From 0c8cad22457218f78c30de239e49187968251613 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 2 Feb 2012 18:29:52 +0000
Subject: [PATCH] - three lemmas on context sensitive parallel reduction closed

---
 .../lambda_delta/Basic_2/reducibility/cpr.ma  |  5 ---
 .../Basic_2/reducibility/cpr_lift.ma          | 40 ++++++++++++++++++
 .../Basic_2/reducibility/cpr_tpss.ma          | 42 +++++++++++++++++++
 3 files changed, 82 insertions(+), 5 deletions(-)
 create mode 100644 matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_tpss.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr.ma b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr.ma
index b3812e942..4edfc6456 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr.ma
@@ -89,8 +89,3 @@ qed-.
    Basic_1: removed local theorems 3:
             pr2_free_free pr2_free_delta pr2_delta_delta
 *)
-
-(*
-pr2/fwd pr2_gen_appl
-pr2/fwd pr2_gen_abbr
-*)
diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_lift.ma
index fcdcdcdd8..4c77444d4 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_lift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_lift.ma
@@ -48,6 +48,46 @@ lemma cpr_inv_abst1: ∀V1,T1,U2. ⓛV1. T1 ➡ U2 →
                      ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛV2. T2.
 /2 width=3/ qed-.
 
+(* Basic_1: was pr2_gen_appl *)
+lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
+                     ∨∨ ∃∃V2,T2.            L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
+                                            U2 = ⓐV2. T2
+                      | ∃∃V2,W,T1,T2.       L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+                                            U0 = ⓛW. T1 &
+                                            U2 = ⓓV2. T2
+                      | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
+                                            ⇧[0,1] V2 ≡ V &
+                                            U0 = ⓓW1. T1 &
+                                            U2 = ⓓW2. ⓐV. T2.
+#L #V1 #U0 #Y * #X #H1 #H2
+elim (tpr_inv_appl1 … H1) -H1 *
+[ #V #U #HV1 #HU0 #H destruct
+  elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
+| #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
+  elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+  lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
+| #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
+  elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
+  elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
+  elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
+  lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒[T1] →
+                            ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+                                     U = ⓐV2. T2.
+#L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5/
+| #V2 #W #W1 #W2 #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+| #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+]
+qed-.
+
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was: pr2_lift *)
diff --git a/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_tpss.ma b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_tpss.ma
new file mode 100644
index 000000000..2add29208
--- /dev/null
+++ b/matita/matita/contribs/lambda_delta/Basic_2/reducibility/cpr_tpss.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/unfold/ltpss_ltpss.ma".
+include "Basic_2/reducibility/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡ U2 →
+                     (∃∃V2,T2,T. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
+                                 L.  ⓓV2 ⊢ T [1, |L|] ▶* T2 &
+                                 U2 = ⓓV2. T
+                      ) ∨
+                      ∃∃T. ⇧[0,1] T ≡ T1 & L ⊢ T ➡ U2.
+#L #V1 #T1 #Y * #X #H1 #H2
+elim (tpr_inv_abbr1 … H1) -H1 *
+[ #V #T0 #T #HV1 #HT10 #HT0 #H destruct
+  elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
+  lapply (tps_lsubs_conf … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
+  elim (ltpss_tps_conf … HT0 (L. ⓓV2) 1 (|L|) ?) -HT0 /2 width=1/ #V0 #HV20 #HV0
+  lapply (tpss_weak_all … HV20) -HV20 #HV20
+  lapply (tpss_lsubs_conf … HV0 (L. ⓓV2) ?) -HV0 /2 width=1/ #HV0
+  elim (tpss_conf_eq … HT2 … HV0) -T #T3 #HT23 #HVT3
+  lapply (tpss_weak_all … HVT3) -HVT3 #HVT3
+  lapply (tpss_trans_eq … HV20 … HVT3) -V0 /4 width=7/
+| /4 width=5/
+]
+qed-.
-- 
2.39.2