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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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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/reducibility/fpr_cpr.ma".
  16 include "basic_2/computation/cprs_lfprs.ma".
  17 include "basic_2/computation/lfprs_ltprs.ma".
  18 include "basic_2/computation/lfprs_fprs.ma".
  19
  20 (* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
  21
  22 (* Advanced inversion lemmas ************************************************)
  23
  24 lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
  25                       ∃∃K2,V2. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  26                                ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄  &
  27                                L2 = K2.ⓑ{I}V2.
  28 #I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
  29 #L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
  30 elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
  31 qed-.
  32
  33 lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
  34                       ∃∃K1,V1. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  35                                ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄  &
  36                                L1 = K1.ⓑ{I}V1.
  37 #I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
  38 #L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
  39 elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
  40 qed-.
  41
  42 (* Advanced forward lemmas **************************************************)
  43
  44 lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
  45                             ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
  46                                      T = -ⓑ{I}V2.T2.
  47 #I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
  48 #L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
  49 elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
  50 qed-.
  51
  52 lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
  53                            ∀b. ∃∃K2,V2. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  54                                         ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
  55                                         L2 = K2.ⓑ{I}V2.
  56 #I #K1 #L2 #V1 #T1 #T2 #H #b
  57 elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
  58 elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
  59 qed-.
  60
  61 (* Properties on context-sensitive parallel computation for terms ***********)
  62
  63 lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
  64 #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
  65 qed.
  66
  67 (* Forward lemmas on context-sensitive parallel computation for terms *******)
  68
  69 lemma fprs_fwd_cprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L1 ⊢ T1 ➡* T2.
  70 #L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 //
  71 #L #L2 #T #T2 #H1 #H2 #IH1
  72 elim (fpr_inv_all … H2) -H2 #L0 #HL0 #HT2 #_ -L2
  73 lapply (lfprs_cpr_trans L1 … HT2) -HT2 /3 width=3/
  74 qed-.
  75 (*
  76 (* Advanced properties ******************************************************)
  77
  78 lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
  79                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
  80 #L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
  81 elim (fpr_inv_all … H) /3 width=4/
  82 qed.
  83
  84 (* Advanced forward lemmas **************************************************)
  85
  86 lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
  87                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
  88                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
  89 * #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
  90 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
  91 [ elim (cpr_inv_abbr1 … H) -H *
  92   [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
  93   | #T #_ #_ #H destruct
  94   ]
  95 | elim (cpr_inv_abst1 … H Abst V2) -H
  96   #V #T #HV1 #_ #H destruct /3 width=4/
  97 ]
  98 qed-.
  99 *)