+(**************************************************************************)
+(* ___ *)
+(* ||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/cnf_lift.ma".
+include "Basic_2/computation/acp.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Advanced properties ******************************************************)
+
+lemma csn_acp: acp cpr (eq …) (csn …).
+@mk_acp
+[ /2 width=1/
+| /2 width=3/
+| /2 width=5/
+| @cnf_lift
+]
+qed.
+
+lemma csn_abst: ∀L,W. L ⊢ ⬇* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬇* T → L ⊢ ⬇* ⓛW. T.
+#L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT
+@csn_intro #X #H1 #H2
+elim (cpr_inv_abst1 … H1 I V) -H1
+#W0 #T0 #HLW0 #HLT0 #H destruct
+elim (eq_false_inv_tpair … H2) -H2
+[ /3 width=5/
+| -HLW0 * #H destruct /3 width=1/
+]
+qed.
+
+(* Relocation properties ****************************************************)
+
+(* Basic_1: was: sn3_lift *)
+lemma csn_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
+ ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → L2 ⊢ ⬇* T2.
+#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+@csn_intro #T #HLT2 #HT2
+elim (cpr_inv_lift … HL21 … HT12 … HLT2) -HLT2 #T0 #HT0 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_mono … HT0 … HT12) in HT2; -T0 /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_lift *)
+lemma csn_inv_lift: ∀L2,L1,T1,d,e. L1 ⊢ ⬇* T1 →
+ ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → L2 ⊢ ⬇* T2.
+#L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+@csn_intro #T #HLT2 #HT2
+elim (lift_total T d e) #T0 #HT0
+lapply (cpr_lift … HL12 … HT21 … HT0 HLT2) -HLT2 #HLT10
+@(IHT1 … HLT10) // -L1 -L2 #H destruct
+>(lift_inj … HT0 … HT21) in HT2; -T0 /2 width=1/
+qed.