(* 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_auto.v                                *)
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auto.
Qed.
intros A B H'; elim H'; auto.
Qed.
intros A B H'; elim H'; auto.
Qed.
auto.
Qed.
auto.
Qed.
intros x y z t H'; elim H'.
intros H'0 H'1 H'2; elim H'2.
auto.
Qed.
auto.
Qed.
intros a b p h; elim h;
[ intro H'; clear h; try exact H' | clear h; intro H' ].
exists a; try assumption.
exists b; try 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.