(* Properties with generic slicing for local environments *******************)
+(* Basic_2A1: uses: drop_lpr_trans *)
lemma drops_lfpr_trans: ∀h,G. dedropable_sn (cpm 0 h G).
-/3 width=6 by lfxs_liftable_dedropable, cpm_lifts/ qed-.
+/3 width=6 by lfxs_liftable_dedropable_sn, cpm_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
+(* Basic_2A1: uses: lpr_drop_conf *)
lemma lfpr_drops_conf: ∀h,G. dropable_sn (cpm 0 h G).
/2 width=5 by lfxs_dropable_sn/ qed-.
+(* Basic_2A1: uses: lpr_drop_trans_O1 *)
lemma lfpr_drops_trans: ∀h,G. dropable_dx (cpm 0 h G).
/2 width=5 by lfxs_dropable_dx/ qed-.
+
+lemma lfpr_inv_lref_pair_sn: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
+
+lemma lfpr_inv_lref_pair_dx: ∀h,G,L1,L2,i. ⦃G, L1⦄ ⊢ ➡[h, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 & ⦃G, K1⦄ ⊢ ➡[h, V1] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2.
+/2 width=3 by lfxs_inv_lref_pair_dx/ qed-.