ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.ComonadicLeftAdjoint.eqv
class ComonadicLeftAdjoint (L : C ⥤ D) where
    R : D ⥤ C
    adj : L ⊣ R
  eqv : (Comonad.comparison adj).IsEquivalence
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {L : CategoryTheory.Functor C D} [self : CategoryTheory.ComonadicLeftAdjoint L]
Goal: (CategoryTheory.Comonad.comparison CategoryTheory.ComonadicLeftAdjoint.adj).IsEquivalence
-- CategoryTheory.Reflective.comparison_full
lemma comparison_full [R.Full] {L : C ⥤ D} (adj : L ⊣ R) :
    (Monad.comparison adj).Full 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {R : CategoryTheory.Functor D C} [R.Full] {L : CategoryTheory.Functor C D} (adj : L ⊣ R)
Goal: (CategoryTheory.Monad.comparison adj).Full
-- CategoryTheory.Coreflective.comparison_full
lemma comparison_full [R.Full] {L : C ⥤ D} (adj : R ⊣ L) :
    (Comonad.comparison adj).Full 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {R : CategoryTheory.Functor D C} [R.Full] {L : CategoryTheory.Functor C D} (adj : R ⊣ L)
Goal: (CategoryTheory.Comonad.comparison adj).Full
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.mk
theorem StableUnderBaseChange.mk {P : MorphismProperty C} [HasPullbacks C] [RespectsIso P]
    (hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) (_ : P g), P (pullback.fst f g)) :
    StableUnderBaseChange P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} [CategoryTheory.Limits.HasPullbacks C] [P.RespectsIso] (hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S), P g → P (CategoryTheory.Limits.pullback.fst f g))
Goal: P.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.respectsIso
theorem StableUnderBaseChange.respectsIso {P : MorphismProperty C} (hP : StableUnderBaseChange P) :
    RespectsIso P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange)
Goal: P.RespectsIso
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.fst
theorem StableUnderBaseChange.fst {P : MorphismProperty C} (hP : StableUnderBaseChange P)
    {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [HasPullback f g] (H : P g) :
    P (pullback.fst f g) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [CategoryTheory.Limits.HasPullback f g] (H : P g)
Goal: P (CategoryTheory.Limits.pullback.fst f g)
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.snd
theorem StableUnderBaseChange.snd {P : MorphismProperty C} (hP : StableUnderBaseChange P)
    {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [HasPullback f g] (H : P f) :
    P (pullback.snd f g) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [CategoryTheory.Limits.HasPullback f g] (H : P f)
Goal: P (CategoryTheory.Limits.pullback.snd f g)
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.baseChange_obj
theorem StableUnderBaseChange.baseChange_obj [HasPullbacks C] {P : MorphismProperty C}
    (hP : StableUnderBaseChange P) {S S' : C} (f : S' ⟶ S) (X : Over S) (H : P X.hom) :
    P ((Over.pullback f).obj X).hom 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) {S S' : C} (f : S' ⟶ S) (X : CategoryTheory.Over S) (H : P X.hom)
Goal: P ((CategoryTheory.Over.pullback f).obj X).hom
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.baseChange_map
theorem StableUnderBaseChange.baseChange_map [HasPullbacks C] {P : MorphismProperty C}
    (hP : StableUnderBaseChange P) {S S' : C} (f : S' ⟶ S) {X Y : Over S} (g : X ⟶ Y)
    (H : P g.left) : P ((Over.pullback f).map g).left 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) {S S' : C} (f : S' ⟶ S) {X Y : CategoryTheory.Over S} (g : X ⟶ Y) (H : P g.left)
Goal: P ((CategoryTheory.Over.pullback f).map g).left
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.pullback_map
theorem StableUnderBaseChange.pullback_map [HasPullbacks C] {P : MorphismProperty C}
    (hP : StableUnderBaseChange P) [P.IsStableUnderComposition] {S X X' Y Y' : C} {f : X ⟶ S}
    {g : Y ⟶ S} {f' : X' ⟶ S} {g' : Y' ⟶ S} {i₁ : X ⟶ X'} {i₂ : Y ⟶ Y'} (h₁ : P i₁) (h₂ : P i₂)
    (e₁ : f = i₁ ≫ f') (e₂ : g = i₂ ≫ g') :
    P (pullback.map f g f' g' i₁ i₂ (𝟙 _) ((Category.comp_id _).trans e₁)
        ((Category.comp_id _).trans e₂)) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) [P.IsStableUnderComposition] {S X X' Y Y' : C} {f : X ⟶ S} {g : Y ⟶ S} {f' : X' ⟶ S} {g' : Y' ⟶ S} {i₁ : X ⟶ X'} {i₂ : Y ⟶ Y'} (h₁ : P i₁) (h₂ : P i₂) (e₁ : f = CategoryTheory.CategoryStruct.comp i₁ f') (e₂ : g = CategoryTheory.CategoryStruct.comp i₂ g')
Goal: P (CategoryTheory.Limits.pullback.map f g f' g' i₁ i₂ (CategoryTheory.CategoryStruct.id S) ⋯ ⋯)
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.mk
theorem StableUnderCobaseChange.mk {P : MorphismProperty C} [HasPushouts C] [RespectsIso P]
    (hP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B) (_ : P f), P (pushout.inr f g)) :
    StableUnderCobaseChange P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} [CategoryTheory.Limits.HasPushouts C] [P.RespectsIso] (hP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P (CategoryTheory.Limits.pushout.inr f g))
Goal: P.StableUnderCobaseChange
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.respectsIso
theorem StableUnderCobaseChange.respectsIso {P : MorphismProperty C}
    (hP : StableUnderCobaseChange P) : RespectsIso P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderCobaseChange)
Goal: P.RespectsIso
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.inl
theorem StableUnderCobaseChange.inl {P : MorphismProperty C} (hP : StableUnderCobaseChange P)
    {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [HasPushout f g] (H : P g) :
    P (pushout.inl f g) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderCobaseChange) {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [CategoryTheory.Limits.HasPushout f g] (H : P g)
Goal: P (CategoryTheory.Limits.pushout.inl f g)
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.inr
theorem StableUnderCobaseChange.inr {P : MorphismProperty C} (hP : StableUnderCobaseChange P)
    {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [HasPushout f g] (H : P f) : P (pushout.inr f g) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderCobaseChange) {A B A' : C} (f : A ⟶ A') (g : A ⟶ B) [CategoryTheory.Limits.HasPushout f g] (H : P f)
Goal: P (CategoryTheory.Limits.pushout.inr f g)
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.op
theorem StableUnderCobaseChange.op {P : MorphismProperty C} (hP : StableUnderCobaseChange P) :
    StableUnderBaseChange P.op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderCobaseChange)
Goal: P.op.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.StableUnderCobaseChange.unop
theorem StableUnderCobaseChange.unop {P : MorphismProperty Cᵒᵖ} (hP : StableUnderCobaseChange P) :
    StableUnderBaseChange P.unop 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty Cᵒᵖ} (hP : P.StableUnderCobaseChange)
Goal: P.unop.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.op
theorem StableUnderBaseChange.op {P : MorphismProperty C} (hP : StableUnderBaseChange P) :
    StableUnderCobaseChange P.op 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange)
Goal: P.op.StableUnderCobaseChange
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.unop
theorem StableUnderBaseChange.unop {P : MorphismProperty Cᵒᵖ} (hP : StableUnderBaseChange P) :
    StableUnderCobaseChange P.unop 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty Cᵒᵖ} (hP : P.StableUnderBaseChange)
Goal: P.unop.StableUnderCobaseChange
-- CategoryTheory.MorphismProperty.IsStableUnderLimitsOfShape.lim_map
lemma IsStableUnderLimitsOfShape.lim_map {J : Type*} [Category J]
