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

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/computation/csx_llpxs.ma".
  16 include "basic_2/computation/llsx_ldrop.ma".
  17 include "basic_2/computation/llsx_llpx.ma".
  18 include "basic_2/computation/llsx_llpxs.ma".
  19 (*
  20 axiom cpx_llpx_trans: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
  21                       ∀L2. ⦃G, L1⦄⊢ ➡[h, g, T2, O] L2 →
  22                       ∃∃L. ⦃G, L1⦄⊢ ➡[h, g, T1, O] L & L ⋕[T2, 0] L2.
  23 (*
  24 fact llsx_cpx_trans_aux: ∀h,g,G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ➡[h, g] T2 →
  25                          ∀L1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
  26                          L0 ⋕[T1, d] L1 → ∀L2. L1 ⋕[T2, d] L2 →
  27                          G ⊢ ⋕⬊*[h, g, T2, d] L2.
  28 #h #g #G #L0 #T1 #T2 #HT12 #L1 #d #H @(llsx_ind … H) -L1
  29 #L1 #_ #IHL1 #Hd #He011 #L2 #He122 @llsx_intro
  30 #L3 #Hx223 #Hn223 destruct
  31 lapply (lleq_cpx_conf_sn … HT12 … He011) #He021
  32 lapply (lleq_cpx_conf … HT12 … He011) -HT12 #HT12
  33 lapply (lleq_llpx_trans … He122 … Hx223) -Hx223 #Hx123
  34 elim (cpx_llpx_trans … HT12 … Hx123) -Hx123 #L4 #Hx114 #He423
  35 (* lapply (lleq_cpx_conf … Hx114 … He011) #He120 *)
  36 @(IHL1 … Hx114) // -IHL1
  37 [ #HL13 @HnL2 -HnL2
  38 *)
  39
  40 fact llsx_cpx_trans_aux: ∀h,g,G,L1,T1,d. G ⊢ ⋕⬊*[h, g, T1, d] L1 → d = 0 →
  41                          ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
  42                          ∀L2. L1 ⋕[T1, d] L2 → G ⊢ ⋕⬊*[h, g, T2, 0] L2.
  43 #h #g #G #L1 #T1 #d #H @(llsx_ind … H) -L1
  44 #L1 #_ #IHL1 #Hd #T2 #HT12 #L2 #He112 @llsx_intro
  45 #L3 #Hx223 #Hn223 destruct
  46 lapply (lleq_cpx_conf_sn … HT12 … He112) #He122
  47 lapply (lleq_cpx_conf … HT12 … He112) -HT12 #HT12
  48 elim (cpx_llpx_trans … HT12 … Hx223) #L4 #Hx214 #He423
  49 @(IHL1 … L4) //
  50 *)
  51 axiom llsx_cpx_trans_O: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
  52                         G ⊢ ⋕⬊*[h, g, T1, 0] L → G ⊢ ⋕⬊*[h, g, T2, 0] L.
  53
  54 (* LAZY SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS *****************)
  55
  56 (* Advanced properties ******************************************************)
  57
  58 lemma llsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,d. d ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
  59                          ∀K2. G ⊢ ⋕⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g, V, 0] K2 →
  60                          ∀L2. ⇩[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L2.
  61 #h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt … H) -V
  62 #V0 #_ #IHV0 #K2 #H @(llsx_ind … H) -K2
  63 #K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @llsx_intro
  64 #L2 #HL02 #HnL02 elim (llpx_inv_lref_ge_sn … HL02 … HLK0) // -HL02
  65 #K2 #V2 #HLK2 #HK02 #HV02 elim (eq_term_dec V0 V2)
  66 #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0 ]
  67 [ /4 width=7 by llpxs_strap1, lleq_lref/
  68 | lapply (llpx_cpx_conf … HV02 … HK02) -HK02 #HK02
  69   @(IHV0 … HnV02 … HLK2) -IHV0 -HnV02 -HLK2
  70   /3 width=3 by llsx_cpx_trans_O, llpxs_cpx_conf_dx, llsx_llpx_trans, llpxs_cpx_trans, llpxs_strap1/
  71 ]
  72 qed.
  73
  74 lemma llsx_lref_be: ∀h,g,I,G,K,V,i,d. d ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
  75                     G ⊢ ⋕⬊*[h, g, V, 0] K →
  76                     ∀L. ⇩[i] L ≡ K.ⓑ{I}V → G ⊢ ⋕⬊*[h, g, #i, d] L.
  77 /2 width=8 by llsx_lref_be_lpxs/ qed.
  78
  79 (* Main properties **********************************************************)
  80
  81 theorem csx_llsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀d. G ⊢ ⋕⬊*[h, g, T, d] L.
  82 #h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
  83 #Z #Y #X #IH #G #L * * //
  84 [ #i #HG #HL #HT #H #d destruct
  85   elim (lt_or_ge i (|L|)) /2 width=1 by llsx_lref_free/
  86   elim (ylt_split i d) /2 width=1 by llsx_lref_skip/
  87   #Hdi #Hi elim (ldrop_O1_lt … Hi) -Hi
  88   #I #K #V #HLK lapply (csx_inv_lref_bind … HLK … H) -H
  89   /4 width=6 by llsx_lref_be, fqup_lref/
  90 | #a #I #V #T #HG #HL #HT #H #d destruct
  91   elim (csx_fwd_bind … H) -H /3 width=1 by llsx_bind/
  92 | #I #V #T #HG #HL #HT #H #d destruct
  93   elim (csx_fwd_flat … H) -H /3 width=1 by llsx_flat/
  94 ]
  95 qed.