--- /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/multiple/lifts_lifts.ma".
+include "basic_2/multiple/drops_drops.ma".
+include "basic_2/static/aaa_lifts.ma".
+include "basic_2/static/aaa_aaa.ma".
+include "basic_2/computation/lsubc_drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem acr_aaa_csubc: ∀RR,RS,RP.
+ gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+ ∀L2. G ⊢ L2 ⫃[RP] L1 → ⦃G, L2, T⦄ ϵ[RP] 〚A〛.
+#RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
+[ #G #L1 #k #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
+ lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ elim (lsubc_drop_O1_trans … HL21 … HLK1) -L1 #X #HLK2 #H
+ elim (lsubc_inv_pair2 … H) -H *
+ [ #K2 #HK21 #H destruct -HKV1B
+ lapply (drop_fwd_drop2 … HLK2) #H
+ elim (lift_total V1 0 (i +1)) #V #HV1
+ lapply (s5 … HB ? G ? ? (◊) … HV1 HLK2) /3 width=7 by s0/
+ | #K2 #V2 #A2 #HVA2 #H1V1A2 #H2V1A2 #_ #H1 #H2 destruct -IHB
+ lapply (aaa_mono … H2V1A2 … HKV1B) #H destruct -H2V1A2 -HKV1B
+ lapply (drop_fwd_drop2 … HLK2) #H
+ elim (lift_total V1 0 (i +1)) #V3 #HV13
+ elim (lift_total V2 0 (i +1)) #V #HV2
+ lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/ -HLK2
+ lapply (s7 … HB G L2 (◊)) /3 width=7 by s0/
+ ]
+| #a #G #L1 #V #T #B #A #_ #_ #IHB #IHA #L2 #HL21
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ lapply (s1 … HB) -HB #HB
+ lapply (s6 … HA G L2 (◊) (◊)) /4 width=1 by lsubc_pair/
+| #a #G #L1 #W #T #B #A #HLWB #_ #IHB #IHA #L2 #HL21
+ @(acr_abst … H1RP H2RP) [ /2 width=5 by/ ]
+ #L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
+ @(gcr_lifts … L2.ⓓⓝW.V3,T … HL32)
+ elim (drops_lsubc_trans … H1RP H2RP … HL32 … HL21) -L2 #L2 #HL32 #HL21
+ lapply (aaa_lifts … L2 W3 … des3 … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
+ @(s0
+
+ @(IHA (L2. ⓛW3) … (des3 + 1)) -IHA
+ /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
+| #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ /3 width=10 by drops_nil, lifts_nil/
+| #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
+ elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (s7 … HA G L2 (◊)) /3 width=5 by/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+/2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
+
+lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
+lapply (acr_gcr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4 by acr_aaa/
+qed.