l3lab/ntp-mathlib huggingface dataset:
protected theorem continuous_inner :
    Continuous (uncurry inner : Completion E × Completion E → 𝕜) := by
  let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
inner' : E →+ E →+ 𝕜 :=
  {
    toZeroHom :=
      { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) },
    map_add' :=
      (_ :
        ∀ (x y : E),
          ZeroHom.toFun
              { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
              (x + y) =
            ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                x +
              ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                y) }
⊢ Continuous (uncurry inner)

Next tactic:
have : Continuous fun p : E × E => inner' p.1 p.2 := continuous_inner
protected theorem continuous_inner :
    Continuous (uncurry inner : Completion E × Completion E → 𝕜) := by
  let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }
  have : Continuous fun p : E × E => inner' p.1 p.2 := continuous_inner
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
inner' : E →+ E →+ 𝕜 :=
  {
    toZeroHom :=
      { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) },
    map_add' :=
      (_ :
        ∀ (x y : E),
          ZeroHom.toFun
              { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
              (x + y) =
            ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                x +
              ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                y) }
this : Continuous fun p => (inner' p.1) p.2
⊢ Continuous (uncurry inner)

Next tactic:
rw [Completion.toInner, inner, uncurry_curry _]
protected theorem continuous_inner :
    Continuous (uncurry inner : Completion E × Completion E → 𝕜) := by
  let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }
  have : Continuous fun p : E × E => inner' p.1 p.2 := continuous_inner
  rw [Completion.toInner, inner, uncurry_curry _]
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
inner' : E →+ E →+ 𝕜 :=
  {
    toZeroHom :=
      { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) },
    map_add' :=
      (_ :
        ∀ (x y : E),
          ZeroHom.toFun
              { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
              (x + y) =
            ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                x +
              ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                y) }
this : Continuous fun p => (inner' p.1) p.2
⊢ Continuous (DenseInducing.extend (_ : DenseInducing fun p => (↑E p.1, ↑E p.2)) (uncurry inner))

Next tactic:
change
    Continuous
      (((denseInducing_toCompl E).prod (denseInducing_toCompl E)).extend fun p : E × E =>
        inner' p.1 p.2)
protected theorem continuous_inner :
    Continuous (uncurry inner : Completion E × Completion E → 𝕜) := by
  let inner' : E →+ E →+ 𝕜 :=
    { toFun := fun x => (innerₛₗ 𝕜 x).toAddMonoidHom
      map_zero' := by ext x; exact inner_zero_left _
      map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }
  have : Continuous fun p : E × E => inner' p.1 p.2 := continuous_inner
  rw [Completion.toInner, inner, uncurry_curry _]
  change
    Continuous
      (((denseInducing_toCompl E).prod (denseInducing_toCompl E)).extend fun p : E × E =>
        inner' p.1 p.2)
  

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
inner' : E →+ E →+ 𝕜 :=
  {
    toZeroHom :=
      { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) },
    map_add' :=
      (_ :
        ∀ (x y : E),
          ZeroHom.toFun
              { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
              (x + y) =
            ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                x +
              ZeroHom.toFun
                { toFun := fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x),
                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom ((innerₛₗ 𝕜) x)) 0 = 0) }
                y) }
this : Continuous fun p => (inner' p.1) p.2
⊢ Continuous (DenseInducing.extend (_ : DenseInducing fun p => (toCompl p.1, toCompl p.2)) fun p => (inner' p.1) p.2)

Next tactic:
exact (denseInducing_toCompl E).extend_Z_bilin (denseInducing_toCompl E) this
instance innerProductSpace : InnerProductSpace 𝕜 (Completion E) where
  norm_sq_eq_inner x :=
    Completion.induction_on x
      (isClosed_eq (continuous_norm.pow 2)
        (continuous_re.comp (Continuous.inner continuous_id' continuous_id')))
      fun a => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
x : Completion E
a : E
⊢ ‖↑E a‖ ^ 2 = re (inner (↑E a) (↑E a))

Next tactic:
simp only [norm_coe, inner_coe, inner_self_eq_norm_sq]
instance innerProductSpace : InnerProductSpace 𝕜 (Completion E) where
  norm_sq_eq_inner x :=
    Completion.induction_on x
      (isClosed_eq (continuous_norm.pow 2)
        (continuous_re.comp (Continuous.inner continuous_id' continuous_id')))
      fun a => by simp only [norm_coe, inner_coe, inner_self_eq_norm_sq]
  conj_symm x y :=
    Completion.induction_on₂ x y
      (isClosed_eq (continuous_conj.comp (Continuous.inner continuous_snd continuous_fst))
        (Continuous.inner continuous_fst continuous_snd))
      fun a b => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
x y : Completion E
a b : E
⊢ (starRingEnd 𝕜) (inner (↑E b) (↑E a)) = inner (↑E a) (↑E b)

