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

Goal state before:
pp: case neg
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
ha : p a
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 = (cfcHom ha) 0

Tactic:
map_zero

Goal state after:
case neg
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
ha : p a
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 = 0
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
⊢ (∀ (i : ℕ),
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)) →
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)

Tactic:
hx

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
u : ℕ → Set α
u_open : ∀ (n : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (u n)
u_univ : ⋃ n, u n = univ
hu : ∀ (n : ℕ), IntegrableOn f (u n) μ
this :
  ∀ (n : ℕ),
    ∀ᵐ (x : α) ∂μ,
      Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
x : α
hx :
  ∀ (i : ℕ),
    Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖(u i).indicator f y - (u i).indicator f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬ContinuousOn f (spectrum R a)
⊢ P (cfc f a)

Tactic:
cfc_apply_of_not_continuousOn _ h

Goal state after:
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
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (spectrum R a)) (ha : p a),
    P ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
h : ¬ContinuousOn f (spectrum R a)
⊢ P 0
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
f g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
⊢ Commute ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ }) (cfc g a)

Tactic:
refine cfc_cases (fun x ↦ Commute _ x) a g (by simp) fun hg _ ↦ ?_

Goal state after:
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 g : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : p a
hg : ContinuousOn g (spectrum R a)
x✝ : p a
⊢ Commute ((cfcHom ha) { toFun := (spectrum R a).domRestrict f, continuous_toFun := ⋯ })
    ((cfcHom x✝) { toFun := (spectrum R a).domRestrict g, continuous_toFun := ⋯ })
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
⊢ Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0) →
    Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)

Tactic:
hx

Goal state after:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ (∫⁻ (y : α) in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0)
⊢ Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
⊢ Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0) →
    (∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a) → Tendsto (fun a ↦ ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (𝓝 (f x))

Tactic:
hx

Goal state after:
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
⊢ (∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a) → Tendsto (fun a ↦ ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (𝓝 (f x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring 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
inst✝ : ContinuousInv₀ R
a : Aˣ
ha : p ↑a
⊢ cfc (fun x ↦ x⁻¹) ↑a = ↑a⁻¹

Tactic:
rw [← Ring.inverse_unit]
  convert! cfc_inv (id : R → R) (a : A) ?_
  · exact (cfc_id R (a : A)).symm
  · rintro x hx rfl
    exact spectrum.zero_notMem R a.isUnit hx

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
a : A
ha : p a
inst✝ : UniqueHom R A
f : C(↑(σₙ R a), R)₀
f' : C(↑(σₙ R a), ↑(σₙ R ((cfcₙHom ha) f)))₀
hff' : ∀ (x : ↑(σₙ R a)), f x = ↑(f' x)
g : C(↑(σₙ R ((cfcₙHom ha) f)), R)₀
⊢ (cfcₙHom ha) (g.comp f') = (cfcₙHom ⋯) g

Tactic:
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R a, R)₀ :=
    { toFun := (ContinuousMapZero.comp · f')
      map_smul' := fun _ _ ↦ rfl
      map_add' := fun _ _ ↦ rfl
      map_mul' := fun _ _ ↦ rfl
      map_zero' := rfl
      map_star' := fun _ ↦ rfl }
  let φ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] A := (cfcₙHom ha).comp ψ
  suffices cfcₙHom (cfcₙHom_predicate ha f) = φ from DFunLike.congr_fun this.symm g
  refine cfcₙHom_eq_of_continuous_of_map_id (cfcₙHom_predicate ha f) φ ?_ ?_
  · refine (cfcₙHom_continuous ha).comp <| continuous_induced_rng.mpr ?_
    exact f'.toContinuousMap.continuous_precomp.comp continuous_induced_dom
  · simp only [φ, ψ, NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_mk',
      NonUnitalAlgHom.coe_mk]
    congr
    ext x
    simp [hff']

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

Goal state before:
pp: case neg
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
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ 0 = (cfcₙHom ha) 0

pp: case neg
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
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ (σₙ R a).domRestrict f 0 ≠ 0

Tactic:
map_zero

Goal state after:
case neg
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
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ 0 = 0

case neg
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
f_cont : ContinuousOn f (σₙ R a)
f_zero : ¬f 0 = 0
⊢ (σₙ R a).domRestrict f 0 ≠ 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
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
f_cont : ¬ContinuousOn f (σₙ R a)
⊢ 0 = (cfcₙHom ha) 0

Tactic:
map_zero

Goal state after:
case neg
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
f_cont : ¬ContinuousOn f (σₙ R a)
⊢ 0 = 0
Module: Mathlib.Topology.ContinuousMap.Polynomial

Goal state before:
pp: case mp.refine_2.convert_2.left
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] := ⋯
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ 0 ≤ (↑x - a) * (b - a)⁻¹

pp: case mp.refine_2.convert_2.right
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] := ⋯
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ≤ 1

