ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Functor.LeftExtension.coconeAt_pt
@[simps]
def coconeAt (Y : D) : Cocone (CostructuredArrow.proj L Y ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D)
Goal: (E.coconeAt Y).pt = E.right.obj Y
-- CategoryTheory.Functor.LeftExtension.coconeAt_ι_app
@[simps]
def coconeAt (Y : D) : Cocone (CostructuredArrow.proj L Y ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) (g : CategoryTheory.CostructuredArrow L Y)
Goal: (E.coconeAt Y).ι.app g = CategoryTheory.CategoryStruct.comp (E.hom.app g.left) (E.right.map g.hom)
-- CategoryTheory.Functor.LeftExtension.coconeAtFunctor_map_hom
@[simps]
def coconeAtFunctor (Y : D) :
    LeftExtension L F ⥤ Cocone (CostructuredArrow.proj L Y ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) {E E' : L.LeftExtension F} (φ : E ⟶ E')
Goal: ((CategoryTheory.Functor.LeftExtension.coconeAtFunctor L F Y).map φ).hom = φ.right.app Y
-- CategoryTheory.Functor.LeftExtension.coconeAtFunctor_obj
@[simps]
def coconeAtFunctor (Y : D) :
    LeftExtension L F ⥤ Cocone (CostructuredArrow.proj L Y ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) (E : L.LeftExtension F)
Goal: (CategoryTheory.Functor.LeftExtension.coconeAtFunctor L F Y).obj E = E.coconeAt Y
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.hasPointwiseLeftKanExtensionAt
lemma IsPointwiseLeftKanExtensionAt.hasPointwiseLeftKanExtensionAt
    {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) :
    HasPointwiseLeftKanExtensionAt L F Y 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y)
Goal: L.HasPointwiseLeftKanExtensionAt F Y
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.isIso_hom_app
lemma IsPointwiseLeftKanExtensionAt.isIso_hom_app
    {X : C} (h : E.IsPointwiseLeftKanExtensionAt (L.obj X)) [L.Full] [L.Faithful] :
    IsIso (E.hom.app X) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) {X : C} (h : E.IsPointwiseLeftKanExtensionAt (L.obj X)) [L.Full] [L.Faithful]
Goal: CategoryTheory.IsIso (E.hom.app X)
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hasPointwiseLeftKanExtension
lemma IsPointwiseLeftKanExtension.hasPointwiseLeftKanExtension :
    HasPointwiseLeftKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension)
Goal: L.HasPointwiseLeftKanExtension F
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hom_ext
lemma IsPointwiseLeftKanExtension.hom_ext (h : E.IsPointwiseLeftKanExtension)
    {G : LeftExtension L F} {f₁ f₂ : E ⟶ G} : f₁ = f₂ 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) {G : L.LeftExtension F} {f₁ f₂ : E ⟶ G}
Goal: f₁ = f₂
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isLeftKanExtension
lemma IsPointwiseLeftKanExtension.isLeftKanExtension :
    E.right.IsLeftKanExtension E.hom 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension)
Goal: E.right.IsLeftKanExtension E.hom
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hasLeftKanExtension
lemma IsPointwiseLeftKanExtension.hasLeftKanExtension :
    HasLeftKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension)
Goal: L.HasLeftKanExtension F
-- CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isIso_hom
lemma IsPointwiseLeftKanExtension.isIso_hom [L.Full] [L.Faithful] :
    IsIso (E.hom) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) [L.Full] [L.Faithful]
Goal: CategoryTheory.IsIso E.hom
-- CategoryTheory.Functor.RightExtension.coneAt_pt
@[simps]
def coneAt (Y : D) : Cone (StructuredArrow.proj Y L ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.RightExtension F) (Y : D)
Goal: (E.coneAt Y).pt = E.left.obj Y
-- CategoryTheory.Functor.RightExtension.coneAt_π_app
