(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
+(* Advanced properties ******************************************************)
+
+axiom lleq_lpxs_trans_nlleq: ∀h,g,G,L1s,L1d,T,d. L1s ⋕[d, T] L1d →
+ ∀L2d. ⦃G, L1d⦄ ⊢ ➡*[h, g] L2d → (L1d ⋕[d, T] L2d → ⊥) →
+ ∃∃L2s. ⦃G, L1s⦄ ⊢ ➡*[h, g] L2s & L2s ⋕[d, T] L2d & L1s ⋕[d, T] L2s → ⊥.
+
+(* Advanced inversion lemmas ************************************************)
+
+axiom lpxs_inv_cpxs_nlleq: ∀h,g,G,L1,L2,T1. ⦃G, L1⦄ ⊢ ➡*[h,g] L2 → (L1 ⋕[O, T1] L2 → ⊥) →
+ ∃∃T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 & T1 = T2 → ⊥ & ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2.
+
(* Properties on lazy equivalence for local environments ********************)
lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →