src/HOL/MicroJava/BV/Effect.thy

 types
   succ_type = "(p_count \<times> state_type option) list"
 
 text {* Program counter of successor instructions: *}
 consts
-  succs :: "instr => p_count => p_count list"
+  succs :: "instr \<Rightarrow> p_count \<Rightarrow> p_count list"
 primrec 
   "succs (Load idx) pc         = [pc+1]"
   "succs (Store idx) pc        = [pc+1]"
   "succs (LitPush v) pc        = [pc+1]"
   "succs (Getfield F C) pc     = [pc+1]"

   "succs (Invoke C mn fpTs) pc = [pc+1]"
   "succs Throw pc              = [pc]"
 
 text "Effect of instruction on the state type:"
 consts 
-eff' :: "instr \<times> jvm_prog \<times> state_type => state_type"
+eff' :: "instr \<times> jvm_prog \<times> state_type \<Rightarrow> state_type"
 
 recdef eff' "{}"
 "eff' (Load idx,  G, (ST, LT))          = (ok_val (LT ! idx) # ST, LT)"
 "eff' (Store idx, G, (ts#ST, LT))       = (ST, LT[idx:= OK ts])"
 "eff' (LitPush v, G, (ST, LT))           = (the (typeof (\<lambda>v. None) v) # ST, LT)"

 lemma match_some_entry:
   "match G X pc et = (if \<exists>e \<in> set et. match_exception_entry G X pc e then [X] else [])"
   by (induct et) auto
 
 consts
-  xcpt_names :: "instr \<times> jvm_prog \<times> p_count \<times> exception_table => cname list" 
+  xcpt_names :: "instr \<times> jvm_prog \<times> p_count \<times> exception_table \<Rightarrow> cname list" 
 recdef xcpt_names "{}"
   "xcpt_names (Getfield F C, G, pc, et) = match G (Xcpt NullPointer) pc et" 
   "xcpt_names (Putfield F C, G, pc, et) = match G (Xcpt NullPointer) pc et" 
   "xcpt_names (New C, G, pc, et)        = match G (Xcpt OutOfMemory) pc et"
   "xcpt_names (Checkcast C, G, pc, et)  = match G (Xcpt ClassCast) pc et"

        (xcpt_names (i,G,pc,et))"
 
   norm_eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> state_type option \<Rightarrow> state_type option"
   "norm_eff i G == option_map (\<lambda>s. eff' (i,G,s))"
 
-  eff :: "instr => jvm_prog => p_count => exception_table => state_type option => succ_type"
+  eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> succ_type"
   "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)"
 
 constdefs
   isPrimT :: "ty \<Rightarrow> bool"
   "isPrimT T == case T of PrimT T' \<Rightarrow> True | RefT T' \<Rightarrow> False"

   by (simp add: list_all2_def) 
 
 
 text "Conditions under which eff is applicable:"
 consts
-app' :: "instr \<times> jvm_prog \<times> p_count \<times> nat \<times> ty \<times> state_type => bool"
+app' :: "instr \<times> jvm_prog \<times> p_count \<times> nat \<times> ty \<times> state_type \<Rightarrow> bool"
 
 recdef app' "{}"
 "app' (Load idx, G, pc, maxs, rT, s) = 
   (idx < length (snd s) \<and> (snd s) ! idx \<noteq> Err \<and> length (fst s) < maxs)"
 "app' (Store idx, G, pc, maxs, rT, (ts#ST, LT)) = 

 
 constdefs
   xcpt_app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> bool"
   "xcpt_app i G pc et \<equiv> \<forall>C\<in>set(xcpt_names (i,G,pc,et)). is_class G C"
 
-  app :: "instr => jvm_prog => nat => ty => nat => exception_table => state_type option => bool"
-  "app i G maxs rT pc et s == case s of None => True | Some t => app' (i,G,pc,maxs,rT,t) \<and> xcpt_app i G pc et"
-
-
-lemma 1: "2 < length a ==> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"
+  app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> bool"
+  "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"
+
+
+lemma 1: "2 < length a \<Longrightarrow> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"
 proof (cases a)
   fix x xs assume "a = x#xs" "2 < length a"
   thus ?thesis by - (cases xs, simp, cases "tl xs", auto)
 qed auto
 
-lemma 2: "\<not>(2 < length a) ==> a = [] \<or> (\<exists> l. a = [l]) \<or> (\<exists> l l'. a = [l,l'])"
+lemma 2: "\<not>(2 < length a) \<Longrightarrow> a = [] \<or> (\<exists> l. a = [l]) \<or> (\<exists> l l'. a = [l,l'])"
 proof -;
   assume "\<not>(2 < length a)"
   hence "length a < (Suc (Suc (Suc 0)))" by simp
   hence * : "length a = 0 \<or> length a = Suc 0 \<or> length a = Suc (Suc 0)" 
     by (auto simp add: less_Suc_eq)

     fix x 
     assume "length x = Suc 0"
     hence "\<exists> l. x = [l]"  by - (cases x, auto)
   } note 0 = this
 
-  have "length a = Suc (Suc 0) ==> \<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
+  have "length a = Suc (Suc 0) \<Longrightarrow> \<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
   with * show ?thesis by (auto dest: 0)
 qed
 
 lemmas [simp] = app_def xcpt_app_def
 

   method (G,C) (mn,fpTs) = Some (mD', rT', b') \<and> 
   (\<forall>C \<in> set (match_any G pc et). is_class G C))" (is "?app s = ?P s")
 proof (cases (open) s)
   note list_all2_def [simp]
   case Pair
-  have "?app (a,b) ==> ?P (a,b)"
+  have "?app (a,b) \<Longrightarrow> ?P (a,b)"
   proof -
     assume app: "?app (a,b)"
     hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) \<and> 
            length fpTs < length a" (is "?a \<and> ?l") 
       by (auto simp add: app_def)