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(* 02110-1301 USA                                                     *)


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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	            Delta Functor (used in Limits)             		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export Functor.

Set Implicit Arguments.
Unset Strict Implicit.

(* Const(b) : A->B 
   Const(b)(a) = b 
   Const(b)(f) = Id(b) *)

Section constFun.

Variables (A B : Category) (b : B).

Definition Const_ob (a : A) := b.

 Section const_map_def.

 Variable a1 a2 : A.

 Definition Const_mor (f : a1 --> a2) := Id b.

 Lemma Const_mor_map_law : Map_law Const_mor.
 Proof.
(* Goal: @Map_law (@Hom A a1 a2) (@Hom B b b) Const_mor *)
 unfold Map_law, Const_mor in |- *.
(* Goal: forall (x y : Carrier (@Hom A a1 a2)) (_ : @Equal (@Hom A a1 a2) x y), @Equal (@Hom B b b) (@Id B b) (@Id B b) *)
 intros f g H; apply Refl.
 Qed.

 Canonical Structure Const_map :
   Map (a1 --> a2) (Const_ob a1 --> Const_ob a1) := Const_mor_map_law.

 End const_map_def.

Lemma Const_comp_law : Fcomp_law Const_map.
Proof.
(* Goal: @Fcomp_law A B Const_ob Const_map *)
unfold Fcomp_law, Const_map, Const_mor in |- *; simpl in |- *.
(* Goal: forall (a b0 c : Ob A) (_ : Carrier (@Hom A a b0)) (_ : Carrier (@Hom A b0 c)), @Equal (@Hom B (Const_ob a) (Const_ob c)) (@Id B b) (@Comp B (Const_ob a) (Const_ob b0) (Const_ob c) (@Id B b) (@Id B b)) *)
intros a1 a2 a3 f g; apply Idr.
Qed.

Lemma Const_id_law : Fid_law Const_map.
Proof.
(* Goal: @Fid_law A B Const_ob Const_map *)
unfold Fid_law, Const_map, Const_mor, Const_ob in |- *; simpl in |- *.
(* Goal: forall _ : Ob A, @Equal (@Hom B b b) (@Id B b) (@Id B b) *)
intros; apply Refl.
Qed.

Canonical Structure Const := Build_Functor Const_comp_law Const_id_law.

End constFun.
