ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- InnerProductSpace.Core.definite
-- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore
structure InnerProductSpace.Core (𝕜 : Type*) (F : Type*) [RCLike 𝕜] [AddCommGroup F]
  [Module 𝕜 F] extends Inner 𝕜 F where
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    nonneg_re : ∀ x, 0 ≤ re (inner x x)
    definite : ∀ x, inner x x = 0 → x = 0
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {F : Type u_5} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] (self : InnerProductSpace.Core 𝕜 F) (x : F)
Goal: ⟪x, x⟫_𝕜 = 0 → x = 0
-- InnerProductSpace.Core.conj_symm
-- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore
structure InnerProductSpace.Core (𝕜 : Type*) (F : Type*) [RCLike 𝕜] [AddCommGroup F]
  [Module 𝕜 F] extends Inner 𝕜 F where
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    nonneg_re : ∀ x, 0 ≤ re (inner x x)
    definite : ∀ x, inner x x = 0 → x = 0
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {F : Type u_5} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] (self : InnerProductSpace.Core 𝕜 F) (x y : F)
Goal: (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_conj_symm
theorem inner_conj_symm (x y : F) : ⟪y, x⟫† = ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_self_nonneg
theorem inner_self_nonneg {x : F} : 0 ≤ re ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] {x : F}
Goal: 0 ≤ RCLike.re ⟪x, x⟫_𝕜
-- InnerProductSpace.Core.inner_self_im
theorem inner_self_im (x : F) : im ⟪x, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: RCLike.im ⟪x, x⟫_𝕜 = 0
-- InnerProductSpace.Core.inner_add_left
theorem inner_add_left (x y z : F) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y z : F)
Goal: ⟪x + y, z⟫_𝕜 = ⟪x, z⟫_𝕜 + ⟪y, z⟫_𝕜
-- InnerProductSpace.Core.inner_add_right
theorem inner_add_right (x y z : F) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y z : F)
Goal: ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜
-- InnerProductSpace.Core.ofReal_normSq_eq_inner_self
theorem ofReal_normSq_eq_inner_self (x : F) : (normSqF x : 𝕜) = ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: ↑(InnerProductSpace.Core.normSq x) = ⟪x, x⟫_𝕜
-- InnerProductSpace.Core.inner_re_symm
theorem inner_re_symm (x y : F) : re ⟪x, y⟫ = re ⟪y, x⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: RCLike.re ⟪x, y⟫_𝕜 = RCLike.re ⟪y, x⟫_𝕜
-- InnerProductSpace.Core.inner_im_symm
theorem inner_im_symm (x y : F) : im ⟪x, y⟫ = -im ⟪y, x⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: RCLike.im ⟪x, y⟫_𝕜 = -RCLike.im ⟪y, x⟫_𝕜
-- InnerProductSpace.Core.inner_smul_left
theorem inner_smul_left (x y : F) {r : 𝕜} : ⟪r • x, y⟫ = r† * ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F) {r : 𝕜}
Goal: ⟪r • x, y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_smul_right
theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F) {r : 𝕜}
Goal: ⟪x, r • y⟫_𝕜 = r * ⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_zero_left
theorem inner_zero_left (x : F) : ⟪0, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: ⟪0, x⟫_𝕜 = 0
-- InnerProductSpace.Core.inner_zero_right
theorem inner_zero_right (x : F) : ⟪x, 0⟫ = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: ⟪x, 0⟫_𝕜 = 0
-- InnerProductSpace.Core.inner_self_eq_zero
theorem inner_self_eq_zero {x : F} : ⟪x, x⟫ = 0 ↔ x = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] {x : F}
Goal: ⟪x, x⟫_𝕜 = 0 ↔ x = 0
-- InnerProductSpace.Core.normSq_eq_zero
theorem normSq_eq_zero {x : F} : normSqF x = 0 ↔ x = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] {x : F}
