src/HOL/IOA/Solve.thy
author haftmann
Mon Aug 14 13:46:06 2006 +0200 (2006-08-14)
changeset 20380 14f9f2a1caa6
parent 19801 b2af2549efd1
child 26342 0f65fa163304
permissions -rw-r--r--
simplified code generator setup
     1 (*  Title:      HOL/IOA/Solve.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow & Konrad Slind
     4     Copyright   1994  TU Muenchen
     5 *)
     6 
     7 header {* Weak possibilities mapping (abstraction) *}
     8 
     9 theory Solve
    10 imports IOA
    11 begin
    12 
    13 constdefs
    14 
    15   is_weak_pmap :: "['c => 'a, ('action,'c)ioa,('action,'a)ioa] => bool"
    16   "is_weak_pmap f C A ==
    17    (!s:starts_of(C). f(s):starts_of(A)) &
    18    (!s t a. reachable C s &
    19             (s,a,t):trans_of(C)
    20             --> (if a:externals(asig_of(C)) then
    21                    (f(s),a,f(t)):trans_of(A)
    22                  else f(s)=f(t)))"
    23 
    24 declare mk_trace_thm [simp] trans_in_actions [simp]
    25 
    26 lemma trace_inclusion: 
    27   "[| IOA(C); IOA(A); externals(asig_of(C)) = externals(asig_of(A));  
    28            is_weak_pmap f C A |] ==> traces(C) <= traces(A)"
    29   apply (unfold is_weak_pmap_def traces_def)
    30 
    31   apply (simp (no_asm) add: has_trace_def)
    32   apply safe
    33   apply (rename_tac ex1 ex2)
    34 
    35   (* choose same trace, therefore same NF *)
    36   apply (rule_tac x = "mk_trace C ex1" in exI)
    37   apply simp
    38 
    39   (* give execution of abstract automata *)
    40   apply (rule_tac x = "(mk_trace A ex1,%i. f (ex2 i))" in bexI)
    41 
    42   (* Traces coincide *)
    43    apply (simp (no_asm_simp) add: mk_trace_def filter_oseq_idemp)
    44 
    45   (* Use lemma *)
    46   apply (frule states_of_exec_reachable)
    47 
    48   (* Now show that it's an execution *)
    49   apply (simp add: executions_def)
    50   apply safe
    51 
    52   (* Start states map to start states *)
    53   apply (drule bspec)
    54   apply assumption
    55 
    56   (* Show that it's an execution fragment *)
    57   apply (simp add: is_execution_fragment_def)
    58   apply safe
    59 
    60   apply (erule_tac x = "ex2 n" in allE)
    61   apply (erule_tac x = "ex2 (Suc n)" in allE)
    62   apply (erule_tac x = a in allE)
    63   apply simp
    64   done
    65 
    66 (* Lemmata *)
    67 
    68 lemma imp_conj_lemma: "(P ==> Q-->R) ==> P&Q --> R"
    69   by blast
    70 
    71 
    72 (* fist_order_tautology of externals_of_par *)
    73 lemma externals_of_par_extra:
    74   "a:externals(asig_of(A1||A2)) =     
    75    (a:externals(asig_of(A1)) & a:externals(asig_of(A2)) |   
    76    a:externals(asig_of(A1)) & a~:externals(asig_of(A2)) |   
    77    a~:externals(asig_of(A1)) & a:externals(asig_of(A2)))"
    78   apply (auto simp add: externals_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def)
    79   done
    80 
    81 lemma comp1_reachable: "[| reachable (C1||C2) s |] ==> reachable C1 (fst s)"
    82   apply (simp add: reachable_def)
    83   apply (erule bexE)
    84   apply (rule_tac x =
    85     "(filter_oseq (%a. a:actions (asig_of (C1))) (fst ex) , %i. fst (snd ex i))" in bexI)
    86 (* fst(s) is in projected execution *)
    87   apply force
    88 (* projected execution is indeed an execution *)
    89   apply (simp cong del: if_weak_cong
    90     add: executions_def is_execution_fragment_def par_def starts_of_def
    91       trans_of_def filter_oseq_def
    92     split add: option.split)
    93   done
    94 
    95 
    96 (* Exact copy of proof of comp1_reachable for the second
    97    component of a parallel composition.     *)
    98 lemma comp2_reachable: "[| reachable (C1||C2) s|] ==> reachable C2 (snd s)"
    99   apply (simp add: reachable_def)
   100   apply (erule bexE)
   101   apply (rule_tac x =
   102     "(filter_oseq (%a. a:actions (asig_of (C2))) (fst ex) , %i. snd (snd ex i))" in bexI)
   103 (* fst(s) is in projected execution *)
   104   apply force
   105 (* projected execution is indeed an execution *)
   106   apply (simp cong del: if_weak_cong
   107     add: executions_def is_execution_fragment_def par_def starts_of_def
