matita/matita/contribs/lambdadelta/basic_2/etc/fpr/cfpr.etc

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   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                  *)
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  13 (**************************************************************************)
  14
  15 notation "hvbox( L ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'FocalizedPRed $L $L1 $T1 $L2 $T2 }.
  18
  19 include "basic_2/reducibility/cpr.ma".
  20 include "basic_2/reducibility/fpr.ma".
  21
  22 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
  23
  24 definition cfpr: lenv → bi_relation lenv term ≝
  25                  λL,L1,T1,L2,T2. |L1| = |L2| ∧ L ⊢ L1 @@ T1 ➡ L2 @@ T2.
  26
  27 interpretation
  28    "context-sensitive parallel reduction (closure)"
  29    'FocalizedPRed L L1 T1 L2 T2 = (cfpr L L1 T1 L2 T2).
  30
  31 (* Basic properties *********************************************************)
  32
  33 lemma cfpr_refl: ∀L. bi_reflexive … (cfpr L).
  34 /2 width=1/ qed.
  35
  36 lemma fpr_cfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⋆ ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
  37 #L1 #L2 #T1 #T2 * /3 width=1/
  38 qed.
  39
  40 (* Basic inversion lemmas ***************************************************)
  41
  42 lemma cfpr_inv_atom1: ∀L,L2,T1,T2. L ⊢ ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → L ⊢ T1 ➡ T2 ∧ L2 = ⋆.
  43 #L #L2 #T1 #T2 * #H >(length_inv_zero_sn … H) /2 width=1/
  44 qed-.
  45
  46 (* Advanced inversion lemmas ************************************************)
  47
  48 lemma fpr_inv_pair1_sn: ∀I,K1,L2,V1,T1,T2. ⦃⋆.ⓑ{I}V1@@K1, T1⦄ ➡ ⦃L2, T2⦄ →
  49                         ∃∃K2,V2. V1 ➡ V2 &
  50                                  ⋆.ⓑ{I}V2 ⊢ ⦃K1, T1⦄ ➡ ⦃K2, T2⦄  &
  51                                  L2 = ⋆.ⓑ{I}V2@@K2.
  52 #I1 #K1 #L2 #V1 #T1 #T2 * >append_length #H
  53 elim (length_inv_pos_sn_append … H) -H #I2 #K2 #V2 #HK12 #H destruct
  54 >shift_append_assoc >shift_append_assoc normalize in ⊢ (%→?); #H
  55 elim (tpr_inv_bind1 … H) -H *
  56 [ #V0 #T #T0 #HV10 #HT1 #HT0 #H destruct /5 width=5/
  57 | #T0 #_ #_ #H destruct
  58 ]
  59 qed-.