--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_drop.ma".
+include "basic_2A/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 →
+ ∀L1. L1 ≡[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) //
+ | * /3 width=7 by cpx_delta/
+ ]
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H
+ /3 width=1 by cpx_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
+ /3 width=1 by cpx_flat/
+| #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_eps/
+| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H
+ /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
+ #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=3 by cpx_theta/
+]
+qed-.
+
+lemma cpx_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L1. L2 ≡[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 →
+ ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
+/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.