--- /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_2/substitution/cpye_lift.ma".
+include "basic_2/reduction/lpx_cpye.ma".
+include "basic_2/computation/cpxs_cpxs.ma".
+include "basic_2/computation/lpxs.ma".
+
+(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+
+(* Forward lemmas on evaluation for extended substitution *******************)
+
+lemma cpx_cpye_fwd_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+ ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀U1,d,e. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] 𝐍⦃U1⦄ →
+ ∀U2. ⦃G, L2⦄ ⊢ T2 ▶*[d, e] 𝐍⦃U2⦄ →
+ ⦃G, L1⦄ ⊢ U1 ➡*[h, g] U2.
+#h #g #G #L1 #L2 #H @(lpxs_ind_dx … H) -L1
+[ /3 width=9 by cpx_cpxs, cpx_cpye_fwd_lpx/
+| #L1 #L #HL1 #_ #IHL2 #T1 #T2 #HT12 #U1 #d #e #HTU1 #U2 #HTU2
+ elim (cpye_total G L T2 d e) #X2 #HTX2
+ lapply (cpx_cpye_fwd_lpx … HT12 … HL1 … HTU1 … HTX2) -T1
+ /4 width=9 by lpx_cpxs_trans, cpxs_strap2/ (**) (* full auto too long: 41s *)
+]
+qed-.