update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / scpes / scpes_scpes.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/scpes/scpes_scpes.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/scpes/scpes_scpes.etc
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/scpes/scpes_scpes.etc
@@ -0,0 +1,69 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/computation/scpds_scpds.ma".
+include "basic_2/equivalence/scpes.ma".
+
+(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma scpes_inv_abst2: ∀h,o,a,G,L,T1,T2,W2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] ⓛ{a}W2.T2 →
+                       ∃∃W,T. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] ⓛ{a}W.T & ⦃G, L⦄ ⊢ W2 ➡* W & 
+                              ⦃G, L.ⓛW2⦄ ⊢ T2 •*➡*[h, o, d2] T.
+#h #o #a #G #L #T1 #T2 #W2 #d1 #d2 * #T0 #HT10 #H
+elim (scpds_inv_abst1 … H) -H #W #T #HW2 #HT2 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma scpes_refl: ∀h,o,G,L,T,d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
+                  ⦃G, L⦄ ⊢ T •*⬌*[h, o, d2, d2] T.
+#h #o #G #L #T #d1 #d2 #Hd21 #Hd1
+elim (da_lstas … Hd1 … d2) #U #HTU #_
+/3 width=3 by scpds_div, lstas_scpds/
+qed.
+
+lemma lstas_scpes_trans: ∀h,o,G,L,T1,d0,d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d0 → d1 ≤ d0 →
+                         ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+                         ∀T2,d,d2. ⦃G, L⦄ ⊢ T •*⬌*[h,o,d,d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,o,d1+d,d2] T2.
+#h #o #G #L #T1 #d0 #d1 #Hd0 #Hd10 #T #HT1 #T2 #d #d2 *
+/3 width=3 by scpds_div, lstas_scpds_trans/ qed-.
+
+(* Properties on parallel computation for terms *****************************)
+
+lemma cprs_scpds_div: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T →
+                      ∀d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d →
+                      ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T →
+                      ⦃G, L⦄⊢ T1 •*⬌*[h, o, 0, d2] T2.
+#h #o #G #L #T1 #T #HT1 #d #Hd elim (da_lstas … Hd 0)
+#X1 #HTX1 #_ elim (cprs_strip … HT1 X1) -HT1
+/3 width=5 by scpds_strap1, scpds_div, lstas_cpr, ex4_2_intro/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem scpes_trans: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
+                     ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
+#h #o #G #L #T1 #T #d1 #d * #X1 #HT1X1 #HTX1 #T2 #d2 * #X2 #HTX2 #HT2X2
+elim (scpds_conf_eq … HTX1 … HTX2) -T -d /3 width=5 by scpds_cprs_trans, scpds_div/
+qed-.
+
+theorem scpes_canc_sn: ∀h,o,G,L,T,T1,d,d1. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d1] T1 →
+                       ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
+/3 width=4 by scpes_trans, scpes_sym/ qed-.
+
+theorem scpes_canc_dx: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
+                       ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*⬌*[h, o, d2, d] T → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
+/3 width=4 by scpes_trans, scpes_sym/ qed-.