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15 include "basic_2/notation/relations/predsubtystarproper_8.ma".
16 include "basic_2/rt_transition/fpb.ma".
17 include "basic_2/rt_computation/fpbs.ma".
19 (* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
21 definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
22 λh,o,G1,L1,T1,G2,L2,T2.
23 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄.
25 interpretation "proper parallel rst-computation (closure)"
26 'PRedSubTyStarProper h o G1 L1 T1 G2 L2 T2 = (fpbg h o G1 L1 T1 G2 L2 T2).
28 (* Basic properties *********************************************************)
30 lemma fpb_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
31 ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
32 /2 width=5 by ex2_3_intro/ qed.
34 lemma fpbg_fpbq_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
35 ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
36 ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
37 #h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
38 /3 width=9 by fpbs_strap1, ex2_3_intro/
41 (* Note: this is used in the closure proof *)
42 lemma fpbg_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
43 ∀G1,L1,T1. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
44 #h #o #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
47 (* Basic_2A1: uses: fpbg_fleq_trans *)
48 lemma fpbg_ffdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ →
49 ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
50 /3 width=5 by fpbg_fpbq_trans, fpbq_ffdeq/ qed-.