milestone update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / gr_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_eq.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_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/xoa/ex_3_2.ma".
+include "ground/notation/relations/ideq_2.ma".
+include "ground/lib/stream_eq.ma".
+include "ground/relocation/gr_map.ma".
+
+(* EXTENSIONAL EQUIVALENCE FOR GENERIC RELOCATION MAPS **********************)
+
+(*** eq *)
+coinductive gr_eq: relation gr_map ≝
+(*** eq_push *)
+| gr_eq_push (f1) (f2) (g1) (g2):
+  gr_eq f1 f2 → ⫯f1 = g1 → ⫯f2 = g2 → gr_eq g1 g2
+(*** eq_next *)
+| gr_eq_next (f1) (f2) (g1) (g2):
+  gr_eq f1 f2 → ↑f1 = g1 → ↑f2 = g2 → gr_eq g1 g2
+.
+
+interpretation
+  "extensional equivalence (generic relocation maps)"
+  'IdEq f1 f2 = (gr_eq f1 f2).
+
+(*** eq_repl *)
+definition gr_eq_repl (R:relation …) ≝
+           ∀f1,f2. f1 ≡ f2 → R f1 f2.
+
+(*** eq_repl_back *)
+definition gr_eq_repl_back (R:predicate …) ≝
+           ∀f1. R f1 → ∀f2. f1 ≡ f2 → R f2.
+
+(*** eq_repl_fwd *)
+definition gr_eq_repl_fwd (R:predicate …) ≝
+           ∀f1. R f1 → ∀f2. f2 ≡ f1 → R f2.
+
+(* Basic properties *********************************************************)
+
+(*** eq_sym *)
+corec lemma gr_eq_sym: symmetric … gr_eq.
+#f1 #f2 * -f1 -f2
+#f1 #f2 #g1 #g2 #Hf #H1 #H2
+[ @(gr_eq_push … H2 H1) | @(gr_eq_next … H2 H1) ] -g2 -g1 /2 width=1 by/
+qed-.
+
+(*** eq_repl_sym *)
+lemma gr_eq_repl_sym (R):
+      gr_eq_repl_back R → gr_eq_repl_fwd R.
+/3 width=3 by gr_eq_sym/ qed-.
+
+(* Alternative definition with stream_eq (specific) ***************************************************)
+
+alias symbol "subseteq" (instance 1) = "relation inclusion".
+
+corec lemma stream_eq_gr_eq: stream_eq … ⊆ gr_eq.
+* #b1 #f1 * #b2 #f2 #H
+cases (stream_eq_inv_cons_bi … H) -H [|*: // ] * -b2 #Hf
+cases b1 /3 width=5 by gr_eq_next, gr_eq_push/
+qed.
+
+corec lemma gr_eq_inv_stream_eq: gr_eq ⊆ stream_eq ….
+#g1 #g2 * -g1 -g2 #f1 #f2 #g1 #g2 #Hf * * -g1 -g2
+/3 width=1 by stream_eq_cons/
+qed-.