- property S6 of stronfly normalizing terms proved

[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 9f8d517e267cfc0e27879e6ebc60e4ba907f53e1..54a94b4963361270c066bc1facbae589db31130b 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
@@ -12,7 +12,9 @@
 (*                                                                        *)
 (**************************************************************************)
 
+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".
 
 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
@@ -22,7 +24,7 @@ include "Basic_2/computation/cprs_lcpr.ma".
 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
-[ /3 width=1/
+[ /3 width=5/
 | #T1 #T #HT1 #_ #IHT1
   @(cprs_strap2 … IHT1) -IHT1 /2 width=1/
 ]
@@ -32,21 +34,58 @@ lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2
                 L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
 /3 width=1/ qed.
 
-(* Basic_1: was only: pr3_pr2_pr3_t *)
-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 /3 width=5/
+lemma cpr_abbr2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV2 ⊢ T1 ➡ T2 →
+                 L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12
+lapply (lcpr_cpr_trans (L. ⓓV1) … HT12) /2 width=1/
 qed.
 
-(* Main propertis ***********************************************************)
+(* Advanced inversion lemmas ************************************************)
 
-(* Basic_1: was: pr3_t *)
-theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
-/2 width=3/ qed.
+(* 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.
+#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 *) 
+  ]
+| /4 width=8/
+| /4 width=10/
+]
+qed-.
+
+(* Main propertis ***********************************************************)
 
 (* Basic_1: was: pr3_confluence *)
 theorem cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ➡* T2 →
                    ∃∃T0. L ⊢ T1 ➡* T0 & L ⊢ T2 ➡* T0.
 /3 width=3/ qed.
 
+(* Basic_1: was: pr3_t *)
+theorem cprs_trans: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+/2 width=3/ qed.
+
+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/ 
+qed.
+
+(* Basic_1: was only: pr3_pr2_pr3_t *)
+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
+@(cprs_trans … IHT2) /2 width=3/
+qed.