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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/unfold/lifts_lifts.ma".
16 include "basic_2/unfold/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 propertis ***********************************************************)
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 (λL,T. RP L T) →
28 ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
29 ∀T0. ⇧*[des] T ≡ T0 → ∀L2. L2 ⊑[RP] L0 →
31 #RR #RS #RP #H1RP #H2RP #L1 #T #A #H elim H -L1 -T -A
32 [ #L #k #L0 #des #HL0 #X #H #L2 #HL20
33 >(lifts_inv_sort1 … H) -H
34 lapply (aacr_acr … H1RP H2RP ⓪) #HAtom
35 @(s2 … HAtom … ◊) // /2 width=2/
36 | #I #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 generalize in match HLK2; generalize in match I; -HLK2 -I * #HLK2
47 [ elim (lift_total V0 0 (i0 +1)) #V #HV0
48 elim (lifts_lift_trans … Hi0 … Hdes0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
49 @(s4 … HB … ◊ … HV0 HLK2) /3 width=7/ (* uses IHB HL20 V2 HV0 *)
50 | @(s2 … HB … ◊) // /2 width=3/
52 | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
53 #K2 #V2 #A2 #HKV2A #HKV0A #_ #H1 #H2 destruct
54 lapply (ldrop_fwd_ldrop2 … HLK2) #HLK2b
55 lapply (aaa_lifts … HK01 … HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
56 >(aaa_mono … HKV0A … HKV0B) in HKV2A; -HKV0A -HKV0B #HKV2B
57 elim (lift_total V2 0 (i0 +1)) #V #HV2
58 @(s4 … HB … ◊ … HV2 HLK2)
61 | #a #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
62 elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
63 lapply (aacr_acr … H1RP H2RP A) #HA
64 lapply (aacr_acr … H1RP H2RP B) #HB
65 lapply (s1 … HB) -HB #HB
66 @(s5 … HA … ◊ ◊) // /3 width=5/
67 | #a #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL02
68 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
69 @(aacr_abst … H1RP H2RP)
70 [ lapply (aacr_acr … H1RP H2RP B) #HB
71 @(s1 … HB) /2 width=5/
73 #L3 #V3 #T3 #des3 #HL32 #HT03 #HB
74 elim (lifts_total des3 W0) #W2 #HW02
75 elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
76 lapply (aaa_lifts … L2 W2 … (des @@ des3) … HLWB) -HLWB /2 width=3/ #HLW2B
77 @(IHA (L2. ⓛW2) … (des + 1 @@ des3 + 1)) -IHA
78 /2 width=3/ /3 width=5/
80 | #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
81 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
83 | #L #V #T #A #_ #_ #IH1A #IH2A #L0 #des #HL0 #X #H #L2 #HL20
84 elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
85 lapply (aacr_acr … H1RP H2RP A) #HA
87 @(s6 … HA … ◊) /2 width=5/ /3 width=5/
91 (* Basic_1: was: sc3_arity *)
92 lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
93 ∀L,T,A. L ⊢ T ⁝ A → ⦃L, T⦄ ϵ[RP] 〚A〛.
96 lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
97 ∀L,T,A. L ⊢ T ⁝ A → RP L T.
98 #RR #RS #RP #H1RP #H2RP #L #T #A #HT
99 lapply (aacr_acr … H1RP H2RP A) #HA
100 @(s1 … HA) /2 width=4/