src/HOL/IOA/Solve.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 36862 952b2b102a0a
child 39159 0dec18004e75
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
     1 (*  Title:      HOL/IOA/Solve.thy
     2     Author:     Tobias Nipkow & Konrad Slind
     3     Copyright   1994  TU Muenchen
     4 *)
     5 
     6 header {* Weak possibilities mapping (abstraction) *}
     7 
     8 theory Solve
     9 imports IOA
    10 begin
    11 
    12 definition is_weak_pmap :: "['c => 'a, ('action,'c)ioa,('action,'a)ioa] => bool" where
    13   "is_weak_pmap f C A ==
    14    (!s:starts_of(C). f(s):starts_of(A)) &
    15    (!s t a. reachable C s &
    16             (s,a,t):trans_of(C)
    17             --> (if a:externals(asig_of(C)) then
    18                    (f(s),a,f(t)):trans_of(A)
    19                  else f(s)=f(t)))"
    20 
    21 declare mk_trace_thm [simp] trans_in_actions [simp]
    22 
    23 lemma trace_inclusion: 
    24   "[| IOA(C); IOA(A); externals(asig_of(C)) = externals(asig_of(A));  
    25            is_weak_pmap f C A |] ==> traces(C) <= traces(A)"
    26   apply (unfold is_weak_pmap_def traces_def)
    27 
    28   apply (simp (no_asm) add: has_trace_def)
    29   apply safe
    30   apply (rename_tac ex1 ex2)
    31 
    32   (* choose same trace, therefore same NF *)
    33   apply (rule_tac x = "mk_trace C ex1" in exI)
    34   apply simp
    35 
    36   (* give execution of abstract automata *)
    37   apply (rule_tac x = "(mk_trace A ex1,%i. f (ex2 i))" in bexI)
    38 
    39   (* Traces coincide *)
    40    apply (simp (no_asm_simp) add: mk_trace_def filter_oseq_idemp)
    41 
    42   (* Use lemma *)
    43   apply (frule states_of_exec_reachable)
    44 
    45   (* Now show that it's an execution *)
    46   apply (simp add: executions_def)
    47   apply safe
    48 
    49   (* Start states map to start states *)
    50   apply (drule bspec)
    51   apply assumption
    52 
    53   (* Show that it's an execution fragment *)
    54   apply (simp add: is_execution_fragment_def)
    55   apply safe
    56 
    57   apply (erule_tac x = "ex2 n" in allE)
    58   apply (erule_tac x = "ex2 (Suc n)" in allE)
    59   apply (erule_tac x = a in allE)
    60   apply simp
    61   done
    62 
    63 (* Lemmata *)
    64 
    65 lemma imp_conj_lemma: "(P ==> Q-->R) ==> P&Q --> R"
    66   by blast
    67 
    68 
    69 (* fist_order_tautology of externals_of_par *)
    70 lemma externals_of_par_extra:
    71   "a:externals(asig_of(A1||A2)) =     
    72    (a:externals(asig_of(A1)) & a:externals(asig_of(A2)) |   
    73    a:externals(asig_of(A1)) & a~:externals(asig_of(A2)) |   
    74    a~:externals(asig_of(A1)) & a:externals(asig_of(A2)))"
    75   apply (auto simp add: externals_def asig_of_par asig_comp_def asig_inputs_def asig_outputs_def)
    76   done
    77 
    78 lemma comp1_reachable: "[| reachable (C1||C2) s |] ==> reachable C1 (fst s)"
    79   apply (simp add: reachable_def)
    80   apply (erule bexE)
    81   apply (rule_tac x =
    82     "(filter_oseq (%a. a:actions (asig_of (C1))) (fst ex) , %i. fst (snd ex i))" in bexI)
    83 (* fst(s) is in projected execution *)
    84   apply force
    85 (* projected execution is indeed an execution *)
    86   apply (simp cong del: if_weak_cong
    87     add: executions_def is_execution_fragment_def par_def starts_of_def
    88       trans_of_def filter_oseq_def
    89     split add: option.split)
    90   done
    91 
    92 
    93 (* Exact copy of proof of comp1_reachable for the second
    94    component of a parallel composition.     *)
    95 lemma comp2_reachable: "[| reachable (C1||C2) s|] ==> reachable C2 (snd s)"
    96   apply (simp add: reachable_def)
    97   apply (erule bexE)
    98   apply (rule_tac x =
    99     "(filter_oseq (%a. a:actions (asig_of (C2))) (fst ex) , %i. snd (snd ex i))" in bexI)
   100 (* fst(s) is in projected execution *)
   101   apply force
   102 (* projected execution is indeed an execution *)
   103   apply (simp cong del: if_weak_cong
   104     add: executions_def is_execution_fragment_def par_def starts_of_def
   105     trans_of_def filter_oseq_def
