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- bug fix in the connection between global and restricted big-tree
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14
15 include "basic_2/notation/relations/btpredstarrestricted_8.ma".
16 include "basic_2/computation/fpbs.ma".
17
18 (* RESTRICTED "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***********)
19
20 inductive fpbr (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
21 | fpbr_inj : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpbr h g G1 L1 T1 G2 L2 T2
22 | fpbr_step: ∀G,G2,L,L2,T,T2. fpbr h g G1 L1 T1 G L T → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
23              fpbr h g G1 L1 T1 G2 L2 T2
24 .
25
26 interpretation "restricted 'big tree' proper parallel computation (closure)"
27    'BTPRedStarRestricted h g G1 L1 T1 G2 L2 T2 = (fpbr h g G1 L1 T1 G2 L2 T2).
28
29 (* Basic inversion lemmas ***************************************************)
30
31 lemma fpbr_inv_fqu_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ →
32                          ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
33 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2 [ /2 width=5 by ex2_3_intro/ ] (**) (* auto fails without brackets *)
34 #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=9 by fpbs_strap1, ex2_3_intro/
35 qed-.
36
37 (* Basic forward lemmas *****************************************************)
38
39 lemma fpbr_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ →
40                      ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
41 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
42 /3 width=5 by fpbs_strap1, fqup_fpbs, fqu_fqup/
43 qed-.
44
45 (* Basic properties *********************************************************)
46
47 lemma fqu_fpbs_fpbr: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
48                      ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
49 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fpbs_ind … H) -G2 -L2 -T2
50 /2 width=5 by fpbr_inj, fpbr_step/
51 qed.
52
53 lemma fpbr_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
54                    ⦃G, L, T⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
55 /3 width=5 by fqu_fpbs_fpbr, fpbr_fwd_fpbs/ qed-.
56
57 (* Note: this is used in the closure proof *)
58 lemma fpbr_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G, L, T⦄ →
59                        ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
60 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(fpbs_ind … HT2) -G2 -L2 -T2
61 /2 width=5 by fpbr_step/
62 qed-.
63
64 lemma fqup_fpbr_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G, L, T⦄ →
65                        ⦃G, L, T⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
66 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 @(fqup_ind … HT1) -G -L -T
67 /3 width=5 by fpbr_strap2/
68 qed-.
69
70 (* Note: this is used in the closure proof *)
71 lemma fqup_fpbr: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
72 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
73 /3 width=5 by fqu_fpbs_fpbr, fqus_fpbs/
74 qed.