (x✝¹ * x✝).toContinuousMap a✝ = (x✝¹.toContinuousMap * x✝.toContinuousMap) a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
a✝ : R
⊢ (toContinuousMap 0) a✝ = 0 a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
x✝¹ x✝ : R[X]
a✝ : R
⊢ (x✝¹ + x✝).toContinuousMap a✝ = (x✝¹.toContinuousMap + x✝.toContinuousMap) a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ (toContinuousMapOn 1 X) a✝ = 1 a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ((x✝¹ * x✝).toContinuousMapOn X) a✝ = (x✝¹.toContinuousMapOn X * x✝.toContinuousMapOn X) a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
X : Set R
a✝ : ↑X
⊢ (toContinuousMapOn 0 X) a✝ = 0 a✝

Tactic:
simp

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

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
X : Set R
x✝¹ x✝ : R[X]
a✝ : ↑X
⊢ ((x✝¹ + x✝).toContinuousMapOn X) a✝ = (x✝¹.toContinuousMapOn X + x✝.toContinuousMapOn X) a✝

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharZero R
n : ℕ
h : 0 ≤ n
⊢ 0 < eval (n - (0 - 1)) (ascPochhammer ℕ 0)

Tactic:
simp

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

Goal state before:
pp: case mp.refine_1
a b : ℝ
h : a < b
f : C(↑(Set.Icc a b), ℝ)
p : ℝ[X]
w : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)
q : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))
⊢ q ∈ ↑⊤.toSubsemiring

Tactic:
simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
ν : ℕ
h : ν ≤ ν
⊢ (-1) ^ (ν - ν) * eval (↑(ν - (ν - ν - 1))) (ascPochhammer R (ν - ν)) =
    (-1) ^ (ν - ν) * eval (↑ν + 1) (ascPochhammer R (ν - ν))

Tactic:
simp

Goal state after: