(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/multiple/lleq_drop.ma".
include "basic_2/reduction/cpx_llpx_sn.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
(* 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 â\86\92 â¦\83G, L1â¦\84 â\8a¢ T1 â\9e¡[h, g] T2.
-#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
+lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+ â\88\80L1. L1 â\89¡[T1, 0] L2 â\86\92 â¦\83G, L1â¦\84 â\8a¢ T1 â\9e¡[h, o] T2.
+#h #o #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) //
| * /3 width=7 by cpx_delta/
| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #H elim (lleq_inv_bind_O … H) -H
/3 width=3 by cpx_zeta/
| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by cpx_tau/
+ /3 width=1 by cpx_eps/
| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by cpx_ti/
+ /3 width=1 by cpx_ct/
| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
#HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=1 by cpx_beta/
| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
]
qed-.
-lemma lleq_cpx_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
-/3 width=6 by llpx_sn_cpx_conf, lift_mono, ex2_intro/ qed-.
+lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+ ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2.
+/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
-lemma lleq_cpx_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1. L1 ⋕[T1, 0] L2 → L1 ⋕[T2, 0] L2.
-/4 width=6 by lleq_cpx_conf_sn, lleq_sym/ qed-.
+lemma cpx_lleq_conf_sn: ∀h,o,G. c_r_confluent1 … (cpx h o G) (lleq 0).
+/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
+
+lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+ ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
+/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.