matita/matita/contribs/lambdadelta/basic_2/etc/cpr/ltpr.etc

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  14
  15 include "basic_2/grammar/lenv_px.ma".
  16 include "basic_2/reducibility/tpr.ma".
  17
  18 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
  19
  20 definition ltpr: relation lenv ≝ lpx tpr.
  21
  22 interpretation
  23   "context-free parallel reduction (environment)"
  24   'PRed L1 L2 = (ltpr L1 L2).
  25
  26 (* Basic properties *********************************************************)
  27
  28 lemma ltpr_refl: reflexive … ltpr.
  29 /2 width=1/ qed.
  30
  31 lemma ltpr_append: ∀K1,K2. K1 ➡ K2 → ∀L1,L2:lenv. L1 ➡ L2 → K1 @@ L1 ➡ K2 @@ L2.
  32 /2 width=1/ qed.
  33
  34 (* Basic inversion lemmas ***************************************************)
  35
  36 (* Basic_1: was: wcpr0_gen_sort *)
  37 lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
  38 /2 width=2 by lpx_inv_atom1/ qed-.
  39
  40 (* Basic_1: was: wcpr0_gen_head *)
  41 lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
  42                       ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
  43 /2 width=1 by lpx_inv_pair1/ qed-.
  44
  45 lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
  46 /2 width=2 by lpx_inv_atom2/ qed-.
  47
  48 lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
  49                       ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
  50 /2 width=1 by lpx_inv_pair2/ qed-.
  51
  52 (* Basic forward lemmas *****************************************************)
  53
  54 lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
  55 /2 width=2 by lpx_fwd_length/ qed-.
  56
  57 (* Advanced inversion lemmas ************************************************)
  58
  59 lemma ltpr_inv_append1: ∀K1,L1. ∀L:lenv. K1 @@ L1 ➡ L →
  60                         ∃∃K2,L2. K1 ➡ K2 & L1 ➡ L2 & L = K2 @@ L2.
  61 /2 width=1 by lpx_inv_append1/ qed-.
  62
  63 lemma ltpr_inv_append2: ∀L:lenv. ∀K2,L2. L ➡ K2 @@ L2 →
  64                         ∃∃K1,L1. K1 ➡ K2 & L1 ➡ L2 & L = K1 @@ L1.
  65 /2 width=1 by lpx_inv_append2/ qed-.
  66
  67 (* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)