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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "ground/xoa/ex_3_6.ma".
16 include "basic_2/notation/relations/predsubtystarproper_6.ma".
17 include "basic_2/rt_transition/fpbc.ma".
18 include "basic_2/rt_computation/fpbs.ma".
20 (* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
22 definition fpbg: tri_relation genv lenv term ≝
24 ∃∃G3,L3,T3,G4,L4,T4. ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ & ❪G3,L3,T3❫ ≻ ❪G4,L4,T4❫ & ❪G4,L4,T4❫ ≥ ❪G2,L2,T2❫.
27 "proper parallel rst-computation (closure)"
28 'PRedSubTyStarProper G1 L1 T1 G2 L2 T2 = (fpbg G1 L1 T1 G2 L2 T2).
30 (* Basic inversion lemmas ***************************************************)
32 lemma fpbg_inv_gen (G1) (G2) (L1) (L2) (T1) (T2):
33 ❪G1,L1,T1❫ > ❪G2,L2,T2❫ →
34 ∃∃G3,L3,T3,G4,L4,T4. ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ & ❪G3,L3,T3❫ ≻ ❪G4,L4,T4❫ & ❪G4,L4,T4❫ ≥ ❪G2,L2,T2❫.
37 (* Basic properties *********************************************************)
39 lemma fpbg_intro (G3) (G4) (L3) (L4) (T3) (T4):
41 ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ → ❪G3,L3,T3❫ ≻ ❪G4,L4,T4❫ →
42 ❪G4,L4,T4❫ ≥ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
43 /2 width=9 by ex3_6_intro/ qed.
45 (* Basic_2A1: was: fpbg_fpbq_trans *)
47 ∀G1,G,G2,L1,L,L2,T1,T,T2.
48 ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ≽ ❪G2,L2,T2❫ →
49 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
50 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
51 elim (fpbg_inv_gen … H1) -H1
52 /3 width=13 by fpbs_strap1, fpbg_intro/
55 (* Basic_2A1: was: fpbq_fpbg_trans *)
57 ∀G1,G,G2,L1,L,L2,T1,T,T2.
58 ❪G1,L1,T1❫ ≽ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ →
59 ❪G1,L1,T1❫ > ❪G2,L2,T2❫.
60 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
61 elim (fpbg_inv_gen … H2) -H2
62 /3 width=13 by fpbs_strap2, fpbg_intro/