matita/matita/contribs/lambda_delta/basic_2/reducibility/fpr_cpr.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
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   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/reducibility/cfpr_cpr.ma".
  16
  17 (* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
  18
  19 (* Properties on context-sensitive parallel reduction for terms *************)
  20
  21 lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
  22 /2 width=4/ qed.
  23
  24 (* Advanced propertis *******************************************************)
  25
  26 lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
  27                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
  28 #L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
  29 elim (fpr_inv_all … H) /3 width=4/
  30 qed.
  31
  32 (* Advanced forward lemmas **************************************************)
  33
  34 lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
  35                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
  36                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
  37 * #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
  38 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
  39 [ elim (cpr_inv_abbr1 … H) -H *
  40   [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
  41   | #T #_ #_ #H destruct
  42   ]
  43 | elim (cpr_inv_abst1 … H Abst V2) -H
  44   #V #T #HV1 #_ #H destruct /3 width=4/
  45 ]
  46 qed-.
  47
  48 (* Advanced inversion lemmas ************************************************)
  49
  50 lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
  51                      ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
  52                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
  53                               L2 = K2.ⓑ{I}V2.
  54 #I1 #K1 #X #V1 #T1 #T2 #H
  55 elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
  56 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
  57 qed-.
  58
  59 lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
  60                      ∃∃K1,V1. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
  61                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄  &
  62                               L1 = K1.ⓑ{I}V1.
  63 #I2 #X #K2 #V2 #T1 #T2 #H
  64 elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
  65 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
  66 qed-.