(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-definition dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-
-definition dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
-
-definition dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
-
-definition lfxs_transitive_next: relation3 … ≝ λR1,R2,R3.
- ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
- lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
+definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+definition f_dropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition f_dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+
+definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
+ ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
+ lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
(* Properties with generic slicing for local environments *******************)
lemma lfxs_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
- d_liftable2_sn … lifts R → dedropable_sn R.
+ d_liftable2_sn … lifts R → f_dedropable_sn R.
#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
elim (frees_total L1 U) #f2 #Hf2
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
qed-.
-lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → lfxs_transitive_next R1 R2 R3.
+lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
generalize in match HLK; -HLK elim H -I1 -I
[ #I #_ #L2 #_ #I2 #H
(* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
(* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma lfxs_dropable_sn: ∀R. dropable_sn R.
+lemma lfxs_dropable_sn: ∀R. f_dropable_sn R.
#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
elim (frees_total K1 T) #f1 #Hf1
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
(* Basic_2A1: was: llpx_sn_drop_trans_O *)
(* Note: the proof might be simplified *)
-lemma lfxs_dropable_dx: ∀R. dropable_dx R.
+lemma lfxs_dropable_dx: ∀R. f_dropable_dx R.
#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
elim (drops_isuni_ex … H1f L1) #K1 #HLK1
elim (frees_total K1 T) #f1 #Hf1