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(* Title: HOL/Bali/State.thy
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12854
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ID: $Id$
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Author: David von Oheimb
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Copyright 1997 Technische Universitaet Muenchen
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*)
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header {* State for evaluation of Java expressions and statements *}
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theory State = DeclConcepts:
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text {*
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design issues:
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\begin{itemize}
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\item all kinds of objects (class instances, arrays, and class objects)
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are handeled via a general object abstraction
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\item the heap and the map for class objects are combined into a single table
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@{text "(recall (loc, obj) table \<times> (qtname, obj) table ~= (loc + qtname, obj) table)"}
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\end{itemize}
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*}
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section "objects"
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datatype obj_tag = (* tag for generic object *)
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CInst qtname (* class instance *)
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| Arr ty int (* array with component type and length *)
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(* | CStat the tag is irrelevant for a class object,
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i.e. the static fields of a class *)
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types vn = "fspec + int" (* variable name *)
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record obj =
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tag :: "obj_tag" (* generalized object *)
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values :: "(vn, val) table"
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translations
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"fspec" <= (type) "vname \<times> qtname"
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"vn" <= (type) "fspec + int"
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"obj" <= (type) "\<lparr>tag::obj_tag, values::vn \<Rightarrow> val option\<rparr>"
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"obj" <= (type) "\<lparr>tag::obj_tag, values::vn \<Rightarrow> val option,\<dots>::'a\<rparr>"
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constdefs
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the_Arr :: "obj option \<Rightarrow> ty \<times> int \<times> (vn, val) table"
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"the_Arr obj \<equiv> \<epsilon>(T,k,t). obj = Some \<lparr>tag=Arr T k,values=t\<rparr>"
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lemma the_Arr_Arr [simp]: "the_Arr (Some \<lparr>tag=Arr T k,values=cs\<rparr>) = (T,k,cs)"
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apply (auto simp: the_Arr_def)
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done
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lemma the_Arr_Arr1 [simp,intro,dest]:
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"\<lbrakk>tag obj = Arr T k\<rbrakk> \<Longrightarrow> the_Arr (Some obj) = (T,k,values obj)"
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apply (auto simp add: the_Arr_def)
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done
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constdefs
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upd_obj :: "vn \<Rightarrow> val \<Rightarrow> obj \<Rightarrow> obj"
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"upd_obj n v \<equiv> \<lambda> obj . obj \<lparr>values:=(values obj)(n\<mapsto>v)\<rparr>"
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lemma upd_obj_def2 [simp]:
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"upd_obj n v obj = obj \<lparr>values:=(values obj)(n\<mapsto>v)\<rparr>"
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apply (auto simp: upd_obj_def)
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done
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constdefs
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obj_ty :: "obj \<Rightarrow> ty"
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"obj_ty obj \<equiv> case tag obj of
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CInst C \<Rightarrow> Class C
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| Arr T k \<Rightarrow> T.[]"
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lemma obj_ty_eq [intro!]: "obj_ty \<lparr>tag=oi,values=x\<rparr> = obj_ty \<lparr>tag=oi,values=y\<rparr>"
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by (simp add: obj_ty_def)
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lemma obj_ty_eq1 [intro!,dest]:
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"tag obj = tag obj' \<Longrightarrow> obj_ty obj = obj_ty obj'"
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by (simp add: obj_ty_def)
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lemma obj_ty_cong [simp]:
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"obj_ty (obj \<lparr>values:=vs\<rparr>) = obj_ty obj"
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by auto
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(*
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lemma obj_ty_cong [simp]:
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"obj_ty (obj \<lparr>values:=vs(n\<mapsto>v)\<rparr>) = obj_ty (obj \<lparr>values:=vs\<rparr>)"
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by auto
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*)
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lemma obj_ty_CInst [simp]:
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"obj_ty \<lparr>tag=CInst C,values=vs\<rparr> = Class C"
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by (simp add: obj_ty_def)
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lemma obj_ty_CInst1 [simp,intro!,dest]:
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"\<lbrakk>tag obj = CInst C\<rbrakk> \<Longrightarrow> obj_ty obj = Class C"
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by (simp add: obj_ty_def)
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lemma obj_ty_Arr [simp]:
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"obj_ty \<lparr>tag=Arr T i,values=vs\<rparr> = T.[]"
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by (simp add: obj_ty_def)
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lemma obj_ty_Arr1 [simp,intro!,dest]:
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"\<lbrakk>tag obj = Arr T i\<rbrakk> \<Longrightarrow> obj_ty obj = T.[]"
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by (simp add: obj_ty_def)
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lemma obj_ty_widenD:
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"G\<turnstile>obj_ty obj\<preceq>RefT t \<Longrightarrow> (\<exists>C. tag obj = CInst C) \<or> (\<exists>T k. tag obj = Arr T k)"
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apply (unfold obj_ty_def)
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apply (auto split add: obj_tag.split_asm)
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done
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constdefs
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obj_class :: "obj \<Rightarrow> qtname"
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"obj_class obj \<equiv> case tag obj of
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CInst C \<Rightarrow> C
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| Arr T k \<Rightarrow> Object"
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lemma obj_class_CInst [simp]: "obj_class \<lparr>tag=CInst C,values=vs\<rparr> = C"
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by (auto simp: obj_class_def)
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lemma obj_class_CInst1 [simp,intro!,dest]:
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"tag obj = CInst C \<Longrightarrow> obj_class obj = C"
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by (auto simp: obj_class_def)
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lemma obj_class_Arr [simp]: "obj_class \<lparr>tag=Arr T k,values=vs\<rparr> = Object"
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by (auto simp: obj_class_def)
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lemma obj_class_Arr1 [simp,intro!,dest]:
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"tag obj = Arr T k \<Longrightarrow> obj_class obj = Object"
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by (auto simp: obj_class_def)
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lemma obj_ty_obj_class: "G\<turnstile>obj_ty obj\<preceq> Class statC = G\<turnstile>obj_class obj \<preceq>\<^sub>C statC"
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apply (case_tac "tag obj")
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apply (auto simp add: obj_ty_def obj_class_def)
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apply (case_tac "statC = Object")
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apply (auto dest: widen_Array_Class)
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done
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section "object references"
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types oref = "loc + qtname" (* generalized object reference *)
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syntax
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Heap :: "loc \<Rightarrow> oref"
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Stat :: "qtname \<Rightarrow> oref"
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translations
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"Heap" => "Inl"
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"Stat" => "Inr"
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"oref" <= (type) "loc + qtname"
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constdefs
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fields_table::
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"prog \<Rightarrow> qtname \<Rightarrow> (fspec \<Rightarrow> field \<Rightarrow> bool) \<Rightarrow> (fspec, ty) table"
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"fields_table G C P
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\<equiv> option_map type \<circ> table_of (filter (split P) (DeclConcepts.fields G C))"
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lemma fields_table_SomeI:
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"\<lbrakk>table_of (DeclConcepts.fields G C) n = Some f; P n f\<rbrakk>
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\<Longrightarrow> fields_table G C P n = Some (type f)"
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apply (unfold fields_table_def)
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apply clarsimp
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apply (rule exI)
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apply (rule conjI)
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apply (erule map_of_filter_in)
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apply assumption
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apply simp
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done
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(* unused *)
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lemma fields_table_SomeD': "fields_table G C P fn = Some T \<Longrightarrow>
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\<exists>f. (fn,f)\<in>set(DeclConcepts.fields G C) \<and> type f = T"
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apply (unfold fields_table_def)
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apply clarsimp
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apply (drule map_of_SomeD)
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apply auto
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done
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lemma fields_table_SomeD:
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"\<lbrakk>fields_table G C P fn = Some T; unique (DeclConcepts.fields G C)\<rbrakk> \<Longrightarrow>
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\<exists>f. table_of (DeclConcepts.fields G C) fn = Some f \<and> type f = T"
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apply (unfold fields_table_def)
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apply clarsimp
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apply (rule exI)
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apply (rule conjI)
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apply (erule table_of_filter_unique_SomeD)
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apply assumption
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apply simp
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done
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constdefs
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in_bounds :: "int \<Rightarrow> int \<Rightarrow> bool" ("(_/ in'_bounds _)" [50, 51] 50)
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"i in_bounds k \<equiv> 0 \<le> i \<and> i < k"
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arr_comps :: "'a \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a option"
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"arr_comps T k \<equiv> \<lambda>i. if i in_bounds k then Some T else None"
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var_tys :: "prog \<Rightarrow> obj_tag \<Rightarrow> oref \<Rightarrow> (vn, ty) table"
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"var_tys G oi r
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\<equiv> case r of
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Heap a \<Rightarrow> (case oi of
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CInst C \<Rightarrow> fields_table G C (\<lambda>n f. \<not>static f) (+) empty
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| Arr T k \<Rightarrow> empty (+) arr_comps T k)
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| Stat C \<Rightarrow> fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f)
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(+) empty"
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lemma var_tys_Some_eq:
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"var_tys G oi r n = Some T
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= (case r of
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Inl a \<Rightarrow> (case oi of
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CInst C \<Rightarrow> (\<exists>nt. n = Inl nt \<and> fields_table G C (\<lambda>n f.
