awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
Goal:
𝕜 : Type u_1
inst : RCLike 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
x y : E
rw :
  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))
⊢ 4⁻¹ *
      (HMul.hMul (α := 𝕜) (↑‖y + x‖ : 𝕜) (↑‖y + x‖ : 𝕜) - HMul.hMul (α := 𝕜) (↑‖y - x‖ : 𝕜) (↑‖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) -
        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))

Tactic:
rw [add_comm y x, norm_sub_rev]

Premises used: add_comm, norm_sub_rev
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
⊢ 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))

Tactic:
simp only [hI, neg_zero, zero_mul]

Premises used: MulZeroClass.zero_mul, neg_zero
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
⊢ HMul.hMul (α := 𝕜) I I = -1

Tactic:
have hI' := I_mul_I_of_nonzero hI
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
v : E
rw : norm.{u_1} (E := 𝕜) I * ‖v‖ = ‖v‖
rw₁ : 1 * ‖v‖ = ‖v‖
rw₂ rw₃ : ‖v‖ = ‖v‖
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I v‖ = ‖v‖

Tactic:
rw [norm_smul, norm_I_of_ne_zero hI, one_mul]

Premises used: RCLike.norm_I_of_ne_zero, norm_smul, one_mul
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
⊢ ∀ (v : E), ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I v‖ = ‖v‖

Tactic:
have I_smul (v : E) : ‖(I : 𝕜) • v‖ = ‖v‖ := by rw [norm_smul, norm_I_of_ne_zero hI, one_mul]
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y)‖ =
    ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y‖

Tactic:
I_smul ((I : 𝕜) • x + y)

Premises used: RCLike.I, HAdd.hAdd, HSMul.hSMul
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
h :
  HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x =
    HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y)
⊢ ‖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‖

Tactic:
convert! I_smul ((I : 𝕜) • x + y) using 2

Premises used: RCLike.I, HAdd.hAdd, HSMul.hSMul
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x + HMul.hMul (α := 𝕜) I I • x -
      (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (-1) x) =
    0

Tactic:
module
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
a :
  HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x + HMul.hMul (α := 𝕜) I I • x -
      (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (-1) x) =
    0
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y - x =
    HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + y)

Tactic:
linear_combination (norm := module) -hI' • x
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ ‖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‖

Tactic:
have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
    convert! I_smul ((I : 𝕜) • x + y) using 2
    linear_combination (norm := module) -hI' • x
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x‖ =
    ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x)‖

Tactic:
(I_smul ((I : 𝕜) • y + x)).symm

Premises used: RCLike.I, HAdd.hAdd, HSMul.hSMul, Eq.symm
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
h :
  HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y =
    HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x)
⊢ ‖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‖

Tactic:
convert! (I_smul ((I : 𝕜) • y + x)).symm using 2

Premises used: RCLike.I, HAdd.hAdd, HSMul.hSMul, Eq.symm
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y + HMul.hMul (α := 𝕜) I I • y -
      (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (-1) y) =
    0

Tactic:
module
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
a :
  HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y + HMul.hMul (α := 𝕜) I I • y -
      (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x) +
        HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (-1) y) =
    0
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - y =
    HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I y + x)

Tactic:
linear_combination (norm := module) -hI' • y
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
⊢ ‖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‖

Tactic:
have h₂ : ‖(I : 𝕜) • y + x‖ = ‖(I : 𝕜) • x - y‖ := by
    convert! (I_smul ((I : 𝕜) • y + x)).symm using 2
    linear_combination (norm := module) -hI' • y
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖
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))
⊢ 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))

Tactic:
rw [h₁, h₂]
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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 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:
ring
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
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))
⊢ inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z

Tactic:
unfold inner_

Premises used: _private.0.inner_
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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
⊢ ‖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)
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖)
⊢ ‖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)
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖)
⊢ ‖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
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖)
⊢ ‖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
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖)
h4 : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ +
      ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ =
    2 *
      (‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ +
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖)

Tactic:
have h5 := parallelogram_identity ((I : 𝕜) • (x + y) + z) ((I : 𝕜) • x - z)
-- Module: Mathlib.Analysis.InnerProductSpace.OfNorm
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‖)
h4 : ‖y - z + z‖ * ‖y - z + z‖ + ‖y - z - z‖ * ‖y - z - z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖)
h5 :
  ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ +
      ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z)‖ =
    2 *
      (‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) + z‖ +
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x - z‖)
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z + (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z)‖ +
      ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z)‖ *
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z - (HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z)‖ =
    2 *
      (‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I (x + y) - z‖ +
        ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖ * ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) I x + z‖)

Tactic:
have h6 := parallelogram_identity ((I : 𝕜) • (x + y) - z) ((I : 𝕜) • x + z)