matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma

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  14
  15 include "basic_2/computation/lpxs_lleq.ma".
  16 include "basic_2/computation/lpxs_lpxs.ma".
  17 include "basic_2/computation/lsx.ma".
  18
  19 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma lsx_leqy_conf: ∀h,g,G,L1,T,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
  24                      ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → G ⊢ ⋕⬊*[h, g, T, d] L2.
  25 #h #g #G #L1 #T #d #H @(lsx_ind … H) -L1
  26 #L1 #_ #IHL1 #L2 #H1L12 #H2L12 @lsx_intro
  27 #L3 #H1L23 #HnL23 lapply (lpxs_fwd_length … H1L23)
  28 #H2L23 elim (lsuby_lpxs_trans_lleq … H1L12 … H1L23) -H1L12 -H1L23
  29 #L0 #H1L03 #H1L10 #H lapply (lpxs_fwd_length … H1L10)
  30 #H2L10 elim (H T) -H //
  31 #_ #H @(IHL1 … H1L10) -IHL1 -H1L10 /3 width=1 by/
  32 qed-.
  33
  34 lemma lsx_lleq_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
  35                       ∀L2. L1 ⋕[T, d] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
  36 #h #g #T #G #L1 #d #H @(lsx_ind … H) -L1
  37 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
  38 #K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
  39 /5 width=4 by lleq_canc_sn, lleq_trans/
  40 qed-.
  41
  42 lemma lsx_lpxs_trans: ∀h,g,T,G,L1,d. G ⊢ ⋕⬊*[h, g, T, d] L1 →
  43                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⋕⬊*[h, g, T, d] L2.
  44 #h #g #T #G #L1 #d #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
  45 elim (lleq_dec T L1 L2 d) /3 width=4 by lsx_lleq_trans/
  46 qed-.
  47
  48 fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
  49                         ∀Y,T. G ⊢ ⋕⬊*[h, g, T, ⫯d] Y →
  50                         ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
  51                         G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L2.
  52 #h #g #a #I #G #L1 #V #d #H @(lsx_ind … H) -L1
  53 #L1 #HL1 #IHL1 #Y #T #H @(lsx_ind … H) -Y
  54 #Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro
  55 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
  56 #HL10 #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
  57 [ #HnV elim (lleq_dec V L1 L2 d)
  58   [ #HV @(IHL1 … L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
  59   | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
  60   ]
  61 | /3 width=4 by lpxs_pair/
  62 ]
  63 qed-.
  64
  65 lemma lsx_bind: ∀h,g,a,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
  66                 ∀T. G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V →
  67                 G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L.
  68 /2 width=3 by lsx_bind_lpxs_aux/ qed.
  69
  70 lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,d. G ⊢ ⋕⬊*[h, g, V, d] L1 →
  71                      ∀L2,T. G ⊢ ⋕⬊*[h, g, T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
  72                      G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L2.
  73 #h #g #I #G #L1 #V #d #H @(lsx_ind … H) -L1
  74 #L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind … H) -L2
  75 #L2 #HL2 #IHL2 #HL12 @lsx_intro
  76 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
  77 #HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
  78 [ #HnV elim (lleq_dec V L1 L2 d)
  79   [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
  80   | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
  81   ]
  82 | /3 width=1 by/
  83 ]
  84 qed-.
  85
  86 lemma lsx_flat: ∀h,g,I,G,L,V,d. G ⊢ ⋕⬊*[h, g, V, d] L →
  87                 ∀T. G ⊢ ⋕⬊*[h, g, T, d] L → G ⊢ ⋕⬊*[h, g, ⓕ{I}V.T, d] L.
  88 /2 width=3 by lsx_flat_lpxs/ qed.
  89
  90 (* Advanced forward lemmas **************************************************)
  91
  92 lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T, d] L →
  93                        G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
  94 #h #g #a #I #G #L #V1 #T #d #H @(lsx_ind … H) -L
  95 #L1 #_ #IHL1 @lsx_intro
  96 #Y #H #HT elim (lpxs_inv_pair1 … H) -H
  97 #L2 #V2 #HL12 #_ #H destruct
  98 @(lsx_leqy_conf … (L2.ⓑ{I}V1)) /2 width=1 by lsuby_succ/
  99 @IHL1 // #H @HT -IHL1 -HL12 -HT
 100 @(lleq_lsuby_trans … (L2.ⓑ{I}V1))
 101 /2 width=2 by lleq_fwd_bind_dx, lsuby_succ/
 102 qed-.
 103
 104 (* Advanced inversion lemmas ************************************************)
 105
 106 lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,d. G ⊢ ⋕⬊*[h, g, ⓑ{a, I}V.T, d] L →
 107                     G ⊢ ⋕⬊*[h, g, V, d] L ∧ G ⊢ ⋕⬊*[h, g, T, ⫯d] L.ⓑ{I}V.
 108 /3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.