    (hW : W.IsStableUnderLimitsOfShape J) {X Y : J ⥤ C}
    (f : X ⟶ Y) [HasLimitsOfShape J C] (hf : W.functorCategory _ f) :
    W (lim.map f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {W : CategoryTheory.MorphismProperty C} {J : Type u_1} [CategoryTheory.Category.{u_2, u_1} J] (hW : W.IsStableUnderLimitsOfShape J) {X Y : CategoryTheory.Functor J C} (f : X ⟶ Y) [CategoryTheory.Limits.HasLimitsOfShape J C] (hf : W.functorCategory J f)
Goal: W (CategoryTheory.Limits.lim.map f)
-- CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk
lemma IsStableUnderProductsOfShape.mk (J : Type*)
    [W.RespectsIso] [HasProductsOfShape J C]
    (hW : ∀ (X₁ X₂ : J → C) (f : ∀ j, X₁ j ⟶ X₂ j) (_ : ∀ (j : J), W (f j)),
      W (Pi.map f)) : W.IsStableUnderProductsOfShape J 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (W : CategoryTheory.MorphismProperty C) (J : Type u_1) [W.RespectsIso] [CategoryTheory.Limits.HasProductsOfShape J C] (hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (CategoryTheory.Limits.Pi.map f))
Goal: W.IsStableUnderProductsOfShape J
-- CategoryTheory.MorphismProperty.IsStableUnderFiniteProducts.isStableUnderProductsOfShape
class IsStableUnderFiniteProducts : Prop :=
  isStableUnderProductsOfShape (J : Type) [Finite J] : W.IsStableUnderProductsOfShape J
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {W : CategoryTheory.MorphismProperty C} [self : W.IsStableUnderFiniteProducts] (J : Type) [Finite J]
Goal: W.IsStableUnderProductsOfShape J
-- CategoryTheory.MorphismProperty.isStableUnderProductsOfShape_of_isStableUnderFiniteProducts
lemma isStableUnderProductsOfShape_of_isStableUnderFiniteProducts
    (J : Type) [Finite J] [W.IsStableUnderFiniteProducts] :
    W.IsStableUnderProductsOfShape J 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (W : CategoryTheory.MorphismProperty C) (J : Type) [Finite J] [W.IsStableUnderFiniteProducts]
Goal: W.IsStableUnderProductsOfShape J
-- CategoryTheory.MorphismProperty.diagonal_iff
theorem diagonal_iff {X Y : C} {f : X ⟶ Y} : P.diagonal f ↔ P (pullback.diagonal f) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} {X Y : C} {f : X ⟶ Y}
Goal: P.diagonal f ↔ P (CategoryTheory.Limits.pullback.diagonal f)
-- CategoryTheory.MorphismProperty.diagonal_isStableUnderComposition
theorem diagonal_isStableUnderComposition [P.IsStableUnderComposition] [RespectsIso P]
    (hP'' : StableUnderBaseChange P) : P.diagonal.IsStableUnderComposition 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} [P.IsStableUnderComposition] [P.RespectsIso] (hP'' : P.StableUnderBaseChange)
Goal: P.diagonal.IsStableUnderComposition
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.diagonal
theorem StableUnderBaseChange.diagonal (hP : StableUnderBaseChange P) [P.RespectsIso] :
    P.diagonal.StableUnderBaseChange 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPullbacks C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange) [P.RespectsIso]
Goal: P.diagonal.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.universally_stableUnderBaseChange
theorem universally_stableUnderBaseChange (P : MorphismProperty C) :
    P.universally.StableUnderBaseChange 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (P : CategoryTheory.MorphismProperty C)
Goal: P.universally.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.universally_le
theorem universally_le (P : MorphismProperty C) : P.universally ≤ P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (P : CategoryTheory.MorphismProperty C)
Goal: P.universally ≤ P
-- CategoryTheory.MorphismProperty.universally_eq_iff
theorem universally_eq_iff {P : MorphismProperty C} :
    P.universally = P ↔ P.StableUnderBaseChange 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C}
Goal: P.universally = P ↔ P.StableUnderBaseChange
-- CategoryTheory.MorphismProperty.StableUnderBaseChange.universally_eq
theorem StableUnderBaseChange.universally_eq {P : MorphismProperty C}
    (hP : P.StableUnderBaseChange) : P.universally = P 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {P : CategoryTheory.MorphismProperty C} (hP : P.StableUnderBaseChange)
Goal: P.universally = P
-- CategoryTheory.MorphismProperty.universally_mono
theorem universally_mono : Monotone (universally : MorphismProperty C → MorphismProperty C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: Monotone CategoryTheory.MorphismProperty.universally
-- CategoryTheory.Monad.ForgetCreatesLimits.γ_app
@[simps]
def γ : D ⋙ T.forget ⋙ ↑T ⟶ D ⋙ T.forget 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (j : J)
Goal: (CategoryTheory.Monad.ForgetCreatesLimits.γ D).app j = (D.obj j).a
-- CategoryTheory.Monad.ForgetCreatesLimits.newCone_π_app
@[simps! π_app]
def newCone : Cone (D ⋙ forget T) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (X : J)
Goal: (CategoryTheory.Monad.ForgetCreatesLimits.newCone D c).π.app X =
  CategoryTheory.CategoryStruct.comp (T.map (c.π.app X)) (D.obj X).a
-- CategoryTheory.Monad.ForgetCreatesLimits.conePoint_A
@[simps]
def conePoint : Algebra T 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (t : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.Monad.ForgetCreatesLimits.conePoint D c t).A = c.pt
-- CategoryTheory.Monad.ForgetCreatesLimits.conePoint_a
@[simps]
def conePoint : Algebra T 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (t : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.Monad.ForgetCreatesLimits.conePoint D c t).a =
  t.lift (CategoryTheory.Monad.ForgetCreatesLimits.newCone D c)
-- CategoryTheory.Monad.ForgetCreatesLimits.liftedCone_pt
@[simps]
def liftedCone : Cone D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (t : CategoryTheory.Limits.IsLimit c)
Goal: (CategoryTheory.Monad.ForgetCreatesLimits.liftedCone D c t).pt =
  CategoryTheory.Monad.ForgetCreatesLimits.conePoint D c t
-- CategoryTheory.Monad.ForgetCreatesLimits.liftedCone_π_app_f
@[simps]
def liftedCone : Cone D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (t : CategoryTheory.Limits.IsLimit c) (j : J)
Goal: ((CategoryTheory.Monad.ForgetCreatesLimits.liftedCone D c t).π.app j).f = c.π.app j
-- CategoryTheory.Monad.ForgetCreatesLimits.liftedConeIsLimit_lift_f
@[simps]
def liftedConeIsLimit : IsLimit (liftedCone D c t) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) (c : CategoryTheory.Limits.Cone (D.comp T.forget)) (t : CategoryTheory.Limits.IsLimit c) (s : CategoryTheory.Limits.Cone D)
Goal: ((CategoryTheory.Monad.ForgetCreatesLimits.liftedConeIsLimit D c t).lift s).f = t.lift (T.forget.mapCone s)
-- CategoryTheory.Monad.hasLimit_of_comp_forget_hasLimit
theorem hasLimit_of_comp_forget_hasLimit (D : J ⥤ Algebra T) [HasLimit (D ⋙ forget T)] :
    HasLimit D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] (D : CategoryTheory.Functor J T.Algebra) [CategoryTheory.Limits.HasLimit (D.comp T.forget)]
Goal: CategoryTheory.Limits.HasLimit D
-- CategoryTheory.Monad.ForgetCreatesColimits.γ_app
@[simps]
def γ : (D ⋙ forget T) ⋙ ↑T ⟶ D ⋙ forget T 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (j : J)
Goal: CategoryTheory.Monad.ForgetCreatesColimits.γ.app j = (D.obj j).a
-- CategoryTheory.Monad.ForgetCreatesColimits.newCocone_pt
@[simps]
def newCocone : Cocone ((D ⋙ forget T) ⋙ (T : C ⥤ C)) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget))
Goal: (CategoryTheory.Monad.ForgetCreatesColimits.newCocone c).pt = c.pt
-- CategoryTheory.Monad.ForgetCreatesColimits.newCocone_ι
@[simps]
def newCocone : Cocone ((D ⋙ forget T) ⋙ (T : C ⥤ C)) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget))
Goal: (CategoryTheory.Monad.ForgetCreatesColimits.newCocone c).ι =
