src/HOL/IOA/Asig.thy

only one internal coinduction rule

(*  Title:      HOL/IOA/Asig.thy
    Author:     Tobias Nipkow & Konrad Slind
    Copyright   1994  TU Muenchen
*)

header {* Action signatures *}

theory Asig
imports Main
begin

type_synonym
  'a signature = "('a set * 'a set * 'a set)"

consts
  "actions" :: "'action signature => 'action set"
  "inputs" :: "'action signature => 'action set"
  "outputs" :: "'action signature => 'action set"
  "internals" :: "'action signature => 'action set"
  externals :: "'action signature => 'action set"

  is_asig       ::"'action signature => bool"
  mk_ext_asig   ::"'action signature => 'action signature"


defs

asig_inputs_def:    "inputs == fst"
asig_outputs_def:   "outputs == (fst o snd)"
asig_internals_def: "internals == (snd o snd)"

actions_def:
   "actions(asig) == (inputs(asig) Un outputs(asig) Un internals(asig))"

externals_def:
   "externals(asig) == (inputs(asig) Un outputs(asig))"

is_asig_def:
  "is_asig(triple) ==
      ((inputs(triple) Int outputs(triple) = {})    &
       (outputs(triple) Int internals(triple) = {}) &
       (inputs(triple) Int internals(triple) = {}))"


mk_ext_asig_def:
  "mk_ext_asig(triple) == (inputs(triple), outputs(triple), {})"


lemmas asig_projections = asig_inputs_def asig_outputs_def asig_internals_def

lemma int_and_ext_is_act: "[| a~:internals(S) ;a~:externals(S)|] ==> a~:actions(S)"
  apply (simp add: externals_def actions_def)
  done

lemma ext_is_act: "[|a:externals(S)|] ==> a:actions(S)"
  apply (simp add: externals_def actions_def)
  done

end