--- /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/reduction/llpr_ldrop.ma".
+include "basic_2/computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma cprs_delta: ∀G,L,K,V,V2,i.
+ ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡* V2 →
+ ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
+#G #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6 by cpr_cprs, cpr_delta/ ]
+#V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
+lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
+elim (lift_total V1 0 (i+1)) /4 width=12 by cpr_lift, cprs_strap1/
+qed.
+
+lemma cprs_llpr_conf: ∀G. s_r_confluent1 … (cprs G) (llpr G 0).
+/3 width=5 by llpr_cpr_conf, s_r_conf1_LTC1/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: pr3_gen_lref *)
+lemma cprs_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
+ T2 = #i ∨
+ ∃∃K,V1,T1. ⇩[i] L ≡ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡* T1 &
+ ⇧[0, i + 1] T1 ≡ T2.
+#G #L #T2 #i #H @(cprs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+ elim (cpr_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+ * /4 width=6 by cpr_cprs, ex3_3_intro, or_intror/
+| * #K #V1 #T1 #HLK #HVT1 #HT1
+ lapply (ldrop_fwd_drop2 … HLK) #H0LK
+ elim (cpr_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+ /4 width=6 by cprs_strap1, ex3_3_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: pr3_lift *)
+lemma cprs_lift: ∀G. l_liftable (cprs G).
+/3 width=10 by l_liftable_LTC, cpr_lift/ qed.
+
+(* Basic_1: was: pr3_gen_lift *)
+lemma cprs_inv_lift1: ∀G. l_deliftable_sn (cprs G).
+/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/
+qed-.