ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrthonormalBasis.sum_repr_symm
protected theorem sum_repr_symm (b : OrthonormalBasis ι 𝕜 E) (v : EuclideanSpace 𝕜 ι) :
    ∑ i, v i • b i = b.repr.symm v 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (v : EuclideanSpace 𝕜 ι)
Goal: ∑ i : ι, v i • b i = b.repr.symm v
-- OrthonormalBasis.sum_inner_mul_inner
protected theorem sum_inner_mul_inner (b : OrthonormalBasis ι 𝕜 E) (x y : E) :
    ∑ i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (x y : E)
Goal: ∑ i : ι, ⟪x, b i⟫_𝕜 * ⟪b i, y⟫_𝕜 = ⟪x, y⟫_𝕜
-- OrthonormalBasis.orthogonalProjection_eq_sum
protected theorem orthogonalProjection_eq_sum {U : Submodule 𝕜 E} [CompleteSpace U]
    (b : OrthonormalBasis ι 𝕜 U) (x : E) :
    orthogonalProjection U x = ∑ i, ⟪(b i : E), x⟫ • b i 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {U : Submodule 𝕜 E} [CompleteSpace ↥U] (b : OrthonormalBasis ι 𝕜 ↥U) (x : E)
Goal: (orthogonalProjection U) x = ∑ i : ι, ⟪↑(b i), x⟫_𝕜 • b i
-- OrthonormalBasis.map_apply
@[simp]
protected theorem map_apply {G : Type*} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G]
    (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι) : b.map L i = L (b i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_8} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι)
Goal: (b.map L) i = L (b i)
-- OrthonormalBasis.toBasis_map
@[simp]
protected theorem toBasis_map {G : Type*} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G]
    (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) :
    (b.map L).toBasis = b.toBasis.map L.toLinearEquiv 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {G : Type u_8} [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (b : OrthonormalBasis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G)
Goal: (b.map L).toBasis = b.toBasis.map L.toLinearEquiv
-- Basis.coe_toOrthonormalBasis_repr
@[simp]
theorem _root_.Basis.coe_toOrthonormalBasis_repr (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
    ((v.toOrthonormalBasis hv).repr : E → EuclideanSpace 𝕜 ι) = v.equivFun 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 ⇑v)
Goal: ⇑(v.toOrthonormalBasis hv).repr = ⇑v.equivFun
-- Basis.coe_toOrthonormalBasis_repr_symm
@[simp]
theorem _root_.Basis.coe_toOrthonormalBasis_repr_symm (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
    ((v.toOrthonormalBasis hv).repr.symm : EuclideanSpace 𝕜 ι → E) = v.equivFun.symm 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 ⇑v)
Goal: ⇑(v.toOrthonormalBasis hv).repr.symm = ⇑v.equivFun.symm
-- Basis.toBasis_toOrthonormalBasis
@[simp]
theorem _root_.Basis.toBasis_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
    (v.toOrthonormalBasis hv).toBasis = v 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 ⇑v)
Goal: (v.toOrthonormalBasis hv).toBasis = v
-- Basis.coe_toOrthonormalBasis
@[simp]
theorem _root_.Basis.coe_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
    (v.toOrthonormalBasis hv : ι → E) = (v : ι → E) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 ⇑v)
Goal: ⇑(v.toOrthonormalBasis hv) = ⇑v
-- Pi.orthonormalBasis.toBasis
theorem _root_.Pi.orthonormalBasis.toBasis {η : Type*} [Fintype η] {ι : η → Type*}
    [∀ i, Fintype (ι i)] {𝕜 : Type*} [RCLike 𝕜] {E : η → Type*} [∀ i, NormedAddCommGroup (E i)]
    [∀ i, InnerProductSpace 𝕜 (E i)] (B : ∀ i, OrthonormalBasis (ι i) 𝕜 (E i)) :
    (Pi.orthonormalBasis B).toBasis =
      ((Pi.basis fun i : η ↦ (B i).toBasis).map (WithLp.linearEquiv 2 _ _).symm) 
Arguments: {η : Type u_8} [Fintype η] {ι : η → Type u_9} [(i : η) → Fintype (ι i)] {𝕜 : Type u_10} [RCLike 𝕜] {E : η → Type u_11} [(i : η) → NormedAddCommGroup (E i)] [(i : η) → InnerProductSpace 𝕜 (E i)] (B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i))
Goal: (Pi.orthonormalBasis B).toBasis = (Pi.basis fun i => (B i).toBasis).map (WithLp.linearEquiv 2 𝕜 ((j : η) → E j)).symm
-- Pi.orthonormalBasis_apply
@[simp]
theorem _root_.Pi.orthonormalBasis_apply {η : Type*} [Fintype η] [DecidableEq η] {ι : η → Type*}
    [∀ i, Fintype (ι i)] {𝕜 : Type*} [RCLike 𝕜] {E : η → Type*} [∀ i, NormedAddCommGroup (E i)]
    [∀ i, InnerProductSpace 𝕜 (E i)] (B : ∀ i, OrthonormalBasis (ι i) 𝕜 (E i))
    (j : (i : η) × (ι i)) :
    Pi.orthonormalBasis B j = (WithLp.equiv _ _).symm (Pi.single _ (B j.fst j.snd)) 
Arguments: {η : Type u_8} [Fintype η] [DecidableEq η] {ι : η → Type u_9} [(i : η) → Fintype (ι i)] {𝕜 : Type u_10} [RCLike 𝕜] {E : η → Type u_11} [(i : η) → NormedAddCommGroup (E i)] [(i : η) → InnerProductSpace 𝕜 (E i)] (B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)) (j : (i : η) × ι i)
Goal: (Pi.orthonormalBasis B) j = (WithLp.equiv 2 ((j : η) → E j)).symm (Pi.single j.fst ((B j.fst) j.snd))
-- Pi.orthonormalBasis_repr
@[simp]
theorem _root_.Pi.orthonormalBasis_repr {η : Type*} [Fintype η] {ι : η → Type*}
    [∀ i, Fintype (ι i)] {𝕜 : Type*} [RCLike 𝕜] {E : η → Type*} [∀ i, NormedAddCommGroup (E i)]
    [∀ i, InnerProductSpace 𝕜 (E i)] (B : ∀ i, OrthonormalBasis (ι i) 𝕜 (E i)) (x : (i : η ) → E i)
    (j : (i : η) × (ι i)) :
    (Pi.orthonormalBasis B).repr x j = (B j.fst).repr (x j.fst) j.snd 
Arguments: {η : Type u_8} [Fintype η] {ι : η → Type u_9} [(i : η) → Fintype (ι i)] {𝕜 : Type u_10} [RCLike 𝕜] {E : η → Type u_11} [(i : η) → NormedAddCommGroup (E i)] [(i : η) → InnerProductSpace 𝕜 (E i)] (B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)) (x : (i : η) → E i) (j : (i : η) × ι i)
Goal: (Pi.orthonormalBasis B).repr x j = (B j.fst).repr (x j.fst) j.snd
-- OrthonormalBasis.coe_mk
@[simp]
protected theorem coe_mk (hon : Orthonormal 𝕜 v) (hsp : ⊤ ≤ Submodule.span 𝕜 (Set.range v)) :
    ⇑(OrthonormalBasis.mk hon hsp) = v 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {v : ι → E} (hon : Orthonormal 𝕜 v) (hsp : ⊤ ≤ Submodule.span 𝕜 (Set.range v))
Goal: ⇑(OrthonormalBasis.mk hon hsp) = v
-- OrthonormalBasis.span_apply
@[simp]
protected theorem span_apply [DecidableEq E] {v' : ι' → E} (h : Orthonormal 𝕜 v') (s : Finset ι')
    (i : s) : (OrthonormalBasis.span h s i : E) = v' i 
Arguments: {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [DecidableEq E] {v' : ι' → E} (h : Orthonormal 𝕜 v') (s : Finset ι') (i : { x // x ∈ s })
Goal: ↑((OrthonormalBasis.span h s) i) = v' ↑i
-- OrthonormalBasis.coe_of_orthogonal_eq_bot_mk
@[simp]
protected theorem coe_of_orthogonal_eq_bot_mk (hon : Orthonormal 𝕜 v)
    (hsp : (span 𝕜 (Set.range v))ᗮ = ⊥) : ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hon hsp) = v 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {v : ι → E} (hon : Orthonormal 𝕜 v) (hsp : (Submodule.span 𝕜 (Set.range v))ᗮ = ⊥)
Goal: ⇑(OrthonormalBasis.mkOfOrthogonalEqBot hon hsp) = v
-- OrthonormalBasis.reindex_apply
protected theorem reindex_apply (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (i' : ι') :
    (b.reindex e) i' = b (e.symm i') 
Arguments: {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [Fintype ι'] (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (i' : ι')
Goal: (b.reindex e) i' = b (e.symm i')
-- OrthonormalBasis.reindex_toBasis
@[simp]
theorem reindex_toBasis (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') :
    (b.reindex e).toBasis = b.toBasis.reindex e 
Arguments: {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [Fintype ι'] (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι')
Goal: (b.reindex e).toBasis = b.toBasis.reindex e
-- OrthonormalBasis.coe_reindex
@[simp]
protected theorem coe_reindex (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') :
    ⇑(b.reindex e) = b ∘ e.symm 
Arguments: {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [Fintype ι'] (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι')
Goal: ⇑(b.reindex e) = ⇑b ∘ ⇑e.symm
-- OrthonormalBasis.repr_reindex
@[simp]
protected theorem repr_reindex (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (x : E) (i' : ι') :
    (b.reindex e).repr x i' = b.repr x (e.symm i') 
Arguments: {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [Fintype ι'] (b : OrthonormalBasis ι 𝕜 E) (e : ι ≃ ι') (x : E) (i' : ι')
Goal: (b.reindex e).repr x i' = b.repr x (e.symm i')
-- EuclideanSpace.basisFun_apply
@[simp]
theorem basisFun_apply [DecidableEq ι] (i : ι) : basisFun ι 𝕜 i = EuclideanSpace.single i 1 
Arguments: (ι : Type u_1) (𝕜 : Type u_3) [RCLike 𝕜] [Fintype ι] [DecidableEq ι] (i : ι)
Goal: (EuclideanSpace.basisFun ι 𝕜) i = EuclideanSpace.single i 1
-- EuclideanSpace.basisFun_repr
@[simp]
theorem basisFun_repr (x : EuclideanSpace 𝕜 ι) (i : ι) : (basisFun ι 𝕜).repr x i = x i 
Arguments: (ι : Type u_1) (𝕜 : Type u_3) [RCLike 𝕜] [Fintype ι] (x : EuclideanSpace 𝕜 ι) (i : ι)
Goal: (EuclideanSpace.basisFun ι 𝕜).repr x i = x i
-- EuclideanSpace.basisFun_toBasis
theorem basisFun_toBasis : (basisFun ι 𝕜).toBasis = PiLp.basisFun _ 𝕜 ι 
Arguments: (ι : Type u_1) (𝕜 : Type u_3) [RCLike 𝕜] [Fintype ι]
Goal: (EuclideanSpace.basisFun ι 𝕜).toBasis = PiLp.basisFun 2 𝕜 ι
-- Complex.orthonormalBasisOneI_repr_apply
@[simp]
theorem Complex.orthonormalBasisOneI_repr_apply (z : ℂ) :
    Complex.orthonormalBasisOneI.repr z = ![z.re, z.im] 
Arguments: (z : ℂ)
Goal: Complex.orthonormalBasisOneI.repr z = ![z.re, z.im]
-- Complex.orthonormalBasisOneI_repr_symm_apply
@[simp]
theorem Complex.orthonormalBasisOneI_repr_symm_apply (x : EuclideanSpace ℝ (Fin 2)) :
    Complex.orthonormalBasisOneI.repr.symm x = x 0 + x 1 * I 
Arguments: (x : EuclideanSpace ℝ (Fin 2))
Goal: Complex.orthonormalBasisOneI.repr.symm x = ↑(x 0) + ↑(x 1) * Complex.I
-- Complex.toBasis_orthonormalBasisOneI
@[simp]
theorem Complex.toBasis_orthonormalBasisOneI :
    Complex.orthonormalBasisOneI.toBasis = Complex.basisOneI 
Goal: Complex.orthonormalBasisOneI.toBasis = Complex.basisOneI
-- Complex.coe_orthonormalBasisOneI
@[simp]
theorem Complex.coe_orthonormalBasisOneI :
    (Complex.orthonormalBasisOneI : Fin 2 → ℂ) = ![1, I] 
Goal: ⇑Complex.orthonormalBasisOneI = ![1, Complex.I]
-- Complex.map_isometryOfOrthonormal
@[simp]
theorem Complex.map_isometryOfOrthonormal (v : OrthonormalBasis (Fin 2) ℝ F) (f : F ≃ₗᵢ[ℝ] F') :
    Complex.isometryOfOrthonormal (v.map f) = (Complex.isometryOfOrthonormal v).trans f 
Arguments: {F : Type u_6} [NormedAddCommGroup F] [InnerProductSpace ℝ F] {F' : Type u_7} [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] (v : OrthonormalBasis (Fin 2) ℝ F) (f : F ≃ₗᵢ[ℝ] F')
Goal: Complex.isometryOfOrthonormal (v.map f) = (Complex.isometryOfOrthonormal v).trans f
-- Complex.isometryOfOrthonormal_symm_apply
theorem Complex.isometryOfOrthonormal_symm_apply (v : OrthonormalBasis (Fin 2) ℝ F) (f : F) :
    (Complex.isometryOfOrthonormal v).symm f =
      (v.toBasis.coord 0 f : ℂ) + (v.toBasis.coord 1 f : ℂ) * I 
Arguments: {F : Type u_6} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (v : OrthonormalBasis (Fin 2) ℝ F) (f : F)
Goal: (Complex.isometryOfOrthonormal v).symm f = ↑((v.toBasis.coord 0) f) + ↑((v.toBasis.coord 1) f) * Complex.I
-- Complex.isometryOfOrthonormal_apply
theorem Complex.isometryOfOrthonormal_apply (v : OrthonormalBasis (Fin 2) ℝ F) (z : ℂ) :
    Complex.isometryOfOrthonormal v z = z.re • v 0 + z.im • v 1 
Arguments: {F : Type u_6} [NormedAddCommGroup F] [InnerProductSpace ℝ F] (v : OrthonormalBasis (Fin 2) ℝ F) (z : ℂ)
Goal: (Complex.isometryOfOrthonormal v) z = z.re • v 0 + z.im • v 1
-- OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary
theorem OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary :
    a.toBasis.toMatrix b ∈ Matrix.unitaryGroup ι 𝕜 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] (a b : OrthonormalBasis ι 𝕜 E)
Goal: a.toBasis.toMatrix ⇑b ∈ Matrix.unitaryGroup ι 𝕜
-- OrthonormalBasis.det_to_matrix_orthonormalBasis
@[simp]
theorem OrthonormalBasis.det_to_matrix_orthonormalBasis : ‖a.toBasis.det b‖ = 1 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] (a b : OrthonormalBasis ι 𝕜 E)
Goal: ‖a.toBasis.det ⇑b‖ = 1
-- OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonal
theorem OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonal :
    a.toBasis.toMatrix b ∈ Matrix.orthogonalGroup ι ℝ 
Arguments: {ι : Type u_1} {F : Type u_6} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [Fintype ι] [DecidableEq ι] (a b : OrthonormalBasis ι ℝ F)
Goal: a.toBasis.toMatrix ⇑b ∈ Matrix.orthogonalGroup ι ℝ
-- OrthonormalBasis.det_to_matrix_orthonormalBasis_real
theorem OrthonormalBasis.det_to_matrix_orthonormalBasis_real :
    a.toBasis.det b = 1 ∨ a.toBasis.det b = -1 
Arguments: {ι : Type u_1} {F : Type u_6} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [Fintype ι] [DecidableEq ι] (a b : OrthonormalBasis ι ℝ F)
Goal: a.toBasis.det ⇑b = 1 ∨ a.toBasis.det ⇑b = -1
-- DirectSum.IsInternal.collectedOrthonormalBasis_mem
theorem DirectSum.IsInternal.collectedOrthonormalBasis_mem [DecidableEq ι]
    (h : DirectSum.IsInternal A) {α : ι → Type*} [∀ i, Fintype (α i)]
    (hV : OrthogonalFamily 𝕜 (fun i => A i) fun i => (A i).subtypeₗᵢ)
    (v : ∀ i, OrthonormalBasis (α i) 𝕜 (A i)) (a : Σi, α i) :
    h.collectedOrthonormalBasis hV v a ∈ A a.1 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] {A : ι → Submodule 𝕜 E} [DecidableEq ι] (h : DirectSum.IsInternal A) {α : ι → Type u_8} [(i : ι) → Fintype (α i)] (hV : OrthogonalFamily 𝕜 (fun i => ↥(A i)) fun i => (A i).subtypeₗᵢ) (v : (i : ι) → OrthonormalBasis (α i) 𝕜 ↥(A i)) (a : (i : ι) × α i)
Goal: (DirectSum.IsInternal.collectedOrthonormalBasis hV h v) a ∈ A a.fst
-- Orthonormal.exists_orthonormalBasis_extension
theorem Orthonormal.exists_orthonormalBasis_extension (hv : Orthonormal 𝕜 ((↑) : v → E)) :
    ∃ (u : Finset E) (b : OrthonormalBasis u 𝕜 E), v ⊆ u ∧ ⇑b = ((↑) : u → E) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] {v : Set E} [FiniteDimensional 𝕜 E] (hv : Orthonormal 𝕜 Subtype.val)
Goal: ∃ u b, v ⊆ ↑u ∧ ⇑b = Subtype.val
-- Orthonormal.exists_orthonormalBasis_extension_of_card_eq
theorem Orthonormal.exists_orthonormalBasis_extension_of_card_eq {ι : Type*} [Fintype ι]
    (card_ι : finrank 𝕜 E = Fintype.card ι) {v : ι → E} {s : Set ι}
    (hv : Orthonormal 𝕜 (s.restrict v)) : ∃ b : OrthonormalBasis ι 𝕜 E, ∀ i ∈ s, b i = v i 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {ι : Type u_8} [Fintype ι] (card_ι : FiniteDimensional.finrank 𝕜 E = Fintype.card ι) {v : ι → E} {s : Set ι} (hv : Orthonormal 𝕜 (s.restrict v))
Goal: ∃ b, ∀ i ∈ s, b i = v i
-- exists_orthonormalBasis
theorem _root_.exists_orthonormalBasis :
    ∃ (w : Finset E) (b : OrthonormalBasis w 𝕜 E), ⇑b = ((↑) : w → E) 
Arguments: (𝕜 : Type u_3) [RCLike 𝕜] (E : Type u_4) [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E]
Goal: ∃ w b, ⇑b = Subtype.val
-- stdOrthonormalBasis_def
irreducible_def stdOrthonormalBasis : OrthonormalBasis (Fin (finrank 𝕜 E)) 𝕜 E 
Arguments: (𝕜 : Type u_8) [RCLike 𝕜] (E : Type u_9) [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E]
Goal: stdOrthonormalBasis 𝕜 E =
  let b := Classical.choose ⋯;
  ⋯.mpr (b.reindex (Fintype.equivFinOfCardEq ⋯))
-- orthonormalBasis_one_dim
theorem orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) :
    (⇑b = fun _ => (1 : ℝ)) ∨ ⇑b = fun _ => (-1 : ℝ) 
Arguments: {ι : Type u_1} [Fintype ι] (b : OrthonormalBasis ι ℝ ℝ)
Goal: (⇑b = fun x => 1) ∨ ⇑b = fun x => -1
-- DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv_def
irreducible_def DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv
    (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
    (Σi, Fin (finrank 𝕜 (V i))) ≃ Fin n 
Arguments: {ι : Type u_8} {𝕜 : Type u_9} [RCLike 𝕜] {E : Type u_10} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv hn hV hV' =
  let b := DirectSum.IsInternal.collectedOrthonormalBasis hV' hV fun i => stdOrthonormalBasis 𝕜 ↥(V i);
  Fintype.equivFinOfCardEq ⋯
-- DirectSum.IsInternal.subordinateOrthonormalBasis_def
irreducible_def DirectSum.IsInternal.subordinateOrthonormalBasis
    (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
    OrthonormalBasis (Fin n) 𝕜 E 
Arguments: {ι : Type u_8} {𝕜 : Type u_9} [RCLike 𝕜] {E : Type u_10} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: DirectSum.IsInternal.subordinateOrthonormalBasis hn hV hV' =
  (DirectSum.IsInternal.collectedOrthonormalBasis hV' hV fun i => stdOrthonormalBasis 𝕜 ↥(V i)).reindex
    (DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv hn hV hV')
-- DirectSum.IsInternal.subordinateOrthonormalBasisIndex_def
irreducible_def DirectSum.IsInternal.subordinateOrthonormalBasisIndex (a : Fin n)
    (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) : ι 
Arguments: {ι : Type u_8} {𝕜 : Type u_9} [RCLike 𝕜] {E : Type u_10} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (a : Fin n) (hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn hV a hV' =
  ((DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv hn hV hV').symm a).fst
-- DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate
theorem DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate (a : Fin n)
    (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
    hV.subordinateOrthonormalBasis hn hV' a ∈ V (hV.subordinateOrthonormalBasisIndex hn a hV') 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [FiniteDimensional 𝕜 E] {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n) [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (a : Fin n) (hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ)
Goal: (DirectSum.IsInternal.subordinateOrthonormalBasis hn hV hV') a ∈
  V (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn hV a hV')
-- LinearIsometry.extend_apply
theorem LinearIsometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S) : L.extend s = L s 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {V : Type u_8} [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V] {S : Submodule 𝕜 V} (L : ↥S →ₗᵢ[𝕜] V) (s : ↥S)
Goal: L.extend ↑s = L s
-- Matrix.toEuclideanLin_piLp_equiv_symm
@[simp]
theorem toEuclideanLin_piLp_equiv_symm (A : Matrix m n 𝕜) (x : n → 𝕜) :
    Matrix.toEuclideanLin A ((WithLp.equiv _ _).symm x) =
      (WithLp.equiv _ _).symm (Matrix.toLin' A x) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜) (x : n → 𝕜)
Goal: (Matrix.toEuclideanLin A) ((WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)).symm x) =
  (WithLp.equiv 2 (m → 𝕜)).symm ((Matrix.toLin' A) x)
-- Matrix.piLp_equiv_toEuclideanLin
@[simp]
theorem piLp_equiv_toEuclideanLin (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n) :
    WithLp.equiv _ _ (Matrix.toEuclideanLin A x) = Matrix.toLin' A (WithLp.equiv _ _ x) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n)
Goal: (WithLp.equiv 2 ((i : m) → (fun x => 𝕜) i)) ((Matrix.toEuclideanLin A) x) =
  (Matrix.toLin' A) ((WithLp.equiv 2 (n → 𝕜)) x)
-- Matrix.toEuclideanLin_apply
theorem toEuclideanLin_apply (M : Matrix m n 𝕜) (v : EuclideanSpace 𝕜 n) :
    toEuclideanLin M v = (WithLp.equiv 2 (m → 𝕜)).symm (M *ᵥ (WithLp.equiv 2 (n → 𝕜)) v) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] (M : Matrix m n 𝕜) (v : EuclideanSpace 𝕜 n)
Goal: (Matrix.toEuclideanLin M) v = (WithLp.equiv 2 (m → 𝕜)).symm (M.mulVec ((WithLp.equiv 2 (n → 𝕜)) v))
-- Matrix.piLp_equiv_toEuclideanLin_apply
@[simp]
theorem piLp_equiv_toEuclideanLin_apply (M : Matrix m n 𝕜) (v : EuclideanSpace 𝕜 n) :
    WithLp.equiv 2 (m → 𝕜) (toEuclideanLin M v) = M *ᵥ WithLp.equiv 2 (n → 𝕜) v 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] (M : Matrix m n 𝕜) (v : EuclideanSpace 𝕜 n)
Goal: (WithLp.equiv 2 (m → 𝕜)) ((Matrix.toEuclideanLin M) v) = M.mulVec ((WithLp.equiv 2 (n → 𝕜)) v)
-- Matrix.toEuclideanLin_apply_piLp_equiv_symm
@[simp]
theorem toEuclideanLin_apply_piLp_equiv_symm (M : Matrix m n 𝕜) (v : n → 𝕜) :
    toEuclideanLin M ((WithLp.equiv 2 (n→ 𝕜)).symm v) = (WithLp.equiv 2 (m → 𝕜)).symm (M *ᵥ v) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] (M : Matrix m n 𝕜) (v : n → 𝕜)
Goal: (Matrix.toEuclideanLin M) ((WithLp.equiv 2 (n → 𝕜)).symm v) = (WithLp.equiv 2 (m → 𝕜)).symm (M.mulVec v)
-- Matrix.toEuclideanLin_eq_toLin
theorem toEuclideanLin_eq_toLin [Finite m] :
    (toEuclideanLin : Matrix m n 𝕜 ≃ₗ[𝕜] _) =
      Matrix.toLin (PiLp.basisFun _ _ _) (PiLp.basisFun _ _ _) 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] [Finite m]
Goal: Matrix.toEuclideanLin = Matrix.toLin (PiLp.basisFun 2 𝕜 n) (PiLp.basisFun 2 𝕜 m)
-- Matrix.toEuclideanLin_eq_toLin_orthonormal
lemma toEuclideanLin_eq_toLin_orthonormal [Fintype m] :
    toEuclideanLin = toLin (basisFun n 𝕜).toBasis (basisFun m 𝕜).toBasis 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] [DecidableEq n] [Fintype m]
Goal: Matrix.toEuclideanLin = Matrix.toLin (EuclideanSpace.basisFun n 𝕜).toBasis (EuclideanSpace.basisFun m 𝕜).toBasis
-- inner_matrix_row_row
theorem inner_matrix_row_row [Fintype n] (A B : Matrix m n 𝕜) (i j : m) :
    ⟪A i, B j⟫ₑ = (B * Aᴴ) j i 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype n] (A B : Matrix m n 𝕜) (i j : m)
Goal: ⟪(WithLp.equiv 2 (n → 𝕜)).symm (A i), (WithLp.equiv 2 (n → 𝕜)).symm (B j)⟫_𝕜 = (B * A.conjTranspose) j i
-- inner_matrix_col_col
theorem inner_matrix_col_col [Fintype m] (A B : Matrix m n 𝕜) (i j : n) :
    ⟪Aᵀ i, Bᵀ j⟫ₑ = (Aᴴ * B) i j 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {m : Type u_8} {n : Type u_9} [Fintype m] (A B : Matrix m n 𝕜) (i j : n)
Goal: ⟪(WithLp.equiv 2 (m → 𝕜)).symm (A.transpose i), (WithLp.equiv 2 (m → 𝕜)).symm (B.transpose j)⟫_𝕜 =
  (A.conjTranspose * B) i j
-- StarConvex.contractibleSpace
protected theorem StarConvex.contractibleSpace (h : StarConvex ℝ x s) (hne : s.Nonempty) :
    ContractibleSpace s 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {s : Set E} {x : E} (h : StarConvex ℝ x s) (hne : s.Nonempty)
Goal: ContractibleSpace ↑s