matita/matita/contribs/lambdadelta/basic_2/etc_2A1/llpx/llsx.etc

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  14
  15 include "basic_2/notation/relations/lazysn_6.ma".
  16 include "basic_2/substitution/lleq.ma".
  17 include "basic_2/reduction/llpx.ma".
  18
  19 (* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
  20
  21 definition llsx: ∀h. sd h → relation4 ynat term genv lenv ≝
  22                  λh,g,d,T,G. SN … (llpx h g G d T) (lleq d T).
  23
  24 interpretation
  25    "lazy extended strong normalization (local environment)"
  26    'LazySN h g d T G L = (llsx h g T d G L).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 lemma llsx_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
  31                 (∀L1. G ⊢ ⋕⬊*[h, g, T, d] L1 →
  32                       (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → R L2) →
  33                       R L1
  34                 ) →
  35                 ∀L. G ⊢ ⋕⬊*[h, g, T, d] L → R L.
  36 #h #g #G #T #d #R #H0 #L1 #H elim H -L1
  37 /5 width=1 by lleq_sym, SN_intro/
  38 qed-.
  39
  40 (* Basic properties *********************************************************)
  41
  42 lemma llsx_intro: ∀h,g,G,L1,T,d.
  43                   (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g, T, d] L2 → (L1 ⋕[T, d] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T, d] L2) →
  44                   G ⊢ ⋕⬊*[h, g, T, d] L1.
  45 /5 width=1 by lleq_sym, SN_intro/ qed.
  46
  47 lemma llsx_sort: ∀h,g,G,L,d,k. G ⊢ ⋕⬊*[h, g, ⋆k, d] L.
  48 #h #g #G #L1 #d #k @llsx_intro
  49 #L2 #HL12 #H elim H -H
  50 /3 width=6 by llpx_fwd_length, lleq_sort/
  51 qed.
  52
  53 lemma llsx_gref: ∀h,g,G,L,d,p. G ⊢ ⋕⬊*[h, g, §p, d] L.
  54 #h #g #G #L1 #d #p @llsx_intro
  55 #L2 #HL12 #H elim H -H
  56 /3 width=6 by llpx_fwd_length, lleq_gref/
  57 qed.