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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 include "ground/lib/star.ma".
16 include "basic_2/notation/relations/predsubtystar_6.ma".
17 include "basic_2/rt_transition/fpbq.ma".
19 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
21 definition fpbs: tri_relation genv lenv term ≝
25 "parallel rst-computation (closure)"
26 'PRedSubTyStar G1 L1 T1 G2 L2 T2 = (fpbs G1 L1 T1 G2 L2 T2).
28 (* Basic eliminators ********************************************************)
31 ∀G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
32 (∀G,G2,L,L2,T,T2. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ → ❪G,L,T❫ ≽ ❪G2,L2,T2❫ → Q G L T → Q G2 L2 T2) →
33 ∀G2,L2,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → Q G2 L2 T2.
34 /3 width=8 by tri_TC_star_ind/ qed-.
37 ∀G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
38 (∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≽ ❪G,L,T❫ → ❪G,L,T❫ ≥ ❪G2,L2,T2❫ → Q G L T → Q G1 L1 T1) →
39 ∀G1,L1,T1. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → Q G1 L1 T1.
40 /3 width=8 by tri_TC_star_ind_dx/ qed-.
42 (* Basic properties *********************************************************)
46 /2 width=1 by tri_inj/ qed.
49 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≽ ❪G2,L2,T2❫ →
50 ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
51 /2 width=1 by tri_inj/ qed.
54 ∀G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ →
55 ❪G,L,T❫ ≽ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
56 /2 width=5 by tri_step/ qed-.
59 ∀G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≽ ❪G,L,T❫ →
60 ❪G,L,T❫ ≥ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
61 /2 width=5 by tri_TC_strap/ qed-.
63 (* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
65 ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
66 /3 width=1 by fpbq_fpbs, fpbq_feqx/ qed.
68 (* Basic_2A1: uses: fpbs_lleq_trans *)
69 lemma fpbs_feqx_trans:
70 ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥ ❪G,L,T❫ →
71 ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
72 /3 width=9 by fpbs_strap1, fpbq_feqx/ qed-.
74 (* Basic_2A1: uses: lleq_fpbs_trans *)
75 lemma feqx_fpbs_trans:
76 ∀G,G2,L,L2,T,T2. ❪G,L,T❫ ≥ ❪G2,L2,T2❫ →
77 ∀G1,L1,T1. ❪G1,L1,T1❫ ≛ ❪G,L,T❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
78 /3 width=5 by fpbs_strap2, fpbq_feqx/ qed-.
80 lemma teqx_reqx_lpx_fpbs:
81 ∀T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
82 ∀G,L2. ❪G,L0❫ ⊢ ⬈ L2 → ❪G,L1,T1❫ ≥ ❪G,L2,T2❫.
83 /4 width=5 by feqx_fpbs, fpbs_strap1, fpbq_lpx, feqx_intro_dx/ qed.
85 (* Basic_2A1: removed theorems 3:
86 fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs