Require Export IdCAT.
Require Export Category_dup1.

Set Implicit Arguments.
Unset Strict Implicit.


Lemma Comp_Functor_congl : Congl_law' Comp_Functor.
Proof.
unfold Congl_law' in |- *; simpl in |- *.
unfold Equal_Functor, Comp_Functor in |- *.
intros A B C F G H eq_FG a1 a2 f.
unfold FMor in |- *; simpl in |- *.
unfold Comp_FOb, Comp_FMor in |- *.
apply eq_FG.
Qed.

Lemma Comp_Functor_congr : Congr_law' Comp_Functor.
Proof.
unfold Congr_law' in |- *; simpl in |- *.
unfold Equal_Functor, Comp_Functor in |- *.
intros A B C F G H eq_FG a1 a2 f.
unfold FMor in |- *; simpl in |- *.
unfold Comp_FOb, Comp_FMor in |- *.
elim (eq_FG a1 a2 f); intros g eq_Ff_g.
apply Build_Equal_hom; apply FPres; assumption.
Qed.

Definition Comp_CAT := Build_Comp' Comp_Functor_congl Comp_Functor_congr.

Lemma Assoc_CAT : Assoc_law' Comp_CAT.
Proof.
unfold Assoc_law', Cat_comp' in |- *; simpl in |- *.
intros C D E F G H X.
unfold Equal_Functor in |- *; simpl in |- *.
unfold Ap2 in |- *; simpl in |- *; auto.
Qed.


Lemma Idl_CAT : Idl_law' Comp_CAT Id_CAT.
Proof.
unfold Idl_law', Cat_comp', Ap2 in |- *; simpl in |- *.
intros C D G; unfold Comp_Functor, Equal_Functor in |- *.
intros a b f; unfold FMor in |- *; simpl in |- *.
unfold Comp_FMor in |- *; unfold Id_CAT, FMor in |- *; simpl in |- *.
auto.
Qed.

Lemma Idr_CAT : Idr_law' Comp_CAT Id_CAT.
Proof.
unfold Idr_law', Cat_comp', Ap2 in |- *; simpl in |- *.
intros C D G; unfold Comp_Functor, Equal_Functor in |- *.
intros a b f; unfold FMor in |- *; simpl in |- *.
unfold Comp_FMor in |- *; unfold Id_CAT, FMor in |- *; simpl in |- *.
auto.
Qed.

Canonical Structure CAT : Category' :=
  Build_Category' Assoc_CAT Idl_CAT Idr_CAT.