awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have pos : Eq.{u_2 + 1} (α := A)
  ((cfcₙHom (p := p) sorry : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := sorry, map_zero' := sorry } :
    A)
  ((cfcₙHom (p := p) sorry : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) g, continuous_toFun := sorry, map_zero' := sorry } :
    A)
Tactic: rw [cfcₙ_apply f a (h.2.1.congr hfg) (hfg (quasispectrum.zero_mem R a) ▸ h.2.2) h.1,
      cfcₙ_apply g a h.2.1 h.2.2 h.1]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0
⊢ 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) ⋯ : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) g, continuous_toFun := ⋯, map_zero' := ⋯ } :
      A)
have pos : (σₙ R a).restrict (π := fun (a : R) ↦ R) f = (σₙ R a).restrict (π := fun (a : R) ↦ R) g
Tactic: congr 3
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (hypothesis)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬(p a ∧ ContinuousOn g (σₙ R a) ∧ g 0 = 0)
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have h₁ : ¬p a ∨ ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
Tactic: simp only [not_and_or] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬p a ∨ ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have neg : ∀ (ha : ¬p a), cfcₙ (p := p) f a = cfcₙ (p := p) g a
Tactic: obtain (ha | hg | h0) := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬p a ∨ ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
neg : ∀ (ha : ¬p a), cfcₙ (p := p) f a = cfcₙ (p := p) g a
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have neg₁ : ∀ (hg : ¬ContinuousOn g (σₙ R a)), cfcₙ (p := p) f a = cfcₙ (p := p) g a
Tactic: obtain (ha | hg | h0) := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h : ¬p a ∨ ¬ContinuousOn g (σₙ R a) ∨ ¬g 0 = 0
neg : ∀ (ha : ¬p a), cfcₙ (p := p) f a = cfcₙ (p := p) g a
neg₁ : ∀ (hg : ¬ContinuousOn g (σₙ R a)), cfcₙ (p := p) f a = cfcₙ (p := p) g a
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have neg₂ : ∀ (h0 : ¬g 0 = 0), cfcₙ (p := p) f a = cfcₙ (p := p) g a
Tactic: obtain (ha | hg | h0) := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
hg : ¬ContinuousOn g (σₙ R a)
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have neg : ¬ContinuousOn f (σₙ R a)
Tactic: rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 g : R → R
a : A
hfg : Set.EqOn f g (σₙ R a)
h0 : ¬g 0 = 0
⊢ cfcₙ (p := p) f a = cfcₙ (p := p) g a
have neg : ¬f 0 = 0
Tactic: rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (hypothesis)
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 g : R → R
a : A
h : cfcₙ (p := p) f a = cfcₙ (p := p) g a
ha : p a
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
hg : ContinuousOn g (σₙ R a)
hg0 : g 0 = 0
hg₁ : ContinuousOn g (σₙ R a)
hg0₁ : g 0 = 0
⊢ Set.EqOn f g (σₙ R a)
have h₁ : 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 := sorry, map_zero' := sorry } :
    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) g, continuous_toFun := sorry, map_zero' := sorry } :
    A)
Tactic: rw [cfcₙ_apply f a, cfcₙ_apply g a] at h
Cleared hypotheses: h, hg, hg0
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
have pos : ∀ (ha : p a), cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
pos : ∀ (ha : p a), cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
have neg : ∀ (ha : ¬p a), cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
ha : ¬p a
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) 0 a = 0
have neg : Eq.{u_2 + 1} (α := A) 0 0
Tactic: rw [cfcₙ_apply_of_not_predicate a ha]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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
⊢ cfcₙ (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
have pos : ∀ (ha : p a), cfcₙ (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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
pos : ∀ (ha : p a), cfcₙ (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
⊢ cfcₙ (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
have neg : ∀ (ha : ¬p a), cfcₙ (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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 (p := p) (fun (x : R) ↦ f x * g x) a = cfcₙ (p := p) f a * cfcₙ (p := p) g a
have 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) fun (x : R) ↦ f x * g x, continuous_toFun := sorry,
