(* Properties on lazy equivalence for local environments ********************)
lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- â\88\80L1. L1 â\8b\95[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+ â\88\80L1. L1 â\89¡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
[ #H elim (ylt_yle_false … H) //
qed-.
lemma cpx_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- â\88\80L1. L2 â\8b\95[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+ â\88\80L1. L2 â\89¡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
lemma cpx_lleq_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
lemma cpx_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- â\88\80L1. L1 â\8b\95[T1, 0] L2 â\86\92 L1 â\8b\95[T2, 0] L2.
+ â\88\80L1. L1 â\89¡[T1, 0] L2 â\86\92 L1 â\89¡[T2, 0] L2.
/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.