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Require Import Coq.Logic.FunctionalExtensionality.
Require Import Coq.Program.Tactics.
Global Set Primitive Projections.
Global Set Universe Polymorphism.
Global Unset Universe Minimization ToSet.
Class Category : Type :=
{
Obj : Type;
Hom : Obj -> Obj -> Type;
compose : forall {a b c : Obj}, (Hom a b) -> (Hom b c) -> (Hom a c);
id : forall {a : Obj}, Hom a a;
}.
Arguments Obj {_}, _.
Arguments id {_ _}, {_} _, _ _.
Arguments Hom {_} _ _, _ _ _.
Arguments compose {_} {_ _ _} _ _, _ {_ _ _} _ _, _ _ _ _ _ _.
Coercion Obj : Category >-> Sortclass.
Definition Opposite (C : Category) : Category :=
{|
Obj := Obj C;
Hom := fun a b => Hom b a;
compose :=
fun a b c (f : Hom b a) (g : Hom c b) => compose C c b a g f;
id := fun c => id C c;
|}.
Record Functor (C C' : Category) : Type :=
{
FO : C -> C';
FA : forall {a b}, Hom a b -> Hom (FO a) (FO b);
}.
Arguments FO {_ _} _ _.
Arguments FA {_ _} _ {_ _} _, {_ _} _ _ _ _.
Section Opposite_Functor.
Context {C D : Category} (F : Functor C D).
Program Definition Opposite_Functor : (Functor (Opposite C) (Opposite D)) :=
{|
FO := FO F;
FA := fun _ _ h => FA F h;
|}.
End Opposite_Functor.
Section Functor_Compose.
Context {C C' C'' : Category} (F : Functor C C') (F' : Functor C' C'').
Program Definition Functor_compose : Functor C C'' :=
{|
FO := fun c => FO F' (FO F c);
FA := fun c d f => FA F' (FA F f)
|}.
End Functor_Compose.
Section Algebras.
Context {C : Category} (T : Functor C C).
Record Algebra : Type :=
{
Alg_Carrier : C;
Constructors : Hom (FO T Alg_Carrier) Alg_Carrier
}.
Record Algebra_Hom (alg alg' : Algebra) : Type :=
{
Alg_map : Hom (Alg_Carrier alg) (Alg_Carrier alg');
Alg_map_com : compose (FA T Alg_map) (Constructors alg')
= compose (Constructors alg) Alg_map
}.
Arguments Alg_map {_ _} _.
Arguments Alg_map_com {_ _} _.
Program Definition Algebra_Hom_compose
{alg alg' alg'' : Algebra}
(h : Algebra_Hom alg alg')
(h' : Algebra_Hom alg' alg'')
: Algebra_Hom alg alg''
:=
{|
Alg_map := compose (Alg_map h) (Alg_map h')
|}.
Next Obligation. Proof. Admitted.
Lemma Algebra_Hom_eq_simplify (alg alg' : Algebra)
(ah ah' : Algebra_Hom alg alg')
: (Alg_map ah) = (Alg_map ah') -> ah = ah'.
Proof. Admitted.
Program Definition Algebra_Hom_id (alg : Algebra) : Algebra_Hom alg alg :=
{|
Alg_map := id
|}.
Next Obligation. Admitted.
Definition Algebra_Cat : Category :=
{|
Obj := Algebra;
Hom := Algebra_Hom;
compose := @Algebra_Hom_compose;
id := Algebra_Hom_id;
|}.
End Algebras.
Arguments Alg_Carrier {_ _} _.
Arguments Constructors {_ _} _.
Arguments Algebra_Hom {_ _} _ _.
Arguments Alg_map {_ _ _ _} _.
Arguments Alg_map_com {_ _ _ _} _.
Arguments Algebra_Hom_id {_ _} _.
Section CoAlgebras.
Context {C : Category}.
Definition CoAlgebra (T : Functor C C) :=
@Algebra (Opposite C) (Opposite_Functor T).
Definition CoAlgebra_Hom {T : Functor C C} :=
@Algebra_Hom (Opposite C) (Opposite_Functor T).
Definition CoAlgebra_Hom_id {T : Functor C C} :=
@Algebra_Hom_id (Opposite C) (Opposite_Functor T).
Definition CoAlgebra_Cat (T : Functor C C) :=
@Algebra_Cat (Opposite C) (Opposite_Functor T).
End CoAlgebras.
