--- /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/reduction/cix.ma".
+include "basic_2A/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Advanced forward lemmas on irreducibility ********************************)
+
+lemma cpx_fwd_cix: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T1⦄ → T2 = T1.
+#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
+[ //
+| #G #L #k #d #Hkd #H elim (cix_inv_sort … Hkd H)
+| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
+ elim (cix_inv_delta … HLK) //
+| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+ [ elim (cix_inv_bind … H) -H #HV1 #HT1 * #H destruct
+ lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+ lapply (IHT1 … HT1) -IHT1 #H destruct //
+ | elim (cix_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/
+ ]
+| * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
+ [ elim (cix_inv_appl … H) -H #HV1 #HT1 #_
+ >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+ | elim (cix_inv_ri2 … H) /2 width=1 by/
+ ]
+| #G #L #V1 #T1 #T #T2 #_ #_ #_ #H
+ elim (cix_inv_ri2 … H) /2 width=1 by or_introl/
+| #G #L #V1 #T1 #T2 #_ #_ #H
+ elim (cix_inv_ri2 … H) /2 width=1 by/
+| #G #L #V1 #V2 #T #_ #_ #H
+ elim (cix_inv_ri2 … H) /2 width=1 by/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H
+ elim (cix_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+| #a #G #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H
+ elim (cix_inv_appl … H) -H #_ #_ #H
+ elim (simple_inv_bind … H)
+]
+qed-.