ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Basis.ext_multilinear
theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
    (e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g 
Arguments: {R : Type u_1} {ι : Type u_2} {M₂ : Type u_4} {M₃ : Type u_5} [CommSemiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [Finite ι] {f g : MultilinearMap R (fun x => M₂) M₃} {ι₁ : Type u_6} (e : Basis ι₁ R M₂) (h : ∀ (v : ι → ι₁), (f fun i => e (v i)) = g fun i => e (v i))
Goal: f = g
-- CliffordAlgebra.contractLeftAux_apply_apply
@[simps!]
def contractLeftAux (d : Module.Dual R M) :
    M →ₗ[R] CliffordAlgebra Q × CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (a : M)
Goal: ∀ (a_1 : CliffordAlgebra Q × CliffordAlgebra Q),
  ((CliffordAlgebra.contractLeftAux Q d) a) a_1 = d a • a_1.1 - (CliffordAlgebra.ι Q) a * a_1.2
-- CliffordAlgebra.contractLeftAux_contractLeftAux
theorem contractLeftAux_contractLeftAux (v : M) (x : CliffordAlgebra Q) (fx : CliffordAlgebra Q) :
    contractLeftAux Q d v (ι Q v * x, contractLeftAux Q d v (x, fx)) = Q v • fx 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (v : M) (x fx : CliffordAlgebra Q)
Goal: ((CliffordAlgebra.contractLeftAux Q d) v)
    ((CliffordAlgebra.ι Q) v * x, ((CliffordAlgebra.contractLeftAux Q d) v) (x, fx)) =
  Q v • fx
-- CliffordAlgebra.contractRight_eq
theorem contractRight_eq (x : CliffordAlgebra Q) :
    contractRight (Q := Q) x d = reverse (contractLeft (R := R) (M := M) d <| reverse x) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractRight x) d =
  CliffordAlgebra.reverse ((CliffordAlgebra.contractLeft d) (CliffordAlgebra.reverse x))
-- CliffordAlgebra.contractLeft_ι_mul
theorem contractLeft_ι_mul (a : M) (b : CliffordAlgebra Q) :
    d⌋(ι Q a * b) = d a • b - ι Q a * (d⌋b) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (a : M) (b : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractLeft d) ((CliffordAlgebra.ι Q) a * b) =
  d a • b - (CliffordAlgebra.ι Q) a * (CliffordAlgebra.contractLeft d) b
-- CliffordAlgebra.contractRight_mul_ι
theorem contractRight_mul_ι (a : M) (b : CliffordAlgebra Q) :
    b * ι Q a⌊d = d a • b - b⌊d * ι Q a 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (a : M) (b : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractRight (b * (CliffordAlgebra.ι Q) a)) d =
  d a • b - (CliffordAlgebra.contractRight b) d * (CliffordAlgebra.ι Q) a
-- CliffordAlgebra.contractLeft_algebraMap_mul
theorem contractLeft_algebraMap_mul (r : R) (b : CliffordAlgebra Q) :
    d⌋(algebraMap _ _ r * b) = algebraMap _ _ r * (d⌋b) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (r : R) (b : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractLeft d) ((algebraMap R (CliffordAlgebra Q)) r * b) =
  (algebraMap R (CliffordAlgebra Q)) r * (CliffordAlgebra.contractLeft d) b
-- CliffordAlgebra.contractLeft_mul_algebraMap
theorem contractLeft_mul_algebraMap (a : CliffordAlgebra Q) (r : R) :
    d⌋(a * algebraMap _ _ r) = d⌋a * algebraMap _ _ r 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (a : CliffordAlgebra Q) (r : R)
Goal: (CliffordAlgebra.contractLeft d) (a * (algebraMap R (CliffordAlgebra Q)) r) =
  (CliffordAlgebra.contractLeft d) a * (algebraMap R (CliffordAlgebra Q)) r
-- CliffordAlgebra.contractRight_algebraMap_mul
theorem contractRight_algebraMap_mul (r : R) (b : CliffordAlgebra Q) :
    algebraMap _ _ r * b⌊d = algebraMap _ _ r * (b⌊d) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (r : R) (b : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractRight ((algebraMap R (CliffordAlgebra Q)) r * b)) d =
  (algebraMap R (CliffordAlgebra Q)) r * (CliffordAlgebra.contractRight b) d
-- CliffordAlgebra.contractRight_mul_algebraMap
theorem contractRight_mul_algebraMap (a : CliffordAlgebra Q) (r : R) :
    a * algebraMap _ _ r⌊d = a⌊d * algebraMap _ _ r 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (a : CliffordAlgebra Q) (r : R)
Goal: (CliffordAlgebra.contractRight (a * (algebraMap R (CliffordAlgebra Q)) r)) d =
  (CliffordAlgebra.contractRight a) d * (algebraMap R (CliffordAlgebra Q)) r
-- CliffordAlgebra.contractLeft_ι
@[simp]
theorem contractLeft_ι (x : M) : d⌋ι Q x = algebraMap R _ (d x) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (x : M)
Goal: (CliffordAlgebra.contractLeft d) ((CliffordAlgebra.ι Q) x) = (algebraMap R (CliffordAlgebra Q)) (d x)
-- CliffordAlgebra.contractRight_ι
@[simp]
theorem contractRight_ι (x : M) : ι Q x⌊d = algebraMap R _ (d x) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (x : M)
Goal: (CliffordAlgebra.contractRight ((CliffordAlgebra.ι Q) x)) d = (algebraMap R (CliffordAlgebra Q)) (d x)
-- CliffordAlgebra.contractLeft_algebraMap
@[simp]
theorem contractLeft_algebraMap (r : R) : d⌋algebraMap R (CliffordAlgebra Q) r = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (r : R)
Goal: (CliffordAlgebra.contractLeft d) ((algebraMap R (CliffordAlgebra Q)) r) = 0
-- CliffordAlgebra.contractRight_algebraMap
@[simp]
theorem contractRight_algebraMap (r : R) : algebraMap R (CliffordAlgebra Q) r⌊d = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M) (r : R)
Goal: (CliffordAlgebra.contractRight ((algebraMap R (CliffordAlgebra Q)) r)) d = 0
-- CliffordAlgebra.contractLeft_one
@[simp]
theorem contractLeft_one : d⌋(1 : CliffordAlgebra Q) = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M)
Goal: (CliffordAlgebra.contractLeft d) 1 = 0
-- CliffordAlgebra.contractRight_one
@[simp]
theorem contractRight_one : (1 : CliffordAlgebra Q)⌊d = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (d : Module.Dual R M)
Goal: (CliffordAlgebra.contractRight 1) d = 0
-- CliffordAlgebra.contractLeft_contractLeft
theorem contractLeft_contractLeft (x : CliffordAlgebra Q) : d⌋(d⌋x) = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractLeft d) ((CliffordAlgebra.contractLeft d) x) = 0
-- CliffordAlgebra.contractRight_contractRight
theorem contractRight_contractRight (x : CliffordAlgebra Q) : x⌊d⌊d = 0 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractRight ((CliffordAlgebra.contractRight x) d)) d = 0
-- CliffordAlgebra.contractLeft_comm
theorem contractLeft_comm (x : CliffordAlgebra Q) : d⌋(d'⌋x) = -(d'⌋(d⌋x)) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d d' : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractLeft d) ((CliffordAlgebra.contractLeft d') x) =
  -(CliffordAlgebra.contractLeft d') ((CliffordAlgebra.contractLeft d) x)
-- CliffordAlgebra.contractRight_comm
theorem contractRight_comm (x : CliffordAlgebra Q) : x⌊d⌊d' = -(x⌊d'⌊d) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (d d' : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.contractRight ((CliffordAlgebra.contractRight x) d)) d' =
  -(CliffordAlgebra.contractRight ((CliffordAlgebra.contractRight x) d')) d
-- CliffordAlgebra.changeFormAux_apply_apply
@[simps!]
