src/HOL/MicroJava/BV/Effect.thy
author wenzelm
Thu Oct 04 20:29:42 2007 +0200 (2007-10-04)
changeset 24850 0cfd722ab579
parent 18576 8d98b7711e47
child 25362 8d06e11a01d1
permissions -rw-r--r--
Name.uu, Name.aT;
     1 (*  Title:      HOL/MicroJava/BV/Effect.thy
     2     ID:         $Id$
     3     Author:     Gerwin Klein
     4     Copyright   2000 Technische Universitaet Muenchen
     5 *)
     6 
     7 header {* \isaheader{Effect of Instructions on the State Type} *}
     8 
     9 theory Effect 
    10 imports JVMType "../JVM/JVMExceptions"
    11 begin
    12 
    13 
    14 types
    15   succ_type = "(p_count \<times> state_type option) list"
    16 
    17 text {* Program counter of successor instructions: *}
    18 consts
    19   succs :: "instr \<Rightarrow> p_count \<Rightarrow> p_count list"
    20 primrec 
    21   "succs (Load idx) pc         = [pc+1]"
    22   "succs (Store idx) pc        = [pc+1]"
    23   "succs (LitPush v) pc        = [pc+1]"
    24   "succs (Getfield F C) pc     = [pc+1]"
    25   "succs (Putfield F C) pc     = [pc+1]"
    26   "succs (New C) pc            = [pc+1]"
    27   "succs (Checkcast C) pc      = [pc+1]"
    28   "succs Pop pc                = [pc+1]"
    29   "succs Dup pc                = [pc+1]"
    30   "succs Dup_x1 pc             = [pc+1]"
    31   "succs Dup_x2 pc             = [pc+1]"
    32   "succs Swap pc               = [pc+1]"
    33   "succs IAdd pc               = [pc+1]"
    34   "succs (Ifcmpeq b) pc        = [pc+1, nat (int pc + b)]"
    35   "succs (Goto b) pc           = [nat (int pc + b)]"
    36   "succs Return pc             = [pc]"  
    37   "succs (Invoke C mn fpTs) pc = [pc+1]"
    38   "succs Throw pc              = [pc]"
    39 
    40 text "Effect of instruction on the state type:"
    41 consts 
    42 eff' :: "instr \<times> jvm_prog \<times> state_type \<Rightarrow> state_type"
    43 
    44 recdef eff' "{}"
    45 "eff' (Load idx,  G, (ST, LT))          = (ok_val (LT ! idx) # ST, LT)"
    46 "eff' (Store idx, G, (ts#ST, LT))       = (ST, LT[idx:= OK ts])"
    47 "eff' (LitPush v, G, (ST, LT))           = (the (typeof (\<lambda>v. None) v) # ST, LT)"
    48 "eff' (Getfield F C, G, (oT#ST, LT))    = (snd (the (field (G,C) F)) # ST, LT)"
    49 "eff' (Putfield F C, G, (vT#oT#ST, LT)) = (ST,LT)"
    50 "eff' (New C, G, (ST,LT))               = (Class C # ST, LT)"
    51 "eff' (Checkcast C, G, (RefT rt#ST,LT)) = (Class C # ST,LT)"
    52 "eff' (Pop, G, (ts#ST,LT))              = (ST,LT)"
    53 "eff' (Dup, G, (ts#ST,LT))              = (ts#ts#ST,LT)"
    54 "eff' (Dup_x1, G, (ts1#ts2#ST,LT))      = (ts1#ts2#ts1#ST,LT)"
    55 "eff' (Dup_x2, G, (ts1#ts2#ts3#ST,LT))  = (ts1#ts2#ts3#ts1#ST,LT)"
    56 "eff' (Swap, G, (ts1#ts2#ST,LT))        = (ts2#ts1#ST,LT)"
    57 "eff' (IAdd, G, (PrimT Integer#PrimT Integer#ST,LT)) 
    58                                          = (PrimT Integer#ST,LT)"
    59 "eff' (Ifcmpeq b, G, (ts1#ts2#ST,LT))   = (ST,LT)"
    60 "eff' (Goto b, G, s)                    = s"
    61   -- "Return has no successor instruction in the same method"
    62 "eff' (Return, G, s)                    = s" 
    63   -- "Throw always terminates abruptly"
    64 "eff' (Throw, G, s)                     = s"
    65 "eff' (Invoke C mn fpTs, G, (ST,LT))    = (let ST' = drop (length fpTs) ST 
    66   in  (fst (snd (the (method (G,C) (mn,fpTs))))#(tl ST'),LT))" 
    67 
    68 
    69 consts
    70   match_any :: "jvm_prog \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> cname list"
    71 primrec
    72   "match_any G pc [] = []"
    73   "match_any G pc (e#es) = (let (start_pc, end_pc, handler_pc, catch_type) = e;
    74                                 es' = match_any G pc es 
    75                             in 
    76                             if start_pc <= pc \<and> pc < end_pc then catch_type#es' else es')"
    77 
