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

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