awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
      ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E) =
    0
⊢ ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
have pos : 0 ≤
  ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
      ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
      ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
Tactic: simp only [h, norm_zero]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  ¬HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E) =
      0
⊢ norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
have rw : ‖HMul.hMul (α := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
      ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2‖ ≤
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
Tactic: rw [map_mul, Prod.snd_mul]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  ¬HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E) =
      0
⊢ ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
have this : HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2 ≤
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
Tactic: have : ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤
          ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ * ‖(Unitization.splitMul 𝕜 E x).snd‖ :=
          (norm_splitMul_snd_sq 𝕜 x).trans <| by
            rw [map_mul, Prod.snd_mul]
            exact norm_mul_le _ _
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  ¬HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E) =
      0
this :
  HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
⊢ ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
have this₁ : norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
Tactic: rw [sq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  ¬HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E) =
      0
this :
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
⊢ ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
have h₁ : 0 <
  ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
      ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖
Tactic: rw [← Ne, ← norm_pos_iff] at h
Cleared hypotheses: h, this
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x_1 x : Unitization 𝕜 E
h :
  0 <
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖
this :
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
⊢ ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖
have this₁ : ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E) ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ *
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖ ≤
  ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1⋆ : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2⋆ : E →L[𝕜] E)‖ *
    ‖HAdd.hAdd (α := E →L[𝕜] E) (β := E →L[𝕜] E)
        ((algebraMap 𝕜 (E →L[𝕜] E) : (a : 𝕜) → E →L[𝕜] E) x.toProd.1 : E →L[𝕜] E)
        ((mul 𝕜 E : (a : E) → E →L[𝕜] E) x.toProd.2 : E →L[𝕜] E)‖
Tactic: simp only [Unitization.splitMul_apply, Unitization.snd_star,
          Unitization.fst_star] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
⊢ max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) *
      max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) ≤
    max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2)
have h₁ : ∀ (x : Unitization 𝕜 E),
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
Tactic: have h₁ : ∀ x : Unitization 𝕜 E,
        ‖(Unitization.splitMul 𝕜 E x).snd‖ ≤ ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ := by
      simp only [Unitization.splitMul_apply, Unitization.snd_star, Unitization.fst_star]
      intro x
      /- split based on whether the term inside the norm is zero or not. If so, it's trivial.
      If not, then apply `norm_splitMul_snd_sq` and cancel one copy of the norm -/
      by_cases h : algebraMap 𝕜 (E →L[𝕜] E) x.fst + mul 𝕜 E x.snd = 0
      · simp only [h, norm_zero]
        exact norm_nonneg _
      · have : ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤
          ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ * ‖(Unitization.splitMul 𝕜 E x).snd‖ :=
          (norm_splitMul_snd_sq 𝕜 x).trans <| by
            rw [map_mul, Prod.snd_mul]
            exact norm_mul_le _ _
        rw [sq] at this
        rw [← Ne, ← norm_pos_iff] at h
        simp only [Unitization.splitMul_apply, Unitization.snd_star,
          Unitization.fst_star] at this
        exact (mul_le_mul_iff_left₀ h).mp this
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
⊢ norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 =
    HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
have refine : norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 ≤
  HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
Tactic: refine le_antisymm ?_ (norm_splitMul_snd_sq 𝕜 x)
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
⊢ norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 ≤
    HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
have rw : ‖HMul.hMul (α := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
      ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2‖ ≤
  HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
Tactic: rw [map_mul, Prod.snd_mul]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
⊢ norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
    HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
have rw : norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 *
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
Tactic: rw [sq]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
⊢ norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 *
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
have hbc : norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2 ≤
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
⊢ max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) *
      max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) ≤
    max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2)
have h₂ : norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 =
  HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
Tactic: have h₂ : ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖
        = ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 := by
      refine le_antisymm ?_ (norm_splitMul_snd_sq 𝕜 x)
      rw [map_mul, Prod.snd_mul]
      exact (norm_mul_le _ _).trans <| by
        rw [sq]
        gcongr
        simpa only [star_star] using h₁ (star x)
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
h₂ :
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 =
    HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
⊢ max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) *
      max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) ≤
    max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2)
have h₃ : norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1 =
  HPow.hPow (β := ℕ) (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1) 2
Tactic: have h₃ : ‖(Unitization.splitMul 𝕜 E (star x * x)).fst‖
        = ‖(Unitization.splitMul 𝕜 E x).fst‖ ^ 2 := by
      simp only [Unitization.splitMul_apply, Unitization.fst_mul, Unitization.fst_star,
        norm_mul, norm_star, sq]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.Unitization (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : DenselyNormedField 𝕜
inst_1 : NonUnitalNormedRing E
inst_2 : StarRing E
inst_3 : CStarRing E
inst_4 : NormedSpace 𝕜 E
inst_5 : IsScalarTower 𝕜 E E
inst_6 : SMulCommClass 𝕜 E E
inst_7 : StarRing 𝕜
inst_8 : StarModule 𝕜 E
inst_9 : CStarRing 𝕜
x : Unitization 𝕜 E
h₁ :
  ∀ (x : Unitization 𝕜 E),
    norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2 ≤
      norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x⋆).2
h₂ :
  norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2 =
    HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2
h₃ :
  norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1 =
    HPow.hPow (β := ℕ) (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1) 2
⊢ max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) *
      max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
        (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) ≤
    max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) (x⋆ * x)).2)
have rw : max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) *
    max (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1)
      (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) ≤
  max (HPow.hPow (β := ℕ) (norm.{u_1} (E := 𝕜) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).1) 2)
    (HPow.hPow (β := ℕ) (norm.{u_2} (E := E →L[𝕜] E) ((splitMul 𝕜 E : (a : Unitization 𝕜 E) → 𝕜 × (E →L[𝕜] E)) x).2) 2)
Tactic: rw [h₂, h₃]