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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 "ground_2/lib/star.ma".
16 include "basic_2/notation/relations/predsubtystar_7.ma".
17 include "basic_2/rt_transition/fpbq.ma".
19 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
21 definition fpbs: ∀h. tri_relation genv lenv term ≝
22 λh. tri_TC … (fpbq h).
24 interpretation "parallel rst-computation (closure)"
25 'PRedSubTyStar h G1 L1 T1 G2 L2 T2 = (fpbs h G1 L1 T1 G2 L2 T2).
27 (* Basic eliminators ********************************************************)
29 lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
30 (∀G,G2,L,L2,T,T2. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ → ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ → Q G L T → Q G2 L2 T2) →
31 ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → Q G2 L2 T2.
32 /3 width=8 by tri_TC_star_ind/ qed-.
34 lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
35 (∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≽[h] ❪G,L,T❫ → ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ → Q G L T → Q G1 L1 T1) →
36 ∀G1,L1,T1. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → Q G1 L1 T1.
37 /3 width=8 by tri_TC_star_ind_dx/ qed-.
39 (* Basic properties *********************************************************)
41 lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h).
42 /2 width=1 by tri_inj/ qed.
44 lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≽[h] ❪G2,L2,T2❫ →
45 ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
46 /2 width=1 by tri_inj/ qed.
48 lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ →
49 ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
50 /2 width=5 by tri_step/ qed-.
52 lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≽[h] ❪G,L,T❫ →
53 ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
54 /2 width=5 by tri_TC_strap/ qed-.
56 (* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
57 lemma feqx_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
58 /3 width=1 by fpbq_fpbs, fpbq_feqx/ qed.
60 (* Basic_2A1: uses: fpbs_lleq_trans *)
61 lemma fpbs_feqx_trans: ∀h,G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ →
62 ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
63 /3 width=9 by fpbs_strap1, fpbq_feqx/ qed-.
65 (* Basic_2A1: uses: lleq_fpbs_trans *)
66 lemma feqx_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ →
67 ∀G1,L1,T1. ❪G1,L1,T1❫ ≛ ❪G,L,T❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
68 /3 width=5 by fpbs_strap2, fpbq_feqx/ qed-.
70 lemma teqx_reqx_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
71 ∀G,L2. ❪G,L0❫ ⊢ ⬈[h] L2 → ❪G,L1,T1❫ ≥[h] ❪G,L2,T2❫.
72 /4 width=5 by feqx_fpbs, fpbs_strap1, fpbq_lpx, feqx_intro_dx/ qed.
74 (* Basic_2A1: removed theorems 3:
75 fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs