(* Properties with generic slicing for local environments *******************)
-(* Basic_2A1: includes: lleq_lift_le lleq_lift_ge *)
lemma lfdeq_lifts_sn: ∀h,o. dedropable_sn (cdeq h o).
/3 width=5 by lfxs_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-(* Basic_2A1: restricts: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
lemma lfdeq_inv_lifts_sn: ∀h,o. dropable_sn (cdeq h o).
/2 width=5 by lfxs_dropable_sn/ qed-.
lemma lfdeq_inv_lifts_dx: ∀h,o. dropable_dx (cdeq h o).
/2 width=5 by lfxs_dropable_dx/ qed-.
-(* Note: missing in basic_2A1 *)
-lemma lfdeq_inv_lifts_bi: â\88\80h,o,L1,L2,U. L1 â\89¡[h, o, U] L2 →
- ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
- ∀T. ⬆*[i] T ≡ U → K1 ≡[h, o, T] K2.
-/2 width=8 by lfxs_inv_lifts_bi/ qed-.
+(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
+lemma lfdeq_inv_lifts_bi: â\88\80h,o,L1,L2,U. L1 â\89\9b[h, o, U] L2 â\86\92 â\88\80b,f. ð\9d\90\94â¦\83fâ¦\84 →
+ ∀K1,K2. ⬇*[b, f] L1 ≡ K1 → ⬇*[b, f] L2 ≡ K2 →
+ ∀T. ⬆*[f] T ≡ U → K1 ≛[h, o, T] K2.
+/2 width=10 by lfxs_inv_lifts_bi/ qed-.
-lemma lfdeq_inv_lref_pair_sn: â\88\80h,o,L1,L2,i. L1 â\89¡[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
- â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89¡ K2.â\93\91{I}V2 & K1 â\89¡[h, o, V1] K2 & V1 â\89¡[h, o] V2.
+lemma lfdeq_inv_lref_pair_sn: â\88\80h,o,L1,L2,i. L1 â\89\9b[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
+ â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89¡ K2.â\93\91{I}V2 & K1 â\89\9b[h, o, V1] K2 & V1 â\89\9b[h, o] V2.
/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
-lemma lfdeq_inv_lref_pair_dx: â\88\80h,o,L1,L2,i. L1 â\89¡[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
- â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89¡ K1.â\93\91{I}V1 & K1 â\89¡[h, o, V1] K2 & V1 â\89¡[h, o] V2.
+lemma lfdeq_inv_lref_pair_dx: â\88\80h,o,L1,L2,i. L1 â\89\9b[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
+ â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89¡ K1.â\93\91{I}V1 & K1 â\89\9b[h, o, V1] K2 & V1 â\89\9b[h, o] V2.
/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.