@[simps]
def coneAt (Y : D) : Cone (StructuredArrow.proj Y L ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.RightExtension F) (Y : D) (g : CategoryTheory.StructuredArrow Y L)
Goal: (E.coneAt Y).π.app g = CategoryTheory.CategoryStruct.comp (E.left.map g.hom) (E.hom.app g.right)
-- CategoryTheory.Functor.RightExtension.coneAtFunctor_obj
@[simps]
def coneAtFunctor (Y : D) :
    RightExtension L F ⥤ Cone (StructuredArrow.proj Y L ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) (E : L.RightExtension F)
Goal: (CategoryTheory.Functor.RightExtension.coneAtFunctor L F Y).obj E = E.coneAt Y
-- CategoryTheory.Functor.RightExtension.coneAtFunctor_map_hom
@[simps]
def coneAtFunctor (Y : D) :
    RightExtension L F ⥤ Cone (StructuredArrow.proj Y L ⋙ F) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) {E E' : L.RightExtension F} (φ : E ⟶ E')
Goal: ((CategoryTheory.Functor.RightExtension.coneAtFunctor L F Y).map φ).hom = φ.left.app Y
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtensionAt.hasPointwiseRightKanExtensionAt
lemma IsPointwiseRightKanExtensionAt.hasPointwiseRightKanExtensionAt
    {Y : D} (h : E.IsPointwiseRightKanExtensionAt Y) :
    HasPointwiseRightKanExtensionAt L F Y 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} {Y : D} (h : E.IsPointwiseRightKanExtensionAt Y)
Goal: L.HasPointwiseRightKanExtensionAt F Y
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtensionAt.isIso_hom_app
lemma IsPointwiseRightKanExtensionAt.isIso_hom_app
    {X : C} (h : E.IsPointwiseRightKanExtensionAt (L.obj X)) [L.Full] [L.Faithful] :
    IsIso (E.hom.app X) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.RightExtension F) {X : C} (h : E.IsPointwiseRightKanExtensionAt (L.obj X)) [L.Full] [L.Faithful]
Goal: CategoryTheory.IsIso (E.hom.app X)
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtension.hasPointwiseRightKanExtension
lemma IsPointwiseRightKanExtension.hasPointwiseRightKanExtension :
    HasPointwiseRightKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} (h : E.IsPointwiseRightKanExtension)
Goal: L.HasPointwiseRightKanExtension F
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtension.hom_ext
lemma IsPointwiseRightKanExtension.hom_ext (h : E.IsPointwiseRightKanExtension)
    {G : RightExtension L F} {f₁ f₂ : G ⟶ E} : f₁ = f₂ 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} (h : E.IsPointwiseRightKanExtension) {G : L.RightExtension F} {f₁ f₂ : G ⟶ E}
Goal: f₁ = f₂
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtension.isRightKanExtension
lemma IsPointwiseRightKanExtension.isRightKanExtension :
    E.left.IsRightKanExtension E.hom 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} (h : E.IsPointwiseRightKanExtension)
Goal: E.left.IsRightKanExtension E.hom
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtension.hasRightKanExtension
lemma IsPointwiseRightKanExtension.hasRightKanExtension :
    HasRightKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] [CategoryTheory.Category.{u_5, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} (h : E.IsPointwiseRightKanExtension)
Goal: L.HasRightKanExtension F
-- CategoryTheory.Functor.RightExtension.IsPointwiseRightKanExtension.isIso_hom
lemma IsPointwiseRightKanExtension.isIso_hom [L.Full] [L.Faithful] :
    IsIso (E.hom) 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.RightExtension F} (h : E.IsPointwiseRightKanExtension) [L.Full] [L.Faithful]
Goal: CategoryTheory.IsIso E.hom
-- CategoryTheory.Functor.pointwiseLeftKanExtension_map
@[simps]
noncomputable def pointwiseLeftKanExtension : D ⥤ H 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F] {Y₁ Y₂ : D} (f : Y₁ ⟶ Y₂)
Goal: (L.pointwiseLeftKanExtension F).map f =
  CategoryTheory.Limits.colimit.desc ((CategoryTheory.CostructuredArrow.proj L Y₁).comp F)
    { pt := CategoryTheory.Limits.colimit ((CategoryTheory.CostructuredArrow.proj L Y₂).comp F),
      ι :=
        {
          app := fun g =>
            CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y₂).comp F)
              ((CategoryTheory.CostructuredArrow.map f).obj g),
          naturality := ⋯ } }
-- CategoryTheory.Functor.pointwiseLeftKanExtension_obj
@[simps]
noncomputable def pointwiseLeftKanExtension : D ⥤ H 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F] (Y : D)
Goal: (L.pointwiseLeftKanExtension F).obj Y =
  CategoryTheory.Limits.colimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)
-- CategoryTheory.Functor.pointwiseLeftKanExtensionUnit_app
@[simps]
noncomputable def pointwiseLeftKanExtensionUnit : F ⟶ L ⋙ pointwiseLeftKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseLeftKanExtension F] (X : C)
Goal: (L.pointwiseLeftKanExtensionUnit F).app X =
  CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L (L.obj X)).comp F)
    (CategoryTheory.CostructuredArrow.mk (CategoryTheory.CategoryStruct.id (L.obj X)))
-- CategoryTheory.Functor.pointwiseRightKanExtension_map
@[simps]
noncomputable def pointwiseRightKanExtension : D ⥤ H 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseRightKanExtension F] {Y₁ Y₂ : D} (f : Y₁ ⟶ Y₂)
Goal: (L.pointwiseRightKanExtension F).map f =
  CategoryTheory.Limits.limit.lift ((CategoryTheory.StructuredArrow.proj Y₂ L).comp F)
    { pt := CategoryTheory.Limits.limit ((CategoryTheory.StructuredArrow.proj Y₁ L).comp F),
      π :=
        {
          app := fun g =>
            CategoryTheory.Limits.limit.π ((CategoryTheory.StructuredArrow.proj Y₁ L).comp F)
              ((CategoryTheory.StructuredArrow.map f).obj g),
          naturality := ⋯ } }
-- CategoryTheory.Functor.pointwiseRightKanExtension_obj
@[simps]
noncomputable def pointwiseRightKanExtension : D ⥤ H 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseRightKanExtension F] (Y : D)
Goal: (L.pointwiseRightKanExtension F).obj Y = CategoryTheory.Limits.limit ((CategoryTheory.StructuredArrow.proj Y L).comp F)
-- CategoryTheory.Functor.pointwiseRightKanExtensionCounit_app
@[simps]
noncomputable def pointwiseRightKanExtensionCounit :
    L ⋙ pointwiseRightKanExtension L F ⟶ F 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) [L.HasPointwiseRightKanExtension F] (X : C)
Goal: (L.pointwiseRightKanExtensionCounit F).app X =
  CategoryTheory.Limits.limit.π ((CategoryTheory.StructuredArrow.proj (L.obj X) L).comp F)
    (CategoryTheory.StructuredArrow.mk (CategoryTheory.CategoryStruct.id (L.obj X)))
-- Polynomial.taylor_apply
theorem taylor_apply : taylor r f = f.comp (X + C r) 
Arguments: {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R)
Goal: (Polynomial.taylor r) f = f.comp (Polynomial.X + Polynomial.C r)
-- Polynomial.taylor_X
@[simp]
theorem taylor_X : taylor r X = X + C r 
Arguments: {R : Type u_1} [Semiring R] (r : R)
Goal: (Polynomial.taylor r) Polynomial.X = Polynomial.X + Polynomial.C r
-- Polynomial.taylor_C
@[simp]
theorem taylor_C (x : R) : taylor r (C x) = C x 
Arguments: {R : Type u_1} [Semiring R] (r x : R)
Goal: (Polynomial.taylor r) (Polynomial.C x) = Polynomial.C x
-- Polynomial.taylor_zero'
@[simp]
theorem taylor_zero' : taylor (0 : R) = LinearMap.id 
Arguments: {R : Type u_1} [Semiring R]
Goal: Polynomial.taylor 0 = LinearMap.id
-- Polynomial.taylor_zero
theorem taylor_zero (f : R[X]) : taylor 0 f = f 
Arguments: {R : Type u_1} [Semiring R] (f : Polynomial R)
Goal: (Polynomial.taylor 0) f = f
-- Polynomial.taylor_one
@[simp]
theorem taylor_one : taylor r (1 : R[X]) = C 1 
Arguments: {R : Type u_1} [Semiring R] (r : R)
Goal: (Polynomial.taylor r) 1 = Polynomial.C 1
-- Polynomial.taylor_monomial
@[simp]
theorem taylor_monomial (i : ℕ) (k : R) : taylor r (monomial i k) = C k * (X + C r) ^ i 
Arguments: {R : Type u_1} [Semiring R] (r : R) (i : ℕ) (k : R)
Goal: (Polynomial.taylor r) ((Polynomial.monomial i) k) = Polynomial.C k * (Polynomial.X + Polynomial.C r) ^ i
-- Polynomial.taylor_coeff
theorem taylor_coeff (n : ℕ) : (taylor r f).coeff n = (hasseDeriv n f).eval r 
Arguments: {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R) (n : ℕ)
Goal: ((Polynomial.taylor r) f).coeff n = Polynomial.eval r ((Polynomial.hasseDeriv n) f)
-- Polynomial.taylor_coeff_zero
@[simp]
theorem taylor_coeff_zero : (taylor r f).coeff 0 = f.eval r 
Arguments: {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R)
Goal: ((Polynomial.taylor r) f).coeff 0 = Polynomial.eval r f
-- Polynomial.taylor_coeff_one
@[simp]
theorem taylor_coeff_one : (taylor r f).coeff 1 = f.derivative.eval r 
Arguments: {R : Type u_1} [Semiring R] (r : R) (f : Polynomial R)
Goal: ((Polynomial.taylor r) f).coeff 1 = Polynomial.eval r (Polynomial.derivative f)
-- Polynomial.natDegree_taylor
@[simp]
theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p 
Arguments: {R : Type u_1} [Semiring R] (p : Polynomial R) (r : R)
Goal: ((Polynomial.taylor r) p).natDegree = p.natDegree
-- Polynomial.taylor_mul
@[simp]
theorem taylor_mul {R} [CommSemiring R] (r : R) (p q : R[X]) :
    taylor r (p * q) = taylor r p * taylor r q 
Arguments: {R : Type u_2} [CommSemiring R] (r : R) (p q : Polynomial R)
Goal: (Polynomial.taylor r) (p * q) = (Polynomial.taylor r) p * (Polynomial.taylor r) q
-- Polynomial.taylorAlgHom_apply
@[simps!]
def taylorAlgHom {R} [CommSemiring R] (r : R) : R[X] →ₐ[R] R[X] 
Arguments: {R : Type u_2} [CommSemiring R] (r : R) (a : Polynomial R)
Goal: (Polynomial.taylorAlgHom r) a = (Polynomial.taylor r) a
-- Polynomial.taylor_taylor
theorem taylor_taylor {R} [CommSemiring R] (f : R[X]) (r s : R) :
    taylor r (taylor s f) = taylor (r + s) f 
Arguments: {R : Type u_2} [CommSemiring R] (f : Polynomial R) (r s : R)
Goal: (Polynomial.taylor r) ((Polynomial.taylor s) f) = (Polynomial.taylor (r + s)) f
-- Polynomial.taylor_eval
theorem taylor_eval {R} [CommSemiring R] (r : R) (f : R[X]) (s : R) :
    (taylor r f).eval s = f.eval (s + r) 
Arguments: {R : Type u_2} [CommSemiring R] (r : R) (f : Polynomial R) (s : R)
Goal: Polynomial.eval s ((Polynomial.taylor r) f) = Polynomial.eval (s + r) f
-- Polynomial.taylor_eval_sub
theorem taylor_eval_sub {R} [CommRing R] (r : R) (f : R[X]) (s : R) :
    (taylor r f).eval (s - r) = f.eval s 
Arguments: {R : Type u_2} [CommRing R] (r : R) (f : Polynomial R) (s : R)
Goal: Polynomial.eval (s - r) ((Polynomial.taylor r) f) = Polynomial.eval s f
-- Polynomial.taylor_injective
theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r) 
Arguments: {R : Type u_2} [CommRing R] (r : R)
Goal: Function.Injective ⇑(Polynomial.taylor r)
-- Polynomial.eq_zero_of_hasseDeriv_eq_zero
theorem eq_zero_of_hasseDeriv_eq_zero {R} [CommRing R] (f : R[X]) (r : R)
    (h : ∀ k, (hasseDeriv k f).eval r = 0) : f = 0 
Arguments: {R : Type u_2} [CommRing R] (f : Polynomial R) (r : R) (h : ∀ (k : ℕ), Polynomial.eval r ((Polynomial.hasseDeriv k) f) = 0)
Goal: f = 0
-- Polynomial.sum_taylor_eq
theorem sum_taylor_eq {R} [CommRing R] (f : R[X]) (r : R) :
