src/HOL/IOA/IOA.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 11482 ec2c382ff4f0
child 17288 aa3833fb7bee
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/IOA/IOA.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow & Konrad Slind
     4     Copyright   1994  TU Muenchen
     5 
     6 The I/O automata of Lynch and Tuttle.
     7 *)
     8 
     9 IOA = Asig + 
    10 
    11 types
    12    'a seq            =   "nat => 'a"
    13    'a oseq           =   "nat => 'a option"
    14    ('a,'b)execution  =   "'a oseq * 'b seq"
    15    ('a,'s)transition =   "('s * 'a * 's)"
    16    ('a,'s)ioa        =   "'a signature * 's set * ('a,'s)transition set"
    17 
    18 consts
    19 
    20   (* IO automata *)
    21   state_trans::"['action signature, ('action,'state)transition set] => bool"
    22   asig_of    ::"('action,'state)ioa => 'action signature"
    23   starts_of  ::"('action,'state)ioa => 'state set"
    24   trans_of   ::"('action,'state)ioa => ('action,'state)transition set"
    25   IOA        ::"('action,'state)ioa => bool"
    26 
    27   (* Executions, schedules, and traces *)
    28 
    29   is_execution_fragment,
    30   has_execution ::"[('action,'state)ioa, ('action,'state)execution] => bool"
    31   executions    :: "('action,'state)ioa => ('action,'state)execution set"
    32   mk_trace  :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
    33   reachable     :: "[('action,'state)ioa, 'state] => bool"
    34   invariant     :: "[('action,'state)ioa, 'state=>bool] => bool"
    35   has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
    36   traces    :: "('action,'state)ioa => 'action oseq set"
    37   NF            :: "'a oseq => 'a oseq"
    38 
    39   (* Composition of action signatures and automata *)
    40   compatible_asigs ::"('a => 'action signature) => bool"
    41   asig_composition ::"('a => 'action signature) => 'action signature"
    42   compatible_ioas  ::"('a => ('action,'state)ioa) => bool"
    43   ioa_composition  ::"('a => ('action, 'state)ioa) =>('action,'a => 'state)ioa"
    44 
    45   (* binary composition of action signatures and automata *)
    46   compat_asigs ::"['action signature, 'action signature] => bool"
    47   asig_comp    ::"['action signature, 'action signature] => 'action signature"
    48   compat_ioas  ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
    49   "||"         ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa"  (infixr 10)
    50 
    51   (* Filtering and hiding *)
    52   filter_oseq  :: "('a => bool) => 'a oseq => 'a oseq"
    53 
    54   restrict_asig :: "['a signature, 'a set] => 'a signature"
    55   restrict      :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
    56 
    57   (* Notions of correctness *)
    58   ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
    59 
    60   (* Instantiation of abstract IOA by concrete actions *)
    61   rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
    62 
    63 defs
    64 
    65 state_trans_def
    66   "state_trans asig R == 
    67      (!triple. triple:R --> fst(snd(triple)):actions(asig)) & 
    68      (!a. (a:inputs(asig)) --> (!s1. ? s2. (s1,a,s2):R))"
    69 
    70 
    71 asig_of_def   "asig_of == fst"
    72 starts_of_def "starts_of == (fst o snd)"
    73 trans_of_def  "trans_of == (snd o snd)"
    74 
    75 ioa_def
    76   "IOA(ioa) == (is_asig(asig_of(ioa))      &                            
    77                 (~ starts_of(ioa) = {})    &                            
    78                 state_trans (asig_of ioa) (trans_of ioa))"
    79 
    80 
    81 (* An execution fragment is modelled with a pair of sequences:
    82  * the first is the action options, the second the state sequence.
    83  * Finite executions have None actions from some point on.
