milestone update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / lprs_cpms.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lprs_cpms.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 "basic_2/rt_computation/lprs_lpr.ma".
+
+(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
+
+(* Properties with t-bound context-sensitive rt-computarion for terms *******)
+
+lemma lprs_cpms_trans (n) (h) (G):
+                      ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 →
+                      ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+#n #h #G #L2 #T1 #T2 #HT12 #L1 #H
+@(lprs_ind_sn … H) -L1 /2 width=3 by lpr_cpms_trans/
+qed-.
+
+lemma lprs_cpm_trans (n) (h) (G):
+                     ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
+                     ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+/3 width=3 by lprs_cpms_trans, cpm_cpms/ qed-.
+
+(* Basic_2A1: includes cprs_bind2 *)
+lemma cpms_bind_dx (n) (h) (G) (L):
+                   ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
+                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
+                   ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+/4 width=5 by lprs_cpms_trans, lprs_pair, cpms_bind/ qed.
+
+(* Inversion lemmas with t-bound context-sensitive rt-computarion for terms *)
+
+(* Basic_1: was: pr3_gen_abst *)
+(* Basic_2A1: includes: cprs_inv_abst1 *)
+(* Basic_2A1: uses: scpds_inv_abst1 *)
+lemma cpms_inv_abst_sn (n) (h) (G) (L):
+                       ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 →
+                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 &
+                                X2 = ⓛ{p}V2.T2.
+#n #h #G #L #p #V1 #T1 #X2 #H
+@(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/
+#n1 #n2 #X #X2 #_ * #V #T #HV1 #HT1 #H1 #H2 destruct
+elim (cpm_inv_abst1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H2 destruct
+/5 width=7 by lprs_cpm_trans, lprs_pair, cprs_step_dx, cpms_trans, ex3_2_intro/
+qed-.
+
+(* Basic_2A1: includes: cprs_inv_abst *)
+lemma cpms_inv_abst_bi (n) (h) (G) (L):
+                       ∀p,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[n, h] ⓛ{p}W2.T2 →
+                       ∧∧ ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
+#n #h #G #L #p #W1 #W2 #T1 #T2 #H
+elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct
+/2 width=1 by conj/
+qed-.
+
+(* Basic_1: was pr3_gen_abbr *)
+(* Basic_2A1: includes: cprs_inv_abbr1 *)
+lemma cpms_inv_abbr_sn (n) (h) (G) (L):
+                       ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n, h] X2 →
+                       ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T2 & X2 = ⓓ{p}V2.T2
+                        | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
+#n #h #G #L #p #V1 #T1 #X2 #H
+@(cpms_ind_dx … H) -X2 -n /3 width=5 by ex3_2_intro, or_introl/
+#n1 #n2 #X #X2 #_ * *
+[ #V #T #HV1 #HT1 #H #HX2 destruct
+  elim (cpm_inv_abbr1 … HX2) -HX2 *
+  [ #V2 #T2 #HV2 #HT2 #H destruct
+    /6 width=7 by lprs_cpm_trans, lprs_pair, cprs_step_dx, cpms_trans, ex3_2_intro, or_introl/
+  | #T2 #HT2 #HXT2 #Hp
+    /6 width=7 by lprs_cpm_trans, lprs_pair, cpms_trans, ex3_intro, or_intror/
+  ]
+| #T #HT1 #HXT #Hp #HX2
+  elim (cpm_lifts_sn … HX2 (Ⓣ) … (L.ⓓV1) … HXT) -X
+  /4 width=3 by cpms_step_dx, drops_refl, drops_drop, ex3_intro, or_intror/
+]
+qed-.
+
+(* Basic_2A1: uses: scpds_inv_abbr_abst *)
+lemma cpms_inv_abbr_abst (n) (h) (G) (L):
+                         ∀p1,p2,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
+                         ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
+#n #h #G #L #p1 #p2 #V1 #W2 #T1 #T2 #H
+elim (cpms_inv_abbr_sn … H) -H *
+[ #V #T #_ #_ #H destruct
+| /2 width=3 by ex3_intro/
+]
+qed-.