minor corrections and updates

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / list_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/list_eq.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/list_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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2/notation/relations/ringeq_3.ma".
+include "ground_2/lib/list.ma".
+
+(* EXTENSIONAL EQUIVALENCE OF LISTS *****************************************)
+
+rec definition eq_list A (l1,l2:list A) on l1 ≝
+match l1 with
+[ nil        ⇒
+  match l2 with
+  [ nil      ⇒ ⊤
+  | cons _ _ ⇒ ⊥
+  ]
+| cons a1 l1 ⇒
+  match l2 with
+  [ nil        ⇒ ⊥
+  | cons a2 l2 ⇒ a1 = a2 ∧ eq_list A l1 l2
+  ]
+].
+
+interpretation "extensional equivalence (list)"
+   'RingEq A l1 l2 = (eq_list A l1 l2).
+
+(* Basic properties *********************************************************)
+
+lemma eq_list_refl (A): reflexive … (eq_list A).
+#A #l elim l -l /2 width=1 by conj/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem eq_eq_list (A,l1,l2): l1 = l2 → l1 ≗{A} l2.
+// qed.
+
+(* Main inversion propertiess ***********************************************)
+
+theorem eq_list_inv_eq (A,l1,l2): l1 ≗{A} l2 → l1 = l2.
+#A #l1 elim l1 -l1 [| #a1 #l1 #IH ] *
+[ //
+| #a2 #l2 #H elim H
+| #H elim H
+| #a2 #l2 * #Ha #Hl /3 width=1 by eq_f2/
+]
+qed-.