Fix {lex,colex}_trans.

Modify {symlex,revlex}_trans.

FEM/Compl/Binary_relation.v

@@ -933,36 +933,50 @@ Lemma revlexT_antisym :
   antisymmetric R -> forall n, antisymmetric (@revlexT _ R n).
 Proof. intros; apply antisym_conv, colexT_antisym; easy. Qed.
 
-(* This seems wrong! Maybe some hypotheses are missing... *)
 Lemma lex_trans :
-  forall P, transitive R -> forall n, transitive (@lex _ R P n).
+  antisymmetric R \/ asymmetric R -> transitive R ->
+  forall P n, transitive (@lex _ R P n).
 Proof.
-intros P H1 n x y z; induction n; [easy |].
-rewrite !lex_S; intros [[H2a H2b] | [H2 H3]] [[H4a H4b] | [H4 H5]].
-left; split; [admit (* WRONG? *) | apply H1 with (y ord0); easy].
-rewrite -H4; left; easy.
-rewrite H2; left; easy.
-right; split; [rewrite H2 | apply IHn with (skipF y ord0)]; easy.
-Admitted.
+intros H1 H2 P n x y z; induction n; [easy |].
+rewrite !lex_S; intros [[H3a H3b] | [H3 H4]] [[H5a H5b] | [H5 H6]].
+(* *)
+left; split; [| apply H2 with (y ord0); easy].
+destruct (HT (x ord0) (z ord0)) as [H4 |]; [exfalso | easy].
+rewrite H4 in H3b; destruct H1 as [H1 | H1].
+contradict H5a; apply H1; easy.
+contradict H5b; apply H1; easy.
+(* *)
+rewrite -H5; left; easy.
+rewrite H3; left; easy.
+right; split; [rewrite H3 | apply IHn with (skipF y ord0)]; easy.
+Qed.
 
-(* This seems wrong! Maybe some hypotheses are missing... *)
 Lemma colex_trans :
-  forall P, transitive R -> forall n, transitive (@colex _ R P n).
+  antisymmetric R \/ asymmetric R -> transitive R ->
+  forall P n, transitive (@colex _ R P n).
 Proof.
-intros P H1 n x y z; induction n; [easy |].
-rewrite !colex_S; intros [[H2a H2b] | [H2 H3]] [[H4a H4b] | [H4 H5]].
-left; split; [admit (* WRONG? *) | apply H1 with (y ord_max); easy].
-rewrite -H4; left; easy.
-rewrite H2; left; easy.
-right; split; [rewrite H2 | apply IHn with (skipF y ord_max)]; easy.
-Admitted.
+intros H1 H2 P n x y z; induction n; [easy |].
+rewrite !colex_S; intros [[H3a H3b] | [H3 H4]] [[H5a H5b] | [H5 H6]].
+(* *)
+left; split; [| apply H2 with (y ord_max); easy].
+destruct (HT (x ord_max) (z ord_max)) as [H4 |]; [exfalso | easy].
+rewrite H4 in H3b; destruct H1 as [H1 | H1].
+contradict H5a; apply H1; easy.
+contradict H5b; apply H1; easy.
+(* *)
+rewrite -H5; left; easy.
+rewrite H3; left; easy.
+right; split; [rewrite H3 | apply IHn with (skipF y ord_max)]; easy.
+Qed.
 
 Lemma symlex_trans :
-  forall P, transitive R -> forall n, transitive (@symlex _ R P n).
+  antisymmetric R \/ asymmetric R -> transitive R ->
+  forall P n, transitive (@symlex _ R P n).
 Proof. intros; apply trans_conv, lex_trans; easy. Qed.
 
 Lemma revlex_trans :
-  forall P, transitive R -> forall n, transitive (@revlex _ R P n).
+  antisymmetric R \/ asymmetric R -> transitive R ->
+  forall P n, transitive (@revlex _ R P n).
 Proof. intros; apply trans_conv, colex_trans; easy. Qed.
 
 Lemma lexF_conn_alt :