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

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  14
  15 include "basic_2/notation/relations/predtysnstrong_5.ma".
  16 include "basic_2/static/lfdeq.ma".
  17 include "basic_2/rt_transition/lfpx.ma".
  18
  19 (* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
  20
  21 definition lfsx: ∀h. sd h → relation3 term genv lenv ≝
  22                  λh,o,T,G. SN … (lfpx h G T) (lfdeq h o T).
  23
  24 interpretation
  25    "strong normalization for uncounted context-sensitive parallel rt-transition on referred entries (local environment)"
  26    'PRedTySNStrong h o T G L = (lfsx h o T G L).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 (* Basic_2A1: was: lsx_ind *)
  31 lemma lfsx_ind: ∀h,o,G,T. ∀R:predicate lenv.
  32                 (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  33                       (∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → R L2) →
  34                       R L1
  35                 ) →
  36                 ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
  37 #h #o #G #T #R #H0 #L1 #H elim H -L1
  38 /5 width=1 by lfdeq_sym, SN_intro/
  39 qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 (* Basic_2A1: was: lsx_intro *)
  44 lemma lfsx_intro: ∀h,o,G,L1,T.
  45                   (∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
  46                   G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
  47 /5 width=1 by lfdeq_sym, SN_intro/ qed.
  48 (*
  49 lemma lfsx_sort: ∀h,o,G,L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
  50 #h #o #G #L1 #s @lfsx_intro
  51 #L2 #H #Hs elim Hs -Hs elim (lfpx_inv_sort … H) -H *
  52 [ #H1 #H2 destruct //
  53 | #I #K1 #K2 #V1 #V2 #HK12 #H1 #H2 destruct
  54   @lfdeq_sort
  55 qed.
  56
  57 lemma lfsx_gref: ∀h,o,G,L,l,p. G ⊢ ⬈*[h, o, §p, l] L.
  58 #h #o #G #L1 #l #p @lfsx_intro
  59 #L2 #HL12 #H elim H -H
  60 /3 width=4 by lfpx_fwd_length, lfdeq_gref/
  61 qed.
  62
  63 (* Basic forward lemmas *****************************************************)
  64
  65 lemma lfsx_fwd_bind_sn: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓑ{a,I}V.T, l] L →
  66                        G ⊢ ⬈*[h, o, V, l] L.
  67 #h #o #a #I #G #L #V #T #l #H @(lfsx_ind … H) -L
  68 #L1 #_ #IHL1 @lfsx_intro
  69 #L2 #HL12 #HV @IHL1 /3 width=4 by lfdeq_fwd_bind_sn/
  70 qed-.
  71
  72 lemma lfsx_fwd_flat_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L →
  73                        G ⊢ ⬈*[h, o, V, l] L.
  74 #h #o #I #G #L #V #T #l #H @(lfsx_ind … H) -L
  75 #L1 #_ #IHL1 @lfsx_intro
  76 #L2 #HL12 #HV @IHL1 /3 width=3 by lfdeq_fwd_flat_sn/
  77 qed-.
  78
  79 lemma lfsx_fwd_flat_dx: ∀h,o,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L →
  80                        G ⊢ ⬈*[h, o, T, l] L.
  81 #h #o #I #G #L #V #T #l #H @(lfsx_ind … H) -L
  82 #L1 #_ #IHL1 @lfsx_intro
  83 #L2 #HL12 #HV @IHL1 /3 width=3 by lfdeq_fwd_flat_dx/
  84 qed-.
  85
  86 lemma lfsx_fwd_pair_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ②{I}V.T, l] L →
  87                        G ⊢ ⬈*[h, o, V, l] L.
  88 #h #o * /2 width=4 by lfsx_fwd_bind_sn, lfsx_fwd_flat_sn/
  89 qed-.
  90
  91 (* Basic inversion lemmas ***************************************************)
  92
  93 lemma lfsx_inv_flat: ∀h,o,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L →
  94                     G ⊢ ⬈*[h, o, V, l] L ∧ G ⊢ ⬈*[h, o, T, l] L.
  95 /3 width=3 by lfsx_fwd_flat_sn, lfsx_fwd_flat_dx, conj/ qed-.
  96
  97 (* Basic_2A1: removed theorems 5:
  98               lsx_atom lsx_sort lsx_gref lsx_ge_up lsx_ge
  99 *)
 100 *)