def changeFormAux (B : BilinForm R M) : M →ₗ[R] CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (B : LinearMap.BilinForm R M) (a : M)
Goal: ∀ (a_1 : CliffordAlgebra Q),
  ((CliffordAlgebra.changeFormAux Q B) a) a_1 = (CliffordAlgebra.ι Q) a * a_1 - (CliffordAlgebra.contractLeft (B a)) a_1
-- CliffordAlgebra.changeFormAux_changeFormAux
theorem changeFormAux_changeFormAux (B : BilinForm R M) (v : M) (x : CliffordAlgebra Q) :
    changeFormAux Q B v (changeFormAux Q B v x) = (Q v - B v v) • x 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (B : LinearMap.BilinForm R M) (v : M) (x : CliffordAlgebra Q)
Goal: ((CliffordAlgebra.changeFormAux Q B) v) (((CliffordAlgebra.changeFormAux Q B) v) x) = (Q v - (B v) v) • x
-- CliffordAlgebra.changeForm.zero_proof
theorem changeForm.zero_proof : (0 : BilinForm R M).toQuadraticMap = Q - Q 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
Goal: LinearMap.BilinMap.toQuadraticMap 0 = Q - Q
-- CliffordAlgebra.changeForm.add_proof
theorem changeForm.add_proof : (B + B').toQuadraticMap = Q'' - Q 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' Q'' : QuadraticForm R M} {B B' : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (h' : LinearMap.BilinMap.toQuadraticMap B' = Q'' - Q')
Goal: LinearMap.BilinMap.toQuadraticMap (B + B') = Q'' - Q
-- CliffordAlgebra.changeForm.neg_proof
theorem changeForm.neg_proof : (-B).toQuadraticMap = Q - Q' 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q)
Goal: LinearMap.BilinMap.toQuadraticMap (-B) = Q - Q'
-- CliffordAlgebra.changeForm.associated_neg_proof
theorem changeForm.associated_neg_proof [Invertible (2 : R)] :
    (QuadraticMap.associated (R := R) (M := M) (-Q)).toQuadraticMap = 0 - Q 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} [Invertible 2]
Goal: (QuadraticMap.associated (-Q)).toQuadraticMap = 0 - Q
-- CliffordAlgebra.changeForm_algebraMap
@[simp]
theorem changeForm_algebraMap (r : R) : changeForm h (algebraMap R _ r) = algebraMap R _ r 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (r : R)
Goal: (CliffordAlgebra.changeForm h) ((algebraMap R (CliffordAlgebra Q)) r) = (algebraMap R (CliffordAlgebra Q')) r
-- CliffordAlgebra.changeForm_one
@[simp]
theorem changeForm_one : changeForm h (1 : CliffordAlgebra Q) = 1 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q)
Goal: (CliffordAlgebra.changeForm h) 1 = 1
-- CliffordAlgebra.changeForm_ι
@[simp]
theorem changeForm_ι (m : M) : changeForm h (ι (M := M) Q m) = ι (M := M) Q' m 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (m : M)
Goal: (CliffordAlgebra.changeForm h) ((CliffordAlgebra.ι Q) m) = (CliffordAlgebra.ι Q') m
-- CliffordAlgebra.changeForm_ι_mul
theorem changeForm_ι_mul (m : M) (x : CliffordAlgebra Q) :
    changeForm h (ι (M := M) Q m * x) = ι (M := M) Q' m * changeForm h x
    - contractLeft (Q := Q') (B m) (changeForm h x) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (m : M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.changeForm h) ((CliffordAlgebra.ι Q) m * x) =