Next tactic:
simp only [inner_coe, inner_conj_symm]
instance innerProductSpace : InnerProductSpace 𝕜 (Completion E) where
  norm_sq_eq_inner x :=
    Completion.induction_on x
      (isClosed_eq (continuous_norm.pow 2)
        (continuous_re.comp (Continuous.inner continuous_id' continuous_id')))
      fun a => by simp only [norm_coe, inner_coe, inner_self_eq_norm_sq]
  conj_symm x y :=
    Completion.induction_on₂ x y
      (isClosed_eq (continuous_conj.comp (Continuous.inner continuous_snd continuous_fst))
        (Continuous.inner continuous_fst continuous_snd))
      fun a b => by simp only [inner_coe, inner_conj_symm]
  add_left x y z :=
    Completion.induction_on₃ x y z
      (isClosed_eq
        (Continuous.inner (continuous_fst.add (continuous_fst.comp continuous_snd))
          (continuous_snd.comp continuous_snd))
        ((Continuous.inner continuous_fst (continuous_snd.comp continuous_snd)).add
          (Continuous.inner (continuous_fst.comp continuous_snd)
            (continuous_snd.comp continuous_snd))))
      fun a b c => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
x y z : Completion E
a b c : E
⊢ inner (↑E a + ↑E b) (↑E c) = inner (↑E a) (↑E c) + inner (↑E b) (↑E c)

Next tactic:
simp only [← coe_add, inner_coe, inner_add_left]
instance innerProductSpace : InnerProductSpace 𝕜 (Completion E) where
  norm_sq_eq_inner x :=
    Completion.induction_on x
      (isClosed_eq (continuous_norm.pow 2)
        (continuous_re.comp (Continuous.inner continuous_id' continuous_id')))
      fun a => by simp only [norm_coe, inner_coe, inner_self_eq_norm_sq]
  conj_symm x y :=
    Completion.induction_on₂ x y
      (isClosed_eq (continuous_conj.comp (Continuous.inner continuous_snd continuous_fst))
        (Continuous.inner continuous_fst continuous_snd))
      fun a b => by simp only [inner_coe, inner_conj_symm]
  add_left x y z :=
    Completion.induction_on₃ x y z
      (isClosed_eq
        (Continuous.inner (continuous_fst.add (continuous_fst.comp continuous_snd))
          (continuous_snd.comp continuous_snd))
        ((Continuous.inner continuous_fst (continuous_snd.comp continuous_snd)).add
          (Continuous.inner (continuous_fst.comp continuous_snd)
            (continuous_snd.comp continuous_snd))))
      fun a b c => by simp only [← coe_add, inner_coe, inner_add_left]
  smul_left x y c :=
    Completion.induction_on₂ x y
      (isClosed_eq (Continuous.inner (continuous_fst.const_smul c) continuous_snd)
        ((continuous_mul_left _).comp (Continuous.inner continuous_fst continuous_snd)))
      fun a b => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
x y : Completion E
c : 𝕜
a b : E
⊢ inner (c • ↑E a) (↑E b) = (starRingEnd 𝕜) c * inner (↑E a) (↑E b)

Next tactic:
simp only [← coe_smul c a, inner_coe, inner_smul_left]
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ Submodule.span ℝ ↑{1, I} = ⊤

Next tactic:
rw [eq_top_iff]
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ ⊤ ≤ Submodule.span ℝ ↑{1, I}

Next tactic:
intro a _
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      intro a _
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
a✝ : a ∈ ⊤
⊢ a ∈ Submodule.span ℝ ↑{1, I}

Next tactic:
rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      intro a _
      rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
a✝ : a ∈ ⊤
⊢ ∃ a_1, ∃ z ∈ Submodule.span ℝ {I}, a = a_1 • 1 + z

Next tactic:
refine' ⟨re a, im a • I, _, _⟩
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      intro a _
      rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
      refine' ⟨re a, im a • I, _, _⟩
      · 

