awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  ∀ (x : (↑(σₙ R a) : Type u_1)),
    (f : (a : (↑(σₙ R a) : Type u_1)) → R) x =
      (σₙ R a).restrict (π := fun (a : R) ↦ R)
        (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
          (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
        x
⊢ (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)

Tactic:
ext
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
x : (↑(σₙ R a) : Type u_1)
⊢ (f : (a : (↑(σₙ R a) : Type u_1)) → R) x =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
      x

Tactic:
simp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
⊢ (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)

Tactic:
have h : f = (σₙ R a).restrict (Function.extend Subtype.val f g) := by
    ext; simp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
⊢ ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)

Tactic:
have hg : ContinuousOn (Function.extend Subtype.val f g) (σₙ R a) :=
    continuousOn_iff_continuous_restrict.mpr <| h ▸ map_continuous f
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
hg :
  ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)
rw : Eq (α := R) ((f : (a : (↑(σₙ R a) : Type u_1)) → R) 0 : R) ((↑0 : R) : R)
⊢ Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g 0 =
    0

Tactic:
rw [← quasispectrum.coe_zero (R := R) a, Subtype.val_injective.extend_apply]

Premises used: quasispectrum.coe_zero, Subtype.val_injective, Function.Injective.extend_apply
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
hg :
  ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)
⊢ Eq (α := R) ((f : (a : (↑(σₙ R a) : Type u_1)) → R) 0 : R) ((↑0 : R) : R)

Tactic:
map_zero f

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
hg :
  ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)
⊢ Eq (α := R) ((f : (a : (↑(σₙ R a) : Type u_1)) → R) 0 : R) ((↑0 : R) : R)

Tactic:
exact map_zero f

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
hg :
  ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)
⊢ Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g 0 =
    0

Tactic:
have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
    rw [← quasispectrum.coe_zero (R := R) a, Subtype.val_injective.extend_apply]
    exact map_zero f
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
h :
  (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) =
    (σₙ R a).restrict (π := fun (a : R) ↦ R)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
hg :
  ContinuousOn
    (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
    (σₙ R a)
hg0 :
  Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
      (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g 0 =
    0
generalize :
  ∀ (f' : R → R) (h : (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) = (σₙ R a).restrict (π := fun (a : R) ↦ R) f')
    (hg : ContinuousOn f' (σₙ R a)) (hg0 : f' 0 = 0),
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f = cfcₙ (p := p) f' a
⊢ (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f =
    cfcₙ (p := p)
      (Function.extend (α := { x : R // x ∈ σₙ R a }) (Subtype.val (p := fun (x : R) ↦ x ∈ σₙ R a))
        (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) g)
      a

Tactic:
generalize Function.extend Subtype.val f g = f' at *

Premises used: Function.extend, Subtype.val, DFunLike.coe
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
f' : R → R
h : (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) = (σₙ R a).restrict (π := fun (a : R) ↦ R) f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
rw :
  Eq.{u_2 + 1} (α := A) ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f : A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f', continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
⊢ (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f = cfcₙ (p := p) f' a

Tactic:
rw [cfcₙ_apply ..]

Premises used: cfcₙ_apply
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
a : A
g : R → R
ha : p a
f : C((↑(σₙ R a) : Type u_1), R)₀
f' : R → R
h : (⇑f : (a : (↑(σₙ R a) : Type u_1)) → R) = (σₙ R a).restrict (π := fun (a : R) ↦ R) f'
hg : ContinuousOn f' (σₙ R a)
hg0 : f' 0 = 0
⊢ Eq.{u_2 + 1} (α := A) ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f : A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f', continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)

Tactic:
congr!
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : 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
rw :
  Eq.{u_2 + 1} (α := A)
    (if h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0 then
        (cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
          { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := ⋯ }
      else 0 :
      A)
    (((cfcₙL (p := p) ha : C((↑(σₙ R a) : Type u_1), R)₀ →L[R] A) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := hf0 } :
      A)
rw₁ :
  Eq.{u_2 + 1} (α := A)
    ((cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
    (((cfcₙL (p := p) ha : C((↑(σₙ R a) : Type u_1), R)₀ →L[R] A) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := hf0 } :
      A)
rw₂ rw₃ :
  Eq.{u_2 + 1} (α := A)
    ((cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := hf0 } :
      A)
⊢ cfcₙ (p := p) f a =
    ((cfcₙL (p := p) ha : C((↑(σₙ R a) : Type u_1), R)₀ →L[R] A) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := hf0 }

