ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- orthonormal_subtype_iff_ite
theorem orthonormal_subtype_iff_ite [DecidableEq E] {s : Set E} :
    Orthonormal 𝕜 (Subtype.val : s → E) ↔ ∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫ = if v = w then 1 else 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [DecidableEq E] {s : Set E}
Goal: Orthonormal 𝕜 Subtype.val ↔ ∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫_𝕜 = if v = w then 1 else 0
-- Orthonormal.inner_right_finsupp
theorem Orthonormal.inner_right_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
    ⟪v i, Finsupp.total ι E 𝕜 v l⟫ = l i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι)
Goal: ⟪v i, (Finsupp.total ι E 𝕜 v) l⟫_𝕜 = l i
-- Orthonormal.inner_right_sum
theorem Orthonormal.inner_right_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι}
    {i : ι} (hi : i ∈ s) : ⟪v i, ∑ i ∈ s, l i • v i⟫ = l i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι} {i : ι} (hi : i ∈ s)
Goal: ⟪v i, ∑ i ∈ s, l i • v i⟫_𝕜 = l i
-- Orthonormal.inner_right_fintype
theorem Orthonormal.inner_right_fintype [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜)
    (i : ι) : ⟪v i, ∑ i : ι, l i • v i⟫ = l i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι)
Goal: ⟪v i, ∑ i : ι, l i • v i⟫_𝕜 = l i
-- Orthonormal.inner_left_finsupp
theorem Orthonormal.inner_left_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
    ⟪Finsupp.total ι E 𝕜 v l, v i⟫ = conj (l i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι)
Goal: ⟪(Finsupp.total ι E 𝕜 v) l, v i⟫_𝕜 = (starRingEnd 𝕜) (l i)
-- Orthonormal.inner_left_sum
theorem Orthonormal.inner_left_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι}
    {i : ι} (hi : i ∈ s) : ⟪∑ i ∈ s, l i • v i, v i⟫ = conj (l i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι} {i : ι} (hi : i ∈ s)
Goal: ⟪∑ i ∈ s, l i • v i, v i⟫_𝕜 = (starRingEnd 𝕜) (l i)
-- Orthonormal.inner_left_fintype
theorem Orthonormal.inner_left_fintype [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜)
    (i : ι) : ⟪∑ i : ι, l i • v i, v i⟫ = conj (l i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [Fintype ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) (i : ι)
Goal: ⟪∑ i : ι, l i • v i, v i⟫_𝕜 = (starRingEnd 𝕜) (l i)
-- Orthonormal.inner_finsupp_eq_sum_left
theorem Orthonormal.inner_finsupp_eq_sum_left {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
    ⟪Finsupp.total ι E 𝕜 v l₁, Finsupp.total ι E 𝕜 v l₂⟫ = l₁.sum fun i y => conj y * l₂ i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜)
Goal: ⟪(Finsupp.total ι E 𝕜 v) l₁, (Finsupp.total ι E 𝕜 v) l₂⟫_𝕜 = l₁.sum fun i y => (starRingEnd 𝕜) y * l₂ i
-- Orthonormal.inner_finsupp_eq_sum_right
theorem Orthonormal.inner_finsupp_eq_sum_right {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
    ⟪Finsupp.total ι E 𝕜 v l₁, Finsupp.total ι E 𝕜 v l₂⟫ = l₂.sum fun i y => conj (l₁ i) * y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜)
Goal: ⟪(Finsupp.total ι E 𝕜 v) l₁, (Finsupp.total ι E 𝕜 v) l₂⟫_𝕜 = l₂.sum fun i y => (starRingEnd 𝕜) (l₁ i) * y
-- Orthonormal.inner_sum
theorem Orthonormal.inner_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜) (s : Finset ι) :
    ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, conj (l₁ i) * l₂ i 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜) (s : Finset ι)
Goal: ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫_𝕜 = ∑ i ∈ s, (starRingEnd 𝕜) (l₁ i) * l₂ i
-- Orthonormal.inner_left_right_finset
theorem Orthonormal.inner_left_right_finset {s : Finset ι} {v : ι → E} (hv : Orthonormal 𝕜 v)
    {a : ι → ι → 𝕜} : (∑ i ∈ s, ∑ j ∈ s, a i j • ⟪v j, v i⟫) = ∑ k ∈ s, a k k 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {s : Finset ι} {v : ι → E} (hv : Orthonormal 𝕜 v) {a : ι → ι → 𝕜}
