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- bug fix in the induction for the closure property
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14
15 include "basic_2/notation/relations/btpredstarproper_8.ma".
16 include "basic_2/reduction/fpbc.ma".
17 include "basic_2/computation/fpbs.ma".
18
19 (* GENEARAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES *************)
20
21 inductive fpbg (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
22 | fpbg_cpxs: ∀L2,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 →
23              fpbg h g G1 L1 T1 G1 L2 T2
24 | fpbg_fqup: ∀G2,L,L2,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊃+ ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 →
25              fpbg h g G1 L1 T1 G2 L2 T2
26 .
27
28 interpretation "'big tree' proper parallel computation (closure)"
29    'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
30
31 (* Basic properties *********************************************************)
32
33 lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
34                  ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
35 #h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
36 /3 width=5 by fpbg_cpxs, fpbg_fqup, fqu_fqup, cpx_cpxs/
37 qed.
38
39 axiom fpbg_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
40                    ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
41
42 axiom fpbg_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
43                    ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
44
45 lemma fpbg_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
46                        ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
47 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fpbs_ind … H) -G2 -L2 -T2
48 /2 width=5 by fpbg_strap1/
49 qed-.
50
51 lemma fpbs_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
52                        ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
53 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
54 /3 width=5 by fpbg_strap2/
55 qed-.