Rename asym_equiv -> asym_equiv_alt.

Add antisym_equiv, asym_equiv, conn_equiv, conn_alt_equiv, str_conn_equiv.

FEM/Compl/Binary_relation.v

@@ -315,6 +315,44 @@ Hypothesis HT : eq_dec T.
 
 Variable R : T -> T -> Prop.
 
+Lemma antisym_equiv :
+  antisymmetric R <-> forall x y, br_and_conv R x y -> x = y.
+Proof.
+split; intros H x y.
+intros [H1 H2]; apply H; easy.
+intros H1 H2; apply H; easy.
+Qed.
+
+Lemma asym_equiv : asymmetric R <-> forall x y, ~ (br_and_conv R x y).
+Proof.
+split; intros H x y.
+intros [H1 H2]; apply (H x y); easy.
+intros H1 H2; apply (H x y); easy.
+Qed.
+
+Lemma conn_equiv : connected R <-> forall x y, x <> y -> comparable R x y.
+Proof. easy. Qed.
+
+Lemma conn_contra_equiv :
+  connected_contra R <->
+  forall x y, br_and_conv (complementary R) x y -> x = y.
+Proof.
+split; intros H x y.
+intros [H1 H2]; apply H; easy.
+intros H1 H2; apply H; easy.
+Qed.
+
+Lemma conn_alt_equiv :
+  connected_alt R <-> forall x y, comparable R x y \/ x = y.
+Proof.
+split; intros H x y.
+destruct (H x y) as [H1 | [H1 | H1]]; [left; left | right | left; right]; easy.
+destruct (H x y) as [[H1 | H1] | H1]; tauto.
+Qed.
+
+Lemma str_conn_equiv : strongly_connected R <-> forall x y, comparable R x y.
+Proof. easy. Qed.
+
 Lemma asym_antisym : asymmetric R -> antisymmetric R.
 Proof. intros H x y H1 H2. exfalso; apply (H _ _ H1 H2). Qed.
 
@@ -336,7 +374,7 @@ Proof.
 intros H; split; [apply irrefl_asym_w_trans; easy | apply asym_irrefl].
 Qed.
 
-Lemma asym_equiv : asymmetric R <-> irreflexive R /\ antisymmetric R.
+Lemma asym_equiv_alt : asymmetric R <-> irreflexive R /\ antisymmetric R.
 Proof.
 split; intros.
 split; [apply asym_irrefl | apply asym_antisym]; easy.
@@ -576,6 +614,13 @@ Lemma trans_incompar_rev :
     transitive (incomparable R) -> transitive R.
 Proof.
 intros R1 H1 H2 H3 x y z; apply imp3_imp_equiv; intros H4.
+destruct (H2 x y) as [H5 | H5], (H2 y z) as [H6 | H6]; try tauto.
+apply not_and_equiv; intros [H7 H8].
+specialize (H3 x y z).
+rewrite imp3_imp_equiv in H3. unfold incomparable in H3.
+rewrite -!not_or_equiv in H3. rewrite !NNPP_equiv in H3.
+apply H5.
+
 Aglopted. *)
 
 (** With the br_and_conv operation. *)