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\<not>static f) nt = Some T)
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| Arr t k \<Rightarrow> (\<exists> i. n = Inr i \<and> i in_bounds k \<and> t = T))
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| Inr C \<Rightarrow> (\<exists>nt. n = Inl nt \<and>
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fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f) nt
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= Some T))"
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apply (unfold var_tys_def arr_comps_def)
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apply (force split add: sum.split_asm sum.split obj_tag.split)
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done
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section "stores"
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types globs (* global variables: heap and static variables *)
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= "(oref , obj) table"
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heap
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= "(loc , obj) table"
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locals
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= "(lname, val) table" (* local variables *)
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translations
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"globs" <= (type) "(oref , obj) table"
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"heap" <= (type) "(loc , obj) table"
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"locals" <= (type) "(lname, val) table"
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datatype st = (* pure state, i.e. contents of all variables *)
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st globs locals
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subsection "access"
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constdefs
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globs :: "st \<Rightarrow> globs"
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"globs \<equiv> st_case (\<lambda>g l. g)"
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locals :: "st \<Rightarrow> locals"
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"locals \<equiv> st_case (\<lambda>g l. l)"
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heap :: "st \<Rightarrow> heap"
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"heap s \<equiv> globs s \<circ> Heap"
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lemma globs_def2 [simp]: " globs (st g l) = g"
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by (simp add: globs_def)
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lemma locals_def2 [simp]: "locals (st g l) = l"
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by (simp add: locals_def)
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lemma heap_def2 [simp]: "heap s a=globs s (Heap a)"
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by (simp add: heap_def)
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syntax
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val_this :: "st \<Rightarrow> val"
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lookup_obj :: "st \<Rightarrow> val \<Rightarrow> obj"
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translations
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"val_this s" == "the (locals s This)"
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"lookup_obj s a'" == "the (heap s (the_Addr a'))"
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subsection "memory allocation"
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constdefs
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new_Addr :: "heap \<Rightarrow> loc option"
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"new_Addr h \<equiv> if (\<forall>a. h a \<noteq> None) then None else Some (\<epsilon>a. h a = None)"
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lemma new_AddrD: "new_Addr h = Some a \<Longrightarrow> h a = None"
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apply (unfold new_Addr_def)
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apply auto
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apply (case_tac "h (SOME a\<Colon>loc. h a = None)")
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apply simp
