+(**************************************************************************)
+(* ___ *)
+(* ||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/reducibility/fpr_cpr.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/computation/fprs.ma".
+
+(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+
+(* Properties on context-sensitive parallel computation for terms ***********)
+
+lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
+#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
+qed.
+(*
+(* Advanced propertis *******************************************************)
+
+lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
+ ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
+elim (fpr_inv_all … H) /3 width=4/
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
+ ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
+ ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
+* #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
+elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
+[ elim (cpr_inv_abbr1 … H) -H *
+ [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
+ | #T #_ #_ #H destruct
+ ]
+| elim (cpr_inv_abst1 … H Abst V2) -H
+ #V #T #HV1 #_ #H destruct /3 width=4/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lamma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I1 #K1 #X #V1 #T1 #T2 #H
+elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+
+lamma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #X #K2 #V2 #T1 #T2 #H
+elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
+elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
+qed-.
+*)