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14
15 include "higher_order_defs/relations.ma".
16
17 theorem reflexive_iff: reflexive ? iff.
18  unfold reflexive;
19  intro;
20  split;
21  intro;
22  assumption.
23 qed.
24
25 theorem symmetric_iff: symmetric ? iff.
26  unfold symmetric;
27  intros;
28  elim H;
29  split;
30  assumption.
31 qed.
32
33 theorem transitive_iff: transitive ? iff.
34  unfold transitive;
35  intros;
36  elim H;
37  elim H1;
38  split;
39  intro;
40  autobatch.
41 qed.
42