Goal: ∑ i ∈ s, ∑ j ∈ s, a i j • ⟪v j, v i⟫_𝕜 = ∑ k ∈ s, a k k
-- Orthonormal.linearIndependent
theorem Orthonormal.linearIndependent {v : ι → E} (hv : Orthonormal 𝕜 v) :
    LinearIndependent 𝕜 v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v)
Goal: LinearIndependent 𝕜 v
-- Orthonormal.comp
theorem Orthonormal.comp {ι' : Type*} {v : ι → E} (hv : Orthonormal 𝕜 v) (f : ι' → ι)
    (hf : Function.Injective f) : Orthonormal 𝕜 (v ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {ι' : Type u_5} {v : ι → E} (hv : Orthonormal 𝕜 v) (f : ι' → ι) (hf : Function.Injective f)
Goal: Orthonormal 𝕜 (v ∘ f)
-- orthonormal_subtype_range
theorem orthonormal_subtype_range {v : ι → E} (hv : Function.Injective v) :
    Orthonormal 𝕜 (Subtype.val : Set.range v → E) ↔ Orthonormal 𝕜 v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Function.Injective v)
Goal: Orthonormal 𝕜 Subtype.val ↔ Orthonormal 𝕜 v
-- Orthonormal.toSubtypeRange
theorem Orthonormal.toSubtypeRange {v : ι → E} (hv : Orthonormal 𝕜 v) :
    Orthonormal 𝕜 (Subtype.val : Set.range v → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v)
Goal: Orthonormal 𝕜 Subtype.val
-- Orthonormal.inner_finsupp_eq_zero
theorem Orthonormal.inner_finsupp_eq_zero {v : ι → E} (hv : Orthonormal 𝕜 v) {s : Set ι} {i : ι}
    (hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ Finsupp.supported 𝕜 𝕜 s) :
    ⟪Finsupp.total ι E 𝕜 v l, v i⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) {s : Set ι} {i : ι} (hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ Finsupp.supported 𝕜 𝕜 s)
Goal: ⟪(Finsupp.total ι E 𝕜 v) l, v i⟫_𝕜 = 0
-- Orthonormal.orthonormal_of_forall_eq_or_eq_neg
theorem Orthonormal.orthonormal_of_forall_eq_or_eq_neg {v w : ι → E} (hv : Orthonormal 𝕜 v)
    (hw : ∀ i, w i = v i ∨ w i = -v i) : Orthonormal 𝕜 w 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v w : ι → E} (hv : Orthonormal 𝕜 v) (hw : ∀ (i : ι), w i = v i ∨ w i = -v i)
Goal: Orthonormal 𝕜 w
-- orthonormal_empty
theorem orthonormal_empty : Orthonormal 𝕜 (fun x => x : (∅ : Set E) → E) 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: Orthonormal 𝕜 fun x => ↑x
-- orthonormal_iUnion_of_directed
theorem orthonormal_iUnion_of_directed {η : Type*} {s : η → Set E} (hs : Directed (· ⊆ ·) s)
    (h : ∀ i, Orthonormal 𝕜 (fun x => x : s i → E)) :
    Orthonormal 𝕜 (fun x => x : (⋃ i, s i) → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {η : Type u_5} {s : η → Set E} (hs : Directed (fun x x_1 => x ⊆ x_1) s) (h : ∀ (i : η), Orthonormal 𝕜 fun x => ↑x)
Goal: Orthonormal 𝕜 fun x => ↑x
-- orthonormal_sUnion_of_directed
theorem orthonormal_sUnion_of_directed {s : Set (Set E)} (hs : DirectedOn (· ⊆ ·) s)
    (h : ∀ a ∈ s, Orthonormal 𝕜 (fun x => ((x : a) : E))) :
    Orthonormal 𝕜 (fun x => x : ⋃₀ s → E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {s : Set (Set E)} (hs : DirectedOn (fun x x_1 => x ⊆ x_1) s) (h : ∀ a ∈ s, Orthonormal 𝕜 fun x => ↑x)
Goal: Orthonormal 𝕜 fun x => ↑x
-- exists_maximal_orthonormal
theorem exists_maximal_orthonormal {s : Set E} (hs : Orthonormal 𝕜 (Subtype.val : s → E)) :
    ∃ w ⊇ s, Orthonormal 𝕜 (Subtype.val : w → E) ∧
      ∀ u ⊇ w, Orthonormal 𝕜 (Subtype.val : u → E) → u = w 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {s : Set E} (hs : Orthonormal 𝕜 Subtype.val)
Goal: ∃ w ⊇ s, Orthonormal 𝕜 Subtype.val ∧ ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w
-- Orthonormal.ne_zero
theorem Orthonormal.ne_zero {v : ι → E} (hv : Orthonormal 𝕜 v) (i : ι) : v i ≠ 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : ι → E} (hv : Orthonormal 𝕜 v) (i : ι)
Goal: v i ≠ 0