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apply (fast intro: someI2)
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done
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lemma new_AddrD2: "new_Addr h = Some a \<Longrightarrow> \<forall>b. h b \<noteq> None \<longrightarrow> b \<noteq> a"
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apply (drule new_AddrD)
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apply auto
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done
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lemma new_Addr_SomeI: "h a = None \<Longrightarrow> \<exists>b. new_Addr h = Some b \<and> h b = None"
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apply (unfold new_Addr_def)
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apply (frule not_Some_eq [THEN iffD2])
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apply auto
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apply (drule not_Some_eq [THEN iffD2])
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apply auto
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apply (fast intro!: someI2)
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done
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subsection "initialization"
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syntax
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init_vals :: "('a, ty) table \<Rightarrow> ('a, val) table"
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translations
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"init_vals vs" == "option_map default_val \<circ> vs"
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lemma init_arr_comps_base [simp]: "init_vals (arr_comps T 0) = empty"
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apply (unfold arr_comps_def in_bounds_def)
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apply (rule ext)
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apply auto
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done
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lemma init_arr_comps_step [simp]:
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"0 < j \<Longrightarrow> init_vals (arr_comps T j ) =
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init_vals (arr_comps T (j - 1))(j - 1\<mapsto>default_val T)"
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apply (unfold arr_comps_def in_bounds_def)
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apply (rule ext)
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apply auto
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done
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subsection "update"
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constdefs
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gupd :: "oref \<Rightarrow> obj \<Rightarrow> st \<Rightarrow> st" ("gupd'(_\<mapsto>_')"[10,10]1000)
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"gupd r obj \<equiv> st_case (\<lambda>g l. st (g(r\<mapsto>obj)) l)"
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lupd :: "lname \<Rightarrow> val \<Rightarrow> st \<Rightarrow> st" ("lupd'(_\<mapsto>_')"[10,10]1000)
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"lupd vn v \<equiv> st_case (\<lambda>g l. st g (l(vn\<mapsto>v)))"
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upd_gobj :: "oref \<Rightarrow> vn \<Rightarrow> val \<Rightarrow> st \<Rightarrow> st"
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"upd_gobj r n v \<equiv> st_case (\<lambda>g l. st (chg_map (upd_obj n v) r g) l)"
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set_locals :: "locals \<Rightarrow> st \<Rightarrow> st"
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"set_locals l \<equiv> st_case (\<lambda>g l'. st g l)"
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init_obj :: "prog \<Rightarrow> obj_tag \<Rightarrow> oref \<Rightarrow> st \<Rightarrow> st"
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"init_obj G oi r \<equiv> gupd(r\<mapsto>\<lparr>tag=oi, values=init_vals (var_tys G oi r)\<rparr>)"
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syntax
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init_class_obj :: "prog \<Rightarrow> qtname \<Rightarrow> st \<Rightarrow> st"
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translations