Goal: InnerProductSpace.Core.normSq x = 0 ↔ x = 0
-- InnerProductSpace.Core.inner_self_ne_zero
theorem inner_self_ne_zero {x : F} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] {x : F}
Goal: ⟪x, x⟫_𝕜 ≠ 0 ↔ x ≠ 0
-- InnerProductSpace.Core.inner_self_ofReal_re
theorem inner_self_ofReal_re (x : F) : (re ⟪x, x⟫ : 𝕜) = ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: ↑(RCLike.re ⟪x, x⟫_𝕜) = ⟪x, x⟫_𝕜
-- InnerProductSpace.Core.norm_inner_symm
theorem norm_inner_symm (x y : F) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ‖⟪x, y⟫_𝕜‖ = ‖⟪y, x⟫_𝕜‖
-- InnerProductSpace.Core.inner_neg_left
theorem inner_neg_left (x y : F) : ⟪-x, y⟫ = -⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ⟪-x, y⟫_𝕜 = -⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_neg_right
theorem inner_neg_right (x y : F) : ⟪x, -y⟫ = -⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ⟪x, -y⟫_𝕜 = -⟪x, y⟫_𝕜
-- InnerProductSpace.Core.inner_sub_left
theorem inner_sub_left (x y z : F) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y z : F)
Goal: ⟪x - y, z⟫_𝕜 = ⟪x, z⟫_𝕜 - ⟪y, z⟫_𝕜
-- InnerProductSpace.Core.inner_sub_right
theorem inner_sub_right (x y z : F) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y z : F)
Goal: ⟪x, y - z⟫_𝕜 = ⟪x, y⟫_𝕜 - ⟪x, z⟫_𝕜
-- InnerProductSpace.Core.inner_mul_symm_re_eq_norm
theorem inner_mul_symm_re_eq_norm (x y : F) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: RCLike.re (⟪x, y⟫_𝕜 * ⟪y, x⟫_𝕜) = ‖⟪x, y⟫_𝕜 * ⟪y, x⟫_𝕜‖
-- InnerProductSpace.Core.inner_add_add_self
theorem inner_add_add_self (x y : F) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, x⟫ + ⟪y, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ⟪x + y, x + y⟫_𝕜 = ⟪x, x⟫_𝕜 + ⟪x, y⟫_𝕜 + ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- InnerProductSpace.Core.inner_sub_sub_self
theorem inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ⟪x - y, x - y⟫_𝕜 = ⟪x, x⟫_𝕜 - ⟪x, y⟫_𝕜 - ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- InnerProductSpace.Core.cauchy_schwarz_aux
theorem cauchy_schwarz_aux (x y : F) :
    normSqF (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSqF x * (normSqF x * normSqF y - ‖⟪x, y⟫‖ ^ 2) 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: InnerProductSpace.Core.normSq (⟪x, y⟫_𝕜 • x - ⟪x, x⟫_𝕜 • y) =
  InnerProductSpace.Core.normSq x * (InnerProductSpace.Core.normSq x * InnerProductSpace.Core.normSq y - ‖⟪x, y⟫_𝕜‖ ^ 2)
-- InnerProductSpace.Core.inner_mul_inner_self_le
theorem inner_mul_inner_self_le (x y : F) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ‖⟪x, y⟫_𝕜‖ * ‖⟪y, x⟫_𝕜‖ ≤ RCLike.re ⟪x, x⟫_𝕜 * RCLike.re ⟪y, y⟫_𝕜
-- InnerProductSpace.Core.norm_eq_sqrt_inner
theorem norm_eq_sqrt_inner (x : F) : ‖x‖ = √(re ⟪x, x⟫) 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: ‖x‖ = √(RCLike.re ⟪x, x⟫_𝕜)
-- InnerProductSpace.Core.inner_self_eq_norm_mul_norm
theorem inner_self_eq_norm_mul_norm (x : F) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: RCLike.re ⟪x, x⟫_𝕜 = ‖x‖ * ‖x‖
-- InnerProductSpace.Core.sqrt_normSq_eq_norm
theorem sqrt_normSq_eq_norm (x : F) : √(normSqF x) = ‖x‖ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x : F)
Goal: √(InnerProductSpace.Core.normSq x) = ‖x‖
-- InnerProductSpace.Core.norm_inner_le_norm
theorem norm_inner_le_norm (x y : F) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {F : Type u_3} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] [c : InnerProductSpace.Core 𝕜 F] (x y : F)
Goal: ‖⟪x, y⟫_𝕜‖ ≤ ‖x‖ * ‖y‖
-- inner_conj_symm
@[simp]
