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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 "ground/xoa/ex_2_3.ma".
16 include "basic_2/notation/relations/predsubtystarproper_6.ma".
17 include "basic_2/rt_transition/fpb.ma".
18 include "basic_2/rt_computation/fpbs.ma".
20 (* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
22 definition fpbg: tri_relation genv lenv term ≝
24 ∃∃G,L,T. ❪G1,L1,T1❫ ≻ ❪G,L,T❫ & ❪G,L,T❫ ≥ ❪G2,L2,T2❫.
26 interpretation "proper parallel rst-computation (closure)"
27 'PRedSubTyStarProper G1 L1 T1 G2 L2 T2 = (fpbg G1 L1 T1 G2 L2 T2).
29 (* Basic properties *********************************************************)
33 ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
34 /2 width=5 by ex2_3_intro/ qed.
36 lemma fpbg_fpbq_trans:
37 ∀G1,G,G2,L1,L,L2,T1,T,T2.
38 ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ≽ ❪G2,L2,T2❫ →
39 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
40 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
41 /3 width=9 by fpbs_strap1, ex2_3_intro/
45 ∀G1,G,G2,L1,L,L2,T1,T,T2.
46 ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ →
47 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
48 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
49 /4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
52 (* Note: this is used in the closure proof *)
53 lemma fpbg_fpbs_trans:
54 ∀G,G2,L,L2,T,T2. ❪G,L,T❫ ≥ ❪G2,L2,T2❫ →
55 ∀G1,L1,T1. ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
56 #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
59 (* Basic_2A1: uses: fpbg_fleq_trans *)
60 lemma fpbg_feqx_trans:
61 ∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ > ❪G,L,T❫ →
62 ∀G2,L2,T2. ❪G,L,T❫ ≅ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
63 /3 width=5 by fpbg_fpbq_trans, fpbq_feqx/ qed-.
65 (* Properties with t-bound rt-transition for terms **************************)
67 lemma cpm_tneqx_cpm_fpbg (h) (G) (L):
68 ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T → (T1 ≅ T → ⊥) →
69 ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 → ❪G,L,T1❫ > ❪G,L,T2❫.
70 /4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.