# Theory LiveIOA

Up to index of Isabelle/HOLCF/IOA

theory LiveIOA
imports TLS
uses [LiveIOA.ML]
begin

    (*  Title:      HOLCF/IOA/meta_theory/LiveIOA.thy
        ID:         $Id: LiveIOA.thy,v 1.8 2005/09/02 15:24:01 wenzelm Exp $
        Author:     Olaf Müller
    *)

    header {* Live I/O automata -- specified by temproal formulas *}

    theory LiveIOA
    imports TLS
    begin

    defaultsort type

    types
      ('a, 's) live_ioa = "('a,'s)ioa * ('a,'s)ioa_temp"

    consts

    validLIOA   :: "('a,'s)live_ioa => ('a,'s)ioa_temp  => bool"

    WF         :: "('a,'s)ioa => 'a set => ('a,'s)ioa_temp"
    SF         :: "('a,'s)ioa => 'a set => ('a,'s)ioa_temp"

    liveexecutions    :: "('a,'s)live_ioa => ('a,'s)execution set"
    livetraces        :: "('a,'s)live_ioa => 'a trace set"
    live_implements   :: "('a,'s1)live_ioa => ('a,'s2)live_ioa => bool"
    is_live_ref_map   :: "('s1 => 's2) => ('a,'s1)live_ioa => ('a,'s2)live_ioa => bool"

    defs

    validLIOA_def:
      "validLIOA AL P == validIOA (fst AL) ((snd AL) .--> P)"

    WF_def:
      "WF A acts ==  <> [] <%(s,a,t) . Enabled A acts s> .--> [] <> <xt2 (plift (%a. a : acts))>"

    SF_def:
      "SF A acts ==  [] <> <%(s,a,t) . Enabled A acts s> .--> [] <> <xt2 (plift (%a. a : acts))>"

    liveexecutions_def:
       "liveexecutions AP == {exec. exec : executions (fst AP) & (exec |== (snd AP))}"

    livetraces_def:
      "livetraces AP == {mk_trace (fst AP)$(snd ex) | ex. ex:liveexecutions AP}"

    live_implements_def:
      "live_implements CL AM == (inp (fst CL) = inp (fst AM)) &
                                (out (fst CL) = out (fst AM)) &
                                livetraces CL <= livetraces AM"

    is_live_ref_map_def:
       "is_live_ref_map f CL AM ==
                is_ref_map f (fst CL ) (fst AM) &
                (! exec : executions (fst CL). (exec |== (snd CL)) -->
                                               ((corresp_ex (fst AM) f exec) |== (snd AM)))"

    ML {* use_legacy_bindings (the_context ()) *}

    end

theorem live_implements_trans:

      [| live_implements (A, LA) (B, LB); live_implements (B, LB) (C, LC) |]
      ==> live_implements (A, LA) (C, LC)

## Correctness of live refmap

theorem live_implements:

      [| inp C = inp A; out C = out A; is_live_ref_map f (C, M) (A, L) |]
      ==> live_implements (C, M) (A, L)