awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
case :
  ∀ (g : C((↑(σₙ R a) : Type u_1), R)₀),
    p ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g)
case₁ :
  ∀ (g : C((↑(σₙ R a) : Type u_1), R)₀),
    q ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g) ∧
      QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
        ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A)
        (⇑f : (a : S) → R)
case₂ :
  ∀ (g : C((↑(σₙ R a) : Type u_1), R)₀),
    QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) (A := A)
      ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A) (⇑f : (a : S) → R)
⊢ ∀ (f_1 : C((↑(σₙ R a) : Type u_1), R)₀),
    p ((nonUnitalStarAlgHom (cfcₙHom (p := q) ⋯) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) f_1)
have case₃ : ∀ (g : C((↑(σₙ R a) : Type u_1), R)₀) (s : S)
  (hs :
    s ∈
      quasispectrum.{u_2, u_3} S (A := A)
        ((nonUnitalStarAlgHom (cfcₙHom (p := q) sorry) sorry : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A) g : A)),
  (algebraMap R S : (a : R) → S) ((f : (a : S) → R) s : R) = s
Tactic: case predicate_hom =>
      intro g
      rw [h]
      refine ⟨cfcₙHom_predicate _ _, ?_⟩
      refine { rightInvOn := fun s hs ↦ ?_, left_inv := ((h a).mp ha).2.left_inv }
      rw [nonUnitalStarAlgHom_apply,
        cfcₙHom_map_quasispectrum] at hs
      obtain ⟨r, rfl⟩ := hs
      simp [((h a).mp ha).2.left_inv _]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
⊢ Eq (α := C((↑(σₙ R a) : Type u_1), R)₀ →⋆ₙₐ[R] A) (cfcₙHom (p := p) hpa)
    (nonUnitalStarAlgHom (cfcₙHom (p := q) hqa) h)
have hφ₁ : Continuous[_, inst_20] (⇑(nonUnitalStarAlgHom (cfcₙHom (p := q) hqa) h) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
Tactic: apply cfcₙHom_eq_of_continuous_of_map_id
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
hφ₁ : Continuous[_, inst_20] (⇑(nonUnitalStarAlgHom (cfcₙHom (p := q) hqa) h) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
⊢ Eq (α := C((↑(σₙ R a) : Type u_1), R)₀ →⋆ₙₐ[R] A) (cfcₙHom (p := p) hpa)
    (nonUnitalStarAlgHom (cfcₙHom (p := q) hqa) h)
have hφ₂ : (nonUnitalStarAlgHom (cfcₙHom (p := q) hqa) h : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
    (ContinuousMapZero.id (σₙ R a) : C((↑(σₙ R a) : Type u_1), R)₀) =
  a
Tactic: apply cfcₙHom_eq_of_continuous_of_map_id
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have pos : ∀ (hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0),
  cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
Tactic: by_cases hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
pos :
  ∀ (hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0),
    cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have neg : ∀ (hg : ¬(ContinuousOn g (σₙ R a) ∧ g 0 = 0)),
  cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
Tactic: by_cases hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (hypothesis)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have hg₁ : ContinuousOn g (σₙ R a)
Tactic: obtain ⟨hg, hg0⟩ := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (hypothesis)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0
hg₁ : ContinuousOn g (σₙ R a)
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have hg0 : g 0 = 0
Tactic: obtain ⟨hg, hg0⟩ := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have pos : cfcₙ (R := S) (p := q)
    (Function.extend (α := { x : S // x ∈ σₙ S a }) (Subtype.val (p := fun (x : S) ↦ x ∈ σₙ S a))
      (⇑(comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry, map_zero' := sorry : C(R, S)₀ }
            (comp
              { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) g, continuous_toFun := sorry, map_zero' := sorry :
                C((↑(σₙ R a) : Type u_1), R)₀ }
              {
                toFun :=
                  Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) sorry,
                continuous_toFun := sorry, map_zero' := sorry :
                ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
        (a : (↑(σₙ S a) : Type u_2)) → S)
      0)
    a =
  cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
Tactic: rw [cfcₙ_apply g a, cfcₙHom_eq_restrict f hpa hqa h, nonUnitalStarAlgHom_apply,
      cfcₙHom_eq_cfcₙ_extend 0]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
⊢ cfcₙ (R := S) (p := q)
      (Function.extend (α := { x : S // x ∈ σₙ S a }) (Subtype.val (p := fun (x : S) ↦ x ∈ σₙ S a))
        (⇑(comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ : C(R, S)₀ }
              (comp
                { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) g, continuous_toFun := ⋯, map_zero' := ⋯ :
                  C((↑(σₙ R a) : Type u_1), R)₀ }
                {
                  toFun :=
                    Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯,
                  continuous_toFun := ⋯, map_zero' := ⋯ :
                  ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
          (a : (↑(σₙ S a) : Type u_2)) → S)
        0)
      a =
    cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have apply : ∀ (x : S) (hx : x ∈ σₙ S a),
  Eq (α := S)
    (Function.extend (α := { x : S // x ∈ σₙ S a }) (Subtype.val (p := fun (x : S) ↦ x ∈ σₙ S a))
        (⇑(comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry, map_zero' := sorry }
              (comp { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) g, continuous_toFun := sorry, map_zero' := sorry }
                {
                  toFun :=
                    Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) sorry,
                  continuous_toFun := sorry, map_zero' := sorry })) :
          (a : (↑(σₙ S a) : Type u_2)) → S)
        0 x :
      S)
    ((algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x)) : S)
Tactic: apply cfcₙ_congr fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (hypothesis)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ¬(ContinuousOn g (σₙ R a) ∧ g 0 = 0)
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have hg₁ : ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
Tactic: simp only [not_and_or] at hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have neg : ∀ (hg : ¬ContinuousOn g (σₙ R a)),
  cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
Tactic: obtain (hg | hg) := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
neg :
  ∀ (hg : ¬ContinuousOn g (σₙ R a)),
    cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
⊢ cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
have neg₁ : ∀ (hg : ¬g 0 = 0),
  cfcₙ (p := p) g a = cfcₙ (p := q) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) a
Tactic: obtain (hg | hg) := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
⊢ ¬ContinuousOn (Y := S) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) (σₙ S a)
have refine : ∀ (hg' : ContinuousOn (Y := S) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) (σₙ S a)),
  ContinuousOn g (σₙ R a)
Tactic: refine fun hg' ↦ hg ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : TopologicalSpace A
inst_21 : NonUnitalContinuousFunctionalCalculus S A q
inst_22 : IsScalarTower R A A
inst_23 : SMulCommClass R A A
inst_24 : NonUnitalContinuousFunctionalCalculus R A p
inst_25 : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding.{u_1, u_2} (X := R) (Y := S) (⇑(algebraMap R S) : (a : R) → S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
hg' : ContinuousOn (Y := S) (fun (x : S) ↦ (algebraMap R S : (a : R) → S) (g ((f : (a : S) → R) x))) (σₙ S a)
⊢ ContinuousOn g (σₙ R a)
have rw : ContinuousOn.{u_1, u_2} (Y := S) ((⇑(algebraMap R S) : (a : R) → S) ∘ g) (σₙ R a)
Tactic: rw [halg.isEmbedding.continuousOn_iff]