--- a/src/HOL/IOA/IOA.thy Mon Jan 11 21:20:44 2016 +0100
+++ b/src/HOL/IOA/IOA.thy Mon Jan 11 21:21:02 2016 +0100
@@ -15,183 +15,148 @@
type_synonym ('a, 's) transition = "('s * 'a * 's)"
type_synonym ('a,'s) ioa = "'a signature * 's set * ('a, 's) transition set"
-consts
+(* IO automata *)
+
+definition state_trans :: "['action signature, ('action,'state)transition set] => bool"
+ where "state_trans asig R ==
+ (!triple. triple:R --> fst(snd(triple)):actions(asig)) &
+ (!a. (a:inputs(asig)) --> (!s1. ? s2. (s1,a,s2):R))"
+
+definition asig_of :: "('action,'state)ioa => 'action signature"
+ where "asig_of == fst"
- (* IO automata *)
- state_trans::"['action signature, ('action,'state)transition set] => bool"
- asig_of ::"('action,'state)ioa => 'action signature"
- starts_of ::"('action,'state)ioa => 'state set"
- trans_of ::"('action,'state)ioa => ('action,'state)transition set"
- IOA ::"('action,'state)ioa => bool"
+definition starts_of :: "('action,'state)ioa => 'state set"
+ where "starts_of == (fst o snd)"
+
+definition trans_of :: "('action,'state)ioa => ('action,'state)transition set"
+ where "trans_of == (snd o snd)"
- (* Executions, schedules, and traces *)
+definition IOA :: "('action,'state)ioa => bool"
+ where "IOA(ioa) == (is_asig(asig_of(ioa)) &
+ (~ starts_of(ioa) = {}) &
+ state_trans (asig_of ioa) (trans_of ioa))"
+
+
+(* Executions, schedules, and traces *)
- is_execution_fragment ::"[('action,'state)ioa, ('action,'state)execution] => bool"
- has_execution ::"[('action,'state)ioa, ('action,'state)execution] => bool"
- executions :: "('action,'state)ioa => ('action,'state)execution set"
- mk_trace :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
- reachable :: "[('action,'state)ioa, 'state] => bool"
- invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
- has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
- traces :: "('action,'state)ioa => 'action oseq set"
- NF :: "'a oseq => 'a oseq"
+(* An execution fragment is modelled with a pair of sequences:
+ the first is the action options, the second the state sequence.
+ Finite executions have None actions from some point on. *)
+definition is_execution_fragment :: "[('action,'state)ioa, ('action,'state)execution] => bool"
+ where "is_execution_fragment A ex ==
+ let act = fst(ex); state = snd(ex)
+ in !n a. (act(n)=None --> state(Suc(n)) = state(n)) &
+ (act(n)=Some(a) --> (state(n),a,state(Suc(n))):trans_of(A))"
+
+definition executions :: "('action,'state)ioa => ('action,'state)execution set"
+ where "executions(ioa) == {e. snd e 0:starts_of(ioa) & is_execution_fragment ioa e}"
- (* Composition of action signatures and automata *)
+
+definition reachable :: "[('action,'state)ioa, 'state] => bool"
+ where "reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
+
+definition invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
+ where "invariant A P == (!s. reachable A s --> P(s))"
+
+
+(* Composition of action signatures and automata *)
+
+consts
compatible_asigs ::"('a => 'action signature) => bool"
asig_composition ::"('a => 'action signature) => 'action signature"
compatible_ioas ::"('a => ('action,'state)ioa) => bool"
ioa_composition ::"('a => ('action, 'state)ioa) =>('action,'a => 'state)ioa"
- (* binary composition of action signatures and automata *)
- compat_asigs ::"['action signature, 'action signature] => bool"
- asig_comp ::"['action signature, 'action signature] => 'action signature"
- compat_ioas ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
- par ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "||" 10)
+
+(* binary composition of action signatures and automata *)
- (* Filtering and hiding *)
- filter_oseq :: "('a => bool) => 'a oseq => 'a oseq"
+definition compat_asigs ::"['action signature, 'action signature] => bool"
+ where "compat_asigs a1 a2 ==
+ (((outputs(a1) Int outputs(a2)) = {}) &
+ ((internals(a1) Int actions(a2)) = {}) &
+ ((internals(a2) Int actions(a1)) = {}))"
- restrict_asig :: "['a signature, 'a set] => 'a signature"
- restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
+definition compat_ioas ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
+ where "compat_ioas ioa1 ioa2 == compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"
- (* Notions of correctness *)
- ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
-
- (* Instantiation of abstract IOA by concrete actions *)
- rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
+definition asig_comp :: "['action signature, 'action signature] => 'action signature"
+ where "asig_comp a1 a2 ==
+ (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)),
+ (outputs(a1) Un outputs(a2)),
+ (internals(a1) Un internals(a2))))"
-defs
-
-state_trans_def:
- "state_trans asig R ==
- (!triple. triple:R --> fst(snd(triple)):actions(asig)) &
- (!a. (a:inputs(asig)) --> (!s1. ? s2. (s1,a,s2):R))"
+definition par :: "[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr "||" 10)
+ where "(ioa1 || ioa2) ==
+ (asig_comp (asig_of ioa1) (asig_of ioa2),
+ {pr. fst(pr):starts_of(ioa1) & snd(pr):starts_of(ioa2)},
+ {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
+ in (a:actions(asig_of(ioa1)) | a:actions(asig_of(ioa2))) &
+ (if a:actions(asig_of(ioa1)) then
+ (fst(s),a,fst(t)):trans_of(ioa1)
+ else fst(t) = fst(s))
+ &
+ (if a:actions(asig_of(ioa2)) then
+ (snd(s),a,snd(t)):trans_of(ioa2)
+ else snd(t) = snd(s))})"
-asig_of_def: "asig_of == fst"
-starts_of_def: "starts_of == (fst o snd)"
-trans_of_def: "trans_of == (snd o snd)"
-
-ioa_def:
- "IOA(ioa) == (is_asig(asig_of(ioa)) &
- (~ starts_of(ioa) = {}) &
- state_trans (asig_of ioa) (trans_of ioa))"
-
-
-(* An execution fragment is modelled with a pair of sequences:
- * the first is the action options, the second the state sequence.
- * Finite executions have None actions from some point on.
- *******)
-is_execution_fragment_def:
- "is_execution_fragment A ex ==
- let act = fst(ex); state = snd(ex)
- in !n a. (act(n)=None --> state(Suc(n)) = state(n)) &
- (act(n)=Some(a) --> (state(n),a,state(Suc(n))):trans_of(A))"
-
-
-executions_def:
- "executions(ioa) == {e. snd e 0:starts_of(ioa) &
- is_execution_fragment ioa e}"
-
-
-reachable_def:
- "reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
-
-
-invariant_def: "invariant A P == (!s. reachable A s --> P(s))"
-
+(* Filtering and hiding *)
(* Restrict the trace to those members of the set s *)
-filter_oseq_def:
- "filter_oseq p s ==
+definition filter_oseq :: "('a => bool) => 'a oseq => 'a oseq"
+ where "filter_oseq p s ==
(%i. case s(i)
of None => None
| Some(x) => if p x then Some x else None)"
-
-mk_trace_def:
- "mk_trace(ioa) == filter_oseq(%a. a:externals(asig_of(ioa)))"
-
+definition mk_trace :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
+ where "mk_trace(ioa) == filter_oseq(%a. a:externals(asig_of(ioa)))"
(* Does an ioa have an execution with the given trace *)
-has_trace_def:
- "has_trace ioa b ==
- (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
+definition has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
+ where "has_trace ioa b == (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
-normal_form_def:
- "NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) &
+definition NF :: "'a oseq => 'a oseq"
+ where "NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) &
(!j. j ~: range(f) --> nf(j)= None) &
- (!i. nf(i)=None --> (nf (Suc i)) = None) "
+ (!i. nf(i)=None --> (nf (Suc i)) = None)"
(* All the traces of an ioa *)
-
- traces_def:
- "traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
-
-(*
- traces_def:
- "traces(ioa) == {tr. has_trace ioa tr}"
-*)
-
-compat_asigs_def:
- "compat_asigs a1 a2 ==
- (((outputs(a1) Int outputs(a2)) = {}) &
- ((internals(a1) Int actions(a2)) = {}) &
- ((internals(a2) Int actions(a1)) = {}))"
-
-
-compat_ioas_def:
- "compat_ioas ioa1 ioa2 == compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"
+definition traces :: "('action,'state)ioa => 'action oseq set"
+ where "traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
-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:
- "(ioa1 || ioa2) ==
- (asig_comp (asig_of ioa1) (asig_of ioa2),
- {pr. fst(pr):starts_of(ioa1) & snd(pr):starts_of(ioa2)},
- {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
- in (a:actions(asig_of(ioa1)) | a:actions(asig_of(ioa2))) &
- (if a:actions(asig_of(ioa1)) then
- (fst(s),a,fst(t)):trans_of(ioa1)
- else fst(t) = fst(s))
- &
- (if a:actions(asig_of(ioa2)) then
- (snd(s),a,snd(t)):trans_of(ioa2)
- else snd(t) = snd(s))})"
-
-
-restrict_asig_def:
- "restrict_asig asig actns ==
+definition restrict_asig :: "['a signature, 'a set] => 'a signature"
+ where "restrict_asig asig actns ==
(inputs(asig) Int actns, outputs(asig) Int actns,
internals(asig) Un (externals(asig) - actns))"
-
-restrict_def:
- "restrict ioa actns ==
+definition restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
+ where "restrict ioa actns ==
(restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"
-ioa_implements_def:
- "ioa_implements ioa1 ioa2 ==
+
+(* Notions of correctness *)
+
+definition ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
+ where "ioa_implements ioa1 ioa2 ==
((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) &
(outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) &
traces(ioa1) <= traces(ioa2))"
-rename_def:
-"rename ioa ren ==
- (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))},
- {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))},
- {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}),
- 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)})"
+
+(* Instantiation of abstract IOA by concrete actions *)
+
+definition rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
+ where "rename ioa ren ==
+ (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))},
+ {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))},
+ {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}),
+ 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)})"
declare Let_def [simp]
@@ -206,7 +171,7 @@
lemma trans_in_actions:
"[| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))"
- apply (unfold ioa_def state_trans_def actions_def is_asig_def)
+ apply (unfold IOA_def state_trans_def actions_def is_asig_def)
apply (erule conjE)+
apply (erule allE, erule impE, assumption)
apply simp