matita/matita/contribs/lambdadelta/basic_2/reducibility/fpr.ma

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