    78 consts
    79   match :: "jvm_prog \<Rightarrow> xcpt \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> cname list"
    80 primrec
    81   "match G X pc [] = []"
    82   "match G X pc (e#es) = 
    83   (if match_exception_entry G (Xcpt X) pc e then [Xcpt X] else match G X pc es)"
    84 
    85 lemma match_some_entry:
    86   "match G X pc et = (if \<exists>e \<in> set et. match_exception_entry G (Xcpt X) pc e then [Xcpt X] else [])"
    87   by (induct et) auto
    88 
    89 consts
    90   xcpt_names :: "instr \<times> jvm_prog \<times> p_count \<times> exception_table \<Rightarrow> cname list" 
    91 recdef xcpt_names "{}"
    92   "xcpt_names (Getfield F C, G, pc, et) = match G NullPointer pc et" 
    93   "xcpt_names (Putfield F C, G, pc, et) = match G NullPointer pc et" 
    94   "xcpt_names (New C, G, pc, et)        = match G OutOfMemory pc et"
    95   "xcpt_names (Checkcast C, G, pc, et)  = match G ClassCast pc et"
    96   "xcpt_names (Throw, G, pc, et)        = match_any G pc et"
    97   "xcpt_names (Invoke C m p, G, pc, et) = match_any G pc et" 
    98   "xcpt_names (i, G, pc, et)            = []" 
    99 
   100 
   101 constdefs
   102   xcpt_eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> p_count \<Rightarrow> state_type option \<Rightarrow> exception_table \<Rightarrow> succ_type"
   103   "xcpt_eff i G pc s et == 
   104    map (\<lambda>C. (the (match_exception_table G C pc et), case s of None \<Rightarrow> None | Some s' \<Rightarrow> Some ([Class C], snd s'))) 
   105        (xcpt_names (i,G,pc,et))"
   106 
   107   norm_eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> state_type option \<Rightarrow> state_type option"
   108   "norm_eff i G == option_map (\<lambda>s. eff' (i,G,s))"
   109 
   110   eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> succ_type"
   111   "eff i G pc et s == (map (\<lambda>pc'. (pc',norm_eff i G s)) (succs i pc)) @ (xcpt_eff i G pc s et)"
   112 
   113 constdefs
   114   isPrimT :: "ty \<Rightarrow> bool"
   115   "isPrimT T == case T of PrimT T' \<Rightarrow> True | RefT T' \<Rightarrow> False"
   116 
   117   isRefT :: "ty \<Rightarrow> bool"
   118   "isRefT T == case T of PrimT T' \<Rightarrow> False | RefT T' \<Rightarrow> True"
   119 
   120 lemma isPrimT [simp]:
   121   "isPrimT T = (\<exists>T'. T = PrimT T')" by (simp add: isPrimT_def split: ty.splits)
   122 
   123 lemma isRefT [simp]:
   124   "isRefT T = (\<exists>T'. T = RefT T')" by (simp add: isRefT_def split: ty.splits)
   125 
   126 
   127 lemma "list_all2 P a b \<Longrightarrow> \<forall>(x,y) \<in> set (zip a b). P x y"
   128   by (simp add: list_all2_def) 
   129 
   130 
   131 text "Conditions under which eff is applicable:"
   132 consts
   133 app' :: "instr \<times> jvm_prog \<times> p_count \<times> nat \<times> ty \<times> state_type \<Rightarrow> bool"
   134 
   135 recdef app' "{}"
   136 "app' (Load idx, G, pc, maxs, rT, s) = 
   137   (idx < length (snd s) \<and> (snd s) ! idx \<noteq> Err \<and> length (fst s) < maxs)"
   138 "app' (Store idx, G, pc, maxs, rT, (ts#ST, LT)) = 
   139   (idx < length LT)"
   140 "app' (LitPush v, G, pc, maxs, rT, s) = 
   141   (length (fst s) < maxs \<and> typeof (\<lambda>t. None) v \<noteq> None)"
   142 "app' (Getfield F C, G, pc, maxs, rT, (oT#ST, LT)) = 
   143   (is_class G C \<and> field (G,C) F \<noteq> None \<and> fst (the (field (G,C) F)) = C \<and> 
   144   G \<turnstile> oT \<preceq> (Class C))"
   145 "app' (Putfield F C, G, pc, maxs, rT, (vT#oT#ST, LT)) = 
   146   (is_class G C \<and> field (G,C) F \<noteq> None \<and> fst (the (field (G,C) F)) = C \<and>
   147   G \<turnstile> oT \<preceq> (Class C) \<and> G \<turnstile> vT \<preceq> (snd (the (field (G,C) F))))" 
   148 "app' (New C, G, pc, maxs, rT, s) = 
   149   (is_class G C \<and> length (fst s) < maxs)"
   150 "app' (Checkcast C, G, pc, maxs, rT, (RefT rt#ST,LT)) = 
   151   (is_class G C)"
   152 "app' (Pop, G, pc, maxs, rT, (ts#ST,LT)) = 
   153   True"
   154 "app' (Dup, G, pc, maxs, rT, (ts#ST,LT)) = 
   155   (1+length ST < maxs)"
   156 "app' (Dup_x1, G, pc, maxs, rT, (ts1#ts2#ST,LT)) = 
   157   (2+length ST < maxs)"
   158 "app' (Dup_x2, G, pc, maxs, rT, (ts1#ts2#ts3#ST,LT)) = 
   159   (3+length ST < maxs)"
   160 "app' (Swap, G, pc, maxs, rT, (ts1#ts2#ST,LT)) = 
   161   True"
   162 "app' (IAdd, G, pc, maxs, rT, (PrimT Integer#PrimT Integer#ST,LT)) =
   163   True"
   164 "app' (Ifcmpeq b, G, pc, maxs, rT, (ts#ts'#ST,LT)) = 
   165   (0 \<le> int pc + b \<and> (isPrimT ts \<and> ts' = ts \<or> isRefT ts \<and> isRefT ts'))"
   166 "app' (Goto b, G, pc, maxs, rT, s) = 
   167   (0 \<le> int pc + b)"
   168 "app' (Return, G, pc, maxs, rT, (T#ST,LT)) = 
   169   (G \<turnstile> T \<preceq> rT)"
   170 "app' (Throw, G, pc, maxs, rT, (T#ST,LT)) = 
   171   isRefT T"
   172 "app' (Invoke C mn fpTs, G, pc, maxs, rT, s) = 
   173   (length fpTs < length (fst s) \<and> 
   174   (let apTs = rev (take (length fpTs) (fst s));
   175        X    = hd (drop (length fpTs) (fst s)) 
   176    in  
   177        G \<turnstile> X \<preceq> Class C \<and> is_class G C \<and> method (G,C) (mn,fpTs) \<noteq> None \<and>
   178        list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) apTs fpTs))"
   179   
   180 "app' (i,G, pc,maxs,rT,s) = False"
   181 
   182 constdefs
   183   xcpt_app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> bool"
   184   "xcpt_app i G pc et \<equiv> \<forall>C\<in>set(xcpt_names (i,G,pc,et)). is_class G C"
   185 
   186   app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> bool"
   187   "app i G maxs rT pc et s == case s of None \<Rightarrow> True | Some t \<Rightarrow> app' (i,G,pc,maxs,rT,t) \<and> xcpt_app i G pc et"
   188 
   189 
   190 lemma match_any_match_table:
   191   "C \<in> set (match_any G pc et) \<Longrightarrow> match_exception_table G C pc et \<noteq> None"
   192   apply (induct et)
   193    apply simp
   194   apply simp
   195   apply clarify
   196   apply (simp split: split_if_asm)
   197   apply (auto simp add: match_exception_entry_def)
   198   done
   199 
   200 lemma match_X_match_table:
   201   "C \<in> set (match G X pc et) \<Longrightarrow> match_exception_table G C pc et \<noteq> None"
   202   apply (induct et)
   203    apply simp
   204   apply (simp split: split_if_asm)
   205   done
   206 
   207 lemma xcpt_names_in_et:
   208   "C \<in> set (xcpt_names (i,G,pc,et)) \<Longrightarrow> 
   209   \<exists>e \<in> set et. the (match_exception_table G C pc et) = fst (snd (snd e))"
   210   apply (cases i)
   211   apply (auto dest!: match_any_match_table match_X_match_table 
   212               dest: match_exception_table_in_et)
   213   done
   214 
   215 
   216 lemma 1: "2 < length a \<Longrightarrow> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"
   217 proof (cases a)
   218   fix x xs assume "a = x#xs" "2 < length a"
   219   thus ?thesis by - (cases xs, simp, cases "tl xs", auto)
   220 qed auto
   221 
   222 lemma 2: "\<not>(2 < length a) \<Longrightarrow> a = [] \<or> (\<exists> l. a = [l]) \<or> (\<exists> l l'. a = [l,l'])"
   223 proof -;
   224   assume "\<not>(2 < length a)"
   225   hence "length a < (Suc (Suc (Suc 0)))" by simp
   226   hence * : "length a = 0 \<or> length a = Suc 0 \<or> length a = Suc (Suc 0)" 
   227     by (auto simp add: less_Suc_eq)
   228 
   229   { 
   230     fix x 
   231     assume "length x = Suc 0"
   232     hence "\<exists> l. x = [l]"  by - (cases x, auto)
   233   } note 0 = this
   234 
   235   have "length a = Suc (Suc 0) \<Longrightarrow> \<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
   236   with * show ?thesis by (auto dest: 0)
   237 qed
   238 
   239 lemmas [simp] = app_def xcpt_app_def
