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

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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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 properties ******************************************************)
  23
  24 lemma fprs_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L2, U2⦄ →
  25                         ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡* ⦃L2.ⓑ{I}V2, T2⦄ &
  26                                  U2 = -ⓑ{I}V2.T2.
  27 #I #L1 #L2 #V1 #T1 #U2 #H @(fprs_ind … H) -L2 -U2 /2 width=4/
  28 #L #L2 #U #U2 #_ #HU2 * #V #T #HT1 #H destruct
  29 elim (fpr_bind2_minus … HU2) -HU2 /3 width=4/
  30 qed-.
  31
  32 (* Advanced inversion lemmas ************************************************)
  33
  34 lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
  35                       ∃∃K2,V2. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  36                                ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄  &
  37                                L2 = K2.ⓑ{I}V2.
  38 #I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
  39 #L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
  40 elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
  41 qed-.
  42
  43 lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
  44                       ∃∃K1,V1. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  45                                ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄  &
  46                                L1 = K1.ⓑ{I}V1.
  47 #I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
  48 #L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
  49 elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
  50 qed-.
  51
  52 (* Advanced forward lemmas **************************************************)
  53
  54 lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
  55                             ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
  56                                      T = -ⓑ{I}V2.T2.
  57 #I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
  58 #L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
  59 elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
  60 qed-.
  61
  62 lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
  63                            ∀b. ∃∃K2,V2. ⦃K1, V1⦄  ➡* ⦃K2, V2⦄ &
  64                                         ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
  65                                         L2 = K2.ⓑ{I}V2.
  66 #I #K1 #L2 #V1 #T1 #T2 #H #b
  67 elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
  68 elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
  69 qed-.
  70
  71 lemma fprs_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡* ⦃L2, U2⦄ → ∀b,I,W.
  72                       ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡* ⦃L2, ⓑ{b,I}W.T2⦄ &
  73                                U2 = ⓛ{a}V2.T2.
  74 #a #L1 #L2 #V1 #T1 #U2 #H #b #I #W @(fprs_ind … H) -L2 -U2 /2 width=4/
  75 #L #L2 #U #U2 #_ #H2 * #V #T #HT1 #H destruct
  76 elim (fpr_fwd_abst2 … H2 b I W) -H2 /3 width=4/
  77 qed-.
  78
  79 (* Properties on context-sensitive parallel computation for terms ***********)
  80
  81 lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
  82 #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
  83 qed.
  84
  85 (* Forward lemmas on context-sensitive parallel computation for terms *******)
  86
  87 lemma fprs_fwd_cprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L1 ⊢ T1 ➡* T2.
  88 #L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 //
  89 #L #L2 #T #T2 #H1 #H2 #IH1
  90 elim (fpr_inv_all … H2) -H2 #L0 #HL0 #HT2 #_ -L2
  91 lapply (lfprs_cpr_trans L1 … HT2) -HT2 /3 width=3/
  92 qed-.
  93 (*
  94 (* Advanced properties ******************************************************)
  95
  96 lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
  97                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
  98 #L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
  99 elim (fpr_inv_all … H) /3 width=4/
 100 qed.
 101
 102 (* Advanced forward lemmas **************************************************)
 103
 104 lamma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
 105                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
 106                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
 107 * #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
 108 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
 109 [ elim (cpr_inv_abbr1 … H) -H *
 110   [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
 111   | #T #_ #_ #H destruct
 112   ]
 113 | elim (cpr_inv_abst1 … H Abst V2) -H
 114   #V #T #HV1 #_ #H destruct /3 width=4/
 115 ]
 116 qed-.
 117 *)