matita/matita/contribs/lambdadelta/basic_2/etc/cpys/lsuby.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/extlrsubeq_2.ma".
  16 include "basic_2/relocation/ldrop.ma".
  17
  18 (* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
  19
  20 inductive lsuby: relation lenv ≝
  21 | lsuby_atom: ∀L. lsuby L (⋆)
  22 | lsuby_pair: ∀I1,I2,L1,L2,V. lsuby L1 L2 → lsuby (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
  23 .
  24
  25 interpretation
  26   "local environment refinement (extended substitution)"
  27   'ExtLRSubEq L1 L2 = (lsuby L1 L2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma lsuby_refl: ∀L. L ⊑× L.
  32 #L elim L -L // /2 width=1/
  33 qed.
  34
  35 (* Basic inversion lemmas ***************************************************)
  36
  37 fact lsuby_inv_atom1_aux: ∀L1,L2. L1 ⊑× L2 → L1 = ⋆ → L2 = ⋆.
  38 #L1 #L2 * -L1 -L2 //
  39 #I1 #I2 #L1 #L2 #V #_ #H destruct
  40 qed-.
  41
  42 lemma lsuby_inv_atom1: ∀L2. ⋆ ⊑× L2 → L2 = ⋆.
  43 /2 width=3 by lsuby_inv_atom1_aux/ qed-.
  44
  45 fact lsuby_inv_pair1_aux: ∀L1,L2. L1 ⊑× L2 → ∀J1,K1,W. L1 = K1.ⓑ{J1}W →
  46                           L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
  47 #L1 #L2 * -L1 -L2
  48 [ #L #J1 #K1 #W #H destruct /2 width=1 by or_introl/
  49 | #I1 #I2 #L1 #L2 #V #HL12 #J1 #K1 #W #H destruct /3 width=4 by ex2_2_intro, or_intror/
  50 ]
  51 qed-.
  52
  53 lemma lsuby_inv_pair1: ∀I1,K1,L2,W. K1.ⓑ{I1}W ⊑× L2 →
  54                        L2 = ⋆ ∨ ∃∃I2,K2. K1 ⊑× K2 & L2 = K2.ⓑ{I2}W.
  55 /2 width=4 by lsuby_inv_pair1_aux/ qed-.
  56
  57 (* Basic forward lemmas *****************************************************)
  58
  59 lemma lsuby_fwd_length: ∀L1,L2. L1 ⊑× L2 → |L2| ≤ |L1|.
  60 #L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
  61 qed-.
  62
  63 lemma lsuby_fwd_ldrop2_pair: ∀L1,L2. L1 ⊑× L2 →
  64                              ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
  65                              ∃∃I1,K1. K1 ⊑× K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
  66 #L1 #L2 #H elim H -L1 -L2
  67 [ #L #J2 #K2 #W #i #H
  68   elim (ldrop_inv_atom1 … H) -H #H destruct
  69 | #I1 #I2 #L1 #L2 #V #HL12 #IHL12 #J2 #K2 #W #i #H
  70   elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
  71   [ /3 width=4 by ldrop_pair, ex2_2_intro/
  72   | elim (IHL12 … HLK2) -IHL12 -HLK2 * /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
  73   ]
  74 ]
  75 qed-.