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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 *)
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15 include "static_2/static/feqg_fqup.ma".
16 include "basic_2/rt_transition/fpb_feqg.ma".
17 include "basic_2/rt_computation/fpbs_fqup.ma".
19 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
21 (* Propreties with generic equivalence on referred closures *****************)
23 (* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
24 lemma feqg_fpbs (S) (G1) (G2) (L1) (L2) (T1) (T2):
25 reflexive … S → symmetric … S →
26 ❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
27 /3 width=5 by fpb_fpbs, feqg_fpb/ qed.
29 (* Basic_2A1: uses: fpbs_lleq_trans *)
30 lemma fpbs_feqg_trans (S) (G) (L) (T):
31 reflexive … S → symmetric … S →
32 ∀G1,L1,T1. ❨G1,L1,T1❩ ≥ ❨G,L,T❩ →
33 ∀G2,L2,T2. ❨G,L,T❩ ≛[S] ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
34 /3 width=9 by fpbs_strap1, feqg_fpb/ qed-.
36 (* Basic_2A1: uses: lleq_fpbs_trans *)
37 lemma feqg_fpbs_trans (S) (G) (L) (T):
38 reflexive … S → symmetric … S →
39 ∀G2,L2,T2. ❨G,L,T❩ ≥ ❨G2,L2,T2❩ →
40 ∀G1,L1,T1. ❨G1,L1,T1❩ ≛[S] ❨G,L,T❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
41 /3 width=5 by fpbs_strap2, feqg_fpb/ qed-.
43 lemma teqg_fpbs_trans (S) (T):
44 reflexive … S → symmetric … S →
46 ∀G1,G2,L1,L2,T2. ❨G1,L1,T❩ ≥ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
47 /3 width=8 by feqg_fpbs_trans, teqg_feqg/ qed-.
49 lemma fpbs_teqg_trans (S) (T):
50 reflexive … S → symmetric … S →
51 ∀G1,G2,L1,L2,T1. ❨G1,L1,T1❩ ≥ ❨G2,L2,T❩ →
52 ∀T2. T ≛[S] T2 → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
53 /3 width=8 by fpbs_feqg_trans, teqg_feqg/ qed-.