include "static_2/static/rex_drops.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Properties with generic slicing for local environments *******************)
-lemma rdeq_lifts_sn: ∀h,o. f_dedropable_sn (cdeq h o).
+lemma rdeq_lifts_sn: f_dedropable_sn cdeq.
/3 width=5 by rex_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma rdeq_inv_lifts_sn: ∀h,o. f_dropable_sn (cdeq h o).
+lemma rdeq_inv_lifts_sn: f_dropable_sn cdeq.
/2 width=5 by rex_dropable_sn/ qed-.
-lemma rdeq_inv_lifts_dx: ∀h,o. f_dropable_dx (cdeq h o).
+lemma rdeq_inv_lifts_dx: f_dropable_dx cdeq.
/2 width=5 by rex_dropable_dx/ qed-.
-lemma rdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≛[h, o, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- â\88\80K1,K2. â¬\87*[b, f] L1 â\89\98 K1 â\86\92 â¬\87*[b, f] L2 ≘ K2 →
- â\88\80T. â¬\86*[f] T â\89\98 U â\86\92 K1 â\89\9b[h, o, T] K2.
+lemma rdeq_inv_lifts_bi: ∀L1,L2,U. L1 ≛[U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ â\88\80K1,K2. â\87©*[b,f] L1 â\89\98 K1 â\86\92 â\87©*[b,f] L2 ≘ K2 →
+ â\88\80T. â\87§*[f] T â\89\98 U â\86\92 K1 â\89\9b[T] K2.
/2 width=10 by rex_inv_lifts_bi/ qed-.
-lemma rdeq_inv_lref_pair_sn: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89\98 K2.â\93\91{I}V2 & K1 â\89\9b[h, o, V1] K2 & V1 â\89\9b[h, o] V2.
+lemma rdeq_inv_lref_pair_sn: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K1,V1. ⇩*[i] L1 ≘ K1.ⓑ{I}V1 →
+ â\88\83â\88\83K2,V2. â\87©*[i] L2 â\89\98 K2.â\93\91{I}V2 & K1 â\89\9b[V1] K2 & V1 â\89\9b V2.
/2 width=3 by rex_inv_lref_pair_sn/ qed-.
-lemma rdeq_inv_lref_pair_dx: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89\98 K1.â\93\91{I}V1 & K1 â\89\9b[h, o, V1] K2 & V1 â\89\9b[h, o] V2.
+lemma rdeq_inv_lref_pair_dx: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K2,V2. ⇩*[i] L2 ≘ K2.ⓑ{I}V2 →
+ â\88\83â\88\83K1,V1. â\87©*[i] L1 â\89\98 K1.â\93\91{I}V1 & K1 â\89\9b[V1] K2 & V1 â\89\9b V2.
/2 width=3 by rex_inv_lref_pair_dx/ qed-.
+
+lemma rdeq_inv_lref_pair_bi (L1) (L2) (i):
+ L1 ≛[#i] L2 →
+ ∀I1,K1,V1. ⇩*[i] L1 ≘ K1.ⓑ{I1}V1 →
+ ∀I2,K2,V2. ⇩*[i] L2 ≘ K2.ⓑ{I2}V2 →
+ ∧∧ K1 ≛[V1] K2 & V1 ≛ V2 & I1 = I2.
+/2 width=6 by rex_inv_lref_pair_bi/ qed-.