1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2/multiple/lifts_lifts.ma".
16 include "basic_2/multiple/drops_drops.ma".
17 include "basic_2/static/aaa_lifts.ma".
18 include "basic_2/static/aaa_aaa.ma".
19 include "basic_2/computation/lsubc_drops.ma".
21 (* GENERIC COMPUTATION PROPERTIES *******************************************)
23 (* Main properties **********************************************************)
25 (* Basic_1: was: sc3_arity_csubc *)
26 theorem acr_aaa_csubc: ∀RR,RS,RP.
27 gcp RR RS RP → gcr RR RS RP RP →
28 ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
29 ∀L2. G ⊢ L2 ⫃[RP] L1 → ⦃G, L2, T⦄ ϵ[RP] 〚A〛.
30 #RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
32 lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
33 lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
34 | #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L2 #HL21
35 lapply (acr_gcr … H1RP H2RP B) #HB
36 elim (lsubc_drop_O1_trans … HL21 … HLK1) -L1 #X #HLK2 #H
37 elim (lsubc_inv_pair2 … H) -H *
38 [ #K2 #HK21 #H destruct -HKV1B
39 lapply (drop_fwd_drop2 … HLK2) #H
40 elim (lift_total V1 0 (i +1)) #V #HV1
41 lapply (s5 … HB ? G ? ? (◊) … HV1 HLK2) /3 width=7 by s0/
42 | #K2 #V2 #A2 #HVA2 #H1V1A2 #H2V1A2 #_ #H1 #H2 destruct -IHB
43 lapply (aaa_mono … H2V1A2 … HKV1B) #H destruct -H2V1A2 -HKV1B
44 lapply (drop_fwd_drop2 … HLK2) #H
45 elim (lift_total V1 0 (i +1)) #V3 #HV13
46 elim (lift_total V2 0 (i +1)) #V #HV2
47 lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/ -HLK2
48 lapply (s7 … HB G L2 (◊)) /3 width=7 by s0/
50 | #a #G #L1 #V #T #B #A #_ #_ #IHB #IHA #L2 #HL21
51 lapply (acr_gcr … H1RP H2RP A) #HA
52 lapply (acr_gcr … H1RP H2RP B) #HB
53 lapply (s1 … HB) -HB #HB
54 lapply (s6 … HA G L2 (◊) (◊)) /4 width=1 by lsubc_pair/
55 | #a #G #L1 #W #T #B #A #HLWB #_ #IHB #IHA #L2 #HL21
56 @(acr_abst … H1RP H2RP) [ /2 width=5 by/ ]
57 #L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
58 @(gcr_lifts … L2.ⓓⓝW.V3,T … HL32)
59 elim (drops_lsubc_trans … H1RP H2RP … HL32 … HL21) -L2 #L2 #HL32 #HL21
60 lapply (aaa_lifts … L2 W3 … des3 … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
63 @(IHA (L2. ⓛW3) … (des3 + 1)) -IHA
64 /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
65 | #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
66 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
67 /3 width=10 by drops_nil, lifts_nil/
68 | #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
69 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
70 lapply (acr_gcr … H1RP H2RP A) #HA
71 lapply (s7 … HA G L2 (◊)) /3 width=5 by/
75 (* Basic_1: was: sc3_arity *)
76 lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
77 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
78 /2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
80 lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
81 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
82 #RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
83 lapply (acr_gcr … H1RP H2RP A) #HA
84 @(s1 … HA) /2 width=4 by acr_aaa/