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(* 02110-1301 USA                                                     *)
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(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
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(*                                                                           *)
(*          Basic Category Theory: epic, monic, iso in SET                   *)
(*                                                                           *)
(*          Amokrane Saibi,  May 1994                                        *)
(*                                                                           *)
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Require Export SET.
Require Export CatProperty.
Require Export MapProperty.
Set Implicit Arguments.
Unset Strict Implicit.
(* Epic <-> Surj *)
SubClass SET_Epic_law (A B : Setoid) (f : Map A B) := Epic_law f.
SubClass SET_Monic_law (A B : Setoid) (f : Map A B) := Monic_law f.
Section set_epic_monic.
Variables (A B : Setoid) (f : Map A B).

(* Specifications: *)
Lemma Surj_epic : forall h : B -> A, Surj_law f h -> SET_Epic_law f.
(* On veut montrer que: \forall b\in B, il existe a\in A tq b =f(a) sachant que (\forall x\in A, g(f(x))=h(f(x))) => (\forall y\in B, g(y)=h(y)). raisonnement par l'absurde: supposons qu'un b n'a aucun ante'cedent. soient g et h tq: g(b) /= h(b) et g(z)=h(z) \forall z\in B. on a alors (\forall x\in A, g(f(x))=h(f(x))) mais (\forall y\in B, g(y)=h(y)) faux, contradiction: b a un antecedent  Lemma epic_surj : (SETisEpic A B f) -> (IsSurj f ?). *)
(* Monic <-> Inj *)
Lemma Inj_monic : Inj_law f -> SET_Monic_law f.
Section map_const_fun.