File: Mathlib/CategoryTheory/Limits/Lattice.lean

full_name: CategoryTheory.Limits.CompleteLattice.finiteLimitCone
code: /-- The limit cone over any functor from a finite diagram into a `SemilatticeInf` with `OrderTop`.
-/
def finiteLimitCone [SemilatticeInf α] [OrderTop α] (F : J ⥤ α) : LimitCone F 
goal: CategoryTheory.Limits.LimitCone F

full_name: CategoryTheory.Limits.CompleteLattice.finiteColimitCocone
code: /--
The colimit cocone over any functor from a finite diagram into a `SemilatticeSup` with `OrderBot`.
-/
def finiteColimitCocone [SemilatticeSup α] [OrderBot α] (F : J ⥤ α) : ColimitCocone F 
goal: CategoryTheory.Limits.ColimitCocone F

full_name: CategoryTheory.Limits.CompleteLattice.hasFiniteLimits_of_semilatticeInf_orderTop
code: instance (priority := 100) hasFiniteLimits_of_semilatticeInf_orderTop [SemilatticeInf α]
    [OrderTop α] : HasFiniteLimits α 
goal: CategoryTheory.Limits.HasFiniteLimits α

full_name: CategoryTheory.Limits.CompleteLattice.hasFiniteColimits_of_semilatticeSup_orderBot
code: instance (priority := 100) hasFiniteColimits_of_semilatticeSup_orderBot [SemilatticeSup α]
    [OrderBot α] : HasFiniteColimits α 
goal: CategoryTheory.Limits.HasFiniteColimits α

full_name: CategoryTheory.Limits.CompleteLattice.finite_limit_eq_finset_univ_inf
code: /-- The limit of a functor from a finite diagram into a `SemilatticeInf` with `OrderTop` is the
infimum of the objects in the image.
-/
theorem finite_limit_eq_finset_univ_inf [SemilatticeInf α] [OrderTop α] (F : J ⥤ α) :
    limit F = Finset.univ.inf F.obj 
goal: CategoryTheory.Limits.limit F = Finset.univ.inf F.obj

full_name: CategoryTheory.Limits.CompleteLattice.finite_colimit_eq_finset_univ_sup
code: /-- The colimit of a functor from a finite diagram into a `SemilatticeSup` with `OrderBot`
is the supremum of the objects in the image.
-/
theorem finite_colimit_eq_finset_univ_sup [SemilatticeSup α] [OrderBot α] (F : J ⥤ α) :
    colimit F = Finset.univ.sup F.obj 
goal: CategoryTheory.Limits.colimit F = Finset.univ.sup F.obj

full_name: CategoryTheory.Limits.CompleteLattice.finite_product_eq_finset_inf
code: /--
A finite product in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum.
-/
theorem finite_product_eq_finset_inf [SemilatticeInf α] [OrderTop α] {ι : Type u} [Fintype ι]
    (f : ι → α) : ∏ᶜ f = Fintype.elems.inf f 
goal: ∏ᶜ f = Fintype.elems.inf f

full_name: CategoryTheory.Limits.CompleteLattice.finite_coproduct_eq_finset_sup
code: /-- A finite coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
    (f : ι → α) : ∐ f = Fintype.elems.sup f 
goal: ∐ f = Fintype.elems.sup f

full_name: CategoryTheory.Limits.CompleteLattice.instHasBinaryProductsOfOrderTop
code: instance (priority := 100) [SemilatticeInf α] [OrderTop α] : HasBinaryProducts α 
goal: CategoryTheory.Limits.HasBinaryProducts α

full_name: CategoryTheory.Limits.CompleteLattice.prod_eq_inf
code: /-- The binary product in the category of a `SemilatticeInf` with `OrderTop` is the same as the
infimum.
-/
@[simp]
theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y 
goal: (x ⨯ y) = x ⊓ y

full_name: CategoryTheory.Limits.CompleteLattice.instHasBinaryCoproductsOfOrderBot
code: instance (priority := 100) [SemilatticeSup α] [OrderBot α] : HasBinaryCoproducts α 
goal: CategoryTheory.Limits.HasBinaryCoproducts α

full_name: CategoryTheory.Limits.CompleteLattice.coprod_eq_sup
code: /-- The binary coproduct in the category of a `SemilatticeSup` with `OrderBot` is the same as the
supremum.
-/
@[simp]
theorem coprod_eq_sup [SemilatticeSup α] [OrderBot α] (x y : α) : Limits.coprod x y = x ⊔ y 
goal: (x ⨿ y) = x ⊔ y

full_name: CategoryTheory.Limits.CompleteLattice.pullback_eq_inf
code: /-- The pullback in the category of a `SemilatticeInf` with `OrderTop` is the same as the infimum
over the objects.
-/
@[simp]
theorem pullback_eq_inf [SemilatticeInf α] [OrderTop α] {x y z : α} (f : x ⟶ z) (g : y ⟶ z) :
    pullback f g = x ⊓ y 
goal: CategoryTheory.Limits.pullback f g = x ⊓ y

full_name: CategoryTheory.Limits.CompleteLattice.pushout_eq_sup
code: /-- The pushout in the category of a `SemilatticeSup` with `OrderBot` is the same as the supremum
over the objects.
-/
@[simp]
theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
    pushout f g = x ⊔ y 
goal: CategoryTheory.Limits.pushout f g = x ⊔ y

full_name: CategoryTheory.Limits.CompleteLattice.limitCone
code: /-- The limit cone over any functor into a complete lattice.
-/
def limitCone (F : J ⥤ α) : LimitCone F 
goal: CategoryTheory.Limits.LimitCone F

full_name: CategoryTheory.Limits.CompleteLattice.colimitCocone
code: /-- The colimit cocone over any functor into a complete lattice.
-/
def colimitCocone (F : J ⥤ α) : ColimitCocone F 
goal: CategoryTheory.Limits.ColimitCocone F

full_name: CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice
code: instance (priority := 100) hasLimits_of_completeLattice : HasLimits α 
goal: CategoryTheory.Limits.HasLimits α

full_name: CategoryTheory.Limits.CompleteLattice.hasColimits_of_completeLattice
code: instance (priority := 100) hasColimits_of_completeLattice : HasColimits α 
goal: CategoryTheory.Limits.HasColimits α

full_name: CategoryTheory.Limits.CompleteLattice.limit_eq_iInf
code: /-- The limit of a functor into a complete lattice is the infimum of the objects in the image.
-/
theorem limit_eq_iInf (F : J ⥤ α) : limit F = iInf F.obj 
goal: CategoryTheory.Limits.limit F = iInf F.obj

full_name: CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup
code: /-- The colimit of a functor into a complete lattice is the supremum of the objects in the image.
-/
theorem colimit_eq_iSup (F : J ⥤ α) : colimit F = iSup F.obj 
goal: CategoryTheory.Limits.colimit F = iSup F.obj