- updated equivalence on referred entries: it nust be degree-based

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / lfdeq_lfdeq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.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 "basic_2/syntax/tdeq_tdeq.ma".
+include "basic_2/static/lfxs_lfxs.ma".
+include "basic_2/static/lfdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Main properties **********************************************************)
+
+theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
+                    L1 ≡[h, o, V1] L2 → L1.ⓑ{I}V1 ≡[h, o, T] L2.ⓑ{I}V2 →
+                    L1 ≡[h, o, ⓑ{p,I}V1.T] L2.
+/2 width=2 by lfxs_bind/ qed.
+
+theorem lfdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, V] L2 → L1 ≡[h, o, T] L2 →
+                    L1 ≡[h, o, ⓕ{I}V.T] L2.
+/2 width=1 by lfxs_flat/ qed.
+
+theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T).
+#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (frees_tdeq_conf_lexs … Hf1 T … HL1) // #H0
+lapply (frees_mono … Hf2 … H0) -Hf2 -H0
+/4 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ex2_intro/
+qed-.
+
+theorem lfdeq_canc_sn: ∀h,o,T. left_cancellable … (lfdeq h o T).
+/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
+
+theorem lfdeq_canc_dx: ∀h,o,T. right_cancellable … (lfdeq h o T).
+/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
+
+(* Advanced properies on negated lazy equivalence *****************************)
+
+(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred ************) 
+lemma lfdeq_nlfdeq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L →
+                          ∀L2. (L ≡[h, o, T] L2 → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
+/3 width=3 by lfdeq_canc_sn/ qed-.
+
+lemma nlfdeq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L →
+                        ∀L1. (L1 ≡[h, o, T] L → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
+/3 width=3 by lfdeq_trans/ qed-.