    84  *******)
    85 is_execution_fragment_def
    86   "is_execution_fragment A ex ==                                        
    87      let act = fst(ex); state = snd(ex)                                 
    88      in !n a. (act(n)=None --> state(Suc(n)) = state(n)) &              
    89               (act(n)=Some(a) --> (state(n),a,state(Suc(n))):trans_of(A))"
    90 
    91 
    92 executions_def
    93   "executions(ioa) == {e. snd e 0:starts_of(ioa) &                      
    94                         is_execution_fragment ioa e}"
    95 
    96   
    97 reachable_def
    98   "reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
    99 
   100 
   101 invariant_def "invariant A P == (!s. reachable A s --> P(s))"
   102 
   103 
   104 (* Restrict the trace to those members of the set s *)
   105 filter_oseq_def
   106   "filter_oseq p s ==                                                   
   107    (%i. case s(i)                                                       
   108          of None => None                                               
   109           | Some(x) => if p x then Some x else None)"
   110 
   111 
   112 mk_trace_def
   113   "mk_trace(ioa) == filter_oseq(%a. a:externals(asig_of(ioa)))"
   114 
   115 
   116 (* Does an ioa have an execution with the given trace *)
   117 has_trace_def
   118   "has_trace ioa b ==                                               
   119      (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
   120 
   121 normal_form_def
   122   "NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) &   
   123                     (!j. j ~: range(f) --> nf(j)= None) &   
   124                     (!i. nf(i)=None --> (nf (Suc i)) = None)   "
   125   
   126 (* All the traces of an ioa *)
   127 
   128   traces_def
   129   "traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
   130 
   131 (*
   132   traces_def
   133   "traces(ioa) == {tr. has_trace ioa tr}"
   134 *)
   135   
   136 compat_asigs_def
   137   "compat_asigs a1 a2 ==                                                
   138    (((outputs(a1) Int outputs(a2)) = {}) &                              
   139     ((internals(a1) Int actions(a2)) = {}) &                            
   140     ((internals(a2) Int actions(a1)) = {}))"
   141 
   142 
   143 compat_ioas_def
   144   "compat_ioas ioa1 ioa2 == compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"
   145 
   146 
   147 asig_comp_def
   148   "asig_comp a1 a2 ==                                                   
   149       (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)),      
   150         (outputs(a1) Un outputs(a2)),                                   
   151         (internals(a1) Un internals(a2))))"
   152 
   153 
   154 par_def
   155   "(ioa1 || ioa2) ==                                                    
   156        (asig_comp (asig_of ioa1) (asig_of ioa2),                        
   157         {pr. fst(pr):starts_of(ioa1) & snd(pr):starts_of(ioa2)},        
   158         {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))        
   159              in (a:actions(asig_of(ioa1)) | a:actions(asig_of(ioa2))) & 
   160                 (if a:actions(asig_of(ioa1)) then                       
   161                    (fst(s),a,fst(t)):trans_of(ioa1)                     
   162                  else fst(t) = fst(s))                                  
   163                 &                                                       
   164                 (if a:actions(asig_of(ioa2)) then                       
   165                    (snd(s),a,snd(t)):trans_of(ioa2)                     
   166                  else snd(t) = snd(s))})"
   167 
   168 
   169 restrict_asig_def
   170   "restrict_asig asig actns ==                                          
   171     (inputs(asig) Int actns, outputs(asig) Int actns,                  
   172      internals(asig) Un (externals(asig) - actns))"
   173 
   174 
   175 restrict_def
   176   "restrict ioa actns ==                                               
   177     (restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"
   178 
   179 
   180 ioa_implements_def
   181   "ioa_implements ioa1 ioa2 ==   
   182   ((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) &   
   183      (outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) &   
   184       traces(ioa1) <= traces(ioa2))"
   185  
   186 rename_def 
   187 "rename ioa ren ==  
   188   (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))},         
   189     {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))},        
   190     {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}),     
   191               starts_of(ioa)   ,                                            
   192    {tr. let s = fst(tr); a = fst(snd(tr));  t = snd(snd(tr))    
   193         in                                                      
   194         ? x. Some(x) = ren(a) & (s,x,t):trans_of(ioa)})"
   195 
   196 end