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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/computation/fpbs_fpbu.ma".
16 include "basic_2/computation/fpbs_fpbc.ma".
17 include "basic_2/computation/fpbs_fpbs.ma".
18 include "basic_2/computation/fpbg_fpbs.ma".
20 (* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
22 (* Advanced inversion lemmas ************************************************)
24 lemma fpbg_inv_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
25 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
26 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind_dx … H) -G1 -L1 -T1
27 [ #G1 #L1 #T1 * /3 width=5 by fleq_fpbs, ex2_3_intro/
28 | #G1 #G #L1 #L #T1 #T *
29 #G0 #L0 #T0 #H10 #H0 #_ *
30 /5 width=9 by fpbu_fpbs, fpbs_trans, fleq_fpbs, ex2_3_intro/
34 (* Advanced forward lemmas **************************************************)
36 lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
37 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
38 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbg_ind … H) -G2 -L2 -T2
39 [2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 @(fpbs_trans … IH1) -IH1 ] (**) (* full auto fails *)
40 /2 width=1 by fpbc_fpbs/
43 (* Advanced properties ******************************************************)
45 lemma fpbs_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
46 ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
47 ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
48 /5 width=5 by fpbg_inv_fpbu_sn, fpbs_fpbg_trans, fpbc_fpbg, fpbu_fpbc/ qed-.
50 (* Man properties ***********************************************************)
52 theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
53 /2 width=5 by tri_TC_transitive/ qed-.