        map_zero' := sorry } :
    A)
  ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      (HMul.hMul (α := C((↑(σₙ R a) : Type u_1), R)₀)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := sorry, map_zero' := sorry :
          C((↑(σₙ R a) : Type u_1), R)₀ }
        { 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 }) :
    A)
Tactic: rw [cfcₙ_apply f a, cfcₙ_apply g a, ← map_mul, cfcₙ_apply _ a]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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
⊢ cfcₙ (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
have pos : ∀ (ha : p a), cfcₙ (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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
pos : ∀ (ha : p a), cfcₙ (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
⊢ cfcₙ (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
have neg : ∀ (ha : ¬p a), cfcₙ (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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 (p := p) (fun (x : R) ↦ f x + g x) a = cfcₙ (p := p) f a + cfcₙ (p := p) g a
have pos : (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) fun (x : R) ↦ f x + g x, continuous_toFun := sorry,
      map_zero' := sorry } =
  HAdd.hAdd (α := A) (β := 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 := sorry, map_zero' := sorry :
          C((↑(σₙ R a) : Type u_1), R)₀ } :
      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) g, continuous_toFun := sorry, map_zero' := sorry :
          C((↑(σₙ R a) : Type u_1), R)₀ } :
      A)
Tactic: rw [cfcₙ_apply f a, cfcₙ_apply g a, cfcₙ_apply _ a]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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 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
⊢ (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) fun (x : R) ↦ f x + g x, continuous_toFun := ⋯,
        map_zero' := ⋯ } =
    HAdd.hAdd (α := A) (β := 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' := ⋯ :
            C((↑(σₙ R a) : Type u_1), R)₀ } :
        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) g, continuous_toFun := ⋯, map_zero' := ⋯ :
            C((↑(σₙ R a) : Type u_1), R)₀ } :
        A)
have 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) fun (x : R) ↦ f x + g x, continuous_toFun := sorry,
        map_zero' := sorry } :
    A)
  ((cfcₙHom (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      (HAdd.hAdd (α := C((↑(σₙ R a) : Type u_1), R)₀)
        { toFun := (σₙ R a).restrict (π := fun (a : R) ↦ R) f, continuous_toFun := sorry, map_zero' := sorry :
          C((↑(σₙ R a) : Type u_1), R)₀ }
        { 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 }) :
    A)
Tactic: simp_rw [← map_add]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
have pos : ∀ (ha : p a), cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
pos : ∀ (ha : p a), cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
have neg : ∀ (ha : ¬p a), cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
Tactic: by_cases ha : p a
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
⊢ s.sum f = fun (z : R) ↦ ∑ i ∈ s, f i z
have h : ∀ (x : R), s.sum f x = ∑ i ∈ s, f i x
Tactic: ext
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (hypothesis)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
have hsum : s.sum f = fun (z : R) ↦ ∑ i ∈ s, f i z
Tactic: have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (goal)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun (z : R) ↦ ∑ i ∈ s, f i z
⊢ ContinuousOn (Y := R) (∑ i : ↥s, f (↑i : ι)) (σₙ R a)
have rw : ContinuousOn (Y := R) (fun (z : R) ↦ ∑ i ∈ s, f i z) (σₙ R a)
Tactic: rw [sum_coe_sort s, hsum]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital (hypothesis)
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
ι : Type u_3
f : ι → R → R
a : A
s : Finset ι
hf : ∀ i ∈ s, ContinuousOn (f i) (σₙ R a)
hf0 : ∀ i ∈ s, f i 0 = 0
ha : p a
hsum : s.sum f = fun (z : R) ↦ ∑ i ∈ s, f i z
⊢ cfcₙ.{u_1, u_2} (R := R) (p := p) (∑ i ∈ s, f i) a = ∑ i ∈ s, cfcₙ (p := p) (f i) a
have hf' : ContinuousOn (Y := R) (∑ i : ↥s, f (↑i : ι)) (σₙ R a)
Tactic: have hf' : ContinuousOn (∑ i : s, f i) (σₙ R a) := by
      rw [sum_coe_sort s, hsum]
      exact continuousOn_finsetSum s fun i hi => hf i hi