   106     split add: option.split)
   107   done
   108 
   109 declare split_if [split del] if_weak_cong [cong del]
   110 
   111 (*Composition of possibility-mappings *)
   112 lemma fxg_is_weak_pmap_of_product_IOA: 
   113      "[| is_weak_pmap f C1 A1;  
   114          externals(asig_of(A1))=externals(asig_of(C1)); 
   115          is_weak_pmap g C2 A2;   
   116          externals(asig_of(A2))=externals(asig_of(C2));  
   117          compat_ioas C1 C2; compat_ioas A1 A2  |]      
   118    ==> is_weak_pmap (%p.(f(fst(p)),g(snd(p)))) (C1||C2) (A1||A2)"
   119   apply (unfold is_weak_pmap_def)
   120   apply (rule conjI)
   121 (* start_states *)
   122   apply (simp add: par_def starts_of_def)
   123 (* transitions *)
   124   apply (rule allI)+
   125   apply (rule imp_conj_lemma)
   126   apply (simp (no_asm) add: externals_of_par_extra)
   127   apply (simp (no_asm) add: par_def)
   128   apply (simp add: trans_of_def)
   129   apply (simplesubst split_if)
   130   apply (rule conjI)
   131   apply (rule impI)
   132   apply (erule disjE)
   133 (* case 1      a:e(A1) | a:e(A2) *)
   134   apply (simp add: comp1_reachable comp2_reachable ext_is_act)
   135   apply (erule disjE)
   136 (* case 2      a:e(A1) | a~:e(A2) *)
   137   apply (simp add: comp1_reachable comp2_reachable ext_is_act ext1_ext2_is_not_act2)
   138 (* case 3      a:~e(A1) | a:e(A2) *)
   139   apply (simp add: comp1_reachable comp2_reachable ext_is_act ext1_ext2_is_not_act1)
   140 (* case 4      a:~e(A1) | a~:e(A2) *)
   141   apply (rule impI)
   142   apply (subgoal_tac "a~:externals (asig_of (A1)) & a~:externals (asig_of (A2))")
   143 (* delete auxiliary subgoal *)
   144   prefer 2
   145   apply force
   146   apply (simp (no_asm) add: conj_disj_distribR cong add: conj_cong split add: split_if)
   147   apply (tactic {*
   148     REPEAT((resolve_tac [conjI,impI] 1 ORELSE etac conjE 1) THEN
   149       asm_full_simp_tac(@{simpset} addsimps[thm "comp1_reachable", thm "comp2_reachable"]) 1) *})
   150   done
   151 
   152 
   153 lemma reachable_rename_ioa: "[| reachable (rename C g) s |] ==> reachable C s"
   154   apply (simp add: reachable_def)
   155   apply (erule bexE)
   156   apply (rule_tac x = "((%i. case (fst ex i) of None => None | Some (x) => g x) ,snd ex)" in bexI)
   157   apply (simp (no_asm))
   158 (* execution is indeed an execution of C *)
   159   apply (simp add: executions_def is_execution_fragment_def par_def
   160     starts_of_def trans_of_def rename_def split add: option.split)
   161   apply force
   162   done
   163 
   164 
   165 lemma rename_through_pmap: "[| is_weak_pmap f C A |] 
   166                        ==> (is_weak_pmap f (rename C g) (rename A g))"
   167   apply (simp add: is_weak_pmap_def)
   168   apply (rule conjI)
   169   apply (simp add: rename_def starts_of_def)
   170   apply (rule allI)+
   171   apply (rule imp_conj_lemma)
   172   apply (simp (no_asm) add: rename_def)
   173   apply (simp add: externals_def asig_inputs_def asig_outputs_def asig_of_def trans_of_def)
   174   apply safe
   175   apply (simplesubst split_if)
   176   apply (rule conjI)
   177   apply (rule impI)
   178   apply (erule disjE)
   179   apply (erule exE)
   180   apply (erule conjE)
   181 (* x is input *)
   182   apply (drule sym)
   183   apply (drule sym)
   184   apply simp
   185   apply hypsubst+
   186   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   187   apply assumption
   188   apply simp
   189 (* x is output *)
   190   apply (erule exE)
   191   apply (erule conjE)
   192   apply (drule sym)
   193   apply (drule sym)
   194   apply simp
   195   apply hypsubst+
   196   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   197   apply assumption
   198   apply simp
   199 (* x is internal *)
   200   apply (simp (no_asm) add: de_Morgan_disj de_Morgan_conj not_ex cong add: conj_cong)
   201   apply (rule impI)
   202   apply (erule conjE)
   203   apply (cut_tac C = "C" and g = "g" and s = "s" in reachable_rename_ioa)
   204   apply auto
   205   done
   206 
   207 declare split_if [split] if_weak_cong [cong]
   208 
   209 end