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"init_class_obj G C" == "init_obj G arbitrary (Inr C)"
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lemma gupd_def2 [simp]: "gupd(r\<mapsto>obj) (st g l) = st (g(r\<mapsto>obj)) l"
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apply (unfold gupd_def)
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apply (simp (no_asm))
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done
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lemma lupd_def2 [simp]: "lupd(vn\<mapsto>v) (st g l) = st g (l(vn\<mapsto>v))"
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apply (unfold lupd_def)
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apply (simp (no_asm))
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done
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lemma globs_gupd [simp]: "globs (gupd(r\<mapsto>obj) s) = globs s(r\<mapsto>obj)"
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apply (induct "s")
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by (simp add: gupd_def)
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lemma globs_lupd [simp]: "globs (lupd(vn\<mapsto>v ) s) = globs s"
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apply (induct "s")
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by (simp add: lupd_def)
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lemma locals_gupd [simp]: "locals (gupd(r\<mapsto>obj) s) = locals s"
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apply (induct "s")
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by (simp add: gupd_def)
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lemma locals_lupd [simp]: "locals (lupd(vn\<mapsto>v ) s) = locals s(vn\<mapsto>v )"
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apply (induct "s")
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367 |
by (simp add: lupd_def)
|
|
368 |
|
|
369 |
lemma globs_upd_gobj_new [rule_format (no_asm), simp]:
|
|
370 |
"globs s r = None \<longrightarrow> globs (upd_gobj r n v s) = globs s"
|
|
371 |
apply (unfold upd_gobj_def)
|
|
372 |
apply (induct "s")
|
|
373 |
apply auto
|
|
374 |
done
|
|
375 |
|
|
376 |
lemma globs_upd_gobj_upd [rule_format (no_asm), simp]:
|
|
377 |
"globs s r=Some obj\<longrightarrow> globs (upd_gobj r n v s) = globs s(r\<mapsto>upd_obj n v obj)"
|
|
378 |
apply (unfold upd_gobj_def)
|
|
379 |
apply (induct "s")
|
|
380 |
apply auto
|
|
381 |
done
|
|
382 |
|
|
383 |
lemma locals_upd_gobj [simp]: "locals (upd_gobj r n v s) = locals s"
|
|
384 |
apply (induct "s")
|
|
385 |
by (simp add: upd_gobj_def)
|
|
386 |
|
|
387 |
|
|
388 |
lemma globs_init_obj [simp]: "globs (init_obj G oi r s) t =
|
|
389 |
(if t=r then Some \<lparr>tag=oi,values=init_vals (var_tys G oi r)\<rparr> else globs s t)"
|
|
390 |
apply (unfold init_obj_def)
|
|
391 |
apply (simp (no_asm))
|
|
392 |
done
|
|
393 |
|
|
394 |
lemma locals_init_obj [simp]: "locals (init_obj G oi r s) = locals s"
|
|
395 |
by (simp add: init_obj_def)
|
|
396 |
|
|
397 |
lemma surjective_st [simp]: "st (globs s) (locals s) = s"
|
|
398 |
apply (induct "s")
|
|
399 |
by auto
|
|
400 |
|
|
401 |
lemma surjective_st_init_obj:
|
|
402 |
"st (globs (init_obj G oi r s)) (locals s) = init_obj G oi r s"
|
|
403 |
apply (subst locals_init_obj [THEN sym])
|
|
404 |
apply (rule surjective_st)
|
|
405 |
done
|
|
406 |
|
|
407 |
lemma heap_heap_upd [simp]:
|
|
408 |
"heap (st (g(Inl a\<mapsto>obj)) l) = heap (st g l)(a\<mapsto>obj)"
|
|
409 |
apply (rule ext)
|
|
410 |
apply (simp (no_asm))
|
|
411 |
done
|
|
412 |
lemma heap_stat_upd [simp]: "heap (st (g(Inr C\<mapsto>obj)) l) = heap (st g l)"
|
|
413 |
apply (rule ext)
|
|
414 |
apply (simp (no_asm))
|
|
415 |
done
|
|
416 |
lemma heap_local_upd [simp]: "heap (st g (l(vn\<mapsto>v))) = heap (st g l)"
|
|
417 |
apply (rule ext)
|
|
418 |
apply (simp (no_asm))
|
|
419 |
done
|
|
420 |
|
|
421 |
lemma heap_gupd_Heap [simp]: "heap (gupd(Heap a\<mapsto>obj) s) = heap s(a\<mapsto>obj)"
|
|
422 |
apply (rule ext)
|
|
423 |
apply (simp (no_asm))
|
|
424 |
done
|
|
425 |
lemma heap_gupd_Stat [simp]: "heap (gupd(Stat C\<mapsto>obj) s) = heap s"
|
|
426 |
apply (rule ext)
|
|
427 |
apply (simp (no_asm))
|
|
428 |
done
|
|
429 |
lemma heap_lupd [simp]: "heap (lupd(vn\<mapsto>v) s) = heap s"
|
|
430 |
apply (rule ext)
|
|
431 |
apply (simp (no_asm))
|
|
432 |
done
|
|
433 |
|
|
434 |
(*
|
|
435 |
