matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpr.etc

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  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ➡ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'FocalizedPRed $L1 $T1 $L2 $T2 }.
  18
  19 include "basic_2/reducibility/tpr.ma".
  20
  21 (* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
  22
  23 definition fpr: bi_relation lenv term ≝
  24                 λL1,T1,L2,T2. |L1| = |L2| ∧ L1 @@ T1 ➡ L2 @@ T2.
  25
  26 interpretation
  27    "context-free parallel reduction (closure)"
  28    'FocalizedPRed L1 T1 L2 T2 = (fpr L1 T1 L2 T2).
  29
  30 (* Basic properties *********************************************************)
  31
  32 lemma fpr_refl: bi_reflexive … fpr.
  33 /2 width=1/ qed.
  34
  35 lemma fpr_shift: ∀I1,I2,L1,L2,V1,V2,T1,T2.
  36                  ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
  37                  ⦃L1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2.ⓑ{I2}V2, T2⦄.
  38 #I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 * #HL12 #HT12
  39 @conj // normalize // (**) (* explicit constructor *)
  40 qed.
  41
  42 (* Basic inversion lemmas ***************************************************)
  43
  44 lemma fpr_inv_atom1: ∀L2,T1,T2. ⦃⋆, T1⦄ ➡ ⦃L2, T2⦄ → T1 ➡ T2 ∧ L2 = ⋆.
  45 #L2 #T1 #T2 * #H
  46 lapply (length_inv_zero_sn … H) -H #H destruct /2 width=1/
  47 qed-.
  48
  49 lemma fpr_inv_atom3: ∀L1,T1,T2. ⦃L1,T1⦄ ➡ ⦃⋆,T2⦄ → T1 ➡ T2 ∧ L1 = ⋆.
  50 #L1 #T1 #T2 * #H
  51 lapply (length_inv_zero_dx … H) -H #H destruct /2 width=1/
  52 qed-.
  53
  54 (* Basic forward lemmas *****************************************************)
  55
  56 lemma fpr_fwd_pair1: ∀I1,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I1}V1, T1⦄ ➡ ⦃L2, T2⦄ →
  57                      ∃∃I2,K2,V2. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄  &
  58                                  L2 = K2.ⓑ{I2}V2.
  59 #I1 #K1 #L2 #V1 #T1 #T2 * #H
  60 elim (length_inv_pos_sn … H) -H #I2 #K2 #V2 #HK12 #H destruct /3 width=5/
  61 qed-.
  62
  63 lemma fpr_fwd_pair3: ∀I2,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I2}V2, T2⦄ →
  64                      ∃∃I1,K1,V1. ⦃K1, -ⓑ{I1}V1.T1⦄ ➡ ⦃K2, -ⓑ{I2}V2.T2⦄  &
  65                                  L1 = K1.ⓑ{I1}V1.
  66 #I2 #L1 #K2 #V2 #T1 #T2 * #H
  67 elim (length_inv_pos_dx … H) -H #I1 #K1 #V1 #HK12 #H destruct /3 width=5/
  68 qed-.