- second precommit for rtmap

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / etc / relocation / trace_sle.etc

diff --git a/matita/matita/contribs/lambdadelta/ground_2/etc/relocation/trace_sle.etc b/matita/matita/contribs/lambdadelta/ground_2/etc/relocation/trace_sle.etc
new file mode 100644 (file)
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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/relocation/trace_at.ma".
+
+(* RELOCATION TRACE *********************************************************)
+
+inductive sle: relation trace ≝
+| sle_empty: sle (◊) (◊)
+| sle_true : ∀t1,t2. sle t1 t2 → sle (Ⓣ @ t1) (Ⓣ @ t2)
+| sle_false: ∀t1,t2,b. sle t1 t2 → sle (Ⓕ @ t1) (b @ t2)
+.
+
+interpretation
+   "inclusion (trace)"
+   'subseteq t1 t2 = (sle t1 t2).
+
+(* Basic properties *********************************************************)
+
+(* Basic forward lemmas *****************************************************)
+
+lemma sle_fwd_length: ∀t1,t2. t1 ⊆ t2 → |t1| = |t2|.
+#t1 #t2 #H elim H -t1 -t2 //
+qed-.
+
+lemma sle_fwd_colength: ∀t1,t2. t1 ⊆ t2 → ∥t1∥ ≤ ∥t2∥.
+#t1 #t2 #H elim H -t1 -t2 /2 width=1 by le_S_S/
+#t1 #t2 * /2 width=1 by le_S/
+qed-.
+
+(* Inversion lemmas on application ******************************************)
+
+lemma sle_inv_at: ∀t1,t2. t1 ⊆ t2 →
+                  ∀i,i1,i2. @⦃i, t1⦄ ≡ i1 → @⦃i, t2⦄  ≡ i2 → i2 ≤ i1.
+#t1 #t2 #H elim H -t1 -t2
+[ #i #i1 #i2 #_ #H2 elim (at_inv_empty … H2) -H2 //
+| #t1 #t2 #_ #IH #i #i1 #i2 #H0 #H2 elim (at_inv_true … H2) -H2 * //
+  #j1 #j2 #H1 #H2 #Hj destruct elim (at_inv_true_succ_sn … H0) -H0
+  /3 width=3 by le_S_S/
+| #t1 #t2 * #_ #IH #i #i1 #i2 #H0 #H2
+  [ elim (at_inv_true … H2) -H2 * //
+    #j #j2 #H1 #H2 #Hj2 destruct elim (at_inv_false … H0) -H0
+    #j1 #H #Hj1 destruct elim (at_monotonic … Hj1 j) -Hj1 //
+    #x #H1x #H2x @le_S_S /4 width=3 by lt_to_le, le_to_lt_to_lt/ (**) (* full auto too slow *)
+  | elim (at_inv_false … H2) elim (at_inv_false … H0) -H0 -H2
+    /3 width=3 by le_S_S/
+  ]
+]  
+qed-.