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

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/computation/lpxs_lleq.ma".
  16 include "basic_2/computation/csx_alt.ma".
  17 include "basic_2/computation/lsx_lpxs.ma".
  18
  19 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  20
  21 (* Advanced forward lemmas **************************************************)
  22
  23 include "basic_2/computation/cpxs_lleq.ma".
  24
  25 (*
  26 lemma lsx_cpxs_conf: ∀h,g,G,L1,T1,T2. G ⊢ ⋕⬊*[h, g, T1] L1 →
  27                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
  28                      G ⊢ ⋕⬊*[h, g, T2] L2.
  29 #h #g #G #L1 #T1 #T2 #H @(lsx_ind … H) -L1
  30 #L1 #HL1 #IHL1 #L2 #HL12 #HT12 @lsx_intro
  31 #L3 #HL23 #HnL23 elim (lleq_dec T2 L1 L2 0) [| -HnL23 ]
  32 [ #HT2 @(IHL1 L3) /2 width=3 by lpxs_trans/
  33
  34  @(lsx_lpxs_trans … HL23)
  35 | #HnT2 @(lsx_lpxs_trans … HL23) @(IHL1 … L2) //
  36   #HT1 @HnT2 @(lleq_cpxs_conf_dx … HT12) //
  37 ]
  38 *)
  39
  40 lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V1. ⦃G, L1⦄ ⊢ ⬊*[h, g] V1 →
  41                             ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V1.T] L2 →
  42                             ∀V2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 →
  43                             G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V2.
  44 #h #g #a #I #G #L1 #V1 #H @(csx_ind_alt … H) -V1
  45 #V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
  46 #L2 #HL2 #IHL2 #V2 #HL12 #HV12 @lsx_intro
  47 #Y #H #HnT elim (lpxs_inv_pair1 … H) -H
  48 #L3 #V3 #HL23 #HV23 #H destruct
  49 lapply (lpxs_trans … HL12 … HL23) #HL13
  50 elim (eq_term_dec V2 V3)
  51 [ (* -HV13 -HL2 -IHV1 -HL12 *) #H destruct
  52   @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
  53 | -IHL2 -HnT #HnV23 elim (eq_term_dec V1 V2)
  54   [ #H -HV12 destruct
  55   (*  @(lsx_lpxs_trans … (L2.ⓑ{I}V2)) /2 width=1 by lpxs_pair/ *)
  56     @(IHV1 … HnV23) -IHV1 -HnV23 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
  57   @(lsx_lpxs_trans … HL23)
  58
  59
  60 lemma lsx_fwd_bind_dx_lpxs: ∀h,g,a,I,G,L1,V. ⦃G, L1⦄ ⊢ ⬊*[h, g] V →
  61                             ∀L2,T. G ⊢ ⋕⬊*[h, g, ⓑ{a,I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
  62                             G ⊢ ⋕⬊*[h, g, T] L2.ⓑ{I}V.
  63 #h #g #a #I #G #L1 #V #H @(csx_ind_alt … H) -V
  64 #V1 #_ #IHV1 #L2 #T #H @(lsx_ind … H) -L2
  65 #L2 #HL2 #IHL2 #HL12 @lsx_intro
  66 #Y #H #HnT elim (lpxs_inv_pair1 … H) -H
  67 #L3 #V3 #HL23 #HV13 #H destruct
  68 lapply (lpxs_trans … HL12 … HL23) #HL13
  69 elim (eq_term_dec V1 V3)
  70 [ -HV13 -HL2 -IHV1 -HL12 #H destruct
  71   @IHL2 -IHL2 // -HL23 -HL13 /3 width=2 by lleq_fwd_bind_O/
  72 | -IHL2 -HnT #HnV13
  73   @(IHV1 … HnV13) -IHV1 -HnV13 /2 width=3 by lpxs_cpxs_trans/ -HL12 -HL13
  74   @(lsx_lpxs_trans … HL23)
  75
  76
  77 (* Advanced inversion lemmas ************************************************)
  78
  79
  80
  81 (* Main properties **********************************************************)
  82
  83 axiom csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → G ⊢ ⋕⬊*[h, g, T] L.
  84 (*
  85 #h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
  86 #G1 #L1 #T1 #IH #G2 #L2 * *
  87 [ #k #HG #HL #HT destruct //
  88 | #i #HG #HL #HT destruct
  89   #H
  90 | #p #HG #HL #HT destruct //
  91 | #a #I #V2 #T2 #HG #HL #HT #H destruct
  92   elim (csx_fwd_bind … H) -H
  93   #HV2 #HT2
  94 | #I #V2 #T2 #HG #HL #HT #H destruct
  95   elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
  96 *)