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[helm.git] / helm / matita / contribs / PREDICATIVE-TOPOLOGY / class_eq.ma

diff --git a/helm/matita/contribs/PREDICATIVE-TOPOLOGY/class_eq.ma b/helm/matita/contribs/PREDICATIVE-TOPOLOGY/class_eq.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_eq".
+
+include "class_le.ma".
+
+theorem ceq_cl: \forall C,c1,c2. ceq ? c1 c2 \to cin C c1 \land cin C c2.
+intros; elim H; clear H.
+lapply cle_cl to H1 using H; clear H1; decompose H;
+lapply cle_cl to H2 using H; clear H2; decompose H.
+auto.
+qed.
+
+theorem ceq_refl: \forall C,c. cin C c \to ceq ? c c.
+intros; apply ceq_intro; auto.
+qed.
+
+theorem ceq_trans: \forall C,c2,c1,c3.
+                   ceq C c2 c3 \to ceq ? c1 c2 \to ceq ? c1 c3.
+intros; elim H; elim H1; clear H; clear H1.
+apply ceq_intro; apply cle_trans; [|auto|auto||auto|auto].
+qed.
+
+theorem ceq_sym: \forall C,c1,c2. ceq C c1 c2 \to ceq C c2 c1.
+intros; elim H; clear H.; auto.
+qed.