    ((taylor r f).sum fun i a => C a * (X - C r) ^ i) = f 
Arguments: {R : Type u_2} [CommRing R] (f : Polynomial R) (r : R)
Goal: (((Polynomial.taylor r) f).sum fun i a => Polynomial.C a * (Polynomial.X - Polynomial.C r) ^ i) = f
-- Polynomial.trailingDegree_lt_wf
theorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q 
Arguments: {R : Type u} [Semiring R]
Goal: WellFounded fun p q => p.trailingDegree < q.trailingDegree
-- Polynomial.TrailingMonic.def
theorem TrailingMonic.def : TrailingMonic p ↔ trailingCoeff p = 1 
Arguments: {R : Type u} [Semiring R] {p : R[X]}
Goal: p.TrailingMonic ↔ p.trailingCoeff = 1
-- Polynomial.TrailingMonic.trailingCoeff
@[simp]
theorem TrailingMonic.trailingCoeff {p : R[X]} (hp : p.TrailingMonic) : trailingCoeff p = 1 
Arguments: {R : Type u} [Semiring R] {p : R[X]} (hp : p.TrailingMonic)
Goal: p.trailingCoeff = 1
-- Polynomial.trailingDegree_zero
@[simp]
theorem trailingDegree_zero : trailingDegree (0 : R[X]) = ⊤ 
Arguments: {R : Type u} [Semiring R]
Goal: Polynomial.trailingDegree 0 = ⊤
-- Polynomial.trailingCoeff_zero
@[simp]
theorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 
Arguments: {R : Type u} [Semiring R]
Goal: Polynomial.trailingCoeff 0 = 0
-- Polynomial.natTrailingDegree_zero
@[simp]
theorem natTrailingDegree_zero : natTrailingDegree (0 : R[X]) = 0 
Arguments: {R : Type u} [Semiring R]
Goal: Polynomial.natTrailingDegree 0 = 0
-- Polynomial.trailingDegree_eq_top
@[simp]
theorem trailingDegree_eq_top : trailingDegree p = ⊤ ↔ p = 0 
Arguments: {R : Type u} [Semiring R] {p : R[X]}
Goal: p.trailingDegree = ⊤ ↔ p = 0
-- Polynomial.trailingDegree_eq_natTrailingDegree
theorem trailingDegree_eq_natTrailingDegree (hp : p ≠ 0) :
    trailingDegree p = (natTrailingDegree p : ℕ∞) 
Arguments: {R : Type u} [Semiring R] {p : R[X]} (hp : p ≠ 0)
Goal: p.trailingDegree = ↑p.natTrailingDegree
-- Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq
theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
    p.trailingDegree = n ↔ p.natTrailingDegree = n 
Arguments: {R : Type u} [Semiring R] {p : R[X]} {n : ℕ} (hp : p ≠ 0)
Goal: p.trailingDegree = ↑n ↔ p.natTrailingDegree = n
-- Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos
theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : ℕ} (hn : n ≠ 0) :
    p.trailingDegree = n ↔ p.natTrailingDegree = n 
Arguments: {R : Type u} [Semiring R] {p : R[X]} {n : ℕ} (hn : n ≠ 0)
Goal: p.trailingDegree = ↑n ↔ p.natTrailingDegree = n
-- Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some
theorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : ℕ}
    (h : trailingDegree p = n) : natTrailingDegree p = n 
Arguments: {R : Type u} [Semiring R] {p : R[X]} {n : ℕ} (h : p.trailingDegree = ↑n)
Goal: p.natTrailingDegree = n
-- Polynomial.natTrailingDegree_le_trailingDegree
@[simp]
theorem natTrailingDegree_le_trailingDegree : ↑(natTrailingDegree p) ≤ trailingDegree p 
Arguments: {R : Type u} [Semiring R] {p : R[X]}
Goal: ↑p.natTrailingDegree ≤ p.trailingDegree
-- Polynomial.natTrailingDegree_eq_of_trailingDegree_eq
theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
    (h : trailingDegree p = trailingDegree q) : natTrailingDegree p = natTrailingDegree q 
Arguments: {R : Type u} {S : Type v} [Semiring R] {p : R[X]} [Semiring S] {q : S[X]} (h : p.trailingDegree = q.trailingDegree)
Goal: p.natTrailingDegree = q.natTrailingDegree
-- Polynomial.trailingDegree_le_of_ne_zero
theorem trailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : trailingDegree p ≤ n 
Arguments: {R : Type u} {n : ℕ} [Semiring R] {p : R[X]} (h : p.coeff n ≠ 0)
Goal: p.trailingDegree ≤ ↑n