lemma heap_upd_gobj_Heap: "!!a. heap (upd_gobj (Heap a) n v s) = ?X"
|
|
436 |
apply (rule ext)
|
|
437 |
apply (simp (no_asm))
|
|
438 |
apply (case_tac "globs s (Heap a)")
|
|
439 |
apply auto
|
|
440 |
*)
|
|
441 |
|
|
442 |
lemma heap_upd_gobj_Stat [simp]: "heap (upd_gobj (Stat C) n v s) = heap s"
|
|
443 |
apply (rule ext)
|
|
444 |
apply (simp (no_asm))
|
|
445 |
apply (case_tac "globs s (Stat C)")
|
|
446 |
apply auto
|
|
447 |
done
|
|
448 |
|
|
449 |
lemma set_locals_def2 [simp]: "set_locals l (st g l') = st g l"
|
|
450 |
apply (unfold set_locals_def)
|
|
451 |
apply (simp (no_asm))
|
|
452 |
done
|
|
453 |
|
|
454 |
lemma set_locals_id [simp]: "set_locals (locals s) s = s"
|
|
455 |
apply (unfold set_locals_def)
|
|
456 |
apply (induct_tac "s")
|
|
457 |
apply (simp (no_asm))
|
|
458 |
done
|
|
459 |
|
|
460 |
lemma set_set_locals [simp]: "set_locals l (set_locals l' s) = set_locals l s"
|
|
461 |
apply (unfold set_locals_def)
|
|
462 |
apply (induct_tac "s")
|
|
463 |
apply (simp (no_asm))
|
|
464 |
done
|
|
465 |
|
|
466 |
lemma locals_set_locals [simp]: "locals (set_locals l s) = l"
|
|
467 |
apply (unfold set_locals_def)
|
|
468 |
apply (induct_tac "s")
|
|
469 |
apply (simp (no_asm))
|
|
470 |
done
|
|
471 |
|
|
472 |
lemma globs_set_locals [simp]: "globs (set_locals l s) = globs s"
|
|
473 |
apply (unfold set_locals_def)
|
|
474 |
apply (induct_tac "s")
|
|
475 |
apply (simp (no_asm))
|
|
476 |
done
|
|
477 |
|
|
478 |
lemma heap_set_locals [simp]: "heap (set_locals l s) = heap s"
|
|
479 |
apply (unfold heap_def)
|
|
480 |
apply (induct_tac "s")
|
|
481 |
apply (simp (no_asm))
|
|
482 |
done
|
|
483 |
|
|
484 |
|
|
485 |
section "abrupt completion"
|
|
486 |
|
|
487 |
|
|
488 |
datatype xcpt (* exception *)
|
|
489 |
= Loc loc (* location of allocated execption object *)
|
|
490 |
| Std xname (* intermediate standard exception, see Eval.thy *)
|
|
491 |
|
|
492 |
datatype abrupt (* abrupt completion *)
|
|
493 |
= Xcpt xcpt (* exception *)
|
|
494 |
| Jump jump (* break, continue, return *)
|
|
495 |
consts
|
|
496 |
|
|
497 |
the_Xcpt :: "abrupt \<Rightarrow> xcpt"
|
|
498 |
the_Jump :: "abrupt => jump"
|
|
499 |
the_Loc :: "xcpt \<Rightarrow> loc"
|
|
500 |
the_Std :: "xcpt \<Rightarrow> xname"
|
|
501 |
|
|
502 |
primrec "the_Xcpt (Xcpt x) = x"
|
|
503 |
primrec "the_Jump (Jump j) = j"
|
|
504 |
primrec "the_Loc (Loc a) = a"
|
|
505 |
primrec "the_Std (Std x) = x"
|
|
506 |
|
|
507 |
types
|
|
508 |
abopt = "abrupt option"
|
|
509 |
|
|
510 |
|
|
511 |
constdefs
|
|
512 |
abrupt_if :: "bool \<Rightarrow> abopt \<Rightarrow> abopt \<Rightarrow> abopt"
|
|
513 |
"abrupt_if c x' x \<equiv> if c \<and> (x = None) then x' else x"
|
|
514 |
|
|
515 |
lemma abrupt_if_True_None [simp]: "abrupt_if True x None = x"
|
|
516 |
by (simp add: abrupt_if_def)
|
|
517 |
|
|
518 |
lemma abrupt_if_True_not_None [simp]: "x \<noteq> None \<Longrightarrow> abrupt_if True x y \<noteq> None"
|
|
519 |
by (simp add: abrupt_if_def)
|
|
520 |
|
|
521 |
lemma abrupt_if_False [simp]: "abrupt_if False x y = y"
|
|
522 |
by (simp add: abrupt_if_def)
|
|
523 |
|
|
524 |
lemma abrupt_if_Some [simp]: "abrupt_if c x (Some y) = Some y"
|
|
525 |
by (simp add: abrupt_if_def)
|
|
526 |
|
|
527 |
lemma abrupt_if_not_None [simp]: "y \<noteq> None \<Longrightarrow> abrupt_if c x y = y"
|
|
528 |
apply (simp add: abrupt_if_def)
|
|
529 |
by auto
|
|
530 |
|
|
531 |
|
|
532 |
lemma split_abrupt_if:
|
|
533 |
"P (abrupt_if c x' x) =
|
|
534 |
((c \<and> x = None \<longrightarrow> P x') \<and> (\<not> (c \<and> x = None) \<longrightarrow> P x))"
|
|
535 |
apply (unfold abrupt_if_def)
|
|
536 |
apply (split split_if)
|
|
537 |
apply auto
|
|
538 |
done
|
|
539 |
|
|
540 |
syntax
|
|
541 |
|
|
542 |
raise_if :: "bool \<Rightarrow> xname \<Rightarrow> abopt \<Rightarrow> abopt"
|
|
543 |
np :: "val \<spacespace> \<Rightarrow> abopt \<Rightarrow> abopt"
|
|
544 |
check_neg:: "val \<spacespace> \<Rightarrow> abopt \<Rightarrow> abopt"
|
|
545 |
|
|
546 |
translations
|
|
547 |
|
|
548 |
"raise_if c xn" == "abrupt_if c (Some (Xcpt (Std xn)))"
|
|
549 |
"np v" == "raise_if (v = Null) NullPointer"
|
|
550 |
"check_neg i'" == "raise_if (the_Intg i'<0) NegArrSize"
|
|
551 |
|
|
552 |
lemma raise_if_None [simp]: "(raise_if c x y = None) = (\<not>c \<and> y = None)"
|
|
553 |
apply (simp add: abrupt_if_def)
|
|
554 |
by auto
|
|
555 |
declare raise_if_None [THEN iffD1, dest!]
