ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearMap.coe_isometryOfOrthonormal
@[simp]
theorem LinearMap.coe_isometryOfOrthonormal (f : E →ₗ[𝕜] E') {v : Basis ι 𝕜 E}
    (hv : Orthonormal 𝕜 v) (hf : Orthonormal 𝕜 (f ∘ v)) : ⇑(f.isometryOfOrthonormal hv hf) = f 
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'] (f : E →ₗ[𝕜] E') {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) (hf : Orthonormal 𝕜 (⇑f ∘ ⇑v))
Goal: ⇑(f.isometryOfOrthonormal hv hf) = ⇑f
-- LinearMap.isometryOfOrthonormal_toLinearMap
@[simp]
theorem LinearMap.isometryOfOrthonormal_toLinearMap (f : E →ₗ[𝕜] E') {v : Basis ι 𝕜 E}
    (hv : Orthonormal 𝕜 v) (hf : Orthonormal 𝕜 (f ∘ v)) :
    (f.isometryOfOrthonormal hv hf).toLinearMap = f 
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'] (f : E →ₗ[𝕜] E') {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) (hf : Orthonormal 𝕜 (⇑f ∘ ⇑v))
Goal: (f.isometryOfOrthonormal hv hf).toLinearMap = f
-- LinearEquiv.coe_isometryOfOrthonormal
@[simp]
theorem LinearEquiv.coe_isometryOfOrthonormal (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜 E}
    (hv : Orthonormal 𝕜 v) (hf : Orthonormal 𝕜 (f ∘ v)) : ⇑(f.isometryOfOrthonormal hv hf) = f 
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'] (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) (hf : Orthonormal 𝕜 (⇑f ∘ ⇑v))
Goal: ⇑(f.isometryOfOrthonormal hv hf) = ⇑f
-- LinearEquiv.isometryOfOrthonormal_toLinearEquiv
@[simp]
theorem LinearEquiv.isometryOfOrthonormal_toLinearEquiv (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜 E}
    (hv : Orthonormal 𝕜 v) (hf : Orthonormal 𝕜 (f ∘ v)) :
    (f.isometryOfOrthonormal hv hf).toLinearEquiv = f 
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'] (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) (hf : Orthonormal 𝕜 (⇑f ∘ ⇑v))
Goal: (f.isometryOfOrthonormal hv hf).toLinearEquiv = f
-- Orthonormal.equiv_toLinearEquiv
@[simp]
theorem Orthonormal.equiv_toLinearEquiv {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v)
    {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') :
    (hv.equiv hv' e).toLinearEquiv = v.equiv v' e 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {ι' : Type u_5} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 ⇑v') (e : ι ≃ ι')
Goal: (hv.equiv hv' e).toLinearEquiv = v.equiv v' e
-- Orthonormal.equiv_apply
@[simp]
theorem Orthonormal.equiv_apply {ι' : Type*} {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v)
    {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') (i : ι) :
    hv.equiv hv' e (v i) = v' (e i) 
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'] {ι' : Type u_9} {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 ⇑v') (e : ι ≃ ι') (i : ι)
Goal: (hv.equiv hv' e) (v i) = v' (e i)
-- Orthonormal.equiv_refl
@[simp]
theorem Orthonormal.equiv_refl {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) :
    hv.equiv hv (Equiv.refl ι) = LinearIsometryEquiv.refl 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v)
Goal: hv.equiv hv (Equiv.refl ι) = LinearIsometryEquiv.refl 𝕜 E
-- Orthonormal.equiv_symm
@[simp]
theorem Orthonormal.equiv_symm {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
