--- a/src/HOLCF/IOA/meta_theory/Automata.thy Fri Sep 02 15:54:47 2005 +0200
+++ b/src/HOLCF/IOA/meta_theory/Automata.thy Fri Sep 02 17:23:59 2005 +0200
@@ -1,22 +1,22 @@
(* Title: HOLCF/IOA/meta_theory/Automata.thy
ID: $Id$
Author: Olaf Müller, Konrad Slind, Tobias Nipkow
+*)
-The I/O automata of Lynch and Tuttle in HOLCF.
-*)
-
-
-Automata = Asig +
+header {* The I/O automata of Lynch and Tuttle in HOLCF *}
-default type
-
+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)"
+ ('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"
@@ -26,10 +26,10 @@
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"
+ 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 *)
@@ -41,7 +41,7 @@
enabled ::"('a,'s)ioa => 'a => 's => bool"
Enabled ::"('a,'s)ioa => 'a set => 's => bool"
- (* action set keeps enabled until probably disabled by itself *)
+ (* action set keeps enabled until probably disabled by itself *)
en_persistent :: "('a,'s)ioa => 'a set => bool"
@@ -61,7 +61,7 @@
(* hiding and restricting *)
hide_asig :: "['a signature, 'a set] => 'a signature"
- hide :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
+ "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"
@@ -70,7 +70,7 @@
rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
-syntax
+syntax
"_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool" ("_ -_--_-> _" [81,81,81,81] 100)
"reachable" :: "[('a,'s)ioa, 's] => bool"
@@ -84,15 +84,15 @@
syntax (xsymbols)
- "_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool"
- ("_ \\<midarrow>_\\<midarrow>_\\<longrightarrow> _" [81,81,81,81] 100)
- "op ||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "\\<parallel>" 10)
+ "_trans_of" :: "'s => 'a => ('a,'s)ioa => 's => bool"
+ ("_ \<midarrow>_\<midarrow>_\<longrightarrow> _" [81,81,81,81] 100)
+ "op ||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "\<parallel>" 10)
-inductive "reachable C"
- intrs
- reachable_0 "s:(starts_of C) ==> s : reachable C"
- reachable_n "[|s:reachable C; (s,a,t):trans_of C|] ==> t:reachable C"
+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
@@ -105,72 +105,68 @@
"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)))"
-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 &
+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))"
+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) = {}) &
+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)),
+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)
+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)"
@@ -178,84 +174,84 @@
(* ------------------------ hiding -------------------------------------------- *)
-restrict_asig_def
- "restrict_asig asig actns ==
- (inputs(asig) Int actns,
- outputs(asig) Int actns,
+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
+(* 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,
+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,
+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,
+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_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
+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) &
+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.
+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_def:
"enabled A a s == ? t. s-a--A-> t"
-Enabled_def
+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
+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_def:
"was_enabled A a t == ? s. s-a--A-> t"
-set_was_enabled_def
+set_was_enabled_def:
"set_was_enabled A W t == ? w:W. was_enabled A w t"
+ML {* use_legacy_bindings (the_context ()) *}
end
-