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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_2A/substitution/lpx_sn_drop.ma".
16 include "basic_2A/substitution/fquq_alt.ma".
17 include "basic_2A/reduction/cpr_lift.ma".
18 include "basic_2A/reduction/lpr.ma".
20 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
22 (* Properties on local environment slicing ***********************************)
24 (* Basic_1: includes: wcpr0_drop *)
25 lemma lpr_drop_conf: ∀G. dropable_sn (lpr G).
26 /3 width=6 by lpx_sn_deliftable_dropable, cpr_inv_lift1/ qed-.
28 (* Basic_1: includes: wcpr0_drop_back *)
29 lemma drop_lpr_trans: ∀G. dedropable_sn (lpr G).
30 /3 width=10 by lpx_sn_liftable_dedropable, cpr_lift/ qed-.
32 lemma lpr_drop_trans_O1: ∀G. dropable_dx (lpr G).
33 /2 width=3 by lpx_sn_dropable/ qed-.
35 (* Properties on context-sensitive parallel reduction for terms *************)
37 lemma fqu_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
38 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
39 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
40 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
41 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
42 #G #L #K #U #T #m #HLK #HUT #U2 #HU2
43 elim (lift_total U2 0 (m+1)) #T2 #HUT2
44 lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
47 lemma fquq_cpr_trans_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
48 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
49 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
50 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
51 [ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
52 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
56 lemma fqu_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
57 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
58 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
59 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
60 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
61 #G #L #K #U #T #m #HLK #HUT #U2 #HU2
62 elim (lift_total U2 0 (m+1)) #T2 #HUT2
63 lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
66 lemma fquq_cpr_trans_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
67 ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡ U2 →
68 ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
69 #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
70 [ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
71 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
75 lemma fqu_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
76 ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
77 ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
78 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
79 /3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpr_pair, ex3_2_intro/
80 [ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 … H) -H
81 #K2 #W2 #HLK2 #HVW2 #H destruct
82 /3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
83 | #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
84 elim (drop_lpr_trans … HLK1 … HK12) -HK12
85 /3 width=7 by fqu_drop, ex3_2_intro/
89 lemma fquq_lpr_trans: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
90 ∀K2. ⦃G2, L2⦄ ⊢ ➡ K2 →
91 ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡ K1 & ⦃G1, L1⦄ ⊢ T1 ➡ T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
92 #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
93 [ #HT12 elim (fqu_lpr_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
94 | * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/