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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 "basic_2/notation/relations/predsubty_8.ma".
16 include "static_2/static/fdeq.ma".
17 include "static_2/s_transition/fquq.ma".
18 include "basic_2/rt_transition/lpr_lpx.ma".
20 (* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
22 (* Basic_2A1: includes: fleq_fpbq fpbq_lleq *)
23 inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
24 | fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
25 | fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
26 | fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h o G1 L1 T1 G1 L2 T1
27 | fpbq_fdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
31 "parallel rst-transition (closure)"
32 'PRedSubTy h o G1 L1 T1 G2 L2 T2 = (fpbq h o G1 L1 T1 G2 L2 T2).
34 (* Basic properties *********************************************************)
36 lemma fpbq_refl (h) (o): tri_reflexive … (fpbq h o).
37 /2 width=1 by fpbq_cpx/ qed.
39 (* Basic_2A1: includes: cpr_fpbq *)
40 lemma cpm_fpbq (n) (h) (o) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
41 /3 width=2 by fpbq_cpx, cpm_fwd_cpx/ qed.
43 lemma lpr_fpbq (h) (o) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1, T⦄ ≽[h, o] ⦃G, L2, T⦄.
44 /3 width=1 by fpbq_lpx, lpr_fwd_lpx/ qed.
46 (* Basic_2A1: removed theorems 2:
47 fpbq_fpbqa fpbqa_inv_fpbq