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

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   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "basic_2A/computation/csx_lpxs.ma".
  16 include "basic_2A/computation/lcosx_cpx.ma".
  17
  18 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
  23                         ∀K2. G ⊢ ⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
  24                         ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L2.
  25 #h #g #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
  26 #V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
  27 #K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
  28 #L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
  29 #Y #H #HLK2 elim (lpx_inv_pair1 … H) -H
  30 #K2 #V2 #HK02 #HV02 #H destruct
  31 elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
  32 [ /4 width=8 by lpxs_strap1, lleq_lref/
  33 | @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
  34   /3 width=4 by lsx_cpx_trans_O, lsx_lpx_trans, lpxs_cpx_trans, lpxs_strap1/ (**) (* full auto too slow *)
  35 ]
  36 qed.
  37
  38 lemma lsx_lref_be: ∀h,g,I,G,K,V,i,l. l ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
  39                    G ⊢ ⬊*[h, g, V, 0] K →
  40                    ∀L. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L.
  41 /2 width=8 by lsx_lref_be_lpxs/ qed.
  42
  43 (* Main properties **********************************************************)
  44
  45 theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀l. G ⊢ ⬊*[h, g, T, l] L.
  46 #h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
  47 #Z #Y #X #IH #G #L * * //
  48 [ #i #HG #HL #HT #H #l destruct
  49   elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
  50   elim (ylt_split i l) /2 width=1 by lsx_lref_skip/
  51   #Hli #Hi elim (drop_O1_lt (Ⓕ) … Hi) -Hi
  52   #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
  53   /4 width=6 by lsx_lref_be, fqup_lref/
  54 | #a #I #V #T #HG #HL #HT #H #l destruct
  55   elim (csx_fwd_bind … H) -H /3 width=1 by lsx_bind/
  56 | #I #V #T #HG #HL #HT #H #l destruct
  57   elim (csx_fwd_flat … H) -H /3 width=1 by lsx_flat/
  58 ]
  59 qed.