--- /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/notation/relations/lazypredsnstar_5.ma".
+include "basic_2/reduction/llpr.ma".
+
+(* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
+
+definition llprs: genv → relation4 ynat term lenv lenv ≝
+ λG,d. LTC … (llpr G d).
+
+interpretation "lazy parallel computation (local environment, sn variant)"
+ 'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
+#G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) //
+qed-.
+
+lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
+#G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+/2 width=1 by inj/ qed.
+
+lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
+/2 width=1 by lpr_llprs/ qed.
+
+lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by step/ qed-.
+
+lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
+#G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
+/3 width=6 by llpr_fwd_length, trans_eq/
+qed-.