Proof state:
case refine'_1
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
a✝ : a ∈ ⊤
⊢ im a • I ∈ Submodule.span ℝ {I}

Next tactic:
rw [Submodule.mem_span_singleton]
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      intro a _
      rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
      refine' ⟨re a, im a • I, _, _⟩
      · rw [Submodule.mem_span_singleton]
        

Proof state:
case refine'_1
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
a✝ : a ∈ ⊤
⊢ ∃ a_1, a_1 • I = im a • I

Next tactic:
use im a
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
  ⟨⟨{1, I}, by
      rw [eq_top_iff]
      intro a _
      rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
      refine' ⟨re a, im a • I, _, _⟩
      · rw [Submodule.mem_span_singleton]
        use im a
      

Proof state:
case refine'_2
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
a✝ : a ∈ ⊤
⊢ a = re a • 1 + im a • I

Next tactic:
simp [re_add_im a, Algebra.smul_def, algebraMap_eq_ofReal]
/-- A finite dimensional vector space over an `IsROrC` is a proper metric space.

This is not an instance because it would cause a search for `FiniteDimensional ?x E` before
`IsROrC ?x`. -/
theorem proper_isROrC [FiniteDimensional K E] : ProperSpace E := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝³ : IsROrC K
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace K E
inst✝ : FiniteDimensional K E
⊢ ProperSpace E

Next tactic:
letI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ K E
/-- A finite dimensional vector space over an `IsROrC` is a proper metric space.

This is not an instance because it would cause a search for `FiniteDimensional ?x E` before
`IsROrC ?x`. -/
theorem proper_isROrC [FiniteDimensional K E] : ProperSpace E := by
  letI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ K E
  

Proof state:
K : Type u_1
E : Type u_2
inst✝³ : IsROrC K
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace K E
inst✝ : FiniteDimensional K E
this : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ K E
⊢ ProperSpace E

Next tactic:
letI : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E
/-- A finite dimensional vector space over an `IsROrC` is a proper metric space.

This is not an instance because it would cause a search for `FiniteDimensional ?x E` before
`IsROrC ?x`. -/
theorem proper_isROrC [FiniteDimensional K E] : ProperSpace E := by
  letI : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ K E
  letI : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E
  

Proof state:
K : Type u_1
E : Type u_2
inst✝³ : IsROrC K
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace K E
inst✝ : FiniteDimensional K E
this✝ : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ K E
this : FiniteDimensional ℝ E := trans ℝ K E
⊢ ProperSpace E

Next tactic:
infer_instance
@[simp, isROrC_simps]
theorem reClm_norm : ‖(reClm : K →L[ℝ] ℝ)‖ = 1 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ ‖reClm‖ = 1

Next tactic:
apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
@[simp, isROrC_simps]
theorem reClm_norm : ‖(reClm : K →L[ℝ] ℝ)‖ = 1 := by
  apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ 1 ≤ ‖LinearMap.mkContinuous reLm 1 (_ : ∀ (x : K), ‖reLm x‖ ≤ 1 * ‖x‖)‖

Next tactic:
convert ContinuousLinearMap.ratio_le_op_norm (reClm : K →L[ℝ] ℝ) (1 : K)
@[simp, isROrC_simps]
theorem reClm_norm : ‖(reClm : K →L[ℝ] ℝ)‖ = 1 := by
  apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
  convert ContinuousLinearMap.ratio_le_op_norm (reClm : K →L[ℝ] ℝ) (1 : K)
  

Proof state:
case h.e'_3
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ 1 = ‖reClm 1‖ / ‖1‖

Next tactic:
simp
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
⊢ InnerProductSpaceable.innerProp' E ↑(-1)

Next tactic:
intro x y
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ inner_ 𝕜 (↑(-1) • x) y = (starRingEnd 𝕜) ↑(-1) * inner_ 𝕜 x y

Next tactic:
simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ 4⁻¹ *
      (𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
        I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖) =
    -4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
rw [neg_mul_comm]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ 4⁻¹ *
      (𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
        I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖) =
    4⁻¹ *
      -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
          I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
congr 1
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
      I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ ‖-x - y‖ = ‖x + y‖

Next tactic:
rw [← neg_add', norm_neg]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
      I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
⊢ ‖-x + y‖ = ‖x - y‖

Next tactic:
rw [← neg_sub, norm_neg, sub_eq_neg_add]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
      I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
⊢ ‖I • -x + y‖ = ‖I • x - y‖

Next tactic:
rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]