-- coe_basisOfOrthonormalOfCardEqFinrank
@[simp]
theorem coe_basisOfOrthonormalOfCardEqFinrank [Fintype ι] [Nonempty ι] {v : ι → E}
    (hv : Orthonormal 𝕜 v) (card_eq : Fintype.card ι = finrank 𝕜 E) :
    (basisOfOrthonormalOfCardEqFinrank hv card_eq : ι → E) = v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [Fintype ι] [Nonempty ι] {v : ι → E} (hv : Orthonormal 𝕜 v) (card_eq : Fintype.card ι = FiniteDimensional.finrank 𝕜 E)
Goal: ⇑(basisOfOrthonormalOfCardEqFinrank hv card_eq) = v
-- norm_eq_sqrt_inner
theorem norm_eq_sqrt_inner (x : E) : ‖x‖ = √(re ⟪x, x⟫) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ‖x‖ = √(RCLike.re ⟪x, x⟫_𝕜)
-- norm_eq_sqrt_real_inner
theorem norm_eq_sqrt_real_inner (x : F) : ‖x‖ = √⟪x, x⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F)
Goal: ‖x‖ = √⟪x, x⟫_ℝ
-- inner_self_eq_norm_mul_norm
theorem inner_self_eq_norm_mul_norm (x : E) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: RCLike.re ⟪x, x⟫_𝕜 = ‖x‖ * ‖x‖
-- inner_self_eq_norm_sq
theorem inner_self_eq_norm_sq (x : E) : re ⟪x, x⟫ = ‖x‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: RCLike.re ⟪x, x⟫_𝕜 = ‖x‖ ^ 2
-- real_inner_self_eq_norm_mul_norm
theorem real_inner_self_eq_norm_mul_norm (x : F) : ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F)
Goal: ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖
-- real_inner_self_eq_norm_sq
theorem real_inner_self_eq_norm_sq (x : F) : ⟪x, x⟫_ℝ = ‖x‖ ^ 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F)
Goal: ⟪x, x⟫_ℝ = ‖x‖ ^ 2
-- norm_add_sq
theorem norm_add_sq (x y : E) : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫ + ‖y‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * RCLike.re ⟪x, y⟫_𝕜 + ‖y‖ ^ 2
-- norm_add_sq_real
theorem norm_add_sq_real (x y : F) : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2
-- norm_add_mul_self
theorem norm_add_mul_self (x y : E) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * RCLike.re ⟪x, y⟫_𝕜 + ‖y‖ * ‖y‖
-- norm_add_mul_self_real
theorem norm_add_mul_self_real (x y : F) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖
-- norm_sub_sq
theorem norm_sub_sq (x y : E) : ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * re ⟪x, y⟫ + ‖y‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * RCLike.re ⟪x, y⟫_𝕜 + ‖y‖ ^ 2
-- norm_sub_sq_real
theorem norm_sub_sq_real (x y : F) : ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2
-- norm_sub_mul_self
theorem norm_sub_mul_self (x y : E) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * RCLike.re ⟪x, y⟫_𝕜 + ‖y‖ * ‖y‖
-- norm_sub_mul_self_real
theorem norm_sub_mul_self_real (x y : F) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖
-- norm_inner_le_norm
theorem norm_inner_le_norm (x y : E) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖⟪x, y⟫_𝕜‖ ≤ ‖x‖ * ‖y‖
-- nnnorm_inner_le_nnnorm
theorem nnnorm_inner_le_nnnorm (x y : E) : ‖⟪x, y⟫‖₊ ≤ ‖x‖₊ * ‖y‖₊ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖⟪x, y⟫_𝕜‖₊ ≤ ‖x‖₊ * ‖y‖₊
-- re_inner_le_norm
theorem re_inner_le_norm (x y : E) : re ⟪x, y⟫ ≤ ‖x‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: RCLike.re ⟪x, y⟫_𝕜 ≤ ‖x‖ * ‖y‖
-- abs_real_inner_le_norm
theorem abs_real_inner_le_norm (x y : F) : |⟪x, y⟫_ℝ| ≤ ‖x‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: |⟪x, y⟫_ℝ| ≤ ‖x‖ * ‖y‖
-- real_inner_le_norm
theorem real_inner_le_norm (x y : F) : ⟪x, y⟫_ℝ ≤ ‖x‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ ≤ ‖x‖ * ‖y‖
-- parallelogram_law_with_norm
theorem parallelogram_law_with_norm (x y : E) :
    ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)
-- parallelogram_law_with_nnnorm
theorem parallelogram_law_with_nnnorm (x y : E) :