|
|
556 |
|
|
557 |
lemma if_raise_if_None [simp]:
|
|
558 |
"((if b then y else raise_if c x y) = None) = ((c \<longrightarrow> b) \<and> y = None)"
|
|
559 |
apply (simp add: abrupt_if_def)
|
|
560 |
apply auto
|
|
561 |
done
|
|
562 |
|
|
563 |
lemma raise_if_SomeD [dest!]:
|
|
564 |
"raise_if c x y = Some z \<Longrightarrow> c \<and> z=(Xcpt (Std x)) \<and> y=None \<or> (y=Some z)"
|
|
565 |
apply (case_tac y)
|
|
566 |
apply (case_tac c)
|
|
567 |
apply (simp add: abrupt_if_def)
|
|
568 |
apply (simp add: abrupt_if_def)
|
|
569 |
apply auto
|
|
570 |
done
|
|
571 |
|
|
572 |
constdefs
|
|
573 |
absorb :: "jump \<Rightarrow> abopt \<Rightarrow> abopt"
|
|
574 |
"absorb j a \<equiv> if a=Some (Jump j) then None else a"
|
|
575 |
|
|
576 |
lemma absorb_SomeD [dest!]: "absorb j a = Some x \<Longrightarrow> a = Some x"
|
|
577 |
by (auto simp add: absorb_def)
|
|
578 |
|
|
579 |
lemma absorb_same [simp]: "absorb j (Some (Jump j)) = None"
|
|
580 |
by (auto simp add: absorb_def)
|
|
581 |
|
|
582 |
lemma absorb_other [simp]: "a \<noteq> Some (Jump j) \<Longrightarrow> absorb j a = a"
|
|
583 |
by (auto simp add: absorb_def)
|
|
584 |
|
|
585 |
|
|
586 |
section "full program state"
|
|
587 |
|
|
588 |
types
|
|
589 |
state = "abopt \<times> st" (* state including exception information *)
|
|
590 |
|
|
591 |
syntax
|
|
592 |
Norm :: "st \<Rightarrow> state"
|
|
593 |
abrupt :: "state \<Rightarrow> abopt"
|
|
594 |
store :: "state \<Rightarrow> st"
|
|
595 |
|
|
596 |
translations
|
|
597 |
|
|
598 |
"Norm s" == "(None,s)"
|
|
599 |
"abrupt" => "fst"
|
|
600 |
"store" => "snd"
|
|
601 |
"abopt" <= (type) "State.abrupt option"
|
|
602 |
"abopt" <= (type) "abrupt option"
|
|
603 |
"state" <= (type) "abopt \<times> State.st"
|
|
604 |
"state" <= (type) "abopt \<times> st"
|
|
605 |
|
|
606 |
|
|
607 |
|
|
608 |
lemma single_stateE: "\<forall>Z. Z = (s::state) \<Longrightarrow> False"
|
|
609 |
apply (erule_tac x = "(Some k,y)" in all_dupE)
|
|
610 |
apply (erule_tac x = "(None,y)" in allE)
|
|
611 |
apply clarify
|
|
612 |
done
|
|
613 |
|
|
614 |
lemma state_not_single: "All (op = (x::state)) \<Longrightarrow> R"
|
|
615 |
apply (drule_tac x = "(if abrupt x = None then Some ?x else None,?y)" in spec)
|
|
616 |
apply clarsimp
|
|
617 |
done
|
|
618 |
|
|
619 |
constdefs
|
|
620 |
|
|
621 |
normal :: "state \<Rightarrow> bool"
|
|
622 |
"normal \<equiv> \<lambda>s. abrupt s = None"
|
|
623 |
|
|
624 |
lemma normal_def2 [simp]: "normal s = (abrupt s = None)"
|
|
625 |
apply (unfold normal_def)
|
|
626 |
apply (simp (no_asm))
|
|
627 |
done
|
|
628 |
|
|
629 |
constdefs
|
|
630 |
heap_free :: "nat \<Rightarrow> state \<Rightarrow> bool"
