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

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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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  13 (**************************************************************************)
  14
  15 include "basic_2/substitution/fqus_fqus.ma".
  16 include "basic_2/unfold/lsstas_lift.ma".
  17 include "basic_2/reduction/llpx_ldrop.ma".
  18 include "basic_2/computation/cpxs.ma".
  19
  20 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
  21
  22 (* Advanced properties ******************************************************)
  23
  24 lemma lsstas_cpxs: ∀h,g,G,L,T1,T2,l1. ⦃G, L⦄ ⊢ T1 •* [h, g, l1] T2 →
  25                    ∀l2. ⦃G, L⦄ ⊢ T1 ▪ [h, g] l2 → l1 ≤ l2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
  26 #h #g #G #L #T1 #T2 #l1 #H @(lsstas_ind_dx … H) -T2 -l1 //
  27 #l1 #T #T2 #HT1 #HT2 #IHT1 #l2 #Hl2 #Hl12
  28 lapply (lsstas_da_conf … HT1 … Hl2) -HT1
  29 >(plus_minus_m_m (l2-l1) 1 ?)
  30 [ /4 width=5 by cpxs_strap1, ssta_cpx, lt_to_le/
  31 | /2 width=1 by monotonic_le_minus_r/
  32 ]
  33 qed.
  34
  35 lemma cpxs_delta: ∀h,g,I,G,L,K,V,V2,i.
  36                   ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, g] V2 →
  37                   ∀W2. ⇧[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, g] W2.
  38 #h #g #I #G #L #K #V #V2 #i #HLK #H elim H -V2
  39 [ /3 width=9 by cpx_cpxs, cpx_delta/
  40 | #V1 lapply (ldrop_fwd_drop2 … HLK) -HLK
  41   elim (lift_total V1 0 (i+1)) /4 width=12 by cpx_lift, cpxs_strap1/
  42 ]
  43 qed.
  44
  45 lemma cpxs_llpx_conf: ∀h,g,G. s_r_confluent1 … (cpxs h g G) (llpx h g G 0).
  46 /3 width=5 by llpx_cpx_conf, s_r_conf1_LTC1/ qed-.
  47
  48 (* Advanced inversion lemmas ************************************************)
  49
  50 lemma cpxs_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, g] T2 →
  51                       T2 = #i ∨
  52                       ∃∃I,K,V1,T1. ⇩[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, g] T1 &
  53                                    ⇧[0, i+1] T1 ≡ T2.
  54 #h #g #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
  55 #T #T2 #_ #HT2 *
  56 [ #H destruct
  57   elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
  58   * /4 width=7 by cpx_cpxs, ex3_4_intro, or_intror/
  59 | * #I #K #V1 #T1 #HLK #HVT1 #HT1
  60   lapply (ldrop_fwd_drop2 … HLK) #H0LK
  61   elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
  62   /4 width=7 by cpxs_strap1, ex3_4_intro, or_intror/
  63 ]
  64 qed-.
  65
  66 (* Relocation properties ****************************************************)
  67
  68 lemma cpxs_lift: ∀h,g,G. l_liftable (cpxs h g G).
  69 /3 width=10 by cpx_lift, cpxs_strap1, l_liftable_LTC/ qed.
  70
  71 lemma cpxs_inv_lift1: ∀h,g,G. l_deliftable_sn (cpxs h g G).
  72 /3 width=6 by l_deliftable_sn_LTC, cpx_inv_lift1/
  73 qed-.
  74
  75 (* Properties on supclosure *************************************************)
  76
  77 lemma fqu_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
  78                       ∀T1. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
  79                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
  80 #h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
  81 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
  82 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
  83 qed-.
  84
  85 lemma fquq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
  86                        ∀T1. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
  87                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
  88 #h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
  89 [ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
  90 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
  91 ]
  92 qed-.
  93
  94 lemma fquq_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
  95                          ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
  96                          ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
  97                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
  98 /3 width=5 by fquq_cpxs_trans, lsstas_cpxs/ qed-.
  99
 100 lemma fqup_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
 101                        ∀T1. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
 102                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
 103 #h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
 104 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
 105 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
 106 qed-.
 107
 108 lemma fqus_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
 109                        ∀T1. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
 110                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
 111 #h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
 112 [ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
 113 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 114 ]
 115 qed-.
 116
 117 lemma fqus_lsstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
 118                          ∀U2,l1. ⦃G2, L2⦄ ⊢ T2 •*[h, g, l1] U2 →
 119                          ∀l2. ⦃G2, L2⦄ ⊢ T2 ▪ [h, g] l2 → l1 ≤ l2 →
 120                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
 121 /3 width=7 by fqus_cpxs_trans, lsstas_cpxs/ qed-.