update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_pat_eq.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_pat_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/relocation/pr_tl_eq.ma".
+include "ground/relocation/pr_pat_lt.ma".
+
+(* POSITIVE APPLICATION FOR PARTIAL RELOCATION MAPS *************************)
+
+(* Constructions with pr_eq *************************************************)
+
+(*** at_eq_repl_back *)
+corec lemma pr_pat_eq_repl_back (i1) (i2):
+            pr_eq_repl_back (λf. @❪i1,f❫ ≘ i2).
+#f1 * -f1 -i1 -i2
+[ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
+  cases (pr_eq_inv_push_sn … H12 … H) -g1 /2 width=2 by pr_pat_refl/
+| #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
+  cases (pr_eq_inv_push_sn … H12 … H) -g1 /3 width=7 by pr_pat_push/
+| #f1 #i1 #i2 #Hf1 #g1 #j2 #H #H2 #f2 #H12
+  cases (pr_eq_inv_next_sn … H12 … H) -g1 /3 width=5 by pr_pat_next/
+]
+qed-.
+
+(*** at_eq_repl_fwd *)
+lemma pr_pat_eq_repl_fwd (i1) (i2):
+      pr_eq_repl_fwd (λf. @❪i1,f❫ ≘ i2).
+#i1 #i2 @pr_eq_repl_sym /2 width=3 by pr_pat_eq_repl_back/
+qed-.
+
+lemma pr_pat_eq (f): ⫯f ≡ f → ∀i. @❪i,f❫ ≘ i.
+#f #Hf #i elim i -i
+[ /3 width=3 by pr_pat_eq_repl_back, pr_pat_refl/
+| /3 width=7 by pr_pat_eq_repl_back, pr_pat_push/
+]
+qed.
+
+(* Inversions with pr_eq ****************************************************)
+
+corec lemma pr_pat_inv_eq (f):
+            (∀i. @❪i,f❫ ≘ i) → ⫯f ≡ f.
+#Hf
+lapply (Hf (𝟏)) #H
+lapply (pr_pat_des_id … H) -H #H
+cases H in Hf; -H #Hf
+@pr_eq_push [3:|*: // ]
+/3 width=7 by pr_pat_inv_succ_push_succ/
+qed-.