matita/matita/contribs/lambdadelta/basic_2A/computation/lsx_lpx.ma

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  14
  15 include "basic_2A/multiple/lleq_lleq.ma".
  16 include "basic_2A/reduction/lpx_lleq.ma".
  17 include "basic_2A/computation/lsx.ma".
  18
  19 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma lsx_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
  24                       ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊*[h, g, T, l] L2.
  25 #h #g #T #G #L1 #l #H @(lsx_ind … H) -L1
  26 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
  27 #K2 #HLK2 #HnLK2 elim (lleq_lpx_trans … HLK2 … HL12) -HLK2
  28 /5 width=4 by lleq_canc_sn, lleq_trans/
  29 qed-.
  30
  31 lemma lsx_lpx_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
  32                      ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
  33 #h #g #T #G #L1 #l #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
  34 elim (lleq_dec T L1 L2 l) /3 width=4 by lsx_lleq_trans/
  35 qed-.
  36
  37 lemma lsx_lreq_conf: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
  38                     ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, g, T, l] L2.
  39 #h #g #G #L1 #T #l #H @(lsx_ind … H) -L1
  40 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
  41 #L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23
  42 #L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
  43 qed-.
  44
  45 (* Advanced forward lemmas **************************************************)
  46
  47 lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
  48                        G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
  49 #h #g #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
  50 #L1 #_ #IHL1 @lsx_intro
  51 #Y #H #HT elim (lpx_inv_pair1 … H) -H
  52 #L2 #V2 #HL12 #_ #H destruct
  53 @(lsx_lreq_conf … (L2.ⓑ{I}V1)) /2 width=1 by lreq_succ/
  54 @IHL1 // #H @HT -IHL1 -HL12 -HT
  55 @(lleq_lreq_trans … (L2.ⓑ{I}V1))
  56 /2 width=2 by lleq_fwd_bind_dx, lreq_succ/
  57 qed-.
  58
  59 (* Advanced inversion lemmas ************************************************)
  60
  61 lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a, I}V.T, l] L →
  62                     G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
  63 /3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.