(* *)
(**************************************************************************)
-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 *********)
]
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/
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
+ @lreq_O2 normalize //
+| #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
+ elim (lift_total W2 l m) #V2 #HWV2
lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -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 #_
+| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_
<plus_n_Sm #H destruct
]
qed-.