    ‖x + y‖₊ * ‖x + y‖₊ + ‖x - y‖₊ * ‖x - y‖₊ = 2 * (‖x‖₊ * ‖x‖₊ + ‖y‖₊ * ‖y‖₊) 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖x + y‖₊ * ‖x + y‖₊ + ‖x - y‖₊ * ‖x - y‖₊ = 2 * (‖x‖₊ * ‖x‖₊ + ‖y‖₊ * ‖y‖₊)
-- re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two
theorem re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : E) :
    re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: RCLike.re ⟪x, y⟫_𝕜 = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2
-- re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two
theorem re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : E) :
    re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: RCLike.re ⟪x, y⟫_𝕜 = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2
-- re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four
theorem re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four (x y : E) :
    re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: RCLike.re ⟪x, y⟫_𝕜 = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4
-- im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four
theorem im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four (x y : E) :
    im ⟪x, y⟫ = (‖x - IK • y‖ * ‖x - IK • y‖ - ‖x + IK • y‖ * ‖x + IK • y‖) / 4 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: RCLike.im ⟪x, y⟫_𝕜 = (‖x - RCLike.I • y‖ * ‖x - RCLike.I • y‖ - ‖x + RCLike.I • y‖ * ‖x + RCLike.I • y‖) / 4
-- inner_eq_sum_norm_sq_div_four
theorem inner_eq_sum_norm_sq_div_four (x y : E) :
    ⟪x, y⟫ = ((‖x + y‖ : 𝕜) ^ 2 - (‖x - y‖ : 𝕜) ^ 2 +
              ((‖x - IK • y‖ : 𝕜) ^ 2 - (‖x + IK • y‖ : 𝕜) ^ 2) * IK) / 4 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ⟪x, y⟫_𝕜 = (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - RCLike.I • y‖ ^ 2 - ↑‖x + RCLike.I • y‖ ^ 2) * RCLike.I) / 4
-- dist_div_norm_sq_smul
theorem dist_div_norm_sq_smul {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ) :
    dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = R ^ 2 / (‖x‖ * ‖y‖) * dist x y 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ)
Goal: dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = R ^ 2 / (‖x‖ * ‖y‖) * dist x y
-- inner_map_polarization
theorem inner_map_polarization (T : V →ₗ[ℂ] V) (x y : V) :
    ⟪T y, x⟫_ℂ =
      (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ +
            Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ -
          Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
        4 
Arguments: {V : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (T : V →ₗ[ℂ] V) (x y : V)
Goal: ⟪T y, x⟫_ℂ =
  (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ -
      Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
    4
-- inner_map_polarization'
theorem inner_map_polarization' (T : V →ₗ[ℂ] V) (x y : V) :
    ⟪T x, y⟫_ℂ =
      (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ -
            Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ +
          Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
        4 
Arguments: {V : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (T : V →ₗ[ℂ] V) (x y : V)
Goal: ⟪T x, y⟫_ℂ =
  (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ - Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ +
      Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /
    4
-- inner_map_self_eq_zero
theorem inner_map_self_eq_zero (T : V →ₗ[ℂ] V) : (∀ x : V, ⟪T x, x⟫_ℂ = 0) ↔ T = 0 
Arguments: {V : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (T : V →ₗ[ℂ] V)
Goal: (∀ (x : V), ⟪T x, x⟫_ℂ = 0) ↔ T = 0
-- ext_inner_map
theorem ext_inner_map (S T : V →ₗ[ℂ] V) : (∀ x : V, ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T 
Arguments: {V : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (S T : V →ₗ[ℂ] V)
Goal: (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S = T
-- LinearIsometry.inner_map_map
@[simp]
theorem LinearIsometry.inner_map_map (f : E →ₗᵢ[𝕜] E') (x y : E) : ⟪f x, f y⟫ = ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E →ₗᵢ[𝕜] E') (x y : E)
Goal: ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜
-- LinearIsometryEquiv.inner_map_map
@[simp]
theorem LinearIsometryEquiv.inner_map_map (f : E ≃ₗᵢ[𝕜] E') (x y : E) : ⟪f x, f y⟫ = ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (x y : E)
Goal: ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜
-- LinearIsometryEquiv.inner_map_eq_flip
theorem LinearIsometryEquiv.inner_map_eq_flip (f : E ≃ₗᵢ[𝕜] E') (x : E) (y : E') :
    ⟪f x, y⟫_𝕜 = ⟪x, f.symm y⟫_𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E') (x : E) (y : E')
Goal: ⟪f x, y⟫_𝕜 = ⟪x, f.symm y⟫_𝕜
-- LinearMap.coe_isometryOfInner
@[simp]
theorem LinearMap.coe_isometryOfInner (f : E →ₗ[𝕜] E') (h) : ⇑(f.isometryOfInner h) = f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E →ₗ[𝕜] E') (h : ∀ (x y : E), ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜)
Goal: ⇑(f.isometryOfInner h) = ⇑f
-- LinearMap.isometryOfInner_toLinearMap
@[simp]
theorem LinearMap.isometryOfInner_toLinearMap (f : E →ₗ[𝕜] E') (h) :
    (f.isometryOfInner h).toLinearMap = f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E →ₗ[𝕜] E') (h : ∀ (x y : E), ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜)
Goal: (f.isometryOfInner h).toLinearMap = f
-- LinearEquiv.coe_isometryOfInner
@[simp]
theorem LinearEquiv.coe_isometryOfInner (f : E ≃ₗ[𝕜] E') (h) : ⇑(f.isometryOfInner h) = f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗ[𝕜] E') (h : ∀ (x y : E), ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜)
Goal: ⇑(f.isometryOfInner h) = ⇑f
-- LinearEquiv.isometryOfInner_toLinearEquiv
@[simp]
theorem LinearEquiv.isometryOfInner_toLinearEquiv (f : E ≃ₗ[𝕜] E') (h) :
    (f.isometryOfInner h).toLinearEquiv = f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E ≃ₗ[𝕜] E') (h : ∀ (x y : E), ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜)
Goal: (f.isometryOfInner h).toLinearEquiv = f
-- LinearMap.norm_map_iff_inner_map_map
theorem LinearMap.norm_map_iff_inner_map_map {F : Type*} [FunLike F E E'] [LinearMapClass F 𝕜 E E']
    (f : F) : (∀ x, ‖f x‖ = ‖x‖) ↔ (∀ x y, ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {F : Type u_9} [FunLike F E E'] [LinearMapClass F 𝕜 E E'] (f : F)
Goal: (∀ (x : E), ‖f x‖ = ‖x‖) ↔ ∀ (x y : E), ⟪f x, f y⟫_𝕜 = ⟪x, y⟫_𝕜
-- LinearIsometry.orthonormal_comp_iff
theorem LinearIsometry.orthonormal_comp_iff {v : ι → E} (f : E →ₗᵢ[𝕜] E') :
    Orthonormal 𝕜 (f ∘ v) ↔ Orthonormal 𝕜 v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : ι → E} (f : E →ₗᵢ[𝕜] E')
Goal: Orthonormal 𝕜 (⇑f ∘ v) ↔ Orthonormal 𝕜 v
-- Orthonormal.comp_linearIsometry
theorem Orthonormal.comp_linearIsometry {v : ι → E} (hv : Orthonormal 𝕜 v) (f : E →ₗᵢ[𝕜] E') :
    Orthonormal 𝕜 (f ∘ v) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : ι → E} (hv : Orthonormal 𝕜 v) (f : E →ₗᵢ[𝕜] E')
Goal: Orthonormal 𝕜 (⇑f ∘ v)
-- Orthonormal.comp_linearIsometryEquiv
theorem Orthonormal.comp_linearIsometryEquiv {v : ι → E} (hv : Orthonormal 𝕜 v) (f : E ≃ₗᵢ[𝕜] E') :
    Orthonormal 𝕜 (f ∘ v) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : ι → E} (hv : Orthonormal 𝕜 v) (f : E ≃ₗᵢ[𝕜] E')
Goal: Orthonormal 𝕜 (⇑f ∘ v)
-- Orthonormal.mapLinearIsometryEquiv
theorem Orthonormal.mapLinearIsometryEquiv {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v)
    (f : E ≃ₗᵢ[𝕜] E') : Orthonormal 𝕜 (v.map f.toLinearEquiv) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) (f : E ≃ₗᵢ[𝕜] E')
Goal: Orthonormal 𝕜 ⇑(v.map f.toLinearEquiv)