  (CliffordAlgebra.ι Q') m * (CliffordAlgebra.changeForm h) x -
    (CliffordAlgebra.contractLeft (B m)) ((CliffordAlgebra.changeForm h) x)
-- CliffordAlgebra.changeForm_ι_mul_ι
theorem changeForm_ι_mul_ι (m₁ m₂ : M) :
    changeForm h (ι Q m₁ * ι Q m₂) = ι Q' m₁ * ι Q' m₂ - algebraMap _ _ (B m₁ m₂) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (m₁ m₂ : M)
Goal: (CliffordAlgebra.changeForm h) ((CliffordAlgebra.ι Q) m₁ * (CliffordAlgebra.ι Q) m₂) =
  (CliffordAlgebra.ι Q') m₁ * (CliffordAlgebra.ι Q') m₂ - (algebraMap R (CliffordAlgebra Q')) ((B m₁) m₂)
-- CliffordAlgebra.changeForm_contractLeft
theorem changeForm_contractLeft (d : Module.Dual R M) (x : CliffordAlgebra Q) :
    -- Porting note: original statement
    --    changeForm h (d⌋x) = d⌋changeForm h x := by
    changeForm h (contractLeft (Q := Q) d x) = contractLeft (Q := Q') d (changeForm h x) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (d : Module.Dual R M) (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.changeForm h) ((CliffordAlgebra.contractLeft d) x) =
  (CliffordAlgebra.contractLeft d) ((CliffordAlgebra.changeForm h) x)
-- CliffordAlgebra.changeForm_self_apply
theorem changeForm_self_apply (x : CliffordAlgebra Q) : changeForm (Q' := Q)
    changeForm.zero_proof x = x 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.changeForm ⋯) x = x
-- CliffordAlgebra.changeForm_self
@[simp]
theorem changeForm_self :
    changeForm changeForm.zero_proof = (LinearMap.id : CliffordAlgebra Q →ₗ[R] _) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
Goal: CliffordAlgebra.changeForm ⋯ = LinearMap.id
-- CliffordAlgebra.changeForm_changeForm
theorem changeForm_changeForm (x : CliffordAlgebra Q) :
    changeForm h' (changeForm h x) = changeForm (changeForm.add_proof h h') x 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' Q'' : QuadraticForm R M} {B B' : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (h' : LinearMap.BilinMap.toQuadraticMap B' = Q'' - Q') (x : CliffordAlgebra Q)
Goal: (CliffordAlgebra.changeForm h') ((CliffordAlgebra.changeForm h) x) = (CliffordAlgebra.changeForm ⋯) x
-- CliffordAlgebra.changeForm_comp_changeForm
theorem changeForm_comp_changeForm :
    (changeForm h').comp (changeForm h) = changeForm (changeForm.add_proof h h') 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' Q'' : QuadraticForm R M} {B B' : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (h' : LinearMap.BilinMap.toQuadraticMap B' = Q'' - Q')
Goal: CliffordAlgebra.changeForm h' ∘ₗ CliffordAlgebra.changeForm h = CliffordAlgebra.changeForm ⋯
-- CliffordAlgebra.changeFormEquiv_apply
@[simps apply]
def changeFormEquiv : CliffordAlgebra Q ≃ₗ[R] CliffordAlgebra Q' 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q) (a : CliffordAlgebra Q)
Goal: (CliffordAlgebra.changeFormEquiv h) a = (CliffordAlgebra.changeForm h) a
-- CliffordAlgebra.changeFormEquiv_symm
@[simp]
theorem changeFormEquiv_symm :
    (changeFormEquiv h).symm = changeFormEquiv (changeForm.neg_proof h) 
