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15 include "basic_2/notation/relations/predsubtystarproper_7.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. tri_relation genv lenv term ≝
23 ∃∃G,L,T. ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ & ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄.
25 interpretation "proper parallel rst-computation (closure)"
26 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
28 (* Basic properties *********************************************************)
30 lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
31 ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
32 /2 width=5 by ex2_3_intro/ qed.
34 lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
35 ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ →
36 ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
37 #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
38 /3 width=9 by fpbs_strap1, ex2_3_intro/
41 lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
42 ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⊐ ⦃G2,L2,T2⦄ →
43 ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
44 #h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
45 /4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
48 (* Note: this is used in the closure proof *)
49 lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
50 ∀G1,L1,T1. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
51 #h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
54 (* Basic_2A1: uses: fpbg_fleq_trans *)
55 lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ →
56 ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
57 /3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-.
59 (* Properties with t-bound rt-transition for terms **************************)
61 lemma cpm_tdneq_cpm_fpbg (h) (G) (L):
62 ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
63 ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
64 /4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.