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15 include "basic_2/reduction/cpx_lleq.ma".
16 include "basic_2/reduction/llpx.ma".
18 (* LAZY SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ***************)
20 (* Properties on lazy equivalence for local environments ********************)
22 lemma llpx_lrefl: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2.
23 /2 width=1 by llpx_sn_lrefl/ qed-.
25 lemma lleq_llpx_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
26 ∀L.⦃G, L2⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L.
27 /3 width=3 by lleq_cpx_trans, lleq_llpx_sn_trans/ qed-.
29 lemma lleq_llpx_conf: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 →
30 ∀L.⦃G, L1⦄ ⊢ ➡[h, g, T, d] L → ⦃G, L2⦄ ⊢ ➡[h, g, T, d] L.
31 /3 width=3 by lleq_cpx_trans, lleq_llpx_sn_conf/ qed-.