# Theory TrivEx

Up to index of Isabelle/HOLCF/IOA/ex

theory TrivEx
imports Abstraction
uses [TrivEx.ML]
begin

    (*  Title:      HOLCF/IOA/TrivEx.thy
        ID:         $Id: TrivEx.thy,v 1.4 2005/09/03 14:50:26 wenzelm Exp $
        Author:     Olaf Müller
    *)

    header {* Trivial Abstraction Example *}

    theory TrivEx
    imports Abstraction
    begin

    datatype action = INC

    consts

    C_asig   ::  "action signature"
    C_trans  :: "(action, nat)transition set"
    C_ioa    :: "(action, nat)ioa"

    A_asig   :: "action signature"
    A_trans  :: "(action, bool)transition set"
    A_ioa    :: "(action, bool)ioa"

    h_abs    :: "nat => bool"

    defs

    C_asig_def:
      "C_asig == ({},{INC},{})"

    C_trans_def: "C_trans ==
     {tr. let s = fst(tr);
              t = snd(snd(tr))
          in case fst(snd(tr))
          of
          INC       => t = Suc(s)}"

    C_ioa_def: "C_ioa ==
     (C_asig, {0}, C_trans,{},{})"

    A_asig_def:
      "A_asig == ({},{INC},{})"

    A_trans_def: "A_trans ==
     {tr. let s = fst(tr);
              t = snd(snd(tr))
          in case fst(snd(tr))
          of
          INC       => t = True}"

    A_ioa_def: "A_ioa ==
     (A_asig, {False}, A_trans,{},{})"

    h_abs_def:
      "h_abs n == n~=0"

    axioms

    MC_result:
      "validIOA A_ioa (<>[] <%(b,a,c). b>)"

    ML {* use_legacy_bindings (the_context ()) *}

    end

theorem h_abs_is_abstraction:

      is_abstraction h_abs C_ioa A_ioa

theorem TrivEx_abstraction:

      validIOA C_ioa (<> [] <%(n, a, m). n ≠ 0>)