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

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  14
  15 include "basic_2/relocation/lleq_lleq.ma".
  16 include "basic_2/computation/lpxs_ldrop.ma".
  17 include "basic_2/computation/lpxs_cpxs.ma".
  18
  19 (* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
  20
  21 (* Inversion lemmas on lazy equivalence for local environments **************)
  22
  23 lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[d, T] L2 → ⊥) →
  24                               ∃∃L. ⦃G, L1⦄ ⊢ ➡*[h, g] L & L1 ⋕[d, T] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L2.
  25 #h #g #G #L1 #L2 #T #d #H @(lpxs_ind_dx … H) -L1
  26 [ #H elim H -H //
  27 | #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L d) #H
  28   [ -H2 elim IH2 -IH2
  29     /4 width=4 by lpxs_strap2, lleq_canc_sn, lleq_trans, ex3_intro/
  30   | -IH2 -H12 /3 width=4 by lpx_lpxs, ex3_intro/ (**) (* auto fails without clear *)
  31   ]
  32 ]
  33 qed-.
  34
  35 (* Properties on lazy equivalence for local environments ********************)
  36
  37 lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
  38                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
  39                            ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
  40 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  41 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
  42   #K0 #V0 #H1KL1 #_ #H destruct
  43   elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
  44   #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
  45   /2 width=4 by fqu_lref_O, ex3_intro/
  46 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
  47   [ elim (lleq_inv_bind … H)
  48   | elim (lleq_inv_flat … H)
  49   ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
  50 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
  51   /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
  52 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
  53   /2 width=4 by fqu_flat_dx, ex3_intro/
  54 | #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
  55   elim (ldrop_O1_le (e+1) K1)
  56   [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
  57     #H2KL elim (lpxs_ldrop_trans_O1 … H1KL1 … HL1) -L1
  58     #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
  59     /3 width=4 by fqu_drop, ex3_intro/
  60   | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g
  61     lapply (lleq_fwd_length … H2KL1) //
  62   ]
  63 ]
  64 qed-.
  65
  66 lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
  67                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
  68                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
  69 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
  70 elim (fquq_inv_gen … H) -H
  71 [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
  72   /3 width=4 by fqu_fquq, ex3_intro/
  73 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
  74 ]
  75 qed-.
  76
  77 lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
  78                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
  79                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
  80 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
  81 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
  82   /3 width=4 by fqu_fqup, ex3_intro/
  83 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
  84   #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
  85   /3 width=5 by fqup_strap1, ex3_intro/
  86 ]
  87 qed-.
  88
  89 lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
  90                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[0, T1] L1 →
  91                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[0, T2] L2.
  92 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
  93 elim (fqus_inv_gen … H) -H
  94 [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
  95   /3 width=4 by fqup_fqus, ex3_intro/
  96 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
  97 ]
  98 qed-.