awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
x y : E
hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0
hI' : HMul.hMul (α := 𝕜) I I = -1
I_smul : ∀ (v : E), ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I v‖ = ‖v‖
h₁ : ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ = ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖
h₂ : ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ = ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (↑‖x + y‖ : 𝕜) (↑‖x + y‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖x - y‖ : 𝕜) (↑‖x - y‖ : 𝕜) +
          HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) *
            (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ : 𝕜) -
        HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜) *
          (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x‖ : 𝕜)) =
    4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
have neg : 4⁻¹ *
    (HMul.hMul (α := 𝕜) (↑‖x + y‖ : 𝕜) (↑‖x + y‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖x - y‖ : 𝕜) (↑‖x - y‖ : 𝕜) +
        HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜) *
          (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜) -
      HMul.hMul (α := 𝕜) (-I) (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) *
        (↑‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜)) =
  4⁻¹ *
    (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
          HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
      HMul.hMul (β := 𝕜)
        (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
Tactic: rw [h₁, h₂]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
⊢ inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z
have unfold : 4⁻¹ *
    (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) -
          HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) +
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜) -
      HMul.hMul (β := 𝕜)
        (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜))
        ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜)) =
  4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜)) +
    4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
Tactic: unfold inner_
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (hypothesis)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜)) =
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜)) +
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
have h1 : ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
  2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
Tactic: have h1 := parallelogram_identity (x + y + z) (x - z)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (hypothesis)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
h1 :
  ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
    2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜)) =
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜)) +
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
have h2 : ‖x + y - z + (x + z)‖ * ‖x + y - z + (x + z)‖ + ‖x + y - z - (x + z)‖ * ‖x + y - z - (x + z)‖ =
  2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖)
Tactic: have h2 := parallelogram_identity (x + y - z) (x + z)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (hypothesis)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
h1 :
  ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
    2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
h2 :
  ‖x + y - z + (x + z)‖ * ‖x + y - z + (x + z)‖ + ‖x + y - z - (x + z)‖ * ‖x + y - z - (x + z)‖ =
    2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖)
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜)) =
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜)) +
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
have h3 : ‖y + z + z‖ * ‖y + z + z‖ + ‖y + z - z‖ * ‖y + z - z‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖)
Tactic: have h3 := parallelogram_identity (y + z) z
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (hypothesis)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y z : E
h1 :
  ‖x + y + z + (x - z)‖ * ‖x + y + z + (x - z)‖ + ‖x + y + z - (x - z)‖ * ‖x + y + z - (x - z)‖ =
    2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖)
h2 :
  ‖x + y - z + (x + z)‖ * ‖x + y - z + (x + z)‖ + ‖x + y - z - (x + z)‖ * ‖x + y - z - (x + z)‖ =
    2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖)
h3 : ‖y + z + z‖ * ‖y + z + z‖ + ‖y + z - z‖ * ‖y + z - z‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖)
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y + z‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y - z‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ : 𝕜)) =
    4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ : 𝕜)) +
      4⁻¹ *
        (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y + z‖ : 𝕜) -
              HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖y - z‖ : 𝕜) +
            HMul.hMul (β := 𝕜)
              (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜)
                ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜))
              ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + z‖ : 𝕜) -
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - z‖ : 𝕜))
have h4 : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
Tactic: have h4 := parallelogram_identity (y - z) z
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
r : ℝ
x y : E
⊢ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y =
    HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
have revert : ∀ (r : ℝ),
  inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y =
    HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
Tactic: revert r
Cleared hypotheses: r
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y : E
⊢ ∀ (r : ℝ),
    inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y =
      HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
have rw : Eq.{max 1 (u_1 + 1)} (α := (x : ℝ) → 𝕜) (fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y)
  fun (r : ℝ) ↦ HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
Tactic: rw [← funext_iff]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y : E
⊢ Eq.{max 1 (u_1 + 1)} (α := (x : ℝ) → 𝕜) (fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y)
    fun (r : ℝ) ↦ HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
have refine_1 : Continuous.{0, u_1} (Y := 𝕜) fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y
Tactic: refine Rat.isDenseEmbedding_coe_real.dense.equalizer ?_ ?_ (funext fun X => ?_)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y : E
refine_1 : Continuous.{0, u_1} (Y := 𝕜) fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y
⊢ Eq.{max 1 (u_1 + 1)} (α := (x : ℝ) → 𝕜) (fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y)
    fun (r : ℝ) ↦ HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
have refine_2 : Continuous.{0, u_1} (Y := 𝕜) fun (r : ℝ) ↦
  HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
Tactic: refine Rat.isDenseEmbedding_coe_real.dense.equalizer ?_ ?_ (funext fun X => ?_)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
x y : E
refine_1 : Continuous.{0, u_1} (Y := 𝕜) fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y
refine_2 :
  Continuous.{0, u_1} (Y := 𝕜) fun (r : ℝ) ↦
    HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
⊢ Eq.{max 1 (u_1 + 1)} (α := (x : ℝ) → 𝕜) (fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y)
    fun (r : ℝ) ↦ HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)
have refine_3 : ∀ (X : ℚ),
  Eq.{u_1 + 1} (α := 𝕜) (((fun (r : ℝ) ↦ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑r : 𝕜) x) y) ∘ Rat.cast) X)
    (((fun (r : ℝ) ↦ HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) (↑r : 𝕜) : 𝕜) (inner_ 𝕜 x y)) ∘ Rat.cast) X)
Tactic: refine Rat.isDenseEmbedding_coe_real.dense.equalizer ?_ ?_ (funext fun X => ?_)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
⊢ innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
have pos : ∀ (hI : Eq.{u_1 + 1} (α := 𝕜) I 0), innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
Tactic: by_cases hI : (I : 𝕜) = 0
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
pos : ∀ (hI : Eq.{u_1 + 1} (α := 𝕜) I 0), innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
⊢ innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
have neg : ∀ (hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0), innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
Tactic: by_cases hI : (I : 𝕜) = 0
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
hI : Eq.{u_1 + 1} (α := 𝕜) I 0
⊢ innerProp'.{u_1, u_2} (𝕜 := 𝕜) E I
have pos : innerProp'.{u_1, u_2} (𝕜 := 𝕜) E 0
Tactic: rw [hI]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (hypothesis)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0
x y : E
⊢ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) y =
    HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) I : 𝕜) (inner_ 𝕜 x y)
have hI' : HMul.hMul (α := 𝕜) I I = -1
Tactic: have hI' := I_mul_I_of_nonzero hI
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0
x y : E
hI' : HMul.hMul (α := 𝕜) I I = -1
⊢ inner_ 𝕜 (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x) y =
    HMul.hMul (α := 𝕜) ((starRingEnd 𝕜 : (a : 𝕜) → 𝕜) I : 𝕜) (inner_ 𝕜 x y)
have neg : 4⁻¹ *
    (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
          HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜)
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜) +
        HMul.hMul (β := 𝕜) (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖-x + y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖-x + y‖ : 𝕜) -
      HMul.hMul (β := 𝕜) (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖-x - y‖ : 𝕜))
        ((𝓚 : (a : ℝ) → 𝕜) ‖-x - y‖ : 𝕜)) =
  4⁻¹ *
    (-I *
      (HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x + y‖ : 𝕜) -
            HMul.hMul (α := 𝕜) (β := 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖x - y‖ : 𝕜) +
          HMul.hMul (β := 𝕜)
            (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜))
            ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖ : 𝕜) -
        HMul.hMul (β := 𝕜)
          (HMul.hMul (α := 𝕜) (β := 𝕜) (I : 𝕜) ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜))
          ((𝓚 : (a : ℝ) → 𝕜) ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y‖ : 𝕜)))
Tactic: rw [conj_I, inner_, inner_, mul_left_comm, smul_smul, hI', neg_one_smul]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0
x y : E
hI' : HMul.hMul (α := 𝕜) I I = -1
⊢ ‖-x - y‖ = ‖x + y‖
have rw : ‖-(x + y)‖ = ‖x + y‖
Tactic: rw [← neg_add', norm_neg]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.OfNorm (goal)
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
inst_3 : InnerProductSpaceable E
hI : ¬Eq.{u_1 + 1} (α := 𝕜) I 0
x y : E
hI' : HMul.hMul (α := 𝕜) I I = -1
rw : ‖-(x + y)‖ = ‖x + y‖
⊢ ‖-x - y‖ = ‖x + y‖
have rw₁ : ‖x + y‖ = ‖x + y‖
Tactic: rw [← neg_add', norm_neg]