milestone update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_coafter_coafter_ist.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_coafter_coafter_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/relocation/gr_pat_tls.ma".
+include "ground/relocation/gr_ist_tls.ma".
+include "ground/relocation/gr_coafter_nat_tls.ma".
+
+(* RELATIONAL CO-COMPOSITION FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(*** H_coafter_inj *)
+definition H_gr_coafter_inj: predicate gr_map ≝
+           λf1. 𝐓❪f1❫ →
+           ∀f,f21,f22. f1 ~⊚ f21 ≘ f → f1 ~⊚ f22 ≘ f → f21 ≡ f22.
+
+(* Main Forward lemmas with istot ************************************************)
+
+(*** coafter_inj_O_aux *)
+corec fact gr_coafter_inj_unit_aux:
+           ∀f1. @❪𝟏, f1❫ ≘ 𝟏 → H_gr_coafter_inj f1.
+#f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+cases (gr_pat_inv_unit_bi … H1f1) -H1f1 [ |*: // ] #g1 #H1
+lapply (gr_ist_inv_push … H2f1 … H1) -H2f1 #H2g1
+cases (H2g1 (𝟏)) #n #Hn
+cases (gr_coafter_inv_push_sn … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
+[ cases (gr_coafter_inv_push_sn_push … H2f … H1 H) -f1 -f #g22 #H2g #H22
+  @(gr_eq_push … H21 H22) -f21 -f22
+| cases (gr_coafter_inv_push_sn_next … H2f … H1 H) -f1 -f #g22 #H2g #H22
+  @(gr_eq_next … H21 H22) -f21 -f22
+]
+@(gr_coafter_inj_unit_aux (⫱*[↓n]g1) … (⫱*[↓n]g)) -gr_coafter_inj_unit_aux
+/2 width=1 by gr_coafter_tls_bi_tls, gr_ist_tls, gr_pat_unit_succ_tls/
+qed-.
+
+(*** coafter_inj_aux *)
+fact gr_coafter_inj_aux:
+     (∀f1. @❪𝟏, f1❫ ≘ 𝟏 → H_gr_coafter_inj f1) →
+     ∀i2,f1. @❪𝟏, f1❫ ≘ i2 → H_gr_coafter_inj f1.
+#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
+#i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
+elim (gr_pat_inv_unit_succ … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
+elim (gr_coafter_inv_next_sn … H1f … H1) -H1f #g #H1g #H
+lapply (gr_coafter_inv_next_sn_push … H2f … H1 H) -f #H2g
+/3 width=6 by gr_ist_inv_next/
+qed-.
+
+(*** coafter_inj *)
+theorem gr_coafter_inj:
+        ∀f1. H_gr_coafter_inj f1.
+#f1 #H cases (H (𝟏)) /3 width=7 by gr_coafter_inj_aux, gr_coafter_inj_unit_aux/
+qed-.