matita/matita/contribs/lambdadelta/basic_2/rt_computation/fpbs.ma

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  14
  15 include "ground_2/lib/star.ma".
  16 include "basic_2/notation/relations/predsubtystar_7.ma".
  17 include "basic_2/rt_transition/fpbq.ma".
  18
  19 (* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
  20
  21 definition fpbs: ∀h. tri_relation genv lenv term ≝
  22                  λh. tri_TC … (fpbq h).
  23
  24 interpretation "parallel rst-computation (closure)"
  25    'PRedSubTyStar  h G1 L1 T1 G2 L2 T2 = (fpbs h G1 L1 T1 G2 L2 T2).
  26
  27 (* Basic eliminators ********************************************************)
  28
  29 lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
  30                 (∀G,G2,L,L2,T,T2. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ → ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ → Q G L T → Q G2 L2 T2) →
  31                 ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → Q G2 L2 T2.
  32 /3 width=8 by tri_TC_star_ind/ qed-.
  33
  34 lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
  35                    (∀G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≽[h] ❪G,L,T❫ → ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ → Q G L T → Q G1 L1 T1) →
  36                    ∀G1,L1,T1. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → Q G1 L1 T1.
  37 /3 width=8 by tri_TC_star_ind_dx/ qed-.
  38
  39 (* Basic properties *********************************************************)
  40
  41 lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h).
  42 /2 width=1 by tri_inj/ qed.
  43
  44 lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≽[h] ❪G2,L2,T2❫ →
  45                  ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  46 /2 width=1 by tri_inj/ qed.
  47
  48 lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ →
  49                    ❪G,L,T❫ ≽[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  50 /2 width=5 by tri_step/ qed-.
  51
  52 lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≽[h] ❪G,L,T❫ →
  53                    ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  54 /2 width=5 by tri_TC_strap/ qed-.
  55
  56 (* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
  57 lemma feqx_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  58 /3 width=1 by fpbq_fpbs, fpbq_feqx/ qed.
  59
  60 (* Basic_2A1: uses: fpbs_lleq_trans *)
  61 lemma fpbs_feqx_trans: ∀h,G1,G,L1,L,T1,T. ❪G1,L1,T1❫ ≥[h] ❪G,L,T❫ →
  62                        ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  63 /3 width=9 by fpbs_strap1, fpbq_feqx/ qed-.
  64
  65 (* Basic_2A1: uses: lleq_fpbs_trans *)
  66 lemma feqx_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ →
  67                        ∀G1,L1,T1. ❪G1,L1,T1❫ ≛ ❪G,L,T❫ → ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫.
  68 /3 width=5 by fpbs_strap2, fpbq_feqx/ qed-.
  69
  70 lemma teqx_reqx_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
  71                           ∀G,L2. ❪G,L0❫ ⊢ ⬈[h] L2 → ❪G,L1,T1❫ ≥[h] ❪G,L2,T2❫.
  72 /4 width=5 by feqx_fpbs, fpbs_strap1, fpbq_lpx, feqx_intro_dx/ qed.
  73
  74 (* Basic_2A1: removed theorems 3:
  75               fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs
  76 *)