update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_after_after_ist.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_after_after_ist.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/arith/nat_pred_succ.ma".
+include "ground/relocation/pr_pat_tls.ma".
+include "ground/relocation/pr_ist_tls.ma".
+include "ground/relocation/pr_after_pat_tls.ma".
+
+(* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
+
+(*** H_after_inj *)
+definition H_pr_after_inj: predicate pr_map ≝
+           λf1. 𝐓❪f1❫ →
+           ∀f,f21,f22. f1 ⊚ f21 ≘ f → f1 ⊚ f22 ≘ f → f21 ≡ f22.
+
+(* Main destructions with pr_ist ********************************************)
+
+(*** after_inj_O_aux *)
+corec fact pr_after_inj_unit_aux:
+           ∀f1. @❪𝟏, f1❫ ≘ 𝟏 → H_pr_after_inj f1.
+#f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+cases (pr_pat_inv_unit_bi … H1f1) -H1f1 [|*: // ] #g1 #H1
+lapply (pr_ist_inv_push … H2f1 … H1) -H2f1 #H2g1
+cases (H2g1 (𝟏)) #p #Hp
+cases (pr_after_inv_push_sn … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
+[ cases (pr_after_inv_push_sn_push … H2f … H1 H) -f1 -f #g22 #H2g #H22
+  @(pr_eq_push … H21 H22) -f21 -f22
+| cases (pr_after_inv_push_sn_next … H2f … H1 H) -f1 -f #g22 #H2g #H22
+  @(pr_eq_next … H21 H22) -f21 -f22
+]
+@(pr_after_inj_unit_aux (⫰*[↓p]g1) … (⫰*[↓p]g)) -pr_after_inj_unit_aux
+/2 width=1 by pr_after_tls_sn_tls, pr_ist_tls, pr_pat_unit_succ_tls/
+qed-.
+
+(*** after_inj_aux *)
+fact pr_after_inj_aux:
+     (∀f1. @❪𝟏, f1❫ ≘ 𝟏 → H_pr_after_inj f1) →
+     ∀i2,f1. @❪𝟏, f1❫ ≘ i2 → H_pr_after_inj f1.
+#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
+#i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+elim (pr_pat_inv_unit_succ … H1f1) -H1f1 [|*: // ] #g1 #H1g1 #H1
+elim (pr_after_inv_next_sn … H1f … H1) -H1f #g #H1g #H
+lapply (pr_after_inv_next_sn_next … H2f … H1 H) -f #H2g
+/3 width=6 by pr_ist_inv_next/
+qed-.
+
+(*** after_inj *)
+theorem pr_after_inj:
+        ∀f1. H_pr_after_inj f1.
+#f1 #H cases (H (𝟏))
+/3 width=7 by pr_after_inj_aux, pr_after_inj_unit_aux/
+qed-.