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

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "basic_2/rt_computation/fpbs_cpxs.ma".
  16 include "basic_2/rt_computation/fpbg_fqup.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: term → nat → term ≝ λW,s. +ⓛW.ⓐ⋆s.ⓐ#0.#0.
  24
  25 definition ApplOmega1: term → nat → term ≝ λW,s. ⓐ(ApplDelta W s).(ApplDelta W s).
  26
  27 definition ApplOmega2: term → nat → term ≝ λW,s. +ⓓⓝW.(ApplDelta W s).ⓐ⋆s.ⓐ#0.#0.
  28
  29 definition ApplOmega3: term → nat → term ≝ λW,s. ⓐ⋆s.(ApplOmega1 W s).
  30
  31 (* Basic properties *********************************************************)
  32
  33 lemma ApplDelta_lifts (f:rtmap):
  34                       ∀W1,W2,s. ⬆*[f] W1 ≘ W2 →
  35                       ⬆*[f] (ApplDelta W1 s) ≘ (ApplDelta W2 s).
  36 /5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
  37
  38 lemma cpr_ApplOmega_12 (h): ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ➡[h] ApplOmega2 W s.
  39 /2 width=1 by cpm_beta/ qed.
  40
  41 lemma cpr_ApplOmega_23 (h): ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega2 W s ➡[h] ApplOmega3 W s.
  42 #h #G #L #W1 #s elim (lifts_total W1 (𝐔❴1❵)) #W2 #HW12
  43 @(cpm_zeta … (ApplOmega3 W2 s)) /4 width=1 by ApplDelta_lifts, lifts_flat/
  44 @cpm_appl // @cpm_appl @(cpm_delta … (ApplDelta W1 s))
  45 /2 width=1 by ApplDelta_lifts, cpm_eps/
  46 qed.
  47
  48 lemma cpxs_ApplOmega_13 (h): ∀G,L,W,s. ⦃G, L⦄ ⊢ ApplOmega1 W s ⬈*[h] ApplOmega3 W s.
  49 /4 width=4 by cpxs_strap1, cpm_fwd_cpx/ qed.
  50
  51 lemma fqup_ApplOmega_13: ∀G,L,W,s. ⦃G, L, ApplOmega3 W s⦄ ⊐+ ⦃G, L, ApplOmega1 W s⦄.
  52 /2 width=1 by/ qed.
  53
  54 (* Main properties **********************************************************)
  55
  56 theorem fpbg_refl (h) (o): ∀G,L,W,s. ⦃G, L, ApplOmega1 W s⦄ >[h,o] ⦃G, L, ApplOmega1 W s⦄.
  57 /3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.