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✝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:
by
  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.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
a : A
ha : p a
⊢ 0 ≤ a ↔ ∀ x ∈ spectrum R a, 0 ≤ x

Tactic:
have := cfc_nonneg_iff (id : R → R) a (by fun_prop) ha

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
a : A
ha : p a
this : 0 ≤ cfc id a ↔ ∀ x ∈ spectrum R a, 0 ≤ id x
⊢ 0 ≤ a ↔ ∀ x ∈ spectrum R a, 0 ≤ x
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 : R → R
a : A
h : ∀ x ∈ spectrum R a, 0 ≤ f x
hf : ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

pp: case neg
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, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a

Tactic:
· simpa using cfc_mono h

Goal state after:
case neg
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, 0 ≤ f x
hf : ¬ContinuousOn f (spectrum R a)
⊢ 0 ≤ cfc f a
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 : R → R
a : A
h : ∀ x ∈ spectrum R a, f x ≤ 0
hf : ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

pp: case neg
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 ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0

Tactic:
· simpa using cfc_mono h

Goal state after:
case neg
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 ≤ 0
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc f a ≤ 0
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
y : α
⊢ eVariationOn f (s ∩ Iic y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ Ici (toDual y))

Tactic:
Ici_toDual,

Goal state after:
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
hf : BoundedVariationOn f s
y : α
⊢ eVariationOn f (s ∩ Iic y) = eVariationOn (f ∘ ⇑ofDual) (⇑ofDual ⁻¹' s ∩ ⇑ofDual ⁻¹' Iic y)
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 g : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
ha : p a
⊢ cfcₙ (fun x ↦ f x + g x) a = cfcₙ f a + cfcₙ g a

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 g : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
ha : ¬p a
⊢ cfcₙ (fun x ↦ f x + g x) a = cfcₙ f a + cfcₙ g a

Tactic:
· rw [cfcₙ_apply f a, cfcₙ_apply g a, cfcₙ_apply _ a]
    simp_rw [← map_add]
    congr

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 g : R → R
a : A
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
ha : ¬p a
⊢ cfcₙ (fun x ↦ f x + g x) a = cfcₙ f a + cfcₙ g 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✝ g : R → R
a✝ : A
hf : ContinuousOn f✝ (σₙ R a✝)
hf0 : f✝ 0 = 0
hg : ContinuousOn g (σₙ R a✝)
hg0 : g 0 = 0
ha : p a✝
f : R → R
a : A
x✝² : ContinuousOn f (σₙ R a)
x✝¹ : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
x✝ : p a
⊢ cfcₙ (fun x ↦ star (f x)) a * cfcₙ f a = cfcₙ f a * cfcₙ (fun x ↦ star (f x)) a

Tactic:
← cfcₙ_mul ..,

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
f✝ g : R → R
a✝ : A
hf : ContinuousOn f✝ (σₙ R a✝)
hf0 : f✝ 0 = 0
hg : ContinuousOn g (σₙ R a✝)
hg0 : g 0 = 0
ha : p a✝
f : R → R
a : A
x✝² : ContinuousOn f (σₙ R a)
x✝¹ : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0
x✝ : p a
⊢ cfcₙ (fun x ↦ star (f x) * f x) a = cfcₙ f a * cfcₙ (fun x ↦ star (f x)) a
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
c : ContinuousOn f (Set.Icc a b)
ε : ℝ
pos : 0 < ε
⊢ ∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε

Tactic:
let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_domRestrict.mp c⟩

Goal state after:
a b : ℝ
f : ℝ → ℝ
c : ContinuousOn f (Set.Icc a b)
ε : ℝ
pos : 0 < ε
f' : C(↑(Set.Icc a b), ℝ) := { toFun := fun x ↦ f ↑x, continuous_toFun := ⋯ }
⊢ ∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁹ : CommSemiring R
inst✝¹⁸ : PartialOrder R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : ContinuousSqrt R
inst✝¹¹ : StarOrderedRing R
inst✝¹⁰ : NoZeroDivisors R
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : PartialOrder A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : NonnegSpectrumClass R A
a : A
ha : p a
⊢ 0 ≤ a ↔ ∀ x ∈ σₙ R a, 0 ≤ x

