ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- QuadraticMap.polarBilin_injective
theorem _root_.QuadraticMap.polarBilin_injective (h : IsUnit (2 : R)) :
    Function.Injective (polarBilin : QuadraticMap R M N → _) 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] (h : IsUnit 2)
Goal: Function.Injective QuadraticMap.polarBilin
-- QuadraticMap.polarBilin_comp
theorem _root_.QuadraticMap.polarBilin_comp (Q : QuadraticMap R N' N) (f : M →ₗ[R] N') :
    polarBilin (Q.comp f) = LinearMap.compl₁₂ (polarBilin Q) f f 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] {N' : Type u_8} [AddCommGroup N'] [Module R N'] (Q : QuadraticMap R N' N) (f : M →ₗ[R] N')
Goal: (Q.comp f).polarBilin = LinearMap.compl₁₂ Q.polarBilin f f
-- LinearMap.compQuadraticMap_polar
theorem _root_.LinearMap.compQuadraticMap_polar [CommSemiring S] [Algebra S R] [Module S N]
    [Module S N'] [IsScalarTower S R N] [Module S M] [IsScalarTower S R M] (f : N →ₗ[S] N')
    (Q : QuadraticMap R M N) (x y : M) : polar (f.compQuadraticMap' Q) x y = f (polar Q x y) 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] {N' : Type u_8} [AddCommGroup N'] [CommSemiring S] [Algebra S R] [Module S N] [Module S N'] [IsScalarTower S R N] [Module S M] [IsScalarTower S R M] (f : N →ₗ[S] N') (Q : QuadraticMap R M N) (x y : M)
Goal: QuadraticMap.polar (⇑(f.compQuadraticMap' Q)) x y = f (QuadraticMap.polar (⇑Q) x y)
-- LinearMap.compQuadraticMap_polarBilin
theorem _root_.LinearMap.compQuadraticMap_polarBilin (f : N →ₗ[R] N') (Q : QuadraticMap R M N) :
    (f.compQuadraticMap' Q).polarBilin = Q.polarBilin.compr₂ f 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] {N' : Type u_8} [AddCommGroup N'] [Module R N'] (f : N →ₗ[R] N') (Q : QuadraticMap R M N)
Goal: (f.compQuadraticMap' Q).polarBilin = LinearMap.compr₂ Q.polarBilin f
-- QuadraticMap.associated_apply
@[simp]
theorem associated_apply (x y : M) : associatedHom S Q x y = ⅟ (2 : R) • (Q (x + y) - Q x - Q y) 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [CommSemiring S] [Algebra S R] [Module S N] [IsScalarTower S R N] [Invertible 2] (Q : QuadraticMap R M N) (x y : M)
Goal: (((QuadraticMap.associatedHom S) Q) x) y = ⅟2 • (Q (x + y) - Q x - Q y)
-- QuadraticMap.two_nsmul_associated
@[simp] theorem two_nsmul_associated : 2 • associatedHom S Q = Q.polarBilin 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [CommSemiring S] [Algebra S R] [Module S N] [IsScalarTower S R N] [Invertible 2] (Q : QuadraticMap R M N)
Goal: 2 • (QuadraticMap.associatedHom S) Q = Q.polarBilin
-- QuadraticMap.associated_isSymm
theorem associated_isSymm (Q : QuadraticMap R M R) : (associatedHom S Q).IsSymm 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (Q : QuadraticMap R M R)
Goal: LinearMap.IsSymm ((QuadraticMap.associatedHom S) Q)
-- QuadraticMap.associated_comp
@[simp]
theorem associated_comp {N' : Type*} [AddCommGroup N'] [Module R N'] (f : N' →ₗ[R] M) :
    associatedHom S (Q.comp f) = (associatedHom S Q).compl₁₂ f f 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [CommSemiring S] [Algebra S R] [Module S N] [IsScalarTower S R N] [Invertible 2] (Q : QuadraticMap R M N) {N' : Type u_8} [AddCommGroup N'] [Module R N'] (f : N' →ₗ[R] M)
Goal: (QuadraticMap.associatedHom S) (Q.comp f) = LinearMap.compl₁₂ ((QuadraticMap.associatedHom S) Q) f f
-- QuadraticMap.associated_toQuadraticMap
theorem associated_toQuadraticMap (B : BilinMap R M R) (x y : M) :
    associatedHom S B.toQuadraticMap x y = ⅟ (2 : R) • (B x y + B y x) 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] (B : LinearMap.BilinMap R M R) (x y : M)
Goal: (((QuadraticMap.associatedHom S) B.toQuadraticMap) x) y = ⅟2 • ((B x) y + (B y) x)
-- QuadraticMap.associated_left_inverse
theorem associated_left_inverse (h : B₁.IsSymm) : associatedHom S B₁.toQuadraticMap = B₁ 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2] {B₁ : LinearMap.BilinMap R M R} (h : LinearMap.IsSymm B₁)
Goal: (QuadraticMap.associatedHom S) B₁.toQuadraticMap = B₁
-- QuadraticMap.associated_eq_self_apply
theorem associated_eq_self_apply (x : M) : associatedHom S Q x x = Q x 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [CommSemiring S] [Algebra S R] [Module S N] [IsScalarTower S R N] [Invertible 2] (Q : QuadraticMap R M N) (x : M)
Goal: (((QuadraticMap.associatedHom S) Q) x) x = Q x
-- QuadraticMap.toQuadraticMap_associated
theorem toQuadraticMap_associated : (associatedHom S Q).toQuadraticMap = Q 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [CommSemiring S] [Algebra S R] [Module S N] [IsScalarTower S R N] [Invertible 2] (Q : QuadraticMap R M N)
Goal: ((QuadraticMap.associatedHom S) Q).toQuadraticMap = Q
-- QuadraticMap.associated_rightInverse
theorem associated_rightInverse :
    Function.RightInverse (associatedHom S) (BilinMap.toQuadraticMap : _ → QuadraticMap R M R) 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [CommSemiring S] [Algebra S R] [Invertible 2]
Goal: Function.RightInverse (⇑(QuadraticMap.associatedHom S)) LinearMap.BilinMap.toQuadraticMap
-- QuadraticMap.exists_quadraticForm_ne_zero
theorem exists_quadraticForm_ne_zero {Q : QuadraticForm R M}
    -- Porting note: added implicit argument
    (hB₁ : associated' (R := R) Q ≠ 0) :
    ∃ x, Q x ≠ 0 
Arguments: {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [Invertible 2] {Q : QuadraticForm R M} (hB₁ : QuadraticMap.associated' Q ≠ 0)
Goal: ∃ x, Q x ≠ 0
-- QuadraticMap.coe_associatedHom
theorem coe_associatedHom :
    ⇑(associatedHom S : QuadraticMap R M N →ₗ[S] BilinMap R M N) = associated 
Arguments: (S : Type u_1) {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring S] [CommRing R] [AddCommGroup M] [Algebra S R] [Module R M] [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower S R N] [Invertible 2]
Goal: ⇑(QuadraticMap.associatedHom S) = ⇑QuadraticMap.associated
-- QuadraticMap.associated_linMulLin
@[simp]
theorem associated_linMulLin (f g : M →ₗ[R] R) :
    associated (R := R) (linMulLin f g) =
      ⅟ (2 : R) • ((mul R R).compl₁₂ f g + (mul R R).compl₁₂ g f) 
Arguments: {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [Invertible 2] (f g : M →ₗ[R] R)
Goal: QuadraticMap.associated (QuadraticMap.linMulLin f g) =
  ⅟2 • ((LinearMap.mul R R).compl₁₂ f g + (LinearMap.mul R R).compl₁₂ g f)
-- QuadraticMap.associated_sq
@[simp]
lemma associated_sq : associated (R := R) sq = mul R R 
Arguments: {R : Type u_3} [CommRing R] [Invertible 2]
Goal: QuadraticMap.associated QuadraticMap.sq = LinearMap.mul R R
-- QuadraticMap.isOrtho_def
theorem isOrtho_def {Q : QuadraticMap R M N} {x y : M} : Q.IsOrtho x y ↔ Q (x + y) = Q x + Q y 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {Q : QuadraticMap R M N} {x y : M}
Goal: Q.IsOrtho x y ↔ Q (x + y) = Q x + Q y
-- QuadraticMap.IsOrtho.all
theorem IsOrtho.all (x y : M) : IsOrtho (0 : QuadraticMap R M N) x y 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (x y : M)
Goal: QuadraticMap.IsOrtho 0 x y
-- QuadraticMap.IsOrtho.zero_left
theorem IsOrtho.zero_left (x : M) : IsOrtho Q (0 : M) x 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {Q : QuadraticMap R M N} (x : M)
Goal: Q.IsOrtho 0 x
-- QuadraticMap.IsOrtho.zero_right
theorem IsOrtho.zero_right (x : M) : IsOrtho Q x (0 : M) 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {Q : QuadraticMap R M N} (x : M)
Goal: Q.IsOrtho x 0
-- QuadraticMap.ne_zero_of_not_isOrtho_self
theorem ne_zero_of_not_isOrtho_self {Q : QuadraticMap R M N} (x : M) (hx₁ : ¬Q.IsOrtho x x) :
    x ≠ 0 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {Q : QuadraticMap R M N} (x : M) (hx₁ : ¬Q.IsOrtho x x)
Goal: x ≠ 0
-- QuadraticMap.isOrtho_comm
theorem isOrtho_comm {x y : M} : IsOrtho Q x y ↔ IsOrtho Q y x 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {Q : QuadraticMap R M N} {x y : M}
Goal: Q.IsOrtho x y ↔ Q.IsOrtho y x
-- LinearMap.BilinForm.toQuadraticMap_isOrtho
theorem _root_.LinearMap.BilinForm.toQuadraticMap_isOrtho [IsCancelAdd R]
    [NoZeroDivisors R] [CharZero R] {B : BilinMap R M R} {x y : M} (h : B.IsSymm) :
    B.toQuadraticMap.IsOrtho x y ↔ B.IsOrtho x y 
Arguments: {R : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [IsCancelAdd R] [NoZeroDivisors R] [CharZero R] {B : LinearMap.BilinMap R M R} {x y : M} (h : LinearMap.IsSymm B)
Goal: B.toQuadraticMap.IsOrtho x y ↔ LinearMap.IsOrtho B x y
-- QuadraticMap.isOrtho_polarBilin
@[simp]
theorem isOrtho_polarBilin {x y : M} : Q.polarBilin.IsOrtho x y ↔ IsOrtho Q x y 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {Q : QuadraticMap R M N} {x y : M}
Goal: LinearMap.IsOrtho Q.polarBilin x y ↔ Q.IsOrtho x y
-- QuadraticMap.IsOrtho.polar_eq_zero
theorem IsOrtho.polar_eq_zero {x y : M} (h : IsOrtho Q x y) : polar Q x y = 0 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {Q : QuadraticMap R M N} {x y : M} (h : Q.IsOrtho x y)
Goal: QuadraticMap.polar (⇑Q) x y = 0
-- QuadraticMap.associated_isOrtho
@[simp]
theorem associated_isOrtho [Invertible (2 : R)] {x y : M} :
    Q.associated.IsOrtho x y ↔ Q.IsOrtho x y 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {Q : QuadraticMap R M N} [Invertible 2] {x y : M}
Goal: LinearMap.IsOrtho (QuadraticMap.associated Q) x y ↔ Q.IsOrtho x y
-- QuadraticMap.not_anisotropic_iff_exists
theorem not_anisotropic_iff_exists (Q : QuadraticMap R M N) :
    ¬Anisotropic Q ↔ ∃ x, x ≠ 0 ∧ Q x = 0 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (Q : QuadraticMap R M N)
Goal: ¬Q.Anisotropic ↔ ∃ x, x ≠ 0 ∧ Q x = 0
-- QuadraticMap.Anisotropic.eq_zero_iff
theorem Anisotropic.eq_zero_iff {Q : QuadraticMap R M N} (h : Anisotropic Q) {x : M} :
    Q x = 0 ↔ x = 0 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {Q : QuadraticMap R M N} (h : Q.Anisotropic) {x : M}
Goal: Q x = 0 ↔ x = 0
-- QuadraticMap.separatingLeft_of_anisotropic
theorem separatingLeft_of_anisotropic [Invertible (2 : R)] (Q : QuadraticMap R M R)
    (hB : Q.Anisotropic) :
    -- Porting note: added implicit argument
    (QuadraticMap.associated' (R := R) Q).SeparatingLeft 
Arguments: {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [Invertible 2] (Q : QuadraticMap R M R) (hB : Q.Anisotropic)
Goal: LinearMap.SeparatingLeft (QuadraticMap.associated' Q)
-- QuadraticMap.PosDef.smul
theorem PosDef.smul {R} [LinearOrderedCommRing R] [Module R M] [Module R N] [PosSMulStrictMono R N]
    {Q : QuadraticMap R M N} (h : PosDef Q) {a : R} (a_pos : 0 < a) : PosDef (a • Q) 
Arguments: {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [PartialOrder N] [AddCommMonoid N] {R : Type u_8} [LinearOrderedCommRing R] [Module R M] [Module R N] [PosSMulStrictMono R N] {Q : QuadraticMap R M N} (h : Q.PosDef) {a : R} (a_pos : 0 < a)
Goal: (a • Q).PosDef
-- QuadraticMap.PosDef.nonneg
theorem PosDef.nonneg {Q : QuadraticMap R₂ M N} (hQ : PosDef Q) (x : M) : 0 ≤ Q x 
Arguments: {M : Type u_4} {N : Type u_5} {R₂ : Type u} [CommSemiring R₂] [AddCommMonoid M] [Module R₂ M] [PartialOrder N] [AddCommMonoid N] [Module R₂ N] {Q : QuadraticMap R₂ M N} (hQ : Q.PosDef) (x : M)
Goal: 0 ≤ Q x
-- QuadraticMap.PosDef.anisotropic
theorem PosDef.anisotropic {Q : QuadraticMap R₂ M N} (hQ : Q.PosDef) : Q.Anisotropic 
Arguments: {M : Type u_4} {N : Type u_5} {R₂ : Type u} [CommSemiring R₂] [AddCommMonoid M] [Module R₂ M] [PartialOrder N] [AddCommMonoid N] [Module R₂ N] {Q : QuadraticMap R₂ M N} (hQ : Q.PosDef)
Goal: Q.Anisotropic
-- QuadraticMap.posDef_of_nonneg
theorem posDef_of_nonneg {Q : QuadraticMap R₂ M N} (h : ∀ x, 0 ≤ Q x) (h0 : Q.Anisotropic) :
    PosDef Q 
Arguments: {M : Type u_4} {N : Type u_5} {R₂ : Type u} [CommSemiring R₂] [AddCommMonoid M] [Module R₂ M] [PartialOrder N] [AddCommMonoid N] [Module R₂ N] {Q : QuadraticMap R₂ M N} (h : ∀ (x : M), 0 ≤ Q x) (h0 : Q.Anisotropic)
Goal: Q.PosDef
-- QuadraticMap.posDef_iff_nonneg
theorem posDef_iff_nonneg {Q : QuadraticMap R₂ M N} : PosDef Q ↔ (∀ x, 0 ≤ Q x) ∧ Q.Anisotropic 
Arguments: {M : Type u_4} {N : Type u_5} {R₂ : Type u} [CommSemiring R₂] [AddCommMonoid M] [Module R₂ M] [PartialOrder N] [AddCommMonoid N] [Module R₂ N] {Q : QuadraticMap R₂ M N}
Goal: Q.PosDef ↔ (∀ (x : M), 0 ≤ Q x) ∧ Q.Anisotropic
-- QuadraticMap.PosDef.add
theorem PosDef.add (Q Q' : QuadraticMap R₂ M N) (hQ : PosDef Q) (hQ' : PosDef Q') :
    PosDef (Q + Q') 
Arguments: {M : Type u_4} {N : Type u_5} {R₂ : Type u} [CommSemiring R₂] [AddCommMonoid M] [Module R₂ M] [PartialOrder N] [AddCommMonoid N] [Module R₂ N] [CovariantClass N N (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (Q Q' : QuadraticMap R₂ M N) (hQ : Q.PosDef) (hQ' : Q'.PosDef)
Goal: (Q + Q').PosDef
-- QuadraticMap.linMulLinSelfPosDef
theorem linMulLinSelfPosDef {R} [LinearOrderedCommRing R] [Module R M] [LinearOrderedSemiring A]
    [ExistsAddOfLE A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (f : M →ₗ[R] A)
    (hf : LinearMap.ker f = ⊥) : PosDef (linMulLin (A := A) f f) 
Arguments: {M : Type u_4} {A : Type u_7} [AddCommMonoid M] {R : Type u_9} [LinearOrderedCommRing R] [Module R M] [LinearOrderedSemiring A] [ExistsAddOfLE A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (f : M →ₗ[R] A) (hf : LinearMap.ker f = ⊥)
Goal: (QuadraticMap.linMulLin f f).PosDef
-- QuadraticMap.toMatrix'_smul
theorem QuadraticMap.toMatrix'_smul (a : R) (Q : QuadraticMap R (n → R) R) :
    (a • Q).toMatrix' = a • Q.toMatrix' 
Arguments: {R : Type u_3} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R] [Invertible 2] (a : R) (Q : QuadraticMap R (n → R) R)
Goal: (a • Q).toMatrix' = a • Q.toMatrix'
-- QuadraticMap.isSymm_toMatrix'
theorem QuadraticMap.isSymm_toMatrix' (Q : QuadraticMap R (n → R) R) : Q.toMatrix'.IsSymm 
Arguments: {R : Type u_3} {n : Type w} [Fintype n] [DecidableEq n] [CommRing R] [Invertible 2] (Q : QuadraticMap R (n → R) R)
Goal: Q.toMatrix'.IsSymm
-- QuadraticMap.toMatrix'_comp
@[simp]
theorem toMatrix'_comp (Q : QuadraticMap R (m → R) R) (f : (n → R) →ₗ[R] m → R) :
    (Q.comp f).toMatrix' = (LinearMap.toMatrix' f)ᵀ * Q.toMatrix' * (LinearMap.toMatrix' f) 
Arguments: {R : Type u_3} {n : Type w} [Fintype n] [CommRing R] [DecidableEq n] [Invertible 2] {m : Type w} [DecidableEq m] [Fintype m] (Q : QuadraticMap R (m → R) R) (f : (n → R) →ₗ[R] m → R)
Goal: (Q.comp f).toMatrix' = (LinearMap.toMatrix' f).transpose * Q.toMatrix' * LinearMap.toMatrix' f
-- QuadraticMap.discr_smul
theorem discr_smul (a : R) : (a • Q).discr = a ^ Fintype.card n * Q.discr 
Arguments: {R : Type u_3} {n : Type w} [Fintype n] [CommRing R] [DecidableEq n] [Invertible 2] {Q : QuadraticMap R (n → R) R} (a : R)
Goal: (a • Q).discr = a ^ Fintype.card n * Q.discr
-- QuadraticMap.discr_comp
theorem discr_comp (f : (n → R) →ₗ[R] n → R) :
    (Q.comp f).discr = f.toMatrix'.det * f.toMatrix'.det * Q.discr 
Arguments: {R : Type u_3} {n : Type w} [Fintype n] [CommRing R] [DecidableEq n] [Invertible 2] {Q : QuadraticMap R (n → R) R} (f : (n → R) →ₗ[R] n → R)
Goal: (Q.comp f).discr = (LinearMap.toMatrix' f).det * (LinearMap.toMatrix' f).det * Q.discr
-- LinearMap.BilinForm.separatingLeft_of_anisotropic
theorem separatingLeft_of_anisotropic {B : BilinForm R M} (hB : B.toQuadraticMap.Anisotropic) :
    B.SeparatingLeft 
Arguments: {R : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (hB : (LinearMap.BilinMap.toQuadraticMap B).Anisotropic)
Goal: LinearMap.SeparatingLeft B
-- LinearMap.BilinForm.exists_bilinForm_self_ne_zero
theorem exists_bilinForm_self_ne_zero [htwo : Invertible (2 : R)] {B : BilinForm R M}
    (hB₁ : B ≠ 0) (hB₂ : B.IsSymm) : ∃ x, ¬B.IsOrtho x x 
Arguments: {R : Type u_3} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [htwo : Invertible 2] {B : LinearMap.BilinForm R M} (hB₁ : B ≠ 0) (hB₂ : LinearMap.IsSymm B)
Goal: ∃ x, ¬LinearMap.IsOrtho B x x
-- LinearMap.BilinForm.exists_orthogonal_basis
theorem exists_orthogonal_basis [hK : Invertible (2 : K)] {B : LinearMap.BilinForm K V}
    (hB₂ : B.IsSymm) : ∃ v : Basis (Fin (finrank K V)) K V, B.IsOrthoᵢ v 
Arguments: {V : Type u} {K : Type v} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] [hK : Invertible 2] {B : LinearMap.BilinForm K V} (hB₂ : LinearMap.IsSymm B)
Goal: ∃ v, LinearMap.IsOrthoᵢ B ⇑v
-- QuadraticMap.basisRepr_apply
@[simp]
theorem basisRepr_apply [Fintype ι] {v : Basis ι R M} (Q : QuadraticMap R M N) (w : ι → R) :
    Q.basisRepr v w = Q (∑ i : ι, w i • v i) 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {ι : Type u_8} [Fintype ι] {v : Basis ι R M} (Q : QuadraticMap R M N) (w : ι → R)
Goal: (Q.basisRepr v) w = Q (∑ i : ι, w i • v i)
-- QuadraticMap.weightedSumSquares_apply
@[simp]
theorem weightedSumSquares_apply [Monoid S] [DistribMulAction S R] [SMulCommClass S R R]
    (w : ι → S) (v : ι → R) :
    weightedSumSquares R w v = ∑ i : ι, w i • (v i * v i) 
Arguments: {S : Type u_1} {R : Type u_3} [CommSemiring R] {ι : Type u_8} [Fintype ι] [Monoid S] [DistribMulAction S R] [SMulCommClass S R R] (w : ι → S) (v : ι → R)
Goal: (QuadraticMap.weightedSumSquares R w) v = ∑ i : ι, w i • (v i * v i)
-- QuadraticMap.basisRepr_eq_of_iIsOrtho
theorem basisRepr_eq_of_iIsOrtho {R M} [CommRing R] [AddCommGroup M] [Module R M]
    [Invertible (2 : R)] (Q : QuadraticMap R M R) (v : Basis ι R M)
    (hv₂ : (associated (R := R) Q).IsOrthoᵢ v) :
    Q.basisRepr v = weightedSumSquares _ fun i => Q (v i) 
Arguments: {ι : Type u_8} [Fintype ι] {R : Type u_9} {M : Type u_10} [CommRing R] [AddCommGroup M] [Module R M] [Invertible 2] (Q : QuadraticMap R M R) (v : Basis ι R M) (hv₂ : LinearMap.IsOrthoᵢ (QuadraticMap.associated Q) ⇑v)
Goal: Q.basisRepr v = QuadraticMap.weightedSumSquares R fun i => Q (v i)
-- LinearMap.pi_apply
@[simp]
theorem pi_apply (f : (i : ι) → M₂ →ₗ[R] φ i) (c : M₂) (i : ι) : pi f c i = f i c 
Arguments: {R : Type u} {M₂ : Type w} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →ₗ[R] φ i) (c : M₂) (i : ι)
Goal: (LinearMap.pi f) c i = (f i) c
-- LinearMap.ker_pi
theorem ker_pi (f : (i : ι) → M₂ →ₗ[R] φ i) : ker (pi f) = ⨅ i : ι, ker (f i) 
Arguments: {R : Type u} {M₂ : Type w} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →ₗ[R] φ i)
Goal: LinearMap.ker (LinearMap.pi f) = ⨅ i, LinearMap.ker (f i)
-- LinearMap.pi_eq_zero
theorem pi_eq_zero (f : (i : ι) → M₂ →ₗ[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 
Arguments: {R : Type u} {M₂ : Type w} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →ₗ[R] φ i)
Goal: LinearMap.pi f = 0 ↔ ∀ (i : ι), f i = 0
-- LinearMap.pi_zero
theorem pi_zero : pi (fun i => 0 : (i : ι) → M₂ →ₗ[R] φ i) = 0 
Arguments: {R : Type u} {M₂ : Type w} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)]
Goal: (LinearMap.pi fun i => 0) = 0
-- LinearMap.pi_comp
theorem pi_comp (f : (i : ι) → M₂ →ₗ[R] φ i) (g : M₃ →ₗ[R] M₂) :
    (pi f).comp g = pi fun i => (f i).comp g 
Arguments: {R : Type u} {M₂ : Type w} {M₃ : Type y} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →ₗ[R] φ i) (g : M₃ →ₗ[R] M₂)
Goal: LinearMap.pi f ∘ₗ g = LinearMap.pi fun i => f i ∘ₗ g
-- LinearMap.coe_proj
@[simp]
theorem coe_proj (i : ι) : ⇑(proj i : ((i : ι) → φ i) →ₗ[R] φ i) = Function.eval i 
Arguments: {R : Type u} {ι : Type x} [Semiring R] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι)
Goal: ⇑(LinearMap.proj i) = Function.eval i
-- LinearMap.proj_apply
theorem proj_apply (i : ι) (b : (i : ι) → φ i) : (proj i : ((i : ι) → φ i) →ₗ[R] φ i) b = b i 
Arguments: {R : Type u} {ι : Type x} [Semiring R] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (i : ι) (b : (i : ι) → φ i)
Goal: (LinearMap.proj i) b = b i
-- LinearMap.proj_pi
theorem proj_pi (f : (i : ι) → M₂ →ₗ[R] φ i) (i : ι) : (proj i).comp (pi f) = f i 
Arguments: {R : Type u} {M₂ : Type w} {ι : Type x} [Semiring R] [AddCommMonoid M₂] [Module R M₂] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (f : (i : ι) → M₂ →ₗ[R] φ i) (i : ι)
Goal: LinearMap.proj i ∘ₗ LinearMap.pi f = f i
-- LinearMap.iInf_ker_proj
theorem iInf_ker_proj : (⨅ i, ker (proj i : ((i : ι) → φ i) →ₗ[R] φ i) :
    Submodule R ((i : ι) → φ i)) = ⊥ 
Arguments: {R : Type u} {ι : Type x} [Semiring R] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)]
Goal: ⨅ i, LinearMap.ker (LinearMap.proj i) = ⊥
-- LinearMap.compLeft_apply
@[simps]
protected def compLeft (f : M₂ →ₗ[R] M₃) (I : Type*) : (I → M₂) →ₗ[R] I → M₃ 
Arguments: {R : Type u} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid M₃] [Module R M₃] (f : M₂ →ₗ[R] M₃) (I : Type u_1) (h : I → M₂)
Goal: ∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) a
-- LinearMap.apply_single
theorem apply_single [AddCommMonoid M] [Module R M] [DecidableEq ι] (f : (i : ι) → φ i →ₗ[R] M)
    (i j : ι) (x : φ i) : f j (Pi.single i x j) = (Pi.single i (f i x) : ι → M) j 
Arguments: {R : Type u} {M : Type v} {ι : Type x} [Semiring R] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] [AddCommMonoid M] [Module R M] [DecidableEq ι] (f : (i : ι) → φ i →ₗ[R] M) (i j : ι) (x : φ i)
Goal: (f j) (Pi.single i x j) = Pi.single i ((f i) x) j
-- LinearMap.coe_single
@[simp]
theorem coe_single [DecidableEq ι] (i : ι) : ⇑(single i : φ i →ₗ[R] (i : ι) → φ i) = Pi.single i 
Arguments: {R : Type u} {ι : Type x} [Semiring R] {φ : ι → Type i} [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] [DecidableEq ι] (i : ι)
Goal: ⇑(LinearMap.single i) = Pi.single i
-- LinearMap.lsum_symm_apply
@[simps symm_apply]
def lsum (S) [AddCommMonoid M] [Module R M] [Fintype ι] [DecidableEq ι] [Semiring S] [Module S M]
    [SMulCommClass R S M] : ((i : ι) → φ i →ₗ[R] M) ≃ₗ[S] ((i : ι) → φ i) →ₗ[R] M 
Arguments: (R : Type u) {M : Type v} {ι : Type x} [Semiring R] (φ : ι → Type i) [(i : ι) → AddCommMonoid (φ i)] [(i : ι) → Module R (φ i)] (S : Type u_1) [AddCommMonoid M] [Module R M] [Fintype ι] [DecidableEq ι] [Semiring S] [Module S M] [SMulCommClass R S M] (f : ((i : ι) → φ i) →ₗ[R] M) (i : ι)
Goal: (LinearMap.lsum R φ S).symm f i = f ∘ₗ LinearMap.single i