ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / lib / ltc_ctc.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/lib/ltc_ctc.ma b/matita/matita/contribs/lambdadelta/ground/lib/ltc_ctc.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/lib/star.ma".
+include "ground/lib/ltc.ma".
+
+(* LABELLED TRANSITIVE CLOSURE **********************************************)
+
+alias symbol "subseteq" = "relation inclusion".
+
+(* Properties with contextual transitive closure ****************************)
+
+lemma ltc_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
+              left_identity … f i →
+              ∀c. CTC … (λc. R c i) c ⊆ ltc … f … (R c) i.
+#C #A #i #f #B #R #Hf #c #b1 #b2 #H elim H -b2 /2 width=1 by ltc_rc/
+#b #b2 #_ #Hb2 #IH >(Hf i) -Hf /2 width=3 by ltc_dx/
+qed.
+
+(* Inversion lemmas with contextual transitive closure **********************)
+
+lemma ltc_inv_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
+                  associative … f → annulment_2 … f i →
+                  ∀c. ltc … f … (R c) i ⊆ CTC … (λc. R c i) c.
+#C #A #i #f #B #R #H1f #H2f #c #b1 #b2
+@(insert_eq_0 … i) #a #H
+@(ltc_ind_dx A f B … H) -a -b2 /2 width=1 by inj/ -H1f
+#a1 #a2 #b #b2 #_ #IH #Hb2 #H <H
+elim (H2f … H) -H2f -H #H1 #H2 destruct
+/3 width=3 by step/
+qed-.