|
|
631 |
"heap_free n \<equiv> \<lambda>s. atleast_free (heap (store s)) n"
|
|
632 |
|
|
633 |
lemma heap_free_def2 [simp]: "heap_free n s = atleast_free (heap (store s)) n"
|
|
634 |
apply (unfold heap_free_def)
|
|
635 |
apply simp
|
|
636 |
done
|
|
637 |
|
|
638 |
subsection "update"
|
|
639 |
|
|
640 |
constdefs
|
|
641 |
|
|
642 |
abupd :: "(abopt \<Rightarrow> abopt) \<Rightarrow> state \<Rightarrow> state"
|
|
643 |
"abupd f \<equiv> prod_fun f id"
|
|
644 |
|
|
645 |
supd :: "(st \<Rightarrow> st) \<Rightarrow> state \<Rightarrow> state"
|
|
646 |
"supd \<equiv> prod_fun id"
|
|
647 |
|
|
648 |
lemma abupd_def2 [simp]: "abupd f (x,s) = (f x,s)"
|
|
649 |
by (simp add: abupd_def)
|
|
650 |
|
|
651 |
lemma abupd_abrupt_if_False [simp]: "\<And> s. abupd (abrupt_if False xo) s = s"
|
|
652 |
by simp
|
|
653 |
|
|
654 |
lemma supd_def2 [simp]: "supd f (x,s) = (x,f s)"
|
|
655 |
by (simp add: supd_def)
|
|
656 |
|
|
657 |
lemma supd_lupd [simp]:
|
|
658 |
"\<And> s. supd (lupd vn v ) s = (abrupt s,lupd vn v (store s))"
|
|
659 |
apply (simp (no_asm_simp) only: split_tupled_all)
|
|
660 |
apply (simp (no_asm))
|
|
661 |
done
|
|
662 |
|
|
663 |
|
|
664 |
lemma supd_gupd [simp]:
|
|
665 |
"\<And> s. supd (gupd r obj) s = (abrupt s,gupd r obj (store s))"
|
|
666 |
apply (simp (no_asm_simp) only: split_tupled_all)
|
|
667 |
apply (simp (no_asm))
|
|
668 |
done
|
|
669 |
|
|
670 |
lemma supd_init_obj [simp]:
|
|
671 |
"supd (init_obj G oi r) s = (abrupt s,init_obj G oi r (store s))"
|
|
672 |
apply (unfold init_obj_def)
|
|
673 |
apply (simp (no_asm))
|
|
674 |
done
|
|
675 |
|
|
676 |
syntax
|
|
677 |
|
|
678 |
set_lvars :: "locals \<Rightarrow> state \<Rightarrow> state"
|
|
679 |
restore_lvars :: "state \<Rightarrow> state \<Rightarrow> state"
|
|
680 |
|
|
681 |
translations
|
|
682 |
|
|
683 |
"set_lvars l" == "supd (set_locals l)"
|
|
684 |
"restore_lvars s' s" == "set_lvars (locals (store s')) s"
|
|
685 |
|
|
686 |
lemma set_set_lvars [simp]: "\<And> s. set_lvars l (set_lvars l' s) = set_lvars l s"
|
|
687 |
apply (simp (no_asm_simp) only: split_tupled_all)
|
|
688 |
apply (simp (no_asm))
|
|
689 |
done
|
|
690 |
|
|
691 |
lemma set_lvars_id [simp]: "\<And> s. set_lvars (locals (store s)) s = s"
|
|
692 |
apply (simp (no_asm_simp) only: split_tupled_all)
|
|
693 |
apply (simp (no_asm))
|
|
694 |
done
|
|
695 |
|
|
696 |
section "initialisation test"
|
|
697 |
|
|
698 |
constdefs
|
|
699 |
|
|
700 |
inited :: "qtname \<Rightarrow> globs \<Rightarrow> bool"
|
|
701 |
"inited C g \<equiv> g (Stat C) \<noteq> None"
|
|
702 |
|
|
703 |
initd :: "qtname \<Rightarrow> state \<Rightarrow> bool"
|
|
704 |
"initd