Theory Asig

```(*  Title:      HOL/IOA/Asig.thy
Author:     Tobias Nipkow & Konrad Slind
*)

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)"