theorem inner_conj_symm (x y : E) : ⟪y, x⟫† = ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜
-- real_inner_comm
theorem real_inner_comm (x y : F) : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ
-- inner_eq_zero_symm
theorem inner_eq_zero_symm {x y : E} : ⟪x, y⟫ = 0 ↔ ⟪y, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E}
Goal: ⟪x, y⟫_𝕜 = 0 ↔ ⟪y, x⟫_𝕜 = 0
-- inner_self_im
@[simp]
theorem inner_self_im (x : E) : im ⟪x, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: RCLike.im ⟪x, x⟫_𝕜 = 0
-- inner_add_left
theorem inner_add_left (x y z : E) : ⟪x + y, z⟫ = ⟪x, z⟫ + ⟪y, z⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y z : E)
Goal: ⟪x + y, z⟫_𝕜 = ⟪x, z⟫_𝕜 + ⟪y, z⟫_𝕜
-- inner_add_right
theorem inner_add_right (x y z : E) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y z : E)
Goal: ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜
-- inner_re_symm
theorem inner_re_symm (x y : E) : re ⟪x, y⟫ = re ⟪y, x⟫ 
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 = RCLike.re ⟪y, x⟫_𝕜
-- inner_im_symm
theorem inner_im_symm (x y : E) : im ⟪x, y⟫ = -im ⟪y, x⟫ 
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 = -RCLike.im ⟪y, x⟫_𝕜
-- inner_smul_left
theorem inner_smul_left (x y : E) (r : 𝕜) : ⟪r • x, y⟫ = r† * ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜)
Goal: ⟪r • x, y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪x, y⟫_𝕜
-- real_inner_smul_left
theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F) (r : ℝ)
Goal: ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ
-- inner_smul_real_left
theorem inner_smul_real_left (x y : E) (r : ℝ) : ⟪(r : 𝕜) • x, y⟫ = r • ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E) (r : ℝ)
Goal: ⟪↑r • x, y⟫_𝕜 = r • ⟪x, y⟫_𝕜
-- inner_smul_right
theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜)
Goal: ⟪x, r • y⟫_𝕜 = r * ⟪x, y⟫_𝕜
-- real_inner_smul_right
theorem real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F) (r : ℝ)
Goal: ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ
-- inner_smul_real_right
theorem inner_smul_real_right (x y : E) (r : ℝ) : ⟪x, (r : 𝕜) • y⟫ = r • ⟪x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E) (r : ℝ)
Goal: ⟪x, ↑r • y⟫_𝕜 = r • ⟪x, y⟫_𝕜
-- sesqFormOfInner_apply_apply
@[simps!]
def sesqFormOfInner : E →ₗ[𝕜] E →ₗ⋆[𝕜] 𝕜 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (m y : E)
Goal: (sesqFormOfInner m) y = ⟪y, m⟫_𝕜
-- bilinFormOfRealInner_apply_apply
@[simps!]
def bilinFormOfRealInner : BilinForm ℝ F 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (m : F)
Goal: ∀ (m_1 : F), (bilinFormOfRealInner m) m_1 = ⟪m, m_1⟫_ℝ
-- sum_inner
theorem sum_inner {ι : Type*} (s : Finset ι) (f : ι → E) (x : E) :
    ⟪∑ i ∈ s, f i, x⟫ = ∑ i ∈ s, ⟪f i, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (s : Finset ι) (f : ι → E) (x : E)
Goal: ⟪∑ i ∈ s, f i, x⟫_𝕜 = ∑ i ∈ s, ⟪f i, x⟫_𝕜
-- inner_sum
theorem inner_sum {ι : Type*} (s : Finset ι) (f : ι → E) (x : E) :
    ⟪x, ∑ i ∈ s, f i⟫ = ∑ i ∈ s, ⟪x, f i⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (s : Finset ι) (f : ι → E) (x : E)
Goal: ⟪x, ∑ i ∈ s, f i⟫_𝕜 = ∑ i ∈ s, ⟪x, f i⟫_𝕜
-- Finsupp.sum_inner
theorem Finsupp.sum_inner {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
    ⟪l.sum fun (i : ι) (a : 𝕜) => a • v i, x⟫ = l.sum fun (i : ι) (a : 𝕜) => conj a • ⟪v i, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (l : ι →₀ 𝕜) (v : ι → E) (x : E)
Goal: ⟪l.sum fun i a => a • v i, x⟫_𝕜 = l.sum fun i a => (starRingEnd 𝕜) a • ⟪v i, x⟫_𝕜
-- Finsupp.inner_sum
theorem Finsupp.inner_sum {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
    ⟪x, l.sum fun (i : ι) (a : 𝕜) => a • v i⟫ = l.sum fun (i : ι) (a : 𝕜) => a • ⟪x, v i⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (l : ι →₀ 𝕜) (v : ι → E) (x : E)
Goal: ⟪x, l.sum fun i a => a • v i⟫_𝕜 = l.sum fun i a => a • ⟪x, v i⟫_𝕜
-- DFinsupp.sum_inner
theorem DFinsupp.sum_inner {ι : Type*} [DecidableEq ι] {α : ι → Type*}
    [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E)
    (l : Π₀ i, α i) (x : E) : ⟪l.sum f, x⟫ = l.sum fun i a => ⟪f i a, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] {α : ι → Type u_5} [(i : ι) → AddZeroClass (α i)] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] (f : (i : ι) → α i → E) (l : Π₀ (i : ι), α i) (x : E)
Goal: ⟪l.sum f, x⟫_𝕜 = l.sum fun i a => ⟪f i a, x⟫_𝕜
-- DFinsupp.inner_sum
theorem DFinsupp.inner_sum {ι : Type*} [DecidableEq ι] {α : ι → Type*}
    [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E)
    (l : Π₀ i, α i) (x : E) : ⟪x, l.sum f⟫ = l.sum fun i a => ⟪x, f i a⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] {α : ι → Type u_5} [(i : ι) → AddZeroClass (α i)] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] (f : (i : ι) → α i → E) (l : Π₀ (i : ι), α i) (x : E)
Goal: ⟪x, l.sum f⟫_𝕜 = l.sum fun i a => ⟪x, f i a⟫_𝕜
-- inner_zero_left
@[simp]
theorem inner_zero_left (x : E) : ⟪0, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ⟪0, x⟫_𝕜 = 0
-- inner_re_zero_left
theorem inner_re_zero_left (x : E) : re ⟪0, x⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: RCLike.re ⟪0, x⟫_𝕜 = 0
-- inner_zero_right
@[simp]
theorem inner_zero_right (x : E) : ⟪x, 0⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ⟪x, 0⟫_𝕜 = 0
-- inner_re_zero_right
theorem inner_re_zero_right (x : E) : re ⟪x, 0⟫ = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: RCLike.re ⟪x, 0⟫_𝕜 = 0
-- inner_self_nonneg
theorem inner_self_nonneg {x : E} : 0 ≤ re ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E}
Goal: 0 ≤ RCLike.re ⟪x, x⟫_𝕜
-- real_inner_self_nonneg
theorem real_inner_self_nonneg {x : F} : 0 ≤ ⟪x, x⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x : F}
Goal: 0 ≤ ⟪x, x⟫_ℝ
-- inner_self_ofReal_re
@[simp]
theorem inner_self_ofReal_re (x : E) : (re ⟪x, x⟫ : 𝕜) = ⟪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⟫_𝕜
-- inner_self_eq_norm_sq_to_K
theorem inner_self_eq_norm_sq_to_K (x : E) : ⟪x, x⟫ = (‖x‖ : 𝕜) ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ⟪x, x⟫_𝕜 = ↑‖x‖ ^ 2
-- inner_self_re_eq_norm
theorem inner_self_re_eq_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⟫_𝕜‖
-- inner_self_ofReal_norm
theorem inner_self_ofReal_norm (x : E) : (‖⟪x, x⟫‖ : 𝕜) = ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ↑‖⟪x, x⟫_𝕜‖ = ⟪x, x⟫_𝕜
-- real_inner_self_abs
theorem real_inner_self_abs (x : F) : |⟪x, x⟫_ℝ| = ⟪x, x⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F)
Goal: |⟪x, x⟫_ℝ| = ⟪x, x⟫_ℝ
-- inner_self_eq_zero
@[simp]
theorem inner_self_eq_zero {x : E} : ⟪x, x⟫ = 0 ↔ x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E}
Goal: ⟪x, x⟫_𝕜 = 0 ↔ x = 0
-- inner_self_ne_zero
theorem inner_self_ne_zero {x : E} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E}
Goal: ⟪x, x⟫_𝕜 ≠ 0 ↔ x ≠ 0
-- inner_self_nonpos
@[simp]
theorem inner_self_nonpos {x : E} : re ⟪x, x⟫ ≤ 0 ↔ x = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E}
Goal: RCLike.re ⟪x, x⟫_𝕜 ≤ 0 ↔ x = 0
-- real_inner_self_nonpos
theorem real_inner_self_nonpos {x : F} : ⟪x, x⟫_ℝ ≤ 0 ↔ x = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x : F}
Goal: ⟪x, x⟫_ℝ ≤ 0 ↔ x = 0
-- norm_inner_symm
theorem norm_inner_symm (x y : E) : ‖⟪x, y⟫‖ = ‖⟪y, x⟫‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖⟪x, y⟫_𝕜‖ = ‖⟪y, x⟫_𝕜‖
-- inner_neg_left
@[simp]
theorem inner_neg_left (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⟫_𝕜
-- inner_neg_right
@[simp]
theorem inner_neg_right (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⟫_𝕜
-- inner_neg_neg
theorem inner_neg_neg (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⟫_𝕜
-- inner_self_conj
theorem inner_self_conj (x : E) : ⟪x, x⟫† = ⟪x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: (starRingEnd 𝕜) ⟪x, x⟫_𝕜 = ⟪x, x⟫_𝕜
-- inner_sub_left
theorem inner_sub_left (x y z : E) : ⟪x - y, z⟫ = ⟪x, z⟫ - ⟪y, z⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y z : E)
Goal: ⟪x - y, z⟫_𝕜 = ⟪x, z⟫_𝕜 - ⟪y, z⟫_𝕜
-- inner_sub_right
theorem inner_sub_right (x y z : E) : ⟪x, y - z⟫ = ⟪x, y⟫ - ⟪x, z⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y z : E)
Goal: ⟪x, y - z⟫_𝕜 = ⟪x, y⟫_𝕜 - ⟪x, z⟫_𝕜
-- inner_mul_symm_re_eq_norm
theorem inner_mul_symm_re_eq_norm (x y : E) : re (⟪x, y⟫ * ⟪y, x⟫) = ‖⟪x, y⟫ * ⟪y, x⟫‖ 
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 * ⟪y, x⟫_𝕜) = ‖⟪x, y⟫_𝕜 * ⟪y, x⟫_𝕜‖
-- inner_add_add_self
theorem inner_add_add_self (x y : E) : ⟪x + y, x + y⟫ = ⟪x, x⟫ + ⟪x, y⟫ + ⟪y, 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⟫_𝕜 = ⟪x, x⟫_𝕜 + ⟪x, y⟫_𝕜 + ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- real_inner_add_add_self
theorem real_inner_add_add_self (x y : F) :
    ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ
-- inner_sub_sub_self
theorem inner_sub_sub_self (x y : E) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, 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⟫_𝕜 = ⟪x, x⟫_𝕜 - ⟪x, y⟫_𝕜 - ⟪y, x⟫_𝕜 + ⟪y, y⟫_𝕜
-- real_inner_sub_sub_self
theorem real_inner_sub_sub_self (x y : F) :
    ⟪x - y, x - y⟫_ℝ = ⟪x, x⟫_ℝ - 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x - y, x - y⟫_ℝ = ⟪x, x⟫_ℝ - 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ
-- ext_inner_left
theorem ext_inner_left {x y : E} (h : ∀ v, ⟪v, x⟫ = ⟪v, y⟫) : x = y 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (h : ∀ (v : E), ⟪v, x⟫_𝕜 = ⟪v, y⟫_𝕜)
Goal: x = y
-- ext_inner_right
theorem ext_inner_right {x y : E} (h : ∀ v, ⟪x, v⟫ = ⟪y, v⟫) : x = y 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (h : ∀ (v : E), ⟪x, v⟫_𝕜 = ⟪y, v⟫_𝕜)
Goal: x = y
-- parallelogram_law
theorem parallelogram_law {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⟫_𝕜 + ⟪x - y, x - y⟫_𝕜 = 2 * (⟪x, x⟫_𝕜 + ⟪y, y⟫_𝕜)
-- inner_mul_inner_self_le
theorem inner_mul_inner_self_le (x y : E) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ‖⟪x, y⟫_𝕜‖ * ‖⟪y, x⟫_𝕜‖ ≤ RCLike.re ⟪x, x⟫_𝕜 * RCLike.re ⟪y, y⟫_𝕜
-- real_inner_mul_inner_self_le
theorem real_inner_mul_inner_self_le (x y : F) : ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ ≤ ⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ
-- linearIndependent_of_ne_zero_of_inner_eq_zero
theorem linearIndependent_of_ne_zero_of_inner_eq_zero {ι : Type*} {v : ι → E} (hz : ∀ i, v i ≠ 0)
    (ho : Pairwise fun i j => ⟪v i, v j⟫ = 0) : 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} (hz : ∀ (i : ι), v i ≠ 0) (ho : Pairwise fun i j => ⟪v i, v j⟫_𝕜 = 0)
Goal: LinearIndependent 𝕜 v
-- orthonormal_iff_ite
theorem orthonormal_iff_ite [DecidableEq ι] {v : ι → E} :
    Orthonormal 𝕜 v ↔ ∀ i j, ⟪v i, v j⟫ = if i = j then (1 : 𝕜) else (0 : 𝕜) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} [DecidableEq ι] {v : ι → E}
Goal: Orthonormal 𝕜 v ↔ ∀ (i j : ι), ⟪v i, v j⟫_𝕜 = if i = j then 1 else 0