Arguments: {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {Q Q' : QuadraticForm R M} {B : LinearMap.BilinForm R M} (h : LinearMap.BilinMap.toQuadraticMap B = Q' - Q)
Goal: (CliffordAlgebra.changeFormEquiv h).symm = CliffordAlgebra.changeFormEquiv ⋯
-- LinearMap.BilinForm.toLin'_apply
@[deprecated (since := "2024-04-26")]
theorem toLin'_apply (A : BilinForm R M) (x : M) : toLinHom (M := M) A x = A x 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (A : LinearMap.BilinForm R M) (x : M)
Goal: (LinearMap.BilinForm.toLinHom A) x = A x
-- LinearMap.BilinForm.sum_left
theorem sum_left {α} (t : Finset α) (g : α → M) (w : M) :
    B (∑ i ∈ t, g i) w = ∑ i ∈ t, B (g i) w 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) {α : Type u_7} (t : Finset α) (g : α → M) (w : M)
Goal: (B (∑ i ∈ t, g i)) w = ∑ i ∈ t, (B (g i)) w
-- LinearMap.BilinForm.sum_right
theorem sum_right {α} (t : Finset α) (w : M) (g : α → M) :
    B w (∑ i ∈ t, g i) = ∑ i ∈ t, B w (g i) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) {α : Type u_7} (t : Finset α) (w : M) (g : α → M)
Goal: (B w) (∑ i ∈ t, g i) = ∑ i ∈ t, (B w) (g i)
-- LinearMap.BilinForm.sum_apply
theorem sum_apply {α} (t : Finset α) (B : α → BilinForm R M) (v w : M) :
    (∑ i ∈ t, B i) v w = ∑ i ∈ t, B i v w 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {α : Type u_7} (t : Finset α) (B : α → LinearMap.BilinForm R M) (v w : M)
Goal: ((∑ i ∈ t, B i) v) w = ∑ i ∈ t, ((B i) v) w
-- LinearMap.BilinForm.toLin'Flip_apply
theorem toLin'Flip_apply (A : BilinForm R M) (x : M) : toLinHomFlip (M := M) A x = fun y => A y x 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (A : LinearMap.BilinForm R M) (x : M)
Goal: ⇑((LinearMap.BilinForm.toLinHomFlip A) x) = fun y => (A y) x
-- LinearMap.toBilinAux_eq
@[deprecated (since := "2024-04-26")]
theorem LinearMap.toBilinAux_eq (f : M →ₗ[R] M →ₗ[R] R) :
    LinearMap.toBilinAux f = f 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] M →ₗ[R] R)
Goal: f.toBilinAux = f
-- LinearMap.toBilin_symm
@[deprecated (since := "2024-04-26")]
theorem LinearMap.toBilin_symm :
    (LinearMap.toBilin.symm : BilinForm R M ≃ₗ[R] _) = BilinForm.toLin 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.toBilin.symm = LinearMap.BilinForm.toLin
-- BilinForm.toLin_symm
@[deprecated (since := "2024-04-26")]
theorem BilinForm.toLin_symm :
    (BilinForm.toLin.symm : _ ≃ₗ[R] BilinForm R M) = LinearMap.toBilin 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M]
Goal: LinearMap.BilinForm.toLin.symm = LinearMap.toBilin
-- LinearMap.toBilin_apply
@[deprecated (since := "2024-04-26")]
theorem LinearMap.toBilin_apply (f : M →ₗ[R] M →ₗ[R] R) (x y : M) :
    toBilin f x y = f x y 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (f : M →ₗ[R] M →ₗ[R] R) (x y : M)
Goal: ((LinearMap.toBilin f) x) y = (f x) y
-- BilinForm.toLin_apply
@[deprecated (since := "2024-04-26")]
theorem BilinForm.toLin_apply (x : M) : BilinForm.toLin B x = B x 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (x : M)
Goal: (LinearMap.BilinForm.toLin B) x = B x
-- LinearMap.compBilinForm_apply_apply
@[simps!]
