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/relocation/ldrop_lpx_sn.ma".
16 include "basic_2/reduction/cpx_lift.ma".
17 include "basic_2/reduction/lpx.ma".
19 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
21 (* Properies on local environment slicing ***********************************)
23 lemma lpx_ldrop_conf: ∀h,g,G. dropable_sn (lpx h g G).
24 /3 width=5 by lpx_sn_deliftable_dropable, cpx_inv_lift1/ qed-.
26 lemma ldrop_lpx_trans: ∀h,g,G. dedropable_sn (lpx h g G).
27 /3 width=9 by lpx_sn_liftable_dedropable, cpx_lift/ qed-.
29 lemma lpx_ldrop_trans_O1: ∀h,g,G. dropable_dx (lpx h g G).
30 /2 width=3 by lpx_sn_dropable/ qed-.
32 (* Properties on supclosure *************************************************)
34 lemma fqu_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
35 ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
36 ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄.
37 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
38 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
39 [ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
40 #K2 #W2 #HLK2 #HVW2 #H destruct
41 /3 width=5 by fqu_fquq, cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
42 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
43 elim (ldrop_lpx_trans … HLK1 … HK12) -HK12
44 /3 width=7 by fqu_drop, ex3_2_intro/
48 lemma fquq_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
49 ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
50 ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄.
51 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
52 [ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
53 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
57 lemma lpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
58 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
59 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
60 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
61 /3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/
62 [ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H
63 #K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
64 /4 width=7 by cpx_delta, fqu_drop, ldrop_ldrop, ex3_2_intro/
65 | #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
66 elim (lpx_ldrop_trans_O1 … HL01 … HLK1) -L1
67 /3 width=5 by fqu_drop, ex3_2_intro/
71 lemma lpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
72 ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
73 ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
74 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
75 [ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
76 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/