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

section ‹Action signatures›

theory Asig
imports Main
begin

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

definition "inputs" :: "'action signature → 'action set"
  where asig_inputs_def: "inputs ≡ fst"

definition "outputs" :: "'action signature → 'action set"
  where asig_outputs_def: "outputs ≡ (fst ∘ snd)"

definition "internals" :: "'action signature → 'action set"
  where asig_internals_def: "internals ≡ (snd ∘ snd)"

definition "actions" :: "'action signature → 'action set"
  where actions_def: "actions(asig) ≡ (inputs(asig) ∪ outputs(asig) ∪ internals(asig))"

definition externals :: "'action signature → 'action set"
  where externals_def: "externals(asig) ≡ (inputs(asig) ∪ outputs(asig))"

definition is_asig :: "'action signature => bool"
  where "is_asig(triple) ≡
    ((inputs(triple) ∩ outputs(triple) = {}) ∧
     (outputs(triple) ∩ internals(triple) = {}) ∧
     (inputs(triple) ∩ internals(triple) = {}))"

definition mk_ext_asig :: "'action signature → 'action signature"
  where "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