def compBilinForm (f : R →ₗ[R'] R') (B : BilinForm R M) : BilinForm R' M 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {R' : Type u_7} [CommSemiring R'] [Algebra R' R] [Module R' M] [IsScalarTower R' R M] (f : R →ₗ[R'] R') (B : LinearMap.BilinForm R M)
Goal: ∀ (a m : M), ((f.compBilinForm B) a) m = f ((B a) m)
-- LinearMap.BilinForm.comp_comp
theorem comp_comp {M'' : Type*} [AddCommMonoid M''] [Module R M''] (B : BilinForm R M'')
    (l r : M →ₗ[R] M') (l' r' : M' →ₗ[R] M'') :
    (B.comp l' r').comp l r = B.comp (l'.comp l) (r'.comp r) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type w} [AddCommMonoid M'] [Module R M'] {M'' : Type u_7} [AddCommMonoid M''] [Module R M''] (B : LinearMap.BilinForm R M'') (l r : M →ₗ[R] M') (l' r' : M' →ₗ[R] M'')
Goal: (B.comp l' r').comp l r = B.comp (l' ∘ₗ l) (r' ∘ₗ r)
-- LinearMap.BilinForm.compLeft_compRight
@[simp]
theorem compLeft_compRight (B : BilinForm R M) (l r : M →ₗ[R] M) :
    (B.compLeft l).compRight r = B.comp l r 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (l r : M →ₗ[R] M)
Goal: (B.compLeft l).compRight r = B.comp l r
-- LinearMap.BilinForm.compRight_compLeft
@[simp]
theorem compRight_compLeft (B : BilinForm R M) (l r : M →ₗ[R] M) :
    (B.compRight r).compLeft l = B.comp l r 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (l r : M →ₗ[R] M)
Goal: (B.compRight r).compLeft l = B.comp l r
-- LinearMap.BilinForm.comp_apply
@[simp]
theorem comp_apply (B : BilinForm R M') (l r : M →ₗ[R] M') (v w) : B.comp l r v w = B (l v) (r w) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type w} [AddCommMonoid M'] [Module R M'] (B : LinearMap.BilinForm R M') (l r : M →ₗ[R] M') (v w : M)
Goal: ((B.comp l r) v) w = (B (l v)) (r w)
-- LinearMap.BilinForm.compLeft_apply
@[simp]
theorem compLeft_apply (B : BilinForm R M) (f : M →ₗ[R] M) (v w) : B.compLeft f v w = B (f v) w 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (f : M →ₗ[R] M) (v w : M)
Goal: ((B.compLeft f) v) w = (B (f v)) w
-- LinearMap.BilinForm.compRight_apply
@[simp]
theorem compRight_apply (B : BilinForm R M) (f : M →ₗ[R] M) (v w) : B.compRight f v w = B v (f w) 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (f : M →ₗ[R] M) (v w : M)
Goal: ((B.compRight f) v) w = (B v) (f w)
-- LinearMap.BilinForm.comp_id_left
@[simp]
theorem comp_id_left (B : BilinForm R M) (r : M →ₗ[R] M) :
    B.comp LinearMap.id r = B.compRight r 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (r : M →ₗ[R] M)
Goal: B.comp LinearMap.id r = B.compRight r
-- LinearMap.BilinForm.comp_id_right
@[simp]
theorem comp_id_right (B : BilinForm R M) (l : M →ₗ[R] M) :
    B.comp l LinearMap.id = B.compLeft l 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M) (l : M →ₗ[R] M)
Goal: B.comp l LinearMap.id = B.compLeft l
-- LinearMap.BilinForm.compLeft_id
@[simp]
theorem compLeft_id (B : BilinForm R M) : B.compLeft LinearMap.id = B 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M)
Goal: B.compLeft LinearMap.id = B
-- LinearMap.BilinForm.compRight_id
@[simp]
theorem compRight_id (B : BilinForm R M) : B.compRight LinearMap.id = B 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M)
Goal: B.compRight LinearMap.id = B
-- LinearMap.BilinForm.comp_id_id
@[simp high]
theorem comp_id_id (B : BilinForm R M) : B.comp LinearMap.id LinearMap.id = B 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (B : LinearMap.BilinForm R M)
Goal: B.comp LinearMap.id LinearMap.id = B
-- LinearMap.BilinForm.comp_inj
theorem comp_inj (B₁ B₂ : BilinForm R M') {l r : M →ₗ[R] M'} (hₗ : Function.Surjective l)
    (hᵣ : Function.Surjective r) : B₁.comp l r = B₂.comp l r ↔ B₁ = B₂ 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {M' : Type w} [AddCommMonoid M'] [Module R M'] (B₁ B₂ : LinearMap.BilinForm R M') {l r : M →ₗ[R] M'} (hₗ : Function.Surjective ⇑l) (hᵣ : Function.Surjective ⇑r)
Goal: B₁.comp l r = B₂.comp l r ↔ B₁ = B₂