matita/matita/contribs/lambdadelta/apps_2/examples/ex_fpbg_refl.ma

   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/rt_computation/fpbg_fqup.ma".
  16 include "basic_2/rt_computation/fpbs_cpxs.ma".
  17 include "basic_2/rt_computation/fpbg_fpbs.ma".
  18
  19 (* EXAMPLES *****************************************************************)
  20
  21 (* Reflexivity of proper rst-computation: the term ApplOmega ****************)
  22
  23 definition ApplDelta (s0) (s): term ≝ +ⓛ⋆s0.ⓐ⋆s.ⓐ#0.#0.
  24
  25 definition ApplOmega1 (s0) (s): term ≝ ⓐ(ApplDelta s0 s).(ApplDelta s0 s).
  26
  27 definition ApplOmega2 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.ⓐ#0.#0.
  28
  29 definition ApplOmega3 (s0) (s): term ≝ +ⓓⓝ⋆s0.(ApplDelta s0 s).ⓐ⋆s.(ApplOmega1 s0 s).
  30
  31 definition ApplOmega4 (s0) (s): term ≝ ⓐ⋆s.(ApplOmega1 s0 s).
  32
  33 (* Basic properties *********************************************************)
  34
  35 lemma ApplDelta_lifts (f) (s0) (s):
  36       ⇧*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
  37 /5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
  38
  39 lemma cpr_ApplOmega_12 (G) (L) (s0) (s):
  40       ❪G,L❫ ⊢ ApplOmega1 s0 s ⬈ ApplOmega2 s0 s.
  41 /2 width=1 by cpx_beta/ qed.
  42
  43 lemma cpr_ApplOmega_23 (G) (L) (s0) (s):
  44       ❪G,L❫ ⊢ ApplOmega2 s0 s ⬈ ApplOmega3 s0 s.
  45 /6 width=3 by cpx_eps, cpx_flat, cpx_bind, cpx_delta, ApplDelta_lifts/ qed.
  46
  47 lemma cpr_ApplOmega_34 (G) (L) (s0) (s):
  48       ❪G,L❫ ⊢ ApplOmega3 s0 s ⬈ ApplOmega4 s0 s.
  49 /4 width=3 by cpx_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
  50
  51 lemma cpxs_ApplOmega_14 (G) (L) (s0) (s):
  52       ❪G,L❫ ⊢ ApplOmega1 s0 s ⬈* ApplOmega4 s0 s.
  53 /5 width=5 by cpxs_strap1, cpx_cpxs/ qed.
  54
  55 lemma fqup_ApplOmega_41 (G) (L) (s0) (s):
  56       ❪G,L,ApplOmega4 s0 s❫ ⬂+ ❪G,L,ApplOmega1 s0 s❫.
  57 /2 width=1 by/ qed.
  58
  59 (* Main properties **********************************************************)
  60
  61 theorem fpbg_refl (G) (L) (s0) (s):
  62         ❪G,L,ApplOmega1 s0 s❫ > ❪G,L,ApplOmega1 s0 s❫.
  63 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.