   240 
   241 text {* 
   242 \medskip
   243 simp rules for @{term app}
   244 *}
   245 lemma appNone[simp]: "app i G maxs rT pc et None = True" by simp
   246 
   247 
   248 lemma appLoad[simp]:
   249 "(app (Load idx) G maxs rT pc et (Some s)) = (\<exists>ST LT. s = (ST,LT) \<and> idx < length LT \<and> LT!idx \<noteq> Err \<and> length ST < maxs)"
   250   by (cases s, simp)
   251 
   252 lemma appStore[simp]:
   253 "(app (Store idx) G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT) \<and> idx < length LT)"
   254   by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
   255 
   256 lemma appLitPush[simp]:
   257 "(app (LitPush v) G maxs rT pc et (Some s)) = (\<exists>ST LT. s = (ST,LT) \<and> length ST < maxs \<and> typeof (\<lambda>v. None) v \<noteq> None)"
   258   by (cases s, simp)
   259 
   260 lemma appGetField[simp]:
   261 "(app (Getfield F C) G maxs rT pc et (Some s)) = 
   262  (\<exists> oT vT ST LT. s = (oT#ST, LT) \<and> is_class G C \<and>  
   263   field (G,C) F = Some (C,vT) \<and> G \<turnstile> oT \<preceq> (Class C) \<and> (\<forall>x \<in> set (match G NullPointer pc et). is_class G x))"
   264   by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
   265 
   266 lemma appPutField[simp]:
   267 "(app (Putfield F C) G maxs rT pc et (Some s)) = 
   268  (\<exists> vT vT' oT ST LT. s = (vT#oT#ST, LT) \<and> is_class G C \<and> 
   269   field (G,C) F = Some (C, vT') \<and> G \<turnstile> oT \<preceq> (Class C) \<and> G \<turnstile> vT \<preceq> vT' \<and>
   270   (\<forall>x \<in> set (match G NullPointer pc et). is_class G x))"
   271   by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
   272 
   273 lemma appNew[simp]:
   274   "(app (New C) G maxs rT pc et (Some s)) = 
   275   (\<exists>ST LT. s=(ST,LT) \<and> is_class G C \<and> length ST < maxs \<and>
   276   (\<forall>x \<in> set (match G OutOfMemory pc et). is_class G x))"
   277   by (cases s, simp)
   278 
   279 lemma appCheckcast[simp]: 
   280   "(app (Checkcast C) G maxs rT pc et (Some s)) =  
   281   (\<exists>rT ST LT. s = (RefT rT#ST,LT) \<and> is_class G C \<and>
   282   (\<forall>x \<in> set (match G ClassCast pc et). is_class G x))"
   283   by (cases s, cases "fst s", simp add: app_def) (cases "hd (fst s)", auto)
   284 
   285 lemma appPop[simp]: 
   286 "(app Pop G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT))"
   287   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   288 
   289 
   290 lemma appDup[simp]:
   291 "(app Dup G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT) \<and> 1+length ST < maxs)" 
   292   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   293 
   294 
   295 lemma appDup_x1[simp]:
   296 "(app Dup_x1 G maxs rT pc et (Some s)) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \<and> 2+length ST < maxs)" 
   297   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   298 
   299 
   300 lemma appDup_x2[simp]:
   301 "(app Dup_x2 G maxs rT pc et (Some s)) = (\<exists>ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT) \<and> 3+length ST < maxs)"
   302   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   303 
   304 
   305 lemma appSwap[simp]:
   306 "app Swap G maxs rT pc et (Some s) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))" 
   307   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   308 
   309 
   310 lemma appIAdd[simp]:
   311 "app IAdd G maxs rT pc et (Some s) = (\<exists> ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))"
   312   (is "?app s = ?P s")
   313 proof (cases (open) s)
   314   case Pair
   315   have "?app (a,b) = ?P (a,b)"
   316   proof (cases "a")
   317     fix t ts assume a: "a = t#ts"
   318     show ?thesis
   319     proof (cases t)
   320       fix p assume p: "t = PrimT p"
   321       show ?thesis
   322       proof (cases p)
   323         assume ip: "p = Integer"
