(* Title: HOL/IOA/meta_theory/Asig.thy ID: $Id: Asig.thy,v 1.9 2007/10/21 14:27:43 wenzelm Exp $ Author: Olaf Müller, Tobias Nipkow & Konrad Slind *) header {* Action signatures *} theory Asig imports Main begin types '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 o snd)" definition internals :: "'action signature => 'action set" where asig_internals_def: "internals = (snd o snd)" definition actions :: "'action signature => 'action set" where "actions(asig) = (inputs(asig) Un outputs(asig) Un internals(asig))" definition externals :: "'action signature => 'action set" where "externals(asig) = (inputs(asig) Un outputs(asig))" definition locals :: "'action signature => 'action set" where "locals asig = ((internals asig) Un (outputs asig))" definition is_asig :: "'action signature => bool" where "is_asig(triple) = ((inputs(triple) Int outputs(triple) = {}) & (outputs(triple) Int internals(triple) = {}) & (inputs(triple) Int 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 asig_triple_proj: "(outputs (a,b,c) = b) & (inputs (a,b,c) = a) & (internals (a,b,c) = c)" apply (simp add: asig_projections) done 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 lemma int_is_act: "[|a:internals S|] ==> a:actions S" apply (simp add: asig_internals_def actions_def) done lemma inp_is_act: "[|a:inputs S|] ==> a:actions S" apply (simp add: asig_inputs_def actions_def) done lemma out_is_act: "[|a:outputs S|] ==> a:actions S" apply (simp add: asig_outputs_def actions_def) done lemma ext_and_act: "(x: actions S & x : externals S) = (x: externals S)" apply (fast intro!: ext_is_act) done lemma not_ext_is_int: "[|is_asig S;x: actions S|] ==> (x~:externals S) = (x: internals S)" apply (simp add: actions_def is_asig_def externals_def) apply blast done lemma not_ext_is_int_or_not_act: "is_asig S ==> (x~:externals S) = (x: internals S | x~:actions S)" apply (simp add: actions_def is_asig_def externals_def) apply blast done lemma int_is_not_ext: "[| is_asig (S); x:internals S |] ==> x~:externals S" apply (unfold externals_def actions_def is_asig_def) apply simp apply blast done end
lemma asig_projections:
inputs = fst
outputs = fst o snd
internals = snd o snd
lemma asig_triple_proj:
outputs (a, b, c) = b ∧ inputs (a, b, c) = a ∧ internals (a, b, c) = c
lemma int_and_ext_is_act:
[| a ∉ internals S; a ∉ externals S |] ==> a ∉ actions S
lemma ext_is_act:
a ∈ externals S ==> a ∈ actions S
lemma int_is_act:
a ∈ internals S ==> a ∈ actions S
lemma inp_is_act:
a ∈ inputs S ==> a ∈ actions S
lemma out_is_act:
a ∈ outputs S ==> a ∈ actions S
lemma ext_and_act:
(x ∈ actions S ∧ x ∈ externals S) = (x ∈ externals S)
lemma not_ext_is_int:
[| is_asig S; x ∈ actions S |] ==> (x ∉ externals S) = (x ∈ internals S)
lemma not_ext_is_int_or_not_act:
is_asig S ==> (x ∉ externals S) = (x ∈ internals S ∨ x ∉ actions S)
lemma int_is_not_ext:
[| is_asig S; x ∈ internals S |] ==> x ∉ externals S