matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lpys.etc

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/notation/relations/extpsubstsnstar_3.ma".
  16 include "basic_2/grammar/lpx_sn.ma".
  17 include "basic_2/substitution/cpys.ma".
  18
  19 (* SN EXTENDED MULTIPLE SUBSTITUTION FOR LOCAL ENVIRONMENTS *****************)
  20
  21 definition lpys: relation3 genv lenv lenv ≝ λG. lpx_sn (cpys G).
  22
  23 interpretation
  24    "extended multiple substitution (local environment, sn variant)"
  25    'ExtPSubstSnStar G L1 L2 = (lpys G L1 L2).
  26
  27 (* Basic inversion lemmas ***************************************************)
  28
  29 lemma lpys_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ▶*× L2 → L2 = ⋆.
  30 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
  31
  32 lemma lpys_inv_pair1: ∀I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶*× L2 →
  33                       ∃∃K2,V2. ⦃G, K1⦄ ⊢ ▶*× K2 & ⦃G, K1⦄ ⊢ V1 ▶*× V2 &
  34                                L2 = K2. ⓑ{I} V2.
  35 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
  36
  37 lemma lpys_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ▶*× ⋆ → L1 = ⋆.
  38 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
  39
  40 lemma lpys_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ▶*× K2.ⓑ{I}V2 →
  41                       ∃∃K1,V1. ⦃G, K1⦄ ⊢ ▶*× K2 & ⦃G, K1⦄ ⊢ V1 ▶*× V2 &
  42                                L1 = K1. ⓑ{I} V1.
  43 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
  44
  45 lemma lpys_inv_pair: ∀I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ▶*× L2.ⓑ{I2}V2 →
  46                      ∧∧ ⦃G, L1⦄ ⊢ ▶*× L2 & ⦃G, L1⦄ ⊢ V1 ▶*× V2 & I1 = I2.
  47 /2 width=1 by lpx_sn_inv_pair/ qed-.
  48
  49 (* Basic properties *********************************************************)
  50
  51 lemma lpys_refl: ∀G,L. ⦃G, L⦄ ⊢ ▶*× L.
  52 /2 width=1 by lpx_sn_refl/ qed.
  53
  54 lemma lpys_pair: ∀I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ▶*× K2 → ⦃G, K1⦄ ⊢ V1 ▶*× V2 →
  55                  ⦃G, K1.ⓑ{I}V1⦄ ⊢ ▶*× K2.ⓑ{I}V2.
  56 /2 width=1 by lpx_sn_pair/ qed.
  57
  58 lemma lpys_append: ∀G,K1,K2. ⦃G, K1⦄ ⊢ ▶*× K2 → ∀L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 →
  59                    ⦃G, L1 @@ K1⦄ ⊢ ▶*× L2 @@ K2.
  60 /3 width=1 by lpx_sn_append, cpys_append/ qed.
  61
  62 (* Basic forward lemmas *****************************************************)
  63
  64 lemma lpys_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ▶*× L2 → |L1| = |L2|.
  65 /2 width=2 by lpx_sn_fwd_length/ qed-.
  66
  67 (* Advanced forward lemmas **************************************************)
  68
  69 lemma lpys_fwd_append1: ∀G,K1,L1,L. ⦃G, K1 @@ L1⦄ ⊢ ▶*× L →
  70                         ∃∃K2,L2. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K2 @@ L2.
  71 /2 width=2 by lpx_sn_fwd_append1/ qed-.
  72
  73 lemma lpys_fwd_append2: ∀G,L,K2,L2. ⦃G, L⦄ ⊢ ▶*× K2 @@ L2 →
  74                         ∃∃K1,L1. ⦃G, K1⦄ ⊢ ▶*× K2 & L = K1 @@ L1.
  75 /2 width=2 by lpx_sn_fwd_append2/ qed-.