matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma

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  14
  15 include "basic_2/unfold/lpqs.ma".
  16 include "basic_2/reduction/cpr.ma".
  17
  18 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
  19
  20 definition lpr: relation lenv ≝ lpx_sn cpr.
  21
  22 interpretation "parallel reduction (local environment, sn variant)"
  23    'PRedSn L1 L2 = (lpr L1 L2).
  24
  25 (* Basic inversion lemmas ***************************************************)
  26
  27 (* Basic_1: includes: wcpr0_gen_sort *)
  28 lemma lpr_inv_atom1: ∀L2. ⋆ ⊢ ➡ L2 → L2 = ⋆.
  29 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  30
  31 (* Basic_1: includes: wcpr0_gen_head *)
  32 lemma lpr_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ➡ L2 →
  33                      ∃∃K2,V2. K1 ⊢ ➡ K2 & K1 ⊢ V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
  34 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  35
  36 lemma lpr_inv_atom2: ∀L1. L1 ⊢ ➡ ⋆ → L1 = ⋆.
  37 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  38
  39 lemma lpr_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➡ K2. ⓑ{I} V2 →
  40                      ∃∃K1,V1. K1 ⊢ ➡ K2 & K1 ⊢ V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
  41 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  42
  43 (* Basic properties *********************************************************)
  44
  45 (* Note: lemma 250 *)
  46 lemma lpr_refl: ∀L. L ⊢ ➡ L.
  47 /2 width=1 by lpx_sn_refl/ qed.
  48
  49 lemma lpr_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ➡ K2 → K1 ⊢ V1 ➡ V2 →
  50                 K1.ⓑ{I}V1 ⊢ ➡ K2.ⓑ{I}V2.
  51 /2 width=1/ qed.
  52
  53 lemma lpr_append: ∀K1,K2. K1 ⊢ ➡ K2 → ∀L1,L2. L1 ⊢ ➡ L2 →
  54                   L1 @@ K1 ⊢ ➡ L2 @@ K2.
  55 /3 width=1 by lpx_sn_append, cpr_append/ qed.
  56
  57 lemma lpqs_lpr: ∀L1,L2. L1 ⊢ ➤* L2 → L1 ⊢ ➡ L2.
  58 #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
  59 qed.
  60
  61 lemma lpss_lpr: ∀L1,L2. L1 ⊢ ▶* L2 → L1 ⊢ ➡ L2.
  62 /3 width=1/ qed.
  63
  64 (* Basic forward lemmas *****************************************************)
  65
  66 lemma lpr_fwd_length: ∀L1,L2. L1 ⊢ ➡ L2 → |L1| = |L2|.
  67 /2 width=2 by lpx_sn_fwd_length/ qed-.
  68
  69 (* Advanced forward lemmas **************************************************)
  70
  71 lemma lpr_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➡ L →
  72                        ∃∃K2,L2. K1 ⊢ ➡ K2 & L = K2 @@ L2.
  73 /2 width=2 by lpx_sn_fwd_append1/ qed-.
  74
  75 lemma lpr_fwd_append2: ∀L,K2,L2. L ⊢ ➡ K2 @@ L2 →
  76                        ∃∃K1,L1. K1 ⊢ ➡ K2 & L = K1 @@ L1.
  77 /2 width=2 by lpx_sn_fwd_append2/ qed-.
  78
  79 (* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
  80                                 pr0_subst1_back
  81 *)