  CategoryTheory.CategoryStruct.comp CategoryTheory.Monad.ForgetCreatesColimits.γ c.ι
-- CategoryTheory.Monad.ForgetCreatesColimits.commuting
theorem commuting (j : J) : (T : C ⥤ C).map (c.ι.app j) ≫ lambda c t = (D.obj j).a ≫ c.ι.app j 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] (j : J)
Goal: CategoryTheory.CategoryStruct.comp (T.map (c.ι.app j)) (CategoryTheory.Monad.ForgetCreatesColimits.lambda c t) =
  CategoryTheory.CategoryStruct.comp (D.obj j).a (c.ι.app j)
-- CategoryTheory.Monad.ForgetCreatesColimits.coconePoint_A
@[simps]
def coconePoint : Algebra T 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] [CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor]
Goal: (CategoryTheory.Monad.ForgetCreatesColimits.coconePoint c t).A = c.pt
-- CategoryTheory.Monad.ForgetCreatesColimits.coconePoint_a
@[simps]
def coconePoint : Algebra T 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] [CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor]
Goal: (CategoryTheory.Monad.ForgetCreatesColimits.coconePoint c t).a = CategoryTheory.Monad.ForgetCreatesColimits.lambda c t
-- CategoryTheory.Monad.ForgetCreatesColimits.liftedCocone_ι_app_f
@[simps]
def liftedCocone : Cocone D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] [CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor] (j : J)
Goal: ((CategoryTheory.Monad.ForgetCreatesColimits.liftedCocone c t).ι.app j).f = c.ι.app j
-- CategoryTheory.Monad.ForgetCreatesColimits.liftedCocone_pt
@[simps]
def liftedCocone : Cocone D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] [CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor]
Goal: (CategoryTheory.Monad.ForgetCreatesColimits.liftedCocone c t).pt =
  CategoryTheory.Monad.ForgetCreatesColimits.coconePoint c t
-- CategoryTheory.Monad.ForgetCreatesColimits.liftedCoconeIsColimit_desc_f
@[simps]
def liftedCoconeIsColimit : IsColimit (liftedCocone c t) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] {D : CategoryTheory.Functor J T.Algebra} (c : CategoryTheory.Limits.Cocone (D.comp T.forget)) (t : CategoryTheory.Limits.IsColimit c) [CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor] [CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor] (s : CategoryTheory.Limits.Cocone D)
Goal: ((CategoryTheory.Monad.ForgetCreatesColimits.liftedCoconeIsColimit c t).desc s).f = t.desc (T.forget.mapCocone s)
-- CategoryTheory.Monad.forget_creates_colimits_of_monad_preserves
theorem forget_creates_colimits_of_monad_preserves [PreservesColimitsOfShape J (T : C ⥤ C)]
    (D : J ⥤ Algebra T) [HasColimit (D ⋙ forget T)] : HasColimit D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {T : CategoryTheory.Monad C} {J : Type u} [CategoryTheory.Category.{v, u} J] [CategoryTheory.Limits.PreservesColimitsOfShape J T.toFunctor] (D : CategoryTheory.Functor J T.Algebra) [CategoryTheory.Limits.HasColimit (D.comp T.forget)]
Goal: CategoryTheory.Limits.HasColimit D
-- CategoryTheory.hasLimit_of_reflective
theorem hasLimit_of_reflective (F : J ⥤ D) (R : D ⥤ C) [HasLimit (F ⋙ R)] [Reflective R] :
    HasLimit F 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : Type u} [CategoryTheory.Category.{v, u} J] (F : CategoryTheory.Functor J D) (R : CategoryTheory.Functor D C) [CategoryTheory.Limits.HasLimit (F.comp R)] [CategoryTheory.Reflective R]
Goal: CategoryTheory.Limits.HasLimit F
-- CategoryTheory.hasLimitsOfShape_of_reflective
theorem hasLimitsOfShape_of_reflective [HasLimitsOfShape J C] (R : D ⥤ C) [Reflective R] :
    HasLimitsOfShape J D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : Type u} [CategoryTheory.Category.{v, u} J] [CategoryTheory.Limits.HasLimitsOfShape J C] (R : CategoryTheory.Functor D C) [CategoryTheory.Reflective R]
Goal: CategoryTheory.Limits.HasLimitsOfShape J D