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

Premises used: cfcₙ_def, dif_pos, cfcₙL_apply
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
pos :
  ∀ (f_cont : ContinuousOn f (σₙ R a)),
    cfcₙ (p := p) f a =
      (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)
neg :
  ∀ (f_cont : ¬ContinuousOn f (σₙ R a)),
    cfcₙ (p := p) f a =
      (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)
⊢ cfcₙ (p := p) f a =
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
by_cases f_cont : ContinuousOn f (quasispectrum R a)

Premises used: quasispectrum, ContinuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ContinuousOn f (σₙ R a)
pos :
  ∀ (f_zero : f 0 = 0),
    cfcₙ (p := p) f a =
      (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)
neg :
  ∀ (f_zero : ¬f 0 = 0),
    cfcₙ (p := p) f a =
      (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)
⊢ cfcₙ (p := p) f a =
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
by_cases f_zero : f 0 = 0

Premises used: OfNat.ofNat
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : 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
pos :
  Eq.{u_2 + 1} (α := A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0) :
      A)
pos₁ pos₂ :
  Eq.{u_2 + 1} (α := A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := f_zero } :
      A)
⊢ cfcₙ (p := p) f a =
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
rw [cfcₙ_apply f a, mkD_of_continuousOn f_cont f_zero]

Premises used: ContinuousMapZero.mkD_of_continuousOn, cfcₙ_apply
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : 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
neg : (σₙ R a).restrict (π := fun (a : R) ↦ R) f 0 ≠ 0
⊢ cfcₙ (p := p) f a =
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
rw [cfcₙ_apply_of_not_map_zero a f_zero, mkD_of_not_zero, map_zero]

Premises used: ContinuousMapZero.mkD_of_not_zero, cfcₙ_apply_of_not_map_zero, map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : 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).restrict (π := fun (a : R) ↦ R) f 0 ≠ 0

Tactic:
f_zero

Premises used: Set.restrict
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : 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).restrict (π := fun (a : R) ↦ R) f 0 ≠ 0

Tactic:
exact f_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
f_cont : ¬ContinuousOn f (σₙ R a)
neg :
  Eq.{u_2 + 1} (α := A) (0 : A)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0) :
      A)
neg₁ : Eq.{u_2 + 1} (α := A) (0 : A) ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) 0 : A)
neg₂ neg₃ : Eq.{u_2 + 1} (α := A) 0 0
⊢ cfcₙ (p := p) f a =
    (cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
rw [cfcₙ_apply_of_not_continuousOn a f_cont, mkD_of_not_continuousOn f_cont, map_zero]

Premises used: map_zero, ContinuousMapZero.mkD_of_not_continuousOn, cfcₙ_apply_of_not_continuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
f : R → R
a : A
ha : p a
⊢ cfcₙ (p := p) f a =
    ((cfcₙL (p := p) ha : C((↑(σₙ R a) : Type u_1), R)₀ →L[R] A) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      (mkD ((σₙ R a).restrict (π := fun (a : R) ↦ R) f) 0)

Tactic:
cfcₙ_apply_mkD _ _

Premises used: cfcₙ_apply_mkD
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : TopologicalSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a))
    (h0 :
      Eq (α := R)
        (({ toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯ :
                C((↑(σₙ R a) : Type u_1), R) } :
              (a : (↑(σₙ R a) : Type u_1)) → R)
            0 :
          R)
        (0 : R))
    (ha : p a),
    P
      ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := ⋯, map_zero' := h0 :
          C((↑(σₙ R a) : Type u_1), R)₀ })
pos : ∀ (h : ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a), P (cfcₙ (p := p) f a)
neg : ∀ (h : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a)), P (cfcₙ (p := p) f a)
⊢ P (cfcₙ (p := p) f a)

Tactic:
by_cases h : ContinuousOn f (σₙ R a) ∧ f 0 = 0 ∧ p a

Premises used: quasispectrum, ContinuousOn, OfNat.ofNat