syntactic components detached from basic_2 become static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / ext2_tc.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/ext2_tc.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/ext2_tc.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_2/lib/star.ma".
+include "static_2/syntax/ext2.ma".
+
+(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
+
+(* Properties with transitive closure ***************************************)
+
+lemma ext2_tc_pair: ∀R,I,V1,V2. TC … R V1 V2 →
+                    TC … (ext2 R) (BPair I V1) (BPair I V2).
+#R #I #V1 #V2 #H elim H -H -V2
+/3 width=3 by ext2_pair, step, inj/
+qed.
+
+lemma ext2_tc_inj: ∀R,I1,I2. ext2 R I1 I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I2 * -I1 -I2
+/3 width=1 by ext2_unit, ext2_pair, inj/
+qed.
+
+(* Main properties with transitive closure **********************************)
+
+theorem ext2_tc_step: ∀R,I1,I. ext2 (TC … R) I1 I →
+                      ∀I2. ext2 R I I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I * -I1 -I
+[ #I #Z #H >(ext2_inv_unit_sn … H) -Z /2 width=1 by ext2_unit/
+| #I #V1 #V #HV1 #Z #H
+  elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
+  /3 width=3 by ext2_pair, step/
+]
+qed-.
+
+(* Advanced properties with transitive closure ******************************)
+
+lemma ext2_tc: ∀R,I1,I2. TC … (ext2 R) I1 I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I2 #H elim H -I2
+/2 width=3 by ext2_tc_step, ext2_tc_inj/
+qed.
+
+(* Advanced inversion lemmas with transitive closure ************************)
+
+lemma ext2_inv_tc: ∀R,I1,I2. ext2 (TC … R) I1 I2 → TC … (ext2 R) I1 I2.
+#R #I1 #I2 * -I1 -I2
+/3 width=1 by ext2_tc_pair, ext2_unit, inj/
+qed-.