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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground/xoa/ex_2_3.ma".
16 include "basic_2/notation/relations/predsubtystarproper_7.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: ∀h. tri_relation genv lenv term ≝
24 ∃∃G,L,T. ❪G1,L1,T1❫ ≻[h] ❪G,L,T❫ & ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫.
26 interpretation "proper parallel rst-computation (closure)"
27 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
29 (* Basic properties *********************************************************)
31 lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2.
32 ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
33 /2 width=5 by ex2_3_intro/ qed.
35 lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
36 ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ →
37 ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
38 #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
39 /3 width=9 by fpbs_strap1, ex2_3_intro/
42 lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
43 ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ →
44 ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
45 #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
46 /4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
49 (* Note: this is used in the closure proof *)
50 lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2.
51 ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ →
52 ∀G1,L1,T1. ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
53 #h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
56 (* Basic_2A1: uses: fpbg_fleq_trans *)
57 lemma fpbg_feqx_trans: ∀h,G1,G,L1,L,T1,T.
58 ❪G1,L1,T1❫ >[h] ❪G,L,T❫ →
59 ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
60 /3 width=5 by fpbg_fpbq_trans, fpbq_feqx/ qed-.
62 (* Properties with t-bound rt-transition for terms **************************)
64 lemma cpm_tneqx_cpm_fpbg (h) (G) (L):
65 ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[h,n1] T → (T1 ≛ T → ⊥) →
66 ∀n2,T2. ❪G,L❫ ⊢ T ➡[h,n2] T2 → ❪G,L,T1❫ >[h] ❪G,L,T2❫.
67 /4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.