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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/grammar/lpx_sn_lpx_sn.ma".
16 include "basic_2/unfold/lpqs_ldrop.ma".
18 (* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************)
20 (* Main properties on context-sensitive rest parallel computation for terms *)
22 theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs.
23 #L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*]
24 [ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct
25 elim (cpqs_inv_atom1 … H1) -H1
27 elim (cpqs_inv_atom1 … HT2) -HT2
29 | * #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
30 elim (lpqs_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
31 elim (lpqs_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
32 lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
34 | * #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
35 elim (lpqs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
36 elim (lpqs_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
37 lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
38 elim (cpqs_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
39 lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) /3 width=9/
41 | #a #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
42 elim (cpqs_inv_bind1 … H1) -H1 *
43 [ #V #T #HV1 #HT1 #H destruct
44 elim (cpqs_inv_bind1 … H2) -H2 *
45 [ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
46 | #T2 #HT2 #HXT2 #H1 #H2 destruct /4 width=5/
48 | #Y1 #HTY1 #HXY1 #H11 #H12 destruct
49 elim (lift_total X2 0 1) #Y2 #HXY2
50 lapply (cpqs_lift … H2 (L2.ⓓV1) … HXY1 … HXY2) /2 width=1/ -X1 /4 width=5/
52 | #I #V1 #T1 #Hn #X1 #H1 #L2 #HL12 #X2 #H2
53 elim (cpqs_inv_flat1 … H1) -H1 *
54 [ #V #T #HV1 #HT1 #H destruct
55 elim (cpqs_inv_flat1 … H2) -H2 *
56 [ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5/
57 | #HX2 #H destruct /3 width=5/
59 | #HX1 #H destruct /3 width=5/
63 theorem cpqs_trans: ∀L. Transitive … (cpqs L).
64 /2 width=5 by cpqs_trans_lpqs/ qed-.
66 (* Properties on context-sensitive rest. parallel computation for terms *****)
68 lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 →
69 ∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2.
70 /2 width=5 by cpqs_trans_lpqs/ qed-.