theory of generic slicing almost completed ....

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / substitution / lpx_sn_drop.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
index b16f8288df53d6a46d3dc9e5245b916534b0fae0..612f16a37b902de62790589bc4c8bf0be9f21166 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
@@ -12,16 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/drop_leq.ma".
+include "basic_2/substitution/drop_lreq.ma".
 include "basic_2/substitution/lpx_sn.ma".
 
 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
-(* Properies on dropping ****************************************************)
+(* Properties on dropping ****************************************************)
 
 lemma lpx_sn_drop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
-                        â\88\80I,K1,V1,i. â\87©[i] L1 ≡ K1.ⓑ{I}V1 →
-                        â\88\83â\88\83K2,V2. â\87©[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
+                        â\88\80I,K1,V1,i. â¬\87[i] L1 ≡ K1.ⓑ{I}V1 →
+                        â\88\83â\88\83K2,V2. â¬\87[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
 #R #L1 #L2 #H elim H -L1 -L2
 [ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
@@ -33,8 +33,8 @@ lemma lpx_sn_drop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
 qed-.
 
 lemma lpx_sn_drop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
-                         â\88\80I,K2,V2,i. â\87©[i] L2 ≡ K2.ⓑ{I}V2 →
-                         â\88\83â\88\83K1,V1. â\87©[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
+                         â\88\80I,K2,V2,i. â¬\87[i] L2 ≡ K2.ⓑ{I}V2 →
+                         â\88\83â\88\83K1,V1. â¬\87[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
 #R #L1 #L2 #H elim H -L1 -L2
 [ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
 | #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
@@ -45,17 +45,17 @@ lemma lpx_sn_drop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
 ]
 qed-.
 
-lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_sn R).
-#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+lemma lpx_sn_deliftable_dropable: ∀R. d_deliftable_sn R → dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 … H) -H
   /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
 | #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
   #L2 #V2 #HL12 #HV12 #H destruct
   /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 … H) -H
   #L2 #V2 #HL12 #HV12 #H destruct
   elim (IHLK1 … HL12) -L1 /3 width=3 by drop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
   elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
   elim (HR … HV12 … HLK1 … HWV1) -V1
   elim (IHLK1 … HL12) -L1 /3 width=5 by drop_skip, lpx_sn_pair, ex2_intro/
@@ -63,40 +63,37 @@ lemma lpx_sn_deliftable_dropable: ∀R. l_deliftable_sn R → dropable_sn (lpx_s
 qed-.
 
 lemma lpx_sn_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
-                                  l_liftable R → dedropable_sn (lpx_sn R).
-#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 … H) -H
+                                  d_liftable R → dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 … H) -H
   /4 width=4 by drop_atom, lpx_sn_atom, ex3_intro/
 | #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 … H) -H
   #K2 #V2 #HK12 #HV12 #H destruct
   lapply (lpx_sn_fwd_length … HK12)
   #H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
-  /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
-  @leq_O2 normalize //
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
-  /3 width=5 by drop_drop, leq_pair, lpx_sn_pair, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+  /3 width=1 by lpx_sn_pair, lreq_O2/
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
+  /3 width=5 by drop_drop, lreq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
   elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
-  elim (lift_total W2 d e) #V2 #HWV2
-  lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+  elim (H2R … HW12 … HLK1 … HWV1) -W1
   elim (IHLK1 … HK12) -K1
-  /3 width=6 by drop_skip, leq_succ, lpx_sn_pair, ex3_intro/
+  /3 width=6 by drop_skip, lreq_succ, lpx_sn_pair, ex3_intro/
 ]
 qed-.
 
-fact lpx_sn_dropable_aux: ∀R,L2,K2,s,d,e. ⇩[s, d, e] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
-                          d = 0 → ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom2 … H) -H 
+fact lpx_sn_dropable_aux: ∀R,L2,K2,s,l,m. ⬇[s, l, m] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+                          l = 0 → ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #l #m #H elim H -L2 -K2 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom2 … H) -H
   /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
 | #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
   #K1 #V1 #HK12 #HV12 #H destruct
   /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+| #I #L2 #K2 #V2 #m #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
   #L1 #V1 #HL12 #HV12 #H destruct
   elim (IHLK2 … HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
-  <plus_n_Sm #H destruct
+| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_ #H elim (ysucc_inv_O_dx … H)
 ]
 qed-.