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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 *)
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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_lifts: ∀RR,RS,RP.
27 gcp RR RS RP → gcr RR RS RP RP →
28 ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L0,cs. ⬇*[Ⓕ, cs] L0 ≡ L1 →
29 ∀T0. ⬆*[cs] T ≡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
30 ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
31 #RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
32 [ #G #L #k #L0 #cs #HL0 #X #H #L2 #HL20
33 >(lifts_inv_sort1 … H) -H
34 lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
35 lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
36 | #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #cs #HL01 #X #H #L2 #HL20
37 lapply (acr_gcr … H1RP H2RP B) #HB
38 elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
39 lapply (drop_fwd_drop2 … HLK1) #HK1b
40 elim (drops_drop_trans … HL01 … HLK1) #X #cs1 #i0 #HL0 #H #Hi0 #Hcs1
41 >(at_mono … Hi1 … Hi0) -i1
42 elim (drops_inv_skip2 … Hcs1 … H) -cs1 #K0 #V0 #cs0 #Hcs0 #HK01 #HV10 #H destruct
43 elim (lsubc_drop_O1_trans … HL20 … HL0) -HL0 #X #HLK2 #H
44 elim (lsubc_inv_pair2 … H) -H *
45 [ #K2 #HK20 #H destruct
46 elim (lift_total V0 0 (i0 +1)) #V #HV0
47 elim (lifts_lift_trans … Hi0 … Hcs0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
48 lapply (s5 … HB ? G ? ? (◊) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
49 | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hcs0
50 #K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
51 lapply (drop_fwd_drop2 … HLK2) #HLK2b
52 lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
53 lapply (aaa_mono … H2KV0A … HKV0B) #H destruct -H2KV0A -HKV0B
54 elim (lift_total V0 0 (i0 +1)) #V3 #HV03
55 elim (lift_total V2 0 (i0 +1)) #V #HV2
56 lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
57 lapply (s7 … HB G L2 (◊)) /3 width=7 by gcr_lift/
59 | #a #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
60 elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
61 lapply (acr_gcr … H1RP H2RP A) #HA
62 lapply (acr_gcr … H1RP H2RP B) #HB
63 lapply (s1 … HB) -HB #HB
64 lapply (s6 … HA G L2 (◊) (◊)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
65 | #a #G #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL02
66 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
67 @(acr_abst … H1RP H2RP) /2 width=5 by/
68 #L3 #V3 #W3 #T3 #cs3 #HL32 #HW03 #HT03 #H1B #H2B
69 elim (drops_lsubc_trans … H1RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
70 lapply (aaa_lifts … L2 W3 … (cs @@ cs3) … HLWB) -HLWB /2 width=4 by drops_trans, lifts_trans/ #HLW2B
71 @(IHA (L2. ⓛW3) … (cs + 1 @@ cs3 + 1)) -IHA
72 /3 width=5 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
73 | #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
74 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
75 /3 width=10 by drops_nil, lifts_nil/
76 | #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #cs #HL0 #X #H #L2 #HL20
77 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
78 lapply (acr_gcr … H1RP H2RP A) #HA
79 lapply (s7 … HA G L2 (◊)) /3 width=5 by/
83 (* Basic_1: was: sc3_arity *)
84 lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
85 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
86 /2 width=8 by drops_nil, lifts_nil, acr_aaa_csubc_lifts/ qed.
88 lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
89 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
90 #RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
91 lapply (acr_gcr … H1RP H2RP A) #HA
92 @(s1 … HA) /2 width=4 by acr_aaa/