   108     trans_of_def filter_oseq_def
   109     split add: option.split)
   110   done
   111 
   112 declare split_if [split del] if_weak_cong [cong del]
   113 
   114 (*Composition of possibility-mappings *)
   115 lemma fxg_is_weak_pmap_of_product_IOA: 
   116      "[| is_weak_pmap f C1 A1;  
   117          externals(asig_of(A1))=externals(asig_of(C1)); 
   118          is_weak_pmap g C2 A2;   
   119          externals(asig_of(A2))=externals(asig_of(C2));  
   120          compat_ioas C1 C2; compat_ioas A1 A2  |]      
   121    ==> is_weak_pmap (%p.(f(fst(p)),g(snd(p)))) (C1||C2) (A1||A2)"
   122   apply (unfold is_weak_pmap_def)
   123   apply (rule conjI)
   124 (* start_states *)
   125   apply (simp add: par_def starts_of_def)
   126 (* transitions *)
   127   apply (rule allI)+
   128   apply (rule imp_conj_lemma)
   129   apply (simp (no_asm) add: externals_of_par_extra)
   130   apply (simp (no_asm) add: par_def)
   131   apply (simp add: trans_of_def)
   132   apply (simplesubst split_if)
   133   apply (rule conjI)
   134   apply (rule impI)
   135   apply (erule disjE)
   136 (* case 1      a:e(A1) | a:e(A2) *)
   137   apply (simp add: comp1_reachable comp2_reachable ext_is_act)
   138   apply (erule disjE)
   139 (* case 2      a:e(A1) | a~:e(A2) *)
   140   apply (simp add: comp1_reachable comp2_reachable ext_is_act ext1_ext2_is_not_act2)
   141 (* case 3      a:~e(A1) | a:e(A2) *)
   142   apply (simp add: comp1_reachable comp2_reachable ext_is_act ext1_ext2_is_not_act1)
   143 (* case 4      a:~e(A1) | a~:e(A2) *)
   144   apply (rule impI)
   145   apply (subgoal_tac "a~:externals (asig_of (A1)) & a~:externals (asig_of (A2))")
   146 (* delete auxiliary subgoal *)
   147   prefer 2
   148   apply force
   149   apply (simp (no_asm) add: conj_disj_distribR cong add: conj_cong split add: split_if)
   150   apply (tactic {*
   151     REPEAT((resolve_tac [conjI,impI] 1 ORELSE etac conjE 1) THEN
   152       asm_full_simp_tac(simpset() addsimps[thm "comp1_reachable", thm "comp2_reachable"]) 1) *})
   153   done
   154 
   155 
   156 lemma reachable_rename_ioa: "[| reachable (rename C g) s |] ==> reachable C s"
   157   apply (simp add: reachable_def)
   158   apply (erule bexE)
   159   apply (rule_tac x = "((%i. case (fst ex i) of None => None | Some (x) => g x) ,snd ex)" in bexI)
   160   apply (simp (no_asm))
   161 (* execution is indeed an execution of C *)
   162   apply (simp add: executions_def is_execution_fragment_def par_def
   163     starts_of_def trans_of_def rename_def split add: option.split)
   164   apply force
   165   done
   166 
   167 
   168 lemma rename_through_pmap: "[| is_weak_pmap f C A |] 
   169                        ==> (is_weak_pmap f (rename C g) (rename A g))"
   170   apply (simp add: is_weak_pmap_def)
   171   apply (rule conjI)
   172   apply (simp add: rename_def starts_of_def)
   173   apply (rule allI)+
   174   apply (rule imp_conj_lemma)
   175   apply (simp (no_asm) add: rename_def)
   176   apply (simp add: externals_def asig_inputs_def asig_outputs_def asig_of_def trans_of_def)
   177   apply safe
   178   apply (simplesubst split_if)
   179   apply (rule conjI)
   180   apply (rule impI)
   181   apply (erule disjE)
   182   apply (erule exE)
   183   apply (erule conjE)
   184 (* x is input *)
   185   apply (drule sym)
   186   apply (drule sym)
   187   apply simp
   188   apply hypsubst+
   189   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   190   apply assumption
   191   apply simp
   192 (* x is output *)
   193   apply (erule exE)
   194   apply (erule conjE)
   195   apply (drule sym)
   196   apply (drule sym)
   197   apply simp
   198   apply hypsubst+
   199   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   200   apply assumption
   201   apply simp
   202 (* x is internal *)
   203   apply (simp (no_asm) add: de_Morgan_disj de_Morgan_conj not_ex cong add: conj_cong)
   204   apply (rule impI)
   205   apply (erule conjE)
   206   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   207   apply auto
   208   done
   209 
   210 declare split_if [split] if_weak_cong [cong]
   211 
   212 end