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

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  13 (**************************************************************************)
  14
  15 include "basic_2/grammar/lenv_px_bi.ma".
  16 include "basic_2/reducibility/fpr.ma".
  17
  18 (* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
  19
  20 (* alternative definition *)
  21 definition lfpra: relation lenv ≝ lpx_bi fpr.
  22
  23 interpretation
  24   "focalized parallel reduction (environment) alternative"
  25   'FocalizedPRedAlt L1 L2 = (lfpra L1 L2).
  26
  27 (* Basic properties *********************************************************)
  28
  29 lemma lfpra_refl: reflexive … lfpra.
  30 /2 width=1/ qed.
  31
  32 (* Basic inversion lemmas ***************************************************)
  33
  34 lemma lfpra_inv_atom1: ∀L2. ⦃⋆⦄ ➡➡ ⦃L2⦄ → L2 = ⋆.
  35 /2 width=2 by lpx_bi_inv_atom1/ qed-.
  36
  37 lemma lfpra_inv_pair1: ∀K1,I,V1,L2. ⦃K1. ⓑ{I} V1⦄ ➡➡ ⦃L2⦄ →
  38                        ∃∃K2,V2. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
  39                                 L2 = K2. ⓑ{I} V2.
  40 /2 width=1 by lpx_bi_inv_pair1/ qed-.
  41
  42 lemma lfpra_inv_atom2: ∀L1. ⦃L1⦄ ➡➡ ⦃⋆⦄ → L1 = ⋆.
  43 /2 width=2 by lpx_bi_inv_atom2/ qed-.
  44
  45 lemma lfpra_inv_pair2: ∀L1,K2,I,V2. ⦃L1⦄ ➡➡ ⦃K2. ⓑ{I} V2⦄ →
  46                        ∃∃K1,V1. ⦃K1⦄ ➡➡ ⦃K2⦄ & ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
  47                                 L1 = K1. ⓑ{I} V1.
  48 /2 width=1 by lpx_bi_inv_pair2/ qed-.
  49
  50 (* Basic forward lemmas *****************************************************)
  51
  52 lemma lfpra_fwd_length: ∀L1,L2. ⦃L1⦄ ➡➡ ⦃L2⦄ → |L1| = |L2|.
  53 /2 width=2 by lpx_bi_fwd_length/ qed-.
  54
  55 (****************************************************************************)
  56 (*
  57 lemma fpr_lfpra: ∀L1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
  58 #L1 #T1 @(cw_wf_ind … L1 T1) -L1 -T1 *
  59 [ #T1 #_ #L2 #T2 #H
  60   elim (fpr_inv_atom1 … H) -H #_ #H destruct //
  61 | #L1 #I #V1 #T1 #IH #Y #X #H
  62   elim (fpr_inv_pair1 … H) -H #L2 #V2 #HL12 #H destruct
  63   @lpx_bi_pair
  64   [ @(IH … HL12)
  65   | @IH
  66
  67   /3 width=4/
  68
  69 lemma fpr_lfpra: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡➡ ⦃L2⦄.
  70 #L1 elim L1 -L1
  71 [ #L2 #T1 #T2 #H
  72   elim (fpr_inv_atom1 … H) -H #_ #H destruct //
  73 | #L1 #I #V1 #IH #L2 #T1 #T2 #H
  74   elim (fpr_inv_pair1 … H) -H #L #V #HL1 #H destruct
  75   @lpx_bi_pair /2 width=3/
  76
  77   /4 width=3/
  78
  79 (*
  80 include "basic_2/reducibility/lcpr.ma".
  81
  82 lemma lcpr_pair2: ∀L1,L2. L1 ⊢ ➡ L2 → ∀V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ →
  83                   ∀I. L1. ⓑ{I} V1 ⊢ ➡ L2. ⓑ{I} V2.
  84 #L1 #L2 * #L #HL1 #HL2 #V1 #V2 *
  85 #H1 #H2 #I
  86 @(ex2_1_intro … (L.ⓑ{I}V1)) /2 width=1/
  87 @tpss_
  88
  89 (*
  90 <(ltpss_fwd_length … HL2) /4 width=5/
  91 qed.
  92 *)
  93 *)
  94 *)