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(* 02110-1301 USA                                                     *)


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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                       Terminal Object in SET          		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export SET.
Require Export Single.
Require Export CatProperty.

Set Implicit Arguments.
Unset Strict Implicit.

(* le(s) setoid(s) singleton  est terminal *)
(* preuve que Single est terminal *)

Section set_terminal.

Variable S : Setoid.

Definition To_Single_fun (x : S) := Elt.
        
Lemma To_Single_map_law : Map_law To_Single_fun.
Proof.
(* Goal: @Map_law S Single To_Single_fun *)
unfold Map_law in |- *.
(* Goal: forall (x y : Carrier S) (_ : @Equal S x y), @Equal Single (To_Single_fun x) (To_Single_fun y) *)
intros; exact I.
Qed.

Definition To_Single := Build_Map To_Single_map_law.

End set_terminal.

Lemma Single_is_Terminal : IsTerminal To_Single.
Proof.
(* Goal: @IsTerminal SET Single To_Single *)
red in |- *; intros a f; simpl in |- *; unfold Ext in |- *; intro x; exact I. 
Defined.

Definition SET_Terminal : Terminal SET := Single_is_Terminal.