- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / computation / cprs_cprs.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma
index 518c2bd07295a88765cbe26ed0a2785cfaf8971e..8f94b0fa7a9641c10abcbac0e4972984c1915494 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/computation/cprs_cprs.ma
@@ -14,30 +14,38 @@
 
 include "basic_2/reducibility/cpr_lift.ma".
 include "basic_2/reducibility/cpr_cpr.ma".
-include "basic_2/reducibility/lcpr_cpr.ma".
-include "basic_2/computation/cprs_lcpr.ma".
+include "basic_2/reducibility/lfpr_cpr.ma".
+include "basic_2/computation/cprs_lfpr.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
 
 (* Advanced properties ******************************************************)
 
-lemma cprs_abbr_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 →
-                    L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
-#L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind_dx … HT12) -T1
+lemma cprs_abst_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
+                    L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2.
+#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HT12) -T2
+[ /3 width=2/
+| /3 width=6 by cprs_strap1, cpr_abst/ (**) (* /3 width=6/ is too slow *)
+]
+qed.
+
+lemma cprs_abbr1_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 →
+                     ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
 [ /3 width=5/
 | #T1 #T #HT1 #_ #IHT1
   @(cprs_strap2 … IHT1) -IHT1 /2 width=1/
 ]
 qed.
 
-lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
-                L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+lemma cpr_abbr1: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
+                 ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
 /3 width=1/ qed.
 
 lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 →
-                 L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+                 ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
 #L #V1 #V2 #HV12 #T1 #T2 #HT12
-lapply (lcpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/
+lapply (lfpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/
 qed.
 
 (* Basic_1: was: pr3_strip *)
@@ -49,23 +57,23 @@ lemma cprs_strip: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡ T2 →
 
 (* Basic_1: was pr3_gen_appl *)
 lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 →
-                      ∨∨ ∃∃V2,T2.     L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
-                                      U2 = ⓐV2. T2
-                       | ∃∃V2,W,T.    L ⊢ V1 ➡* V2 &
-                                      L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ➡* U2
-                       | ∃∃V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
-                                      L ⊢ T1 ➡* ⓓV. T & L ⊢ ⓓV. ⓐV2. T ➡* U2.
+                      ∨∨ ∃∃V2,T2.       L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
+                                        U2 = ⓐV2. T2
+                       | ∃∃a,V2,W,T.    L ⊢ V1 ➡* V2 &
+                                        L ⊢ T1 ➡* ⓛ{a}W. T & L ⊢ ⓓ{a}V2. T ➡* U2
+                       | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
+                                        L ⊢ T1 ➡* ⓓ{a}V. T & L ⊢ ⓓ{a}V. ⓐV2. T ➡* U2.
 #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
 #U #U2 #_ #HU2 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
   elim (cpr_inv_appl1 … HU2) -HU2 *
   [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
-  | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/
-  | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
-    @or3_intro2 @(ex4_4_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) 
+  | #a #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct /4 width=7/
+  | #a #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
+    @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *) 
   ]
-| /4 width=8/
-| /4 width=10/
+| /4 width=9/
+| /4 width=11/
 ]
 qed-.
 
@@ -85,18 +93,56 @@ lemma cprs_flat: ∀I,L,T1,T2. L ⊢ T1 ➡* T2 → ∀V1,V2. L ⊢ V1 ➡* V2 
                  L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
 #I #L #T1 #T2 #HT12 #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
-@(cprs_trans … IHV1) -IHV1 /2 width=1/ 
+@(cprs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
-lemma cprs_abbr: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 →
-                 L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
-#L #V1 #T1 #T2 #HT12 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/
+lemma cprs_abst: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀V,T1,T2.
+                 L.ⓛV ⊢ T1 ➡* T2 → ∀a. L ⊢ ⓛ{a}V1. T1 ➡* ⓛ{a}V2. T2.
+#L #V1 #V2 #HV12 #V #T1 #T2 #HT12 #a @(cprs_ind … HV12) -V2
+[ lapply (cprs_lsubs_trans … HT12 (L.ⓛV1) ?) -HT12 /2 width=2/
+| #V0 #V2 #_ #HV02 #IHV01
+  @(cprs_trans … IHV01) -V1 /2 width=2/
+]
+qed.
+
+lemma cprs_abbr1: ∀L,V1,T1,T2. L. ⓓV1 ⊢ T1 ➡* T2 → ∀V2. L ⊢ V1 ➡* V2 →
+                  ∀a.L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
+#L #V1 #T1 #T2 #HT12 #V2 #HV12 #a @(cprs_ind … HV12) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
+lemma cprs_abbr2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 →
+                     ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
+[ /2 width=1/
+| #T1 #T #HT1 #_ #IHT1
+  lapply (lfpr_cpr_trans (L. ⓓV1) … HT1) -HT1 /2 width=1/ #HT1
+  @(cprs_trans … IHT1) -IHT1 /2 width=1/
+]
+qed.
+
+lemma cprs_abbr2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡* T2 →
+                  ∀a. L ⊢ ⓓ{a}V1. T1 ➡* ⓓ{a}V2. T2.
+#L #V1 #V2 #HV12 @(cprs_ind … HV12) -V2 /2 width=1/
+#V #V2 #_ #HV2 #IHV1 #T1 #T2 #HT12 #a
+lapply (IHV1 T1 T1 ? a) -IHV1 // #HV1
+@(cprs_trans … HV1) -HV1 /2 width=1/
+qed.
+
+lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2.
+                    L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 →
+                    ∀a.L ⊢ ⓐV1.ⓛ{a}W.T1 ➡* ⓓ{a}V2.T2.
+#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 #a @(cprs_ind … HT12) -T2
+[ /3 width=1/
+| -HV12 #T #T2 #_ #HT2 #IHT1
+  lapply (cpr_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+  @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/
+]
+qed.
+
 (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 →
+lemma lcpr_cprs_trans: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ →
                        ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
 #L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
 #T #T2 #_ #HT2 #IHT2