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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/predsubtyproper_6.ma".
16 include "static_2/s_transition/fqu.ma".
17 include "static_2/static/reqx.ma".
18 include "basic_2/rt_transition/lpr_lpx.ma".
20 (* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
22 inductive fpb (G1) (L1) (T1): relation3 genv lenv term ≝
23 | fpb_fqu: ∀G2,L2,T2. ❪G1,L1,T1❫ ⬂ ❪G2,L2,T2❫ → fpb G1 L1 T1 G2 L2 T2
24 | fpb_cpx: ∀T2. ❪G1,L1❫ ⊢ T1 ⬈ T2 → (T1 ≛ T2 → ⊥) → fpb G1 L1 T1 G1 L1 T2
25 | fpb_lpx: ∀L2. ❪G1,L1❫ ⊢ ⬈ L2 → (L1 ≛[T1] L2 → ⊥) → fpb G1 L1 T1 G1 L2 T1
29 "proper parallel rst-transition (closure)"
30 'PRedSubTyProper G1 L1 T1 G2 L2 T2 = (fpb G1 L1 T1 G2 L2 T2).
32 (* Basic properties *********************************************************)
34 (* Basic_2A1: includes: cpr_fpb *)
35 lemma cpm_fpb (h) (n) (G) (L):
36 ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h,n] T2 → (T1 ≛ T2 → ⊥) → ❪G,L,T1❫ ≻ ❪G,L,T2❫.
37 /3 width=3 by fpb_cpx, cpm_fwd_cpx/ qed.
39 lemma lpr_fpb (h) (G) (T):
40 ∀L1,L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → (L1 ≛[T] L2 → ⊥) → ❪G,L1,T❫ ≻ ❪G,L2,T❫.
41 /3 width=2 by fpb_lpx, lpr_fwd_lpx/ qed.