    (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') : (hv.equiv hv' e).symm = hv'.equiv hv e.symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {ι' : Type u_5} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 ⇑v') (e : ι ≃ ι')
Goal: (hv.equiv hv' e).symm = hv'.equiv hv e.symm
-- Orthonormal.equiv_trans
@[simp]
theorem Orthonormal.equiv_trans {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
    (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') {v'' : Basis ι'' 𝕜 E''} (hv'' : Orthonormal 𝕜 v'')
    (e' : ι' ≃ ι'') : (hv.equiv hv' e).trans (hv'.equiv hv'' e') = hv.equiv hv'' (e.trans e') 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {ι' : Type u_5} {ι'' : Type u_6} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {E'' : Type u_8} [NormedAddCommGroup E''] [InnerProductSpace 𝕜 E''] {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 ⇑v') (e : ι ≃ ι') {v'' : Basis ι'' 𝕜 E''} (hv'' : Orthonormal 𝕜 ⇑v'') (e' : ι' ≃ ι'')
Goal: (hv.equiv hv' e).trans (hv'.equiv hv'' e') = hv.equiv hv'' (e.trans e')
-- Orthonormal.map_equiv
theorem Orthonormal.map_equiv {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
    (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') :
    v.map (hv.equiv hv' e).toLinearEquiv = v'.reindex e.symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {ι' : Type u_5} {E' : Type u_7} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 ⇑v) {v' : Basis ι' 𝕜 E'} (hv' : Orthonormal 𝕜 ⇑v') (e : ι ≃ ι')
Goal: v.map (hv.equiv hv' e).toLinearEquiv = v'.reindex e.symm
-- real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two
theorem real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : F) :
    ⟪x, y⟫_ℝ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2
-- real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two
theorem real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : F) :
    ⟪x, y⟫_ℝ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2
-- norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero
theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero (x y : F) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0
-- norm_add_eq_sqrt_iff_real_inner_eq_zero
theorem norm_add_eq_sqrt_iff_real_inner_eq_zero {x y : F} :
    ‖x + y‖ = √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F}
Goal: ‖x + y‖ = √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0
-- norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero
theorem norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (x y : E) (h : ⟪x, y⟫ = 0) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E) (h : ⟪x, y⟫_𝕜 = 0)
Goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
-- norm_add_sq_eq_norm_sq_add_norm_sq_real
theorem norm_add_sq_eq_norm_sq_add_norm_sq_real {x y : F} (h : ⟪x, y⟫_ℝ = 0) :
    ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F} (h : ⟪x, y⟫_ℝ = 0)
Goal: ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
-- norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero
theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero (x y : F) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ ⟪x, y⟫_ℝ = 0
-- norm_sub_eq_sqrt_iff_real_inner_eq_zero
theorem norm_sub_eq_sqrt_iff_real_inner_eq_zero {x y : F} :
    ‖x - y‖ = √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F}
Goal: ‖x - y‖ = √(‖x‖ * ‖x‖ + ‖y‖ * ‖y‖) ↔ ⟪x, y⟫_ℝ = 0
-- norm_sub_sq_eq_norm_sq_add_norm_sq_real
theorem norm_sub_sq_eq_norm_sq_add_norm_sq_real {x y : F} (h : ⟪x, y⟫_ℝ = 0) :
    ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F} (h : ⟪x, y⟫_ℝ = 0)
Goal: ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖
-- real_inner_add_sub_eq_zero_iff
theorem real_inner_add_sub_eq_zero_iff (x y : F) : ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖
-- norm_sub_eq_norm_add
theorem norm_sub_eq_norm_add {v w : E} (h : ⟪v, w⟫ = 0) : ‖w - v‖ = ‖w + v‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v w : E} (h : ⟪v, w⟫_𝕜 = 0)
Goal: ‖w - v‖ = ‖w + v‖
-- abs_real_inner_div_norm_mul_norm_le_one
theorem abs_real_inner_div_norm_mul_norm_le_one (x y : F) : |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| ≤ 1 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| ≤ 1
-- real_inner_smul_self_left
theorem real_inner_smul_self_left (x : F) (r : ℝ) : ⟪r • x, x⟫_ℝ = r * (‖x‖ * ‖x‖) 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F) (r : ℝ)
Goal: ⟪r • x, x⟫_ℝ = r * (‖x‖ * ‖x‖)
-- real_inner_smul_self_right
theorem real_inner_smul_self_right (x : F) (r : ℝ) : ⟪x, r • x⟫_ℝ = r * (‖x‖ * ‖x‖) 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x : F) (r : ℝ)
Goal: ⟪x, r • x⟫_ℝ = r * (‖x‖ * ‖x‖)
-- norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul
theorem norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul {x : E} {r : 𝕜} (hx : x ≠ 0)
    (hr : r ≠ 0) : ‖⟪x, r • x⟫‖ / (‖x‖ * ‖r • x‖) = 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x : E} {r : 𝕜} (hx : x ≠ 0) (hr : r ≠ 0)
Goal: ‖⟪x, r • x⟫_𝕜‖ / (‖x‖ * ‖r • x‖) = 1
-- abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul
theorem abs_real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul {x : F} {r : ℝ}
    (hx : x ≠ 0) (hr : r ≠ 0) : |⟪x, r • x⟫_ℝ| / (‖x‖ * ‖r • x‖) = 1 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : r ≠ 0)
Goal: |⟪x, r • x⟫_ℝ| / (‖x‖ * ‖r • x‖) = 1
-- real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul
theorem real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul {x : F} {r : ℝ} (hx : x ≠ 0)
    (hr : 0 < r) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : 0 < r)
Goal: ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1
-- real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul
theorem real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul {x : F} {r : ℝ} (hx : x ≠ 0)
    (hr : r < 0) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = -1 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x : F} {r : ℝ} (hx : x ≠ 0) (hr : r < 0)
Goal: ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = -1
-- norm_inner_eq_norm_tfae
theorem norm_inner_eq_norm_tfae (x y : E) :
    List.TFAE [‖⟪x, y⟫‖ = ‖x‖ * ‖y‖,
      x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x,
      x = 0 ∨ ∃ r : 𝕜, y = r • x,
      x = 0 ∨ y ∈ 𝕜 ∙ x] 
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‖, x = 0 ∨ y = (⟪x, y⟫_𝕜 / ⟪x, x⟫_𝕜) • x, x = 0 ∨ ∃ r, y = r • x,
    x = 0 ∨ y ∈ Submodule.span 𝕜 {x}].TFAE
-- norm_inner_eq_norm_iff
theorem norm_inner_eq_norm_iff {x y : E} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) :
    ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖ ↔ ∃ r : 𝕜, r ≠ 0 ∧ y = r • x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0)
Goal: ‖⟪x, y⟫_𝕜‖ = ‖x‖ * ‖y‖ ↔ ∃ r, r ≠ 0 ∧ y = r • x
-- norm_inner_div_norm_mul_norm_eq_one_iff
theorem norm_inner_div_norm_mul_norm_eq_one_iff (x y : E) :
    ‖⟪x, y⟫ / (‖x‖ * ‖y‖)‖ = 1 ↔ x ≠ 0 ∧ ∃ r : 𝕜, r ≠ 0 ∧ y = r • x 
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‖)‖ = 1 ↔ x ≠ 0 ∧ ∃ r, r ≠ 0 ∧ y = r • x
-- abs_real_inner_div_norm_mul_norm_eq_one_iff
theorem abs_real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :
    |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| = 1 ↔ x ≠ 0 ∧ ∃ r : ℝ, r ≠ 0 ∧ y = r • x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| = 1 ↔ x ≠ 0 ∧ ∃ r, r ≠ 0 ∧ y = r • x
-- inner_eq_norm_mul_iff_div
theorem inner_eq_norm_mul_iff_div {x y : E} (h₀ : x ≠ 0) :
    ⟪x, y⟫ = (‖x‖ : 𝕜) * ‖y‖ ↔ (‖y‖ / ‖x‖ : 𝕜) • x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (h₀ : x ≠ 0)
Goal: ⟪x, y⟫_𝕜 = ↑‖x‖ * ↑‖y‖ ↔ (↑‖y‖ / ↑‖x‖) • x = y
-- inner_eq_norm_mul_iff
theorem inner_eq_norm_mul_iff {x y : E} :
    ⟪x, y⟫ = (‖x‖ : 𝕜) * ‖y‖ ↔ (‖y‖ : 𝕜) • x = (‖x‖ : 𝕜) • 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‖ ↔ ↑‖y‖ • x = ↑‖x‖ • y
-- inner_eq_norm_mul_iff_real
theorem inner_eq_norm_mul_iff_real {x y : F} : ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F}
Goal: ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ ↔ ‖y‖ • x = ‖x‖ • y
-- real_inner_div_norm_mul_norm_eq_one_iff
theorem real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :
    ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x
-- real_inner_div_norm_mul_norm_eq_neg_one_iff
theorem real_inner_div_norm_mul_norm_eq_neg_one_iff (x y : F) :
    ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ x ≠ 0 ∧ ∃ r : ℝ, r < 0 ∧ y = r • x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (x y : F)
Goal: ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x
-- inner_eq_one_iff_of_norm_one
theorem inner_eq_one_iff_of_norm_one {x y : E} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
    ⟪x, y⟫ = 1 ↔ x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: ⟪x, y⟫_𝕜 = 1 ↔ x = y
-- inner_lt_norm_mul_iff_real
theorem inner_lt_norm_mul_iff_real {x y : F} : ⟪x, y⟫_ℝ < ‖x‖ * ‖y‖ ↔ ‖y‖ • x ≠ ‖x‖ • y 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F}
Goal: ⟪x, y⟫_ℝ < ‖x‖ * ‖y‖ ↔ ‖y‖ • x ≠ ‖x‖ • y
-- inner_lt_one_iff_real_of_norm_one
theorem inner_lt_one_iff_real_of_norm_one {x y : F} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
    ⟪x, y⟫_ℝ < 1 ↔ x ≠ y 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {x y : F} (hx : ‖x‖ = 1) (hy : ‖y‖ = 1)
Goal: ⟪x, y⟫_ℝ < 1 ↔ x ≠ y
-- eq_of_norm_le_re_inner_eq_norm_sq
theorem eq_of_norm_le_re_inner_eq_norm_sq {x y : E} (hle : ‖x‖ ≤ ‖y‖) (h : re ⟪x, y⟫ = ‖y‖ ^ 2) :
    x = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {x y : E} (hle : ‖x‖ ≤ ‖y‖) (h : RCLike.re ⟪x, y⟫_𝕜 = ‖y‖ ^ 2)
Goal: x = y
-- inner_sum_smul_sum_smul_of_sum_eq_zero
theorem inner_sum_smul_sum_smul_of_sum_eq_zero {ι₁ : Type*} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ}
    (v₁ : ι₁ → F) (h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ}
    (v₂ : ι₂ → F) (h₂ : ∑ i ∈ s₂, w₂ i = 0) :
    ⟪∑ i₁ ∈ s₁, w₁ i₁ • v₁ i₁, ∑ i₂ ∈ s₂, w₂ i₂ • v₂ i₂⟫_ℝ =
      (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {ι₁ : Type u_4} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ} (v₁ : ι₁ → F) (h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type u_5} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ} (v₂ : ι₂ → F) (h₂ : ∑ i ∈ s₂, w₂ i = 0)
Goal: ⟪∑ i₁ ∈ s₁, w₁ i₁ • v₁ i₁, ∑ i₂ ∈ s₂, w₂ i₂ • v₂ i₂⟫_ℝ =
  (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2
-- innerₛₗ_apply_coe
@[simp]
theorem innerₛₗ_apply_coe (v : E) : ⇑(innerₛₗ 𝕜 v) = fun w => ⟪v, w⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v : E)
Goal: ⇑((innerₛₗ 𝕜) v) = fun w => ⟪v, w⟫_𝕜
-- innerₛₗ_apply
@[simp]
theorem innerₛₗ_apply (v w : E) : innerₛₗ 𝕜 v w = ⟪v, w⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v w : E)
Goal: ((innerₛₗ 𝕜) v) w = ⟪v, w⟫_𝕜
-- flip_innerₗ
@[simp] lemma flip_innerₗ : (innerₗ F).flip = innerₗ F 
Arguments: (F : Type u_3) [NormedAddCommGroup F] [InnerProductSpace ℝ F]
Goal: (innerₗ F).flip = innerₗ F
-- innerₗ_apply
@[simp] lemma innerₗ_apply (v w : F) : innerₗ F v w = ⟪v, w⟫_ℝ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (v w : F)
Goal: ((innerₗ F) v) w = ⟪v, w⟫_ℝ
-- innerSL_apply_coe
@[simp]
theorem innerSL_apply_coe (v : E) : ⇑(innerSL 𝕜 v) = fun w => ⟪v, w⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v : E)
Goal: ⇑((innerSL 𝕜) v) = fun w => ⟪v, w⟫_𝕜
-- innerSL_apply
@[simp]
theorem innerSL_apply (v w : E) : innerSL 𝕜 v w = ⟪v, w⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (v w : E)
Goal: ((innerSL 𝕜) v) w = ⟪v, w⟫_𝕜
-- innerSL_apply_norm
@[simp]
theorem innerSL_apply_norm (x : E) : ‖innerSL 𝕜 x‖ = ‖x‖ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x : E)
Goal: ‖(innerSL 𝕜) x‖ = ‖x‖
-- norm_innerSL_le
lemma norm_innerSL_le : ‖innerSL 𝕜 (E := E)‖ ≤ 1 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ‖innerSL 𝕜‖ ≤ 1
-- innerSLFlip_apply
@[simp]
theorem innerSLFlip_apply (x y : E) : innerSLFlip 𝕜 x y = ⟪y, x⟫ 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (x y : E)
Goal: ((innerSLFlip 𝕜) x) y = ⟪y, x⟫_𝕜
-- innerSL_real_flip
@[simp] lemma innerSL_real_flip : (innerSL ℝ (E := F)).flip = innerSL ℝ 
Arguments: (F : Type u_3) [NormedAddCommGroup F] [InnerProductSpace ℝ F]
Goal: (innerSL ℝ).flip = innerSL ℝ
-- ContinuousLinearMap.toSesqForm_apply_coe
@[simp]
theorem toSesqForm_apply_coe (f : E →L[𝕜] E') (x : E') : toSesqForm f x = (innerSL 𝕜 x).comp f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] (f : E →L[𝕜] E') (x : E')
Goal: (ContinuousLinearMap.toSesqForm f) x = ((innerSL 𝕜) x).comp f
-- ContinuousLinearMap.toSesqForm_apply_norm_le
theorem toSesqForm_apply_norm_le {f : E →L[𝕜] E'} {v : E'} : ‖toSesqForm f v‖ ≤ ‖f‖ * ‖v‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {E' : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E'] {f : E →L[𝕜] E'} {v : E'}
Goal: ‖(ContinuousLinearMap.toSesqForm f) v‖ ≤ ‖f‖ * ‖v‖
-- isBoundedBilinearMap_inner
theorem _root_.isBoundedBilinearMap_inner [NormedSpace ℝ E] :
    IsBoundedBilinearMap ℝ fun p : E × E => ⟪p.1, p.2⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedSpace ℝ E]
Goal: IsBoundedBilinearMap ℝ fun p => ⟪p.1, p.2⟫_𝕜
-- Orthonormal.sum_inner_products_le
theorem Orthonormal.sum_inner_products_le {s : Finset ι} (hv : Orthonormal 𝕜 v) :
    ∑ i ∈ s, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (x : E) {v : ι → E} {s : Finset ι} (hv : Orthonormal 𝕜 v)
Goal: ∑ i ∈ s, ‖⟪v i, x⟫_𝕜‖ ^ 2 ≤ ‖x‖ ^ 2
-- Orthonormal.tsum_inner_products_le
theorem Orthonormal.tsum_inner_products_le (hv : Orthonormal 𝕜 v) :
    ∑' i, ‖⟪v i, x⟫‖ ^ 2 ≤ ‖x‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (x : E) {v : ι → E} (hv : Orthonormal 𝕜 v)
Goal: ∑' (i : ι), ‖⟪v i, x⟫_𝕜‖ ^ 2 ≤ ‖x‖ ^ 2
-- Orthonormal.inner_products_summable
theorem Orthonormal.inner_products_summable (hv : Orthonormal 𝕜 v) :
    Summable fun i => ‖⟪v i, x⟫‖ ^ 2 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} (x : E) {v : ι → E} (hv : Orthonormal 𝕜 v)
Goal: Summable fun i => ‖⟪v i, x⟫_𝕜‖ ^ 2
-- RCLike.inner_apply
@[simp]
theorem RCLike.inner_apply (x y : 𝕜) : ⟪x, y⟫ = conj x * y 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] (x y : 𝕜)
Goal: ⟪x, y⟫_𝕜 = (starRingEnd 𝕜) x * y
-- Submodule.coe_inner
@[simp]
theorem Submodule.coe_inner (W : Submodule 𝕜 E) (x y : W) : ⟪x, y⟫ = ⟪(x : E), ↑y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (W : Submodule 𝕜 E) (x y : ↥W)
Goal: ⟪x, y⟫_𝕜 = ⟪↑x, ↑y⟫_𝕜
-- Orthonormal.codRestrict
theorem Orthonormal.codRestrict {ι : Type*} {v : ι → E} (hv : Orthonormal 𝕜 v) (s : Submodule 𝕜 E)
    (hvs : ∀ i, v i ∈ s) : @Orthonormal 𝕜 s _ _ _ ι (Set.codRestrict v s hvs) 
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 : Submodule 𝕜 E) (hvs : ∀ (i : ι), v i ∈ s)
Goal: Orthonormal 𝕜 (Set.codRestrict v (↑s) hvs)
-- orthonormal_span
theorem orthonormal_span {ι : Type*} {v : ι → E} (hv : Orthonormal 𝕜 v) :
    @Orthonormal 𝕜 (Submodule.span 𝕜 (Set.range v)) _ _ _ ι fun i : ι =>
      ⟨v i, Submodule.subset_span (Set.mem_range_self 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)
Goal: Orthonormal 𝕜 fun i => ⟨v i, ⋯⟩
-- Orthonormal.orthogonalFamily
theorem Orthonormal.orthogonalFamily {v : ι → E} (hv : Orthonormal 𝕜 v) :
    OrthogonalFamily 𝕜 (fun _i : ι => 𝕜) fun i => LinearIsometry.toSpanSingleton 𝕜 E (hv.1 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)
Goal: OrthogonalFamily 𝕜 (fun _i => 𝕜) fun i => LinearIsometry.toSpanSingleton 𝕜 E ⋯
-- OrthogonalFamily.eq_ite
theorem OrthogonalFamily.eq_ite [DecidableEq ι] {i j : ι} (v : G i) (w : G j) :
    ⟪V i v, V j w⟫ = ite (i = j) ⟪V i v, V j w⟫ 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [DecidableEq ι] {i j : ι} (v : G i) (w : G j)
Goal: ⟪(V i) v, (V j) w⟫_𝕜 = if i = j then ⟪(V i) v, (V j) w⟫_𝕜 else 0
-- OrthogonalFamily.inner_right_dfinsupp
theorem OrthogonalFamily.inner_right_dfinsupp [DecidableEq ι] (l : ⨁ i, G i) (i : ι) (v : G i) :
    ⟪V i v, l.sum fun j => V j⟫ = ⟪v, l i⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)] [DecidableEq ι] (l : DirectSum ι fun i => G i) (i : ι) (v : G i)
Goal: ⟪(V i) v, DFinsupp.sum l fun j => ⇑(V j)⟫_𝕜 = ⟪v, l i⟫_𝕜
-- OrthogonalFamily.inner_right_fintype
theorem OrthogonalFamily.inner_right_fintype [Fintype ι] (l : ∀ i, G i) (i : ι) (v : G i) :
    ⟪V i v, ∑ j : ι, V j (l j)⟫ = ⟪v, l i⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {ι : Type u_4} {G : ι → Type u_5} [(i : ι) → NormedAddCommGroup (G i)] [(i : ι) → InnerProductSpace 𝕜 (G i)] {V : (i : ι) → G i →ₗᵢ[𝕜] E} (hV : OrthogonalFamily 𝕜 G V) [Fintype ι] (l : (i : ι) → G i) (i : ι) (v : G i)
Goal: ⟪(V i) v, ∑ j : ι, (V j) (l j)⟫_𝕜 = ⟪v, l i⟫_𝕜