Program Definition Type_Cat : Category :=
{|
Obj := Type;
Hom := (fun A B => A -> B);
compose := fun A B C (g : A -> B) (h : B -> C) => fun (x : A) => h (g x);
id := fun A => fun x => x
|}.
Local Obligation Tactic := idtac.
Program Definition Prod_Cat (C C' : Category) : Category :=
{|
Obj := C * C';
Hom :=
fun a b =>
((Hom (fst a) (fst b)) * (Hom (snd a) (snd b)))%type;
compose :=
fun a b c f g =>
((compose (fst f) (fst g)), (compose (snd f)(snd g)));
id := fun c => (id, id)
|}.
Class Terminal (C : Category) : Type :=
{
terminal : C;
t_morph : forall (d : Obj), Hom d terminal;
t_morph_unique : forall (d : Obj) (f g : (Hom d terminal)), f = g
}.
Arguments terminal {_} _.
Arguments t_morph {_} _ _.
Arguments t_morph_unique {_} _ _ _ _.
Coercion terminal : Terminal >-> Obj.
Definition Initial (C : Category) := Terminal (Opposite C).
Existing Class Initial.
Record Product {C : Category} (c d : C) : Type :=
{
product : C;
Pi_1 : Hom product c;
Pi_2 : Hom product d;
Prod_morph_ex : forall (p' : Obj) (r1 : Hom p' c) (r2 : Hom p' d), (Hom p' product);
}.
Arguments Product _ _ _, {_} _ _.
Arguments Pi_1 {_ _ _ _}, {_ _ _} _.
Arguments Pi_2 {_ _ _ _}, {_ _ _} _.
Arguments Prod_morph_ex {_ _ _} _ _ _ _.
Coercion product : Product >-> Obj.
Definition Has_Products (C : Category) : Type := forall a b, Product a b.
Existing Class Has_Products.
Program Definition Prod_Func (C : Category) {HP : Has_Products C}
: Functor (Prod_Cat C C) C :=
{|
FO := fun x => HP (fst x) (snd x);
FA := fun a b f => Prod_morph_ex _ _ (compose Pi_1 (fst f)) (compose Pi_2 (snd f))
|}.
Arguments Prod_Func _ _, _ {_}.
Definition Sum (C : Category) := @Product (Opposite C).
Arguments Sum _ _ _, {_} _ _.
Definition Has_Sums (C : Category) : Type := forall (a b : C), (Sum a b).
Existing Class Has_Sums.
Program Definition sum_Sum (A B : Type) : (@Sum Type_Cat A B) :=
{|
product := (A + B)%type;
Prod_morph_ex :=
fun (p' : Type)
(r1 : A -> p')
(r2 : B -> p')
(X : A + B) =>
match X return p' with
| inl a => r1 a
| inr b => r2 b
end
|}.
Next Obligation. simpl; auto. Defined.
Next Obligation. simpl; auto. Defined.
Program Instance Type_Cat_Has_Sums : Has_Sums Type_Cat := sum_Sum.
Definition Sum_Func {C : Category} {HS : Has_Sums C} :
Functor (Prod_Cat C C) C := Opposite_Functor (Prod_Func (Opposite C) HS).
Arguments Sum_Func _ _, _ {_}.
Program Instance unit_Type_term : Terminal Type_Cat :=
{
terminal := unit;
t_morph := fun _ _=> tt
}.
Next Obligation. Proof. Admitted.
Program Definition term_id : Functor Type_Cat (Prod_Cat Type_Cat Type_Cat) :=
{|
FO := fun a => (@terminal Type_Cat _, a);
FA := fun a b f => (@id _ (@terminal Type_Cat _), f)
|}.
Definition S_nat_func : Functor Type_Cat Type_Cat :=
Functor_compose term_id (Sum_Func Type_Cat _).
Definition S_nat_alg_cat := Algebra_Cat S_nat_func.
CoInductive CoNat : Set :=
| CoO : CoNat
| CoS : CoNat -> CoNat
.
Definition S_nat_coalg_cat := @CoAlgebra_Cat Type_Cat S_nat_func.
Set Printing Universes.
Program Definition CoNat_alg_term : Initial S_nat_coalg_cat :=
{|
terminal := _;
t_morph := _
|}.
Next Obligation. Admitted.
Next Obligation. Admitted.
Axiom Admit : False.
Next Obligation.
Proof.
intros d f g.
assert(H1 := (@Alg_map_com _ _ _ _ f)). clear.
assert (inl tt = inr tt) by (exfalso; apply Admit).
discriminate.
all: exfalso; apply Admit.
Show Universes.
Qed.
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