Tactic:
· exact mul_nonneg w₂ (le_of_lt w₁)

Goal state after:
case mp.refine_2.convert_2.right
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] := ⋯
x : ↑(Set.Icc a b)
w₁ : 0 < (b - a)⁻¹
w₂ : 0 ≤ ↑x - a
w₃ : ↑x - a ≤ b - a
⊢ (↑x - a) * (b - a)⁻¹ ≤ 1
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case refine_3
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
x y : ℚ[X]
hx✝ : x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hy✝ : y ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
⊢ eval 1 ((⇑derivative)^[n - k] x) = 0 →
    eval 1 ((⇑derivative)^[n - k] y) = 0 → eval 1 ((⇑derivative)^[n - k] (x + y)) = 0

Tactic:
hx

Goal state after:
case refine_3
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
x y : ℚ[X]
hx✝ : x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hy✝ : y ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
hx : eval 1 ((⇑derivative)^[n - k] x) = 0
⊢ eval 1 ((⇑derivative)^[n - k] y) = 0 → eval 1 ((⇑derivative)^[n - k] (x + y)) = 0
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: f : C(↑I, ℝ)
n : ℕ
i : Fin (n + 1)
⊢ bernstein n ↑i • const (↑I) (f (bernstein.z i)) ∈ Set.range ⇑(Polynomial.toContinuousMapOnAlgHom I)

Tactic:
use bernsteinPolynomial ℝ n i * .C (f (bernstein.z i))

Goal state after:
case h
f : C(↑I, ℝ)
n : ℕ
i : Fin (n + 1)
⊢ (Polynomial.toContinuousMapOnAlgHom I) (bernsteinPolynomial ℝ n ↑i * Polynomial.C (f (bernstein.z i))) =
    bernstein n ↑i • const (↑I) (f (bernstein.z i))
Module: Mathlib.Analysis.SpecialFunctions.Bernstein

Goal state before:
pp: E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
⊢ gauge U ((bernsteinApproximation n f) x - f x) < 1

Tactic:
set S : Finset (Fin (n + 1)) := {k : Fin (n + 1) | dist k/ₙ x < δ}

Goal state after:
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
f : C(↑I, E)
this✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
U : Set E
hU₀ : U ∈ 𝓝 0
hUc : Convex ℝ U
C : ℝ
hC : ∀ (x y : ↑I), gauge U (f x - f y) ≤ C
hC₀ : 0 ≤ C
δ : ℝ
hδ₀ : δ > 0
hδ : ∀ (x y : ↑I), dist x y < δ → gauge U (f x - f y) < 1 / 2
nhds_zero : Tendsto (fun n ↦ C / δ ^ 2 / ↑n) atTop (𝓝 0)
n : ℕ
nh : C / δ ^ 2 / ↑n < 1 / 2
hn₀ : n ≠ 0
x : ↑I
S : Finset (Fin (n + 1)) := {k | dist k/ₙ x < δ}
⊢ gauge U ((bernsteinApproximation n f) x - f x) < 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case mpr.inr
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 : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
x : ↑(σₙ R a)
hx : ↑x = 0
⊢ f 0 = 0

Tactic:
map_zero

Goal state after:
case mpr.inr
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 : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R A p
a : A
ha : p a
f : C(↑(σₙ R a), R)₀
x : ↑(σₙ R a)
hx : ↑x = 0
⊢ 0 = 0
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := ⋯
y : MvPolynomial Bool R := ⋯
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := ⋯
⊢ (MvPolynomial.aeval e) ((pderiv true) ((pderiv true) ((x + y) ^ n))) * X ^ 2 = (n * (n - 1)) • X ^ 2

Tactic:
· simp only [x, y, e, (pderiv _).leibniz_pow,
      (pderiv true).map_add, pderiv_true_x, pderiv_true_y, smul_eq_mul, add_zero,
      mul_one, map_nsmul, map_pow, map_add, Bool.cond_true,
      Bool.cond_false, MvPolynomial.aeval_X, add_sub_cancel, one_pow, smul_smul,
      smul_one_mul]

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

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
h : Function.Injective ⇑(mul 𝕜 A)
x : Unitization 𝕜 A
⊢ (splitMul 𝕜 A) x = 0 → x = 0

Tactic:
hx

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
h : Function.Injective ⇑(mul 𝕜 A)
x : Unitization 𝕜 A
hx : (splitMul 𝕜 A) x = 0
⊢ x = 0
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E).flip a) b‖ ≤ ‖a‖ * ‖b‖

Tactic:
by simpa only [mul_comm] using! norm_mul_le b a

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

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
⊢ x₁ ≠ x₂ →
    ∃ f ∈ (fun f ↦ ⇑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra)),
      f x₁ ≠ f x₂

