(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(* This contribution was updated for Coq V5.10 by the COQ workgroup.        *)
(* January 1995                                                             *)
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(*                                  ex1.v                                   *)
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intros A H'; exact H'.
Qed.
intros A B H'; split.
elim H'; intros H'0 H'1; clear H'; exact H'1.
elim H'; intros H'0 H'1; clear H'; exact H'0.
Qed.
intros A B h; elim h; [ intro H'; clear h; try exact H' | clear h; intro H' ].
right; assumption.
left; assumption.
Qed.
intros A B H' H'0.
apply H'0.
exact H'.
Qed.
intros A B C H' H'0 H'1.
apply H'.
exact H'1.
apply H'0.
exact H'1.
Qed.
intros x y z t h; elim h; intros H' H'0; clear h.
intro h; elim h; intros H'1 H'2; clear h.
split.
apply H'; assumption.
apply H'0; assumption.
Qed.
intros p q a H' H'0.
apply H'0.
exact H'.
Qed.
intros a b p h; elim h; intro H'; clear h.
exists a; assumption.
exists b; assumption.
Qed.

Theorem Simple :
forall (A : Set) (R : A -> A -> Prop),
(forall x y z : A, R x y /\ R y z -> R x z) ->
(forall x y : A, R x y -> R y x) ->
forall x : A, (exists y : A, R x y) -> R x x.