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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/substitution/lifts_lifts.ma".
16 include "basic_2/substitution/ldrops_ldrops.ma".
17 include "basic_2/static/aaa_lifts.ma".
18 include "basic_2/static/aaa_aaa.ma".
19 include "basic_2/computation/lsubc_ldrops.ma".
21 (* ABSTRACT COMPUTATION PROPERTIES ******************************************)
23 (* Main properties **********************************************************)
25 (* Basic_1: was: sc3_arity_csubc *)
26 theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP.
27 acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
28 ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
29 ∀T0. ⇧*[des] 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 #des #HL0 #X #H #L2 #HL20
33 >(lifts_inv_sort1 … H) -H
34 lapply (aacr_acr … H1RP H2RP (⓪)) #HAtom
35 @(s4 … HAtom … (◊)) //
36 | #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #des #HL01 #X #H #L2 #HL20
37 lapply (aacr_acr … H1RP H2RP B) #HB
38 elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
39 lapply (ldrop_fwd_ldrop2 … HLK1) #HK1b
40 elim (ldrops_ldrop_trans … HL01 … HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1
41 >(at_mono … Hi1 … Hi0) -i1
42 elim (ldrops_inv_skip2 … Hdes1 … H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct
43 elim (lsubc_ldrop_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 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
48 @(s5 … HB … (◊) … HV0 HLK2) /3 width=7 by ldrops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
49 | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
50 #K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
51 lapply (ldrop_fwd_ldrop2 … 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 @(s5 … HB … (◊) … (ⓝV3.V) … HLK2) [2: /2 width=1 by lift_flat/ ]
57 @(s7 … HB … (◊)) [ @(s8 … HB … HKV2A) // | @(s8 … HB … H1KV0A) // ]
59 | #a #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
60 elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
61 lapply (aacr_acr … H1RP H2RP A) #HA
62 lapply (aacr_acr … H1RP H2RP B) #HB
63 lapply (s1 … HB) -HB #HB
64 @(s6 … HA … (◊) (◊)) /3 width=5 by lsubc_pair, ldrops_skip, liftv_nil/
65 | #a #G #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
66 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
67 @(aacr_abst … H1RP H2RP) [ /2 width=5 by/ ]
68 #L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
69 elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
70 lapply (aaa_lifts … L2 W3 … (des @@ des3) … HLWB) -HLWB /2 width=3 by ldrops_trans, lifts_trans/ #HLW2B
71 @(IHA (L2. ⓛW3) … (des + 1 @@ des3 + 1)) -IHA /2 width=3/ /3 width=5 by lsubc_abbr, ldrops_trans, ldrops_skip/
72 | #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
73 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
74 /3 width=10 by ldrops_nil, lifts_nil/
75 | #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
76 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
77 lapply (aacr_acr … H1RP H2RP A) #HA
78 @(s7 … HA … (◊)) /2 width=5 by/
82 (* Basic_1: was: sc3_arity *)
83 lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
84 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
85 /2 width=8 by ldrops_nil, lifts_nil, aacr_aaa_csubc_lifts/ qed.
87 lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
88 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
89 #RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
90 lapply (aacr_acr … H1RP H2RP A) #HA
91 @(s1 … HA) /2 width=4 by aacr_aaa/