   324         show ?thesis
   325         proof (cases ts)
   326           fix t' ts' assume t': "ts = t' # ts'"
   327           show ?thesis
   328           proof (cases t')
   329             fix p' assume "t' = PrimT p'"
   330             with t' ip p a
   331             show ?thesis by - (cases p', auto)
   332           qed (auto simp add: a p ip t')
   333         qed (auto simp add: a p ip)
   334       qed (auto simp add: a p)
   335     qed (auto simp add: a)
   336   qed auto
   337   with Pair show ?thesis by simp
   338 qed
   339 
   340 
   341 lemma appIfcmpeq[simp]:
   342 "app (Ifcmpeq b) G maxs rT pc et (Some s) = 
   343   (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \<and> 0 \<le> int pc + b \<and>
   344   ((\<exists> p. ts1 = PrimT p \<and> ts2 = PrimT p) \<or> (\<exists>r r'. ts1 = RefT r \<and> ts2 = RefT r')))" 
   345   by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
   346 
   347 lemma appReturn[simp]:
   348 "app Return G maxs rT pc et (Some s) = (\<exists>T ST LT. s = (T#ST,LT) \<and> (G \<turnstile> T \<preceq> rT))" 
   349   by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
   350 
   351 lemma appGoto[simp]:
   352 "app (Goto b) G maxs rT pc et (Some s) = (0 \<le> int pc + b)"
   353   by simp
   354 
   355 lemma appThrow[simp]:
   356   "app Throw G maxs rT pc et (Some s) = 
   357   (\<exists>T ST LT r. s=(T#ST,LT) \<and> T = RefT r \<and> (\<forall>C \<in> set (match_any G pc et). is_class G C))"
   358   by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
   359 
   360 lemma appInvoke[simp]:
   361 "app (Invoke C mn fpTs) G maxs rT pc et (Some s) = (\<exists>apTs X ST LT mD' rT' b'.
   362   s = ((rev apTs) @ (X # ST), LT) \<and> length apTs = length fpTs \<and> is_class G C \<and>
   363   G \<turnstile> X \<preceq> Class C \<and> (\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and> 
   364   method (G,C) (mn,fpTs) = Some (mD', rT', b') \<and> 
   365   (\<forall>C \<in> set (match_any G pc et). is_class G C))" (is "?app s = ?P s")
   366 proof (cases (open) s)
   367   note list_all2_def [simp]
   368   case Pair
   369   have "?app (a,b) \<Longrightarrow> ?P (a,b)"
   370   proof -
   371     assume app: "?app (a,b)"
   372     hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) \<and> 
   373            length fpTs < length a" (is "?a \<and> ?l") 
   374       by (auto simp add: app_def)
   375     hence "?a \<and> 0 < length (drop (length fpTs) a)" (is "?a \<and> ?l") 
   376       by auto
   377     hence "?a \<and> ?l \<and> length (rev (take (length fpTs) a)) = length fpTs" 
   378       by (auto simp add: min_def)
   379     hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> 0 < length ST" 
   380       by blast
   381     hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> ST \<noteq> []" 
   382       by blast
   383     hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> 
   384            (\<exists>X ST'. ST = X#ST')" 
   385       by (simp add: neq_Nil_conv)
   386     hence "\<exists>apTs X ST. a = rev apTs @ X # ST \<and> length apTs = length fpTs" 
   387       by blast
   388     with app
   389     show ?thesis by (unfold app_def, clarsimp) blast
   390   qed
   391   with Pair 
   392   have "?app s \<Longrightarrow> ?P s" by (simp only:)
   393   moreover
   394   have "?P s \<Longrightarrow> ?app s" by (unfold app_def) (clarsimp simp add: min_def)
   395   ultimately
   396   show ?thesis by (rule iffI) 
   397 qed 
   398 
   399 lemma effNone: 
   400   "(pc', s') \<in> set (eff i G pc et None) \<Longrightarrow> s' = None"
   401   by (auto simp add: eff_def xcpt_eff_def norm_eff_def)
   402 
   403 
   404 lemma xcpt_app_lemma [code]:
   405   "xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))"
   406   by (simp add: list_all_iff)
   407 
   408 lemmas [simp del] = app_def xcpt_app_def
   409 
   410 end