mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case refine_4
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
⊢ ∀ (a : ℚ),
    ∀ x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1),
      eval 1 ((⇑derivative)^[n - k] x) = 0 → eval 1 ((⇑derivative)^[n - k] (a • x)) = 0

Tactic:
·

Goal state after:
case refine_4
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)
⊢ ∀ (a : ℚ),
    ∀ x ∈ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1),
      eval 1 ((⇑derivative)^[n - k] x) = 0 → eval 1 ((⇑derivative)^[n - k] (a • x)) = 0
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: case inl
a b : ℝ
h : a < b
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤

pp: case inr
a b : ℝ
h : b ≤ a
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤

Tactic:
·

Goal state after:
case inl
a b : ℝ
h : a < b
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤
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
⊢ star ((cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ }) *
      (cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ } =
    (cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ } *
      star ((cfcₙHom x✝) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := x✝¹ })

Tactic:
rw [← cfcₙ_apply f a, ← cfcₙ_star, ← cfcₙ_mul .., ← 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 (fun x ↦ f x * star (f x)) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : 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✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : 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:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : 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.RingTheory.Polynomial.Bernstein

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

Tactic:
let x : MvPolynomial Bool R := MvPolynomial.X true

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
⊢ ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν = n • X
Module: Mathlib.Topology.ContinuousMap.Weierstrass

Goal state before:
pp: case inr
a b : ℝ
h : b ≤ a
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤

Tactic:
·

Goal state after:
case inr
a b : ℝ
h : b ≤ a
⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : 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:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : 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.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] := ⋯
⊢ ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν = (MvPolynomial.aeval e) ((pderiv true) ((x + y) ^ n)) * X

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) ((x + y) ^ n)) * X = n • X

Tactic:
·

Goal state after:
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] := ⋯
⊢ ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν = (MvPolynomial.aeval e) ((pderiv true) ((x + y) ^ n)) * X
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case zero
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
⊢ 0 • bernsteinPolynomial R n 0 = ↑0 * X ^ (0 - 1) * (1 - X) ^ (n - 0) * ↑(n.choose 0) * X

pp: case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ (k + 1) • bernsteinPolynomial R n (k + 1) =
    ↑(k + 1) * X ^ (k + 1 - 1) * (1 - X) ^ (n - (k + 1)) * ↑(n.choose (k + 1)) * X

Tactic:
·

Goal state after:
case zero
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
⊢ 0 • bernsteinPolynomial R n 0 = ↑0 * X ^ (0 - 1) * (1 - X) ^ (n - 0) * ↑(n.choose 0) * X
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a
Module: Mathlib.RingTheory.Polynomial.Bernstein

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ (k + 1) • bernsteinPolynomial R n (k + 1) =
    ↑(k + 1) * X ^ (k + 1 - 1) * (1 - X) ^ (n - (k + 1)) * ↑(n.choose (k + 1)) * X

Tactic:
·

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
x : MvPolynomial Bool R := MvPolynomial.X true
y : MvPolynomial Bool R := MvPolynomial.X false
pderiv_true_x : (pderiv true) x = 1
pderiv_true_y : (pderiv true) y = 0
e : Bool → R[X] := fun i ↦ bif i then X else 1 - X
k : ℕ
⊢ (k + 1) • bernsteinPolynomial R n (k + 1) =
    ↑(k + 1) * X ^ (k + 1 - 1) * (1 - X) ^ (n - (k + 1)) * ↑(n.choose (k + 1)) * X
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : ¬p a
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
·

Goal state after:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : ¬p a
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a
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✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
·

Goal state after:
case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f 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✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a

Tactic:
·

Goal state after:
case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x ↦ -f x) a = -cfcₙ f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case mp
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
f : C(↑(σₙ R a), R)₀
⊢ 0 ≤ (cfcₙHom ha) f → 0 ≤ f

pp: case mpr
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
f : C(↑(σₙ R a), R)₀
⊢ 0 ≤ f → 0 ≤ (cfcₙHom ha) f

Tactic:
·

Goal state after:
case mp
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
f : C(↑(σₙ R a), R)₀
⊢ 0 ≤ (cfcₙHom ha) f → 0 ≤ f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case mpr
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
f : C(↑(σₙ R a), R)₀
⊢ 0 ≤ f → 0 ≤ (cfcₙHom ha) f

Tactic:
·

Goal state after:
case mpr
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
f : C(↑(σₙ R a), R)₀
⊢ 0 ≤ f → 0 ≤ (cfcₙHom ha) f
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : p a
⊢ cfcₙ f a ≤ cfcₙ g a

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : ¬p a
⊢ cfcₙ f a ≤ cfcₙ g a

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : PartialOrder R
inst✝¹⁶ : Nontrivial R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ContinuousSqrt R
inst✝¹⁰ : StarOrderedRing R
inst✝⁹ : NoZeroDivisors R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
f g : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ g x
hf : ContinuousOn f (σₙ R a)
hg : ContinuousOn g (σₙ R a)
hf0 : f 0 = 0
hg0 : g 0 = 0
ha : p a
⊢ cfcₙ f a ≤ cfcₙ g a