Tactic:
hx

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
⊢ ∃ f ∈ (fun f ↦ ⇑f) '' ↑(comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra)),
    f x₁ ≠ f x₂
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : TopologicalSpace A
inst✝¹ : Module 𝕜 A
inst✝ : Subsingleton A
φ : ↑(characterSpace 𝕜 A)
x : A
⊢ 0 = 0 0

Tactic:
map_zero

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommSemiring 𝕜
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : ContinuousAdd 𝕜
inst✝⁴ : ContinuousConstSMul 𝕜 𝕜
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : TopologicalSpace A
inst✝¹ : Module 𝕜 A
inst✝ : Subsingleton A
φ : ↑(characterSpace 𝕜 A)
x : A
⊢ 0 = 0
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: F : Type u_1
R : Type u_2
S : Type u_3
A : Type u_4
B : Type u_5
p : A → Prop
q : B → Prop
inst✝²⁴ : CommSemiring R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Ring A
inst✝¹⁸ : StarRing A
inst✝¹⁷ : TopologicalSpace A
inst✝¹⁶ : Algebra R A
inst✝¹⁵ : Ring B
inst✝¹⁴ : StarRing B
inst✝¹³ : TopologicalSpace B
inst✝¹² : Algebra R B
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra S B
inst✝⁷ : IsScalarTower R S A
inst✝⁶ : IsScalarTower R S B
inst✝⁵ : ContinuousFunctionalCalculus R A p
inst✝⁴ : ContinuousFunctionalCalculus R B q
inst✝³ : ContinuousMap.UniqueHom R B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F S A B
inst✝ : StarHomClass F A B
φ : F
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
hφ : Continuous[inst✝¹⁷, inst✝¹³] ⇑φ
ha : p a
hφa : q (φ a)
ψ : A →⋆ₐ[R] B := StarAlgHom.restrictScalars R ↑φ
this✝ : Continuous[inst✝¹⁷, inst✝¹³] ⇑ψ
h_spec : spectrum R (ψ a) ⊆ spectrum R a
hψa : q (ψ a)
ι : C(↑(spectrum R (ψ a)), ↑(spectrum R a)) := { toFun := Set.inclusion h_spec, continuous_toFun := ⋯ }
this : ψ.comp (cfcHom ha) = (cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι)
⊢ φ (cfc f a) = cfc f (φ a)

Tactic:
by
    have hf' : ContinuousOn f (spectrum R (ψ a)) := hf.mono h_spec
    rw [cfc_apply .., cfc_apply ..]
    congrm($(this) ⟨_, hf.domRestrict⟩)

Goal state after:
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
| (∀ (x : α), ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' Iic u ∈ 𝓝 x) ↔ ∀ (u : Set β), IsOpen[inst✝] u → ∀ x ∈ f ⁻¹' Iic u, f ⁻¹' Iic u ∈ 𝓝 x

Tactic:
enter [1, x]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
f : α → Set β
inst✝ : TopologicalSpace β
x : α
| ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' Iic u ∈ 𝓝 x
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case a.refine_1
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
x : α
⊢ x ∈ s → x ∈ U.preimage (U.image s ∩ V.preimage t)

Tactic:
hx

Goal state after:
case a.refine_1
α : Type u_1
U V : SetRel α α
s t : Set α
hst : s ⊆ (U ○ V).preimage t
hts : t ⊆ (U ○ V).image s
x : α
hx : x ∈ s
⊢ x ∈ U.preimage (U.image s ∩ V.preimage t)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
U : SetRel α α
inst✝ : U.IsTrans
⊢ (hausdorffEntourage U).IsTrans

Tactic:
grw [isTrans_iff_comp_subset_self, ← hausdorffEntourage_comp, comp_subset_self]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] {∅}

Tactic:
rw [← Set.powerset_empty]

Goal state after:
case left
α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] (𝒫 ∅)
Module: Mathlib.Topology.UniformSpace.Closeds

Goal state before:
pp: α : Type u_1
inst✝ : UniformSpace α
⊢ IsClosed[(UniformSpace.hausdorff α).toTopologicalSpace] {s | TotallyBounded s}

Tactic:
by
  simp_rw [isClosed_iff_frequently, nhds_eq_comap_uniformity]
  intro s hs U hU
  obtain ⟨V : SetRel α α, hV, hVU⟩ := comp_mem_uniformity_sets hU
  rw [(𝓤 α).basis_sets.uniformity_hausdorff.comap _ |>.frequently_iff] at hs
  obtain ⟨t, ⟨hst : s ⊆ V.preimage t, -⟩, ht⟩ := hs V hV
  obtain ⟨u, hu, htu⟩ := ht V hV
  refine ⟨u, hu, ?_⟩
  grw [hst, htu, ← hVU]
  simp [Set.subset_def]
  grind

Goal state after: