Up to index of Isabelle/HOLCF/IOA
theory Automata(* Title: HOLCF/IOA/meta_theory/Automata.thy
ID: $Id: Automata.thy,v 1.13 2005/09/02 15:24:00 wenzelm Exp $
Author: Olaf Müller, Konrad Slind, Tobias Nipkow
*)
header {* The I/O automata of Lynch and Tuttle in HOLCF *}
theory Automata
imports Asig
begin
defaultsort type
types
('a, 's) transition = "'s * 'a * 's"
('a, 's) ioa = "'a signature * 's set * ('a,'s)transition set * ('a set set) * ('a set set)"
consts
(* IO automata *)
asig_of ::"('a,'s)ioa => 'a signature"
starts_of ::"('a,'s)ioa => 's set"
trans_of ::"('a,'s)ioa => ('a,'s)transition set"
wfair_of ::"('a,'s)ioa => ('a set) set"
sfair_of ::"('a,'s)ioa => ('a set) set"
is_asig_of ::"('a,'s)ioa => bool"
is_starts_of ::"('a,'s)ioa => bool"
is_trans_of ::"('a,'s)ioa => bool"
input_enabled ::"('a,'s)ioa => bool"
IOA ::"('a,'s)ioa => bool"
(* constraints for fair IOA *)
fairIOA ::"('a,'s)ioa => bool"
input_resistant::"('a,'s)ioa => bool"
(* enabledness of actions and action sets *)
enabled ::"('a,'s)ioa => 'a => 's => bool"
Enabled ::"('a,'s)ioa => 'a set => 's => bool"
(* action set keeps enabled until probably disabled by itself *)
en_persistent :: "('a,'s)ioa => 'a set => bool"
(* post_conditions for actions and action sets *)
was_enabled ::"('a,'s)ioa => 'a => 's => bool"
set_was_enabled ::"('a,'s)ioa => 'a set => 's => bool"
(* reachability and invariants *)
reachable :: "('a,'s)ioa => 's set"
invariant :: "[('a,'s)ioa, 's=>bool] => bool"
(* binary composition of action signatures and automata *)
asig_comp ::"['a signature, 'a signature] => 'a signature"
compatible ::"[('a,'s)ioa, ('a,'t)ioa] => bool"
"||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr 10)
(* hiding and restricting *)
hide_asig :: "['a signature, 'a set] => 'a signature"
"hide" :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
restrict_asig :: "['a signature, 'a set] => 'a signature"
restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
(* renaming *)
rename_set :: "'a set => ('c => 'a option) => 'c set"
rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
syntax
"_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool" ("_ -_--_-> _" [81,81,81,81] 100)
"reachable" :: "[('a,'s)ioa, 's] => bool"
"act" :: "('a,'s)ioa => 'a set"
"ext" :: "('a,'s)ioa => 'a set"
"int" :: "('a,'s)ioa => 'a set"
"inp" :: "('a,'s)ioa => 'a set"
"out" :: "('a,'s)ioa => 'a set"
"local" :: "('a,'s)ioa => 'a set"
syntax (xsymbols)
"_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool"
("_ \<midarrow>_\<midarrow>_--> _" [81,81,81,81] 100)
"op ||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "\<parallel>" 10)
inductive "reachable C"
intros
reachable_0: "s:(starts_of C) ==> s : reachable C"
reachable_n: "[|s:reachable C; (s,a,t):trans_of C|] ==> t:reachable C"
translations
"s -a--A-> t" == "(s,a,t):trans_of A"
"reachable A s" == "s:reachable A"
"act A" == "actions (asig_of A)"
"ext A" == "externals (asig_of A)"
"int A" == "internals (asig_of A)"
"inp A" == "inputs (asig_of A)"
"out A" == "outputs (asig_of A)"
"local A" == "locals (asig_of A)"
defs
(* --------------------------------- IOA ---------------------------------*)
asig_of_def: "asig_of == fst"
starts_of_def: "starts_of == (fst o snd)"
trans_of_def: "trans_of == (fst o snd o snd)"
wfair_of_def: "wfair_of == (fst o snd o snd o snd)"
sfair_of_def: "sfair_of == (snd o snd o snd o snd)"
is_asig_of_def:
"is_asig_of A == is_asig (asig_of A)"
is_starts_of_def:
"is_starts_of A == (~ starts_of A = {})"
is_trans_of_def:
"is_trans_of A ==
(!triple. triple:(trans_of A) --> fst(snd(triple)):actions(asig_of A))"
input_enabled_def:
"input_enabled A ==
(!a. (a:inputs(asig_of A)) --> (!s1. ? s2. (s1,a,s2):(trans_of A)))"
ioa_def:
"IOA A == (is_asig_of A &
is_starts_of A &
is_trans_of A &
input_enabled A)"
invariant_def: "invariant A P == (!s. reachable A s --> P(s))"
(* ------------------------- parallel composition --------------------------*)
compatible_def:
"compatible A B ==
(((out A Int out B) = {}) &
((int A Int act B) = {}) &
((int B Int act A) = {}))"
asig_comp_def:
"asig_comp a1 a2 ==
(((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)),
(outputs(a1) Un outputs(a2)),
(internals(a1) Un internals(a2))))"
par_def:
"(A || B) ==
(asig_comp (asig_of A) (asig_of B),
{pr. fst(pr):starts_of(A) & snd(pr):starts_of(B)},
{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
in (a:act A | a:act B) &
(if a:act A then
(fst(s),a,fst(t)):trans_of(A)
else fst(t) = fst(s))
&
(if a:act B then
(snd(s),a,snd(t)):trans_of(B)
else snd(t) = snd(s))},
wfair_of A Un wfair_of B,
sfair_of A Un sfair_of B)"
(* ------------------------ hiding -------------------------------------------- *)
restrict_asig_def:
"restrict_asig asig actns ==
(inputs(asig) Int actns,
outputs(asig) Int actns,
internals(asig) Un (externals(asig) - actns))"
(* Notice that for wfair_of and sfair_of nothing has to be changed, as
changes from the outputs to the internals does not touch the locals as
a whole, which is of importance for fairness only *)
restrict_def:
"restrict A actns ==
(restrict_asig (asig_of A) actns,
starts_of A,
trans_of A,
wfair_of A,
sfair_of A)"
hide_asig_def:
"hide_asig asig actns ==
(inputs(asig) - actns,
outputs(asig) - actns,
internals(asig) Un actns)"
hide_def:
"hide A actns ==
(hide_asig (asig_of A) actns,
starts_of A,
trans_of A,
wfair_of A,
sfair_of A)"
(* ------------------------- renaming ------------------------------------------- *)
rename_set_def:
"rename_set A ren == {b. ? x. Some x = ren b & x : A}"
rename_def:
"rename ioa ren ==
((rename_set (inp ioa) ren,
rename_set (out ioa) ren,
rename_set (int ioa) ren),
starts_of ioa,
{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
in
? x. Some(x) = ren(a) & (s,x,t):trans_of ioa},
{rename_set s ren | s. s: wfair_of ioa},
{rename_set s ren | s. s: sfair_of ioa})"
(* ------------------------- fairness ----------------------------- *)
fairIOA_def:
"fairIOA A == (! S : wfair_of A. S<= local A) &
(! S : sfair_of A. S<= local A)"
input_resistant_def:
"input_resistant A == ! W : sfair_of A. ! s a t.
reachable A s & reachable A t & a:inp A &
Enabled A W s & s -a--A-> t
--> Enabled A W t"
enabled_def:
"enabled A a s == ? t. s-a--A-> t"
Enabled_def:
"Enabled A W s == ? w:W. enabled A w s"
en_persistent_def:
"en_persistent A W == ! s a t. Enabled A W s &
a ~:W &
s -a--A-> t
--> Enabled A W t"
was_enabled_def:
"was_enabled A a t == ? s. s-a--A-> t"
set_was_enabled_def:
"set_was_enabled A W t == ? w:W. was_enabled A w t"
ML {* use_legacy_bindings (the_context ()) *}
end
theorem ioa_triple_proj:
asig_of (x, y, z, w, s) = x ∧ starts_of (x, y, z, w, s) = y ∧ trans_of (x, y, z, w, s) = z ∧ wfair_of (x, y, z, w, s) = w ∧ sfair_of (x, y, z, w, s) = s
theorem trans_in_actions:
[| is_trans_of A; s1.0 -a--A-> s2.0 |] ==> a ∈ act A
theorem starts_of_par:
starts_of (A || B) = {p. fst p ∈ starts_of A ∧ snd p ∈ starts_of B}
theorem trans_of_par:
trans_of (A || B) = {tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr) in (a ∈ act A ∨ a ∈ act B) ∧ (if a ∈ act A then fst s -a--A-> fst t else fst t = fst s) ∧ (if a ∈ act B then snd s -a--B-> snd t else snd t = snd s)}
theorem actions_asig_comp:
actions (asig_comp a b) = actions a ∪ actions b
theorem asig_of_par:
asig_of (A || B) = asig_comp (asig_of A) (asig_of B)
theorem externals_of_par:
ext (A1.0 || A2.0) = ext A1.0 ∪ ext A2.0
theorem actions_of_par:
act (A1.0 || A2.0) = act A1.0 ∪ act A2.0
theorem inputs_of_par:
inp (A1.0 || A2.0) = inp A1.0 ∪ inp A2.0 - (out A1.0 ∪ out A2.0)
theorem outputs_of_par:
out (A1.0 || A2.0) = out A1.0 ∪ out A2.0
theorem internals_of_par:
int (A1.0 || A2.0) = int A1.0 ∪ int A2.0
theorem compat_commute:
compatible A B = compatible B A
theorem ext1_is_not_int2:
[| compatible A1.0 A2.0; a ∈ ext A1.0 |] ==> a ∉ int A2.0
theorem ext2_is_not_int1:
[| compatible A2.0 A1.0; a ∈ ext A1.0 |] ==> a ∉ int A2.0
theorem ext1_ext2_is_not_act2:
[| compatible A1.0 A2.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
theorem ext1_ext2_is_not_act1:
[| compatible A2.0 A1.0; a ∈ ext A1.0; a ∉ ext A2.0 |] ==> a ∉ act A2.0
theorem intA_is_not_extB:
[| compatible A B; x ∈ int A |] ==> x ∉ ext B
theorem intA_is_not_actB:
[| compatible A B; a ∈ int A |] ==> a ∉ act B
theorem outAactB_is_inpB:
[| compatible A B; a ∈ out A; a ∈ act B |] ==> a ∈ inp B
theorem inpAAactB_is_inpBoroutB:
[| compatible A B; a ∈ inp A; a ∈ act B |] ==> a ∈ inp B ∨ a ∈ out B
theorem input_enabled_par:
[| compatible A B; input_enabled A; input_enabled B |] ==> input_enabled (A || B)
theorem invariantI:
[| !!s. s ∈ starts_of A ==> P s; !!s t a. [| reachable A s; P s |] ==> s -a--A-> t --> P t |] ==> invariant A P
theorem invariantI1:
[| !!s. s ∈ starts_of A ==> P s; !!s t a. reachable A s ==> P s --> s -a--A-> t --> P t |] ==> invariant A P
theorem invariantE:
[| invariant A P; reachable A s |] ==> P s
theorem cancel_restrict_a:
starts_of (restrict ioa acts) = starts_of ioa ∧ trans_of (restrict ioa acts) = trans_of ioa
theorem cancel_restrict_b:
reachable (restrict ioa acts) s = reachable ioa s
theorem acts_restrict:
act (restrict A acts) = act A
theorem cancel_restrict:
starts_of (restrict ioa acts) = starts_of ioa ∧ trans_of (restrict ioa acts) = trans_of ioa ∧ reachable (restrict ioa acts) s = reachable ioa s ∧ act (restrict A acts) = act A
theorem trans_rename:
s -a--rename C f-> t ==> ∃x. Some x = f a ∧ s -x--C-> t
theorem reachable_rename:
reachable (rename C g) s ==> reachable C s
theorem trans_A_proj:
[| s -a--(A || B)-> t; a ∈ act A |] ==> fst s -a--A-> fst t
theorem trans_B_proj:
[| s -a--(A || B)-> t; a ∈ act B |] ==> snd s -a--B-> snd t
theorem trans_A_proj2:
[| s -a--(A || B)-> t; a ∉ act A |] ==> fst s = fst t
theorem trans_B_proj2:
[| s -a--(A || B)-> t; a ∉ act B |] ==> snd s = snd t
theorem trans_AB_proj:
s -a--(A || B)-> t ==> a ∈ act A ∨ a ∈ act B
theorem trans_AB:
[| a ∈ act A; a ∈ act B; fst s -a--A-> fst t; snd s -a--B-> snd t |] ==> s -a--(A || B)-> t
theorem trans_A_notB:
[| a ∈ act A; a ∉ act B; fst s -a--A-> fst t; snd s = snd t |] ==> s -a--(A || B)-> t
theorem trans_notA_B:
[| a ∉ act A; a ∈ act B; snd s -a--B-> snd t; fst s = fst t |] ==> s -a--(A || B)-> t
theorem trans_of_par4:
s -a--(A || B || C || D)-> t = ((a ∈ act A ∨ a ∈ act B ∨ a ∈ act C ∨ a ∈ act D) ∧ (if a ∈ act A then fst s -a--A-> fst t else fst t = fst s) ∧ (if a ∈ act B then fst (snd s) -a--B-> fst (snd t) else fst (snd t) = fst (snd s)) ∧ (if a ∈ act C then fst (snd (snd s)) -a--C-> fst (snd (snd t)) else fst (snd (snd t)) = fst (snd (snd s))) ∧ (if a ∈ act D then snd (snd (snd s)) -a--D-> snd (snd (snd t)) else snd (snd (snd t)) = snd (snd (snd s))))
theorem is_trans_of_par:
is_trans_of (A || B)
theorem is_trans_of_restrict:
is_trans_of A ==> is_trans_of (restrict A acts)
theorem is_trans_of_rename:
is_trans_of A ==> is_trans_of (rename A f)
theorem is_asig_of_par:
[| is_asig_of A; is_asig_of B; compatible A B |] ==> is_asig_of (A || B)
theorem is_asig_of_restrict:
is_asig_of A ==> is_asig_of (restrict A f)
theorem is_asig_of_rename:
is_asig_of A ==> is_asig_of (rename A f)
theorem compatible_par:
[| compatible A B; compatible A C |] ==> compatible A (B || C)
theorem compatible_par2:
[| compatible A C; compatible B C |] ==> compatible (A || B) C
theorem compatible_restrict:
[| compatible A B; (ext B - S) ∩ ext A = {} |] ==> compatible A (restrict B S)