Tactic:
have := cfcₙ_nonneg_iff (id : R → R) a (by fun_prop)
  simpa [cfcₙ_id _ a ha] using this

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
| ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
      (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
    n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν

Tactic:
Finset.mul_sum,

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
| ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
      ∑ i ∈ Finset.range (n + 1), (1 - (2 * n) • X) * i • bernsteinPolynomial R n i +
    n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
| ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
      ∑ i ∈ Finset.range (n + 1), (1 - (2 * n) • X) * i • bernsteinPolynomial R n i +
    n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν

Tactic:
Finset.mul_sum,

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
p :
  ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
        (1 - (2 * n) • X) * ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν +
      n ^ 2 • X ^ 2 * ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν
| ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν +
      ∑ i ∈ Finset.range (n + 1), (1 - (2 * n) • X) * i • bernsteinPolynomial R n i +
    ∑ i ∈ Finset.range (n + 1), n ^ 2 • X ^ 2 * bernsteinPolynomial R n i
Module: Mathlib.Topology.ContinuousMap.StoneWeierstrass

Goal state before:
pp: case pos.left
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := ⋯
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := ⋯
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := ⋯
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := ⋯
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := ⋯
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯
xs_w : ⋃ x ∈ xs, W x = ⊤ := ⋯
xs_nonempty : xs.Nonempty
k : ↑L := ⋯
z : X
⊢ ↑k z < f z + ε

pp: case pos.right
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := ⋯
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := ⋯
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := ⋯
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := ⋯
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := ⋯
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯
xs_w : ⋃ x ∈ xs, W x = ⊤ := ⋯
xs_nonempty : xs.Nonempty
k : ↑L := ⋯
z : X
⊢ f z - ε < ↑k z

Tactic:
· simp only [k, Finset.inf'_lt_iff, ContinuousMap.inf'_apply]
    exact Set.exists_set_mem_of_union_eq_top _ _ xs_w z

Goal state after:
case pos.right
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y : X), g x y ∈ L
w₁ : ∀ (x y : X), (g x y) x = f x
w₂ : ∀ (x y : X), (g x y) y = f y
U : X → X → Set X := ⋯
U_nhds_y : ∀ (x y : X), U x y ∈ 𝓝 y
ys : X → Finset X := ⋯
ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := ⋯
ys_nonempty : ∀ (x : X), (ys x).Nonempty
h : X → ↑L := ⋯
lt_h : ∀ (x z : X), f z - ε < ↑(h x) z
h_eq : ∀ (x : X), ↑(h x) x = f x
W : X → Set X := ⋯
W_nhds : ∀ (x : X), W x ∈ 𝓝 x
xs : Finset X := ⋯
xs_w : ⋃ x ∈ xs, W x = ⊤ := ⋯
xs_nonempty : xs.Nonempty
k : ↑L := ⋯
z : X
⊢ f z - ε < ↑k z
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : TopologicalSpace A
inst✝⁵ : T2Space A
inst✝⁴ : NonUnitalRing A
inst✝³ : StarRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
s : Set 𝕜
hs : CompactSpace ↑s
h0 : Fact (0 ∈ s)
φ ψ : ContinuousMapZero (↑s) 𝕜 →⋆ₙₐ[𝕜] A
hφ : Continuous[_, inst✝⁶] ⇑φ
hψ : Continuous[_, inst✝⁶] ⇑ψ
h : φ (ContinuousMapZero.id s) = ψ (ContinuousMapZero.id s)
⊢ φ = ψ

Tactic:
rw [DFunLike.ext'_iff, ← Set.eqOn_univ, ← (ContinuousMapZero.adjoin_id_dense _).closure_eq]
    refine Set.EqOn.closure (fun f hf ↦ ?_) hφ hψ
    rw [← NonUnitalStarAlgHom.mem_equalizer]
    apply adjoin_le ?_ hf
    rw [Set.singleton_subset_iff]
    exact h

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitization

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖b⋆‖ * ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖ ≤
    ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖

Tactic:
nth_rewrite 2 [← one_mul ‖_ + _‖]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : NonUnitalNormedRing E
inst✝⁶ : StarRing E
inst✝⁵ : CStarRing E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : IsScalarTower 𝕜 E E
inst✝² : SMulCommClass 𝕜 E E
inst✝¹ : StarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
b : E
hb : b ∈ Metric.closedBall 0 1
⊢ ‖b⋆‖ * ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖ ≤
    1 * ‖x.toProd.1⋆ • (x.toProd.1 • b + x.toProd.2 * b) + x.toProd.2⋆ * (x.toProd.1 • b + x.toProd.2 * b)‖
Module: Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
⊢ { toFun := fun φ ↦ φ 1, continuous_toFun := ⋯ } = 1

Tactic:
ext a

Goal state after:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : CommRing 𝕜
inst✝⁵ : NoZeroDivisors 𝕜
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : IsTopologicalRing 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Semiring A
inst✝ : Algebra 𝕜 A
a : ↑(characterSpace 𝕜 A)
⊢ { toFun := fun φ ↦ φ 1, continuous_toFun := ⋯ } a = 1 a
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
⊢ IsOpenMap f ↔ LowerHemicontinuous fun x ↦ f ⁻¹' {x}

Tactic:
rw [isOpenMap_iff_kernImage, lowerHemicontinuous_iff_isClosed_preimage_Iic]
  aesop

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
    {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}

Tactic:
simp_rw [ofPred_and, ofPred_forall]

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
u : Set (Set α)
hu₁ : u.Finite
hu₂ : ∀ U ∈ u, IsOpen[inst✝] U
⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]
    ({a | a ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x | (x ∩ i).Nonempty})
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → Set β
ι : Type u_5
inst✝ : RegularSpace β
x₀ : α
l : Filter ι
hf : UpperHemicontinuousAt f x₀
hf_closed : IsClosed[inst✝¹] (f x₀)
x : ι → α
hx : Tendsto x l (𝓝 x₀)
y : ι → β
hy : ∃ᶠ (n : ι) in l, y n ∈ f (x n)
y₀ : β
hy₀ : Tendsto y l (𝓝 y₀)
this : y₀ ∉ f x₀
s : Set β
hs : s ∈ 𝓝ˢ (f x₀)
t : Set β
ht : t ∈ 𝓝 y₀
hst : Disjoint s t
n : ι
hyn : y n ∈ f (x n)
hn : f (x n) ⊆ interior s
⊢ y n ∈ s

Tactic:
exact interior_subset <| hn hyn

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] (V, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
right✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
⊢ IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ (V, u).1 ∧ ∀ U ∈ (V, u).2, (s ∩ U).Nonempty}

Tactic:
simp_rw [ofPred_and, ofPred_forall]

Goal state after:
case refine_1
α : Type u_1
inst✝ : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
V : Set α
u : Set (Set α)
hV : IsOpen[inst✝] (V, u).1
hu : (V, u).2.Finite
huB : (V, u).2 ⊆ B
right✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1
⊢ IsOpen[TopologicalSpace.vietoris α] ({a | a ⊆ V} ∩ ⋂ i ∈ u, {x | (x ∩ i).Nonempty})
Module: Mathlib.Topology.Semicontinuity.Hemicontinuity

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
⊢ UpperHemicontinuousOn f s

Tactic:
rw [upperHemicontinuousOn_iff_forall_isOpen]

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
f : α → Set β
s : Set α
ι : Type u_3
F : ι → α → Set β
l : Filter ι
inst✝¹ : l.NeBot
inst✝ : UniformSpace β
htendsto : TendstoUniformlyOn F f l s
hF : ∀ (n : ι), UpperHemicontinuousOn (F n) s
hf_compact : ∀ x ∈ s, IsCompact (f x)
⊢ ∀ x ∈ s, ∀ (u : Set β), IsOpen[inst✝.toTopologicalSpace] u → f x ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ u
Module: Mathlib.Analysis.Normed.Algebra.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
ε : ℝ≥0∞
hε : ε < 1
h : ¬ε = 0
i : ℕ
⊢ ε⁻¹ = ∞ → ⨅ i_1, ↑‖a ^ (i_1 + i)‖₊ ^ (1 / ↑(i_1 + i)) = 0 → ∃ i_1, ↑‖a ^ (i_1 + i)‖₊ ^ (1 / ↑(i_1 + i)) = 0

Tactic:
by simp [h]

Goal state after: