ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- QuadraticMap.map_zero
protected theorem map_zero : Q 0 = 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)
Goal: Q 0 = 0
-- QuadraticMap.map_smul_of_tower
theorem map_smul_of_tower [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M]
    [Module S N] [IsScalarTower S R N] (a : S)
    (x : M) : Q (a • x) = (a * a) • Q x 
Arguments: {S : Type u_1} {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) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] [Module S N] [IsScalarTower S R N] (a : S) (x : M)
Goal: Q (a • x) = (a * a) • Q x
-- QuadraticMap.map_neg
@[simp]
theorem map_neg (x : M) : Q (-x) = Q x 
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] (Q : QuadraticMap R M N) (x : M)
Goal: Q (-x) = Q x
-- QuadraticMap.map_sub
theorem map_sub (x y : M) : Q (x - y) = Q (y - x) 
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] (Q : QuadraticMap R M N) (x y : M)
Goal: Q (x - y) = Q (y - x)
-- QuadraticMap.polar_zero_left
@[simp]
theorem polar_zero_left (y : M) : polar Q 0 y = 0 
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] (Q : QuadraticMap R M N) (y : M)
Goal: QuadraticMap.polar (⇑Q) 0 y = 0
-- QuadraticMap.polar_add_left
@[simp]
theorem polar_add_left (x x' y : M) : polar Q (x + x') y = polar Q x y + polar Q x' y 
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] (Q : QuadraticMap R M N) (x x' y : M)
Goal: QuadraticMap.polar (⇑Q) (x + x') y = QuadraticMap.polar (⇑Q) x y + QuadraticMap.polar (⇑Q) x' y
-- QuadraticMap.polar_smul_left
@[simp]
theorem polar_smul_left (a : R) (x y : M) : polar Q (a • x) y = a • polar Q x y 
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] (Q : QuadraticMap R M N) (a : R) (x y : M)
Goal: QuadraticMap.polar (⇑Q) (a • x) y = a • QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.polar_neg_left
@[simp]
theorem polar_neg_left (x y : M) : polar Q (-x) y = -polar Q x y 
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] (Q : QuadraticMap R M N) (x y : M)
Goal: QuadraticMap.polar (⇑Q) (-x) y = -QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.polar_sub_left
@[simp]
theorem polar_sub_left (x x' y : M) : polar Q (x - x') y = polar Q x y - polar Q x' y 
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] (Q : QuadraticMap R M N) (x x' y : M)
Goal: QuadraticMap.polar (⇑Q) (x - x') y = QuadraticMap.polar (⇑Q) x y - QuadraticMap.polar (⇑Q) x' y
-- QuadraticMap.polar_zero_right
@[simp]
theorem polar_zero_right (y : M) : polar Q y 0 = 0 
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] (Q : QuadraticMap R M N) (y : M)
Goal: QuadraticMap.polar (⇑Q) y 0 = 0
-- QuadraticMap.polar_add_right
@[simp]
theorem polar_add_right (x y y' : M) : polar Q x (y + y') = polar Q x y + polar Q x y' 
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] (Q : QuadraticMap R M N) (x y y' : M)
Goal: QuadraticMap.polar (⇑Q) x (y + y') = QuadraticMap.polar (⇑Q) x y + QuadraticMap.polar (⇑Q) x y'
-- QuadraticMap.polar_smul_right
@[simp]
theorem polar_smul_right (a : R) (x y : M) : polar Q x (a • y) = a • polar Q x y 
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] (Q : QuadraticMap R M N) (a : R) (x y : M)
Goal: QuadraticMap.polar (⇑Q) x (a • y) = a • QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.polar_neg_right
@[simp]
theorem polar_neg_right (x y : M) : polar Q x (-y) = -polar Q x y 
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] (Q : QuadraticMap R M N) (x y : M)
Goal: QuadraticMap.polar (⇑Q) x (-y) = -QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.polar_sub_right
@[simp]
theorem polar_sub_right (x y y' : M) : polar Q x (y - y') = polar Q x y - polar Q x y' 
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] (Q : QuadraticMap R M N) (x y y' : M)
Goal: QuadraticMap.polar (⇑Q) x (y - y') = QuadraticMap.polar (⇑Q) x y - QuadraticMap.polar (⇑Q) x y'
-- QuadraticMap.polar_self
@[simp]
theorem polar_self (x : M) : polar Q x x = 2 • Q x 
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] (Q : QuadraticMap R M N) (x : M)
Goal: QuadraticMap.polar (⇑Q) x x = 2 • Q x
-- QuadraticMap.polarBilin_apply_apply
@[simps!]
def polarBilin : BilinMap 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] (Q : QuadraticMap R M N) (m y : M)
Goal: (Q.polarBilin m) y = QuadraticMap.polar (⇑Q) m y
-- QuadraticMap.polar_smul_left_of_tower
@[simp]
theorem polar_smul_left_of_tower (a : S) (x y : M) : polar Q (a • x) y = a • 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] (Q : QuadraticMap R M N) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] [Module S N] [IsScalarTower S R N] (a : S) (x y : M)
Goal: QuadraticMap.polar (⇑Q) (a • x) y = a • QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.polar_smul_right_of_tower
@[simp]
theorem polar_smul_right_of_tower (a : S) (x y : M) : polar Q x (a • y) = a • 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] (Q : QuadraticMap R M N) [CommSemiring S] [Algebra S R] [Module S M] [IsScalarTower S R M] [Module S N] [IsScalarTower S R N] (a : S) (x y : M)
Goal: QuadraticMap.polar (⇑Q) x (a • y) = a • QuadraticMap.polar (⇑Q) x y
-- QuadraticMap.ofPolar_apply
@[simps]
def ofPolar (toFun : M → N) (toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x)
    (polar_add_left : ∀ x x' y : M, polar toFun (x + x') y = polar toFun x y + polar toFun x' y)
    (polar_smul_left : ∀ (a : R) (x y : M), polar toFun (a • x) y = a • polar toFun x y) :
    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] (toFun : M → N) (toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x) (polar_add_left :
  ∀ (x x' y : M), QuadraticMap.polar toFun (x + x') y = QuadraticMap.polar toFun x y + QuadraticMap.polar toFun x' y) (polar_smul_left : ∀ (a : R) (x y : M), QuadraticMap.polar toFun (a • x) y = a • QuadraticMap.polar toFun x y)
Goal: ∀ (a : M), (QuadraticMap.ofPolar toFun toFun_smul polar_add_left polar_smul_left) a = toFun a
-- QuadraticMap.choose_exists_companion
theorem choose_exists_companion : Q.exists_companion.choose = polarBilin Q 
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] (Q : QuadraticMap R M N)
Goal: ⋯.choose = Q.polarBilin
-- QuadraticMap.map_sum
protected theorem map_sum {ι} [DecidableEq ι] (Q : QuadraticMap R M N) (s : Finset ι) (f : ι → M) :
    Q (∑ i ∈ s, f i) = ∑ i ∈ s, Q (f i) +
      ∑ ij in s.sym2.filter (¬ ·.IsDiag),
        Sym2.lift ⟨fun i j => polar Q (f i) (f j), fun _ _ => polar_comm _ _ _⟩ ij 
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] {ι : Type u_8} [DecidableEq ι] (Q : QuadraticMap R M N) (s : Finset ι) (f : ι → M)
Goal: Q (∑ i ∈ s, f i) =
  ∑ i ∈ s, Q (f i) +
    ∑ ij ∈ Finset.filter (fun x => ¬x.IsDiag) s.sym2, Sym2.lift ⟨fun i j => QuadraticMap.polar (⇑Q) (f i) (f j), ⋯⟩ ij
-- QuadraticMap.map_sum'
protected theorem map_sum' {ι} (Q : QuadraticMap R M N) (s : Finset ι) (f : ι → M) :
    Q (∑ i ∈ s, f i) =
      ∑ ij in s.sym2,
        Sym2.lift ⟨fun i j => polar Q (f i) (f j), fun _ _ => polar_comm _ _ _⟩ ij
      - ∑ i ∈ s, Q (f i) 
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] {ι : Type u_8} (Q : QuadraticMap R M N) (s : Finset ι) (f : ι → M)
Goal: Q (∑ i ∈ s, f i) = ∑ ij ∈ s.sym2, Sym2.lift ⟨fun i j => QuadraticMap.polar (⇑Q) (f i) (f j), ⋯⟩ ij - ∑ i ∈ s, Q (f i)
-- QuadraticMap.coeFn_smul
@[simp]
theorem coeFn_smul (a : S) (Q : QuadraticMap R M N) : ⇑(a • Q) = a • ⇑Q 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Monoid S] [DistribMulAction S N] [SMulCommClass S R N] (a : S) (Q : QuadraticMap R M N)
Goal: ⇑(a • Q) = a • ⇑Q
-- QuadraticMap.smul_apply
@[simp]
theorem smul_apply (a : S) (Q : QuadraticMap R M N) (x : M) : (a • Q) x = a • Q x 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Monoid S] [DistribMulAction S N] [SMulCommClass S R N] (a : S) (Q : QuadraticMap R M N) (x : M)
Goal: (a • Q) x = a • Q x
-- QuadraticMap.coeFn_zero
@[simp]
theorem coeFn_zero : ⇑(0 : QuadraticMap R M N) = 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]
Goal: ⇑0 = 0
-- QuadraticMap.zero_apply
@[simp]
theorem zero_apply (x : M) : (0 : QuadraticMap R M N) 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] (x : M)
Goal: 0 x = 0
-- QuadraticMap.coeFn_add
@[simp]
theorem coeFn_add (Q Q' : QuadraticMap R M N) : ⇑(Q + Q') = Q + Q' 
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 Q' : QuadraticMap R M N)
Goal: ⇑(Q + Q') = ⇑Q + ⇑Q'
-- QuadraticMap.add_apply
@[simp]
theorem add_apply (Q Q' : QuadraticMap R M N) (x : M) : (Q + Q') x = Q x + Q' 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 Q' : QuadraticMap R M N) (x : M)
Goal: (Q + Q') x = Q x + Q' x
-- QuadraticMap.coeFnAddMonoidHom_apply
@[simps apply]
def coeFnAddMonoidHom : QuadraticMap R M N →+ M → N 
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]
Goal: ∀ (a : QuadraticMap R M N) (a_1 : M), QuadraticMap.coeFnAddMonoidHom a a_1 = a a_1
-- QuadraticMap.evalAddMonoidHom_apply
@[simps! apply]
def evalAddMonoidHom (m : M) : QuadraticMap R M N →+ N 
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] (m : M)
Goal: ∀ (a : QuadraticMap R M N), (QuadraticMap.evalAddMonoidHom m) a = a m
-- QuadraticMap.coeFn_sum
@[simp]
theorem coeFn_sum {ι : Type*} (Q : ι → QuadraticMap R M N) (s : Finset ι) :
    ⇑(∑ i ∈ s, Q i) = ∑ i ∈ s, ⇑(Q 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} (Q : ι → QuadraticMap R M N) (s : Finset ι)
Goal: ⇑(∑ i ∈ s, Q i) = ∑ i ∈ s, ⇑(Q i)
-- QuadraticMap.sum_apply
@[simp]
theorem sum_apply {ι : Type*} (Q : ι → QuadraticMap R M N) (s : Finset ι) (x : M) :
    (∑ i ∈ s, Q i) x = ∑ i ∈ s, Q i 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] {ι : Type u_8} (Q : ι → QuadraticMap R M N) (s : Finset ι) (x : M)
Goal: (∑ i ∈ s, Q i) x = ∑ i ∈ s, (Q i) x
-- QuadraticMap.coeFn_neg
@[simp]
theorem coeFn_neg (Q : QuadraticMap R M N) : ⇑(-Q) = -Q 
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)
Goal: ⇑(-Q) = -⇑Q
-- QuadraticMap.neg_apply
@[simp]
theorem neg_apply (Q : QuadraticMap R M N) (x : M) : (-Q) x = -Q x 
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 : M)
Goal: (-Q) x = -Q x
-- QuadraticMap.coeFn_sub
@[simp]
theorem coeFn_sub (Q Q' : QuadraticMap R M N) : ⇑(Q - Q') = Q - Q' 
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 Q' : QuadraticMap R M N)
Goal: ⇑(Q - Q') = ⇑Q - ⇑Q'
-- QuadraticMap.sub_apply
@[simp]
theorem sub_apply (Q Q' : QuadraticMap R M N) (x : M) : (Q - Q') x = Q x - Q' x 
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 Q' : QuadraticMap R M N) (x : M)
Goal: (Q - Q') x = Q x - Q' x
-- QuadraticMap.restrictScalars_apply
@[simps!]
def restrictScalars (Q : QuadraticMap R M N) : QuadraticMap S M N 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [CommSemiring S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module S M] [Module S N] [Algebra S R] [IsScalarTower S R M] [IsScalarTower S R N] (Q : QuadraticMap R M N) (x : M)
Goal: Q.restrictScalars x = Q x
-- QuadraticMap.comp_apply
@[simp]
theorem comp_apply (Q : QuadraticMap R N P) (f : M →ₗ[R] N) (x : M) : (Q.comp f) x = Q (f x) 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} {P : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] (Q : QuadraticMap R N P) (f : M →ₗ[R] N) (x : M)
Goal: (Q.comp f) x = Q (f x)
-- LinearMap.compQuadraticMap_apply
@[simps (config := { simpRhs := true })]
def _root_.LinearMap.compQuadraticMap (f : N →ₗ[R] P) (Q : QuadraticMap R M N) :
    QuadraticMap R M P 
Arguments: {R : Type u_3} {M : Type u_4} {N : Type u_5} {P : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] (f : N →ₗ[R] P) (Q : QuadraticMap R M N) (x : M)
Goal: (f.compQuadraticMap Q) x = f (Q x)
-- LinearMap.compQuadraticMap'_apply
@[simps! (config := { simpRhs := true })]
def _root_.LinearMap.compQuadraticMap' [CommSemiring S] [Algebra S R] [Module S N] [Module S M]
    [IsScalarTower S R N] [IsScalarTower S R M] [Module S P]
    (f : N →ₗ[S] P) (Q : QuadraticMap R M N) : QuadraticMap S M P 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} {P : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [CommSemiring S] [Algebra S R] [Module S N] [Module S M] [IsScalarTower S R N] [IsScalarTower S R M] [Module S P] (f : N →ₗ[S] P) (Q : QuadraticMap R M N) (x : M)
Goal: (f.compQuadraticMap' Q) x = f (Q x)
-- QuadraticMap.linMulLin_apply
@[simp]
theorem linMulLin_apply (f g : M →ₗ[R] A) (x) : linMulLin f g x = f x * g x 
Arguments: {R : Type u_3} {M : Type u_4} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [AddCommMonoid M] [Module R M] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (f g : M →ₗ[R] A) (x : M)
Goal: (QuadraticMap.linMulLin f g) x = f x * g x
-- QuadraticMap.add_linMulLin
@[simp]
theorem add_linMulLin (f g h : M →ₗ[R] A) : linMulLin (f + g) h = linMulLin f h + linMulLin g h 
Arguments: {R : Type u_3} {M : Type u_4} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [AddCommMonoid M] [Module R M] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (f g h : M →ₗ[R] A)
Goal: QuadraticMap.linMulLin (f + g) h = QuadraticMap.linMulLin f h + QuadraticMap.linMulLin g h
-- QuadraticMap.linMulLin_add
@[simp]
theorem linMulLin_add (f g h : M →ₗ[R] A) : linMulLin f (g + h) = linMulLin f g + linMulLin f h 
Arguments: {R : Type u_3} {M : Type u_4} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [AddCommMonoid M] [Module R M] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (f g h : M →ₗ[R] A)
Goal: QuadraticMap.linMulLin f (g + h) = QuadraticMap.linMulLin f g + QuadraticMap.linMulLin f h
-- QuadraticMap.linMulLin_comp
@[simp]
theorem linMulLin_comp (f g : M →ₗ[R] A) (h : N' →ₗ[R] M) :
    (linMulLin f g).comp h = linMulLin (f.comp h) (g.comp h) 
Arguments: {R : Type u_3} {M : Type u_4} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [AddCommMonoid M] [Module R M] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {N' : Type u_8} [AddCommMonoid N'] [Module R N'] (f g : M →ₗ[R] A) (h : N' →ₗ[R] M)
Goal: (QuadraticMap.linMulLin f g).comp h = QuadraticMap.linMulLin (f ∘ₗ h) (g ∘ₗ h)
-- QuadraticMap.sq_apply
@[simps!]
def sq : QuadraticMap R A A 
Arguments: {R : Type u_3} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a : A)
Goal: QuadraticMap.sq a = a * a
-- QuadraticMap.proj_apply
@[simp]
theorem proj_apply (i j : n) (x : n → A) : proj (R := R) i j x = x i * x j 
Arguments: {R : Type u_3} {A : Type u_7} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {n : Type u_9} (i j : n) (x : n → A)
Goal: (QuadraticMap.proj i j) x = x i * x j
-- LinearMap.BilinMap.toQuadraticMap_apply
@[simp]
theorem toQuadraticMap_apply (B : BilinMap R M N) (x : M) : B.toQuadraticMap x = B x 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] (B : LinearMap.BilinMap R M N) (x : M)
Goal: B.toQuadraticMap x = (B x) x
-- LinearMap.BilinMap.toQuadraticMap_comp_same
theorem toQuadraticMap_comp_same (B : BilinMap R M N) (f : N' →ₗ[R] M) :
    BilinMap.toQuadraticMap (B.compl₁₂ f f) = B.toQuadraticMap.comp f 
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] {N' : Type u_8} [AddCommMonoid N'] [Module R N'] (B : LinearMap.BilinMap R M N) (f : N' →ₗ[R] M)
Goal: LinearMap.BilinMap.toQuadraticMap (LinearMap.compl₁₂ B f f) = B.toQuadraticMap.comp f
-- LinearMap.BilinMap.toQuadraticMap_zero
@[simp]
theorem toQuadraticMap_zero : (0 : BilinMap R M N).toQuadraticMap = 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]
Goal: LinearMap.BilinMap.toQuadraticMap 0 = 0
-- LinearMap.BilinMap.toQuadraticMap_add
@[simp]
theorem toQuadraticMap_add (B₁ B₂ : BilinMap R M N) :
    (B₁ + B₂).toQuadraticMap = B₁.toQuadraticMap + B₂.toQuadraticMap 
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] (B₁ B₂ : LinearMap.BilinMap R M N)
Goal: (B₁ + B₂).toQuadraticMap = B₁.toQuadraticMap + B₂.toQuadraticMap
-- LinearMap.BilinMap.toQuadraticMap_smul
@[simp]
theorem toQuadraticMap_smul [Monoid S] [DistribMulAction S N] [SMulCommClass S R N]
    [SMulCommClass R S N] (a : S)
    (B : BilinMap R M N) : (a • B).toQuadraticMap = a • B.toQuadraticMap 
Arguments: {S : Type u_1} {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Monoid S] [DistribMulAction S N] [SMulCommClass S R N] [SMulCommClass R S N] (a : S) (B : LinearMap.BilinMap R M N)
Goal: (a • B).toQuadraticMap = a • B.toQuadraticMap
-- LinearMap.BilinMap.toQuadraticMapAddMonoidHom_apply
@[simps]
def toQuadraticMapAddMonoidHom : (BilinMap R M N) →+ QuadraticMap R M N 
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] (B : LinearMap.BilinMap R M N)
Goal: (LinearMap.BilinMap.toQuadraticMapAddMonoidHom R M) B = B.toQuadraticMap
-- LinearMap.BilinMap.toQuadraticMapLinearMap_apply_apply
@[simps!]
def toQuadraticMapLinearMap [Semiring S] [Module S N] [SMulCommClass S R N] [SMulCommClass R S N] :
    (BilinMap R M N) →ₗ[S] QuadraticMap R M N 
Arguments: (S : Type u_1) (R : Type u_3) (M : Type u_4) {N : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Module S N] [SMulCommClass S R N] [SMulCommClass R S N] (B : LinearMap.BilinMap R M N) (x : M)
Goal: ((LinearMap.BilinMap.toQuadraticMapLinearMap S R M) B) x = (B x) x
-- LinearMap.BilinMap.toQuadraticMap_list_sum
@[simp]
theorem toQuadraticMap_list_sum (B : List (BilinMap R M N)) :
    B.sum.toQuadraticMap = (B.map toQuadraticMap).sum 
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] (B : List (LinearMap.BilinMap R M N))
Goal: B.sum.toQuadraticMap = (List.map LinearMap.BilinMap.toQuadraticMap B).sum
-- LinearMap.BilinMap.toQuadraticMap_multiset_sum
@[simp]
theorem toQuadraticMap_multiset_sum (B : Multiset (BilinMap R M N)) :
    B.sum.toQuadraticMap = (B.map toQuadraticMap).sum 
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] (B : Multiset (LinearMap.BilinMap R M N))
Goal: B.sum.toQuadraticMap = (Multiset.map LinearMap.BilinMap.toQuadraticMap B).sum
-- LinearMap.BilinMap.toQuadraticMap_sum
@[simp]
theorem toQuadraticMap_sum {ι : Type*} (s : Finset ι) (B : ι → (BilinMap R M N)) :
    (∑ i ∈ s, B i).toQuadraticMap = ∑ i ∈ s, (B i).toQuadraticMap 
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_9} (s : Finset ι) (B : ι → LinearMap.BilinMap R M N)
Goal: (∑ i ∈ s, B i).toQuadraticMap = ∑ i ∈ s, (B i).toQuadraticMap
-- LinearMap.BilinMap.toQuadraticMap_eq_zero
@[simp]
theorem toQuadraticMap_eq_zero {B : BilinMap R M N} :
    B.toQuadraticMap = 0 ↔ B.IsAlt 
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] {B : LinearMap.BilinMap R M N}
Goal: B.toQuadraticMap = 0 ↔ LinearMap.IsAlt B
-- LinearMap.BilinMap.toQuadraticMap_neg
@[simp]
theorem toQuadraticMap_neg (B : BilinMap R M N) : (-B).toQuadraticMap = -B.toQuadraticMap 
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] (B : LinearMap.BilinMap R M N)
Goal: (-B).toQuadraticMap = -B.toQuadraticMap
-- LinearMap.BilinMap.toQuadraticMap_sub
@[simp]
theorem toQuadraticMap_sub (B₁ B₂ : BilinMap R M N) :
    (B₁ - B₂).toQuadraticMap = B₁.toQuadraticMap - B₂.toQuadraticMap 
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] (B₁ B₂ : LinearMap.BilinMap R M N)
Goal: (B₁ - B₂).toQuadraticMap = B₁.toQuadraticMap - B₂.toQuadraticMap
-- LinearMap.BilinMap.polar_toQuadraticMap
theorem polar_toQuadraticMap (x y : M) : polar (toQuadraticMap B) x y = B x y + B y x 
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] {B : LinearMap.BilinMap R M N} (x y : M)
Goal: QuadraticMap.polar (⇑B.toQuadraticMap) x y = (B x) y + (B y) x
-- LinearMap.BilinMap.polarBilin_toQuadraticMap
theorem polarBilin_toQuadraticMap : polarBilin (toQuadraticMap B) = B + flip B 
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] {B : LinearMap.BilinMap R M N}
Goal: B.toQuadraticMap.polarBilin = B + LinearMap.flip B
-- QuadraticMap.toQuadraticMap_polarBilin
@[simp] theorem _root_.QuadraticMap.toQuadraticMap_polarBilin (Q : QuadraticMap R M N) :
    toQuadraticMap (polarBilin Q) = 2 • Q 
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] (Q : QuadraticMap R M N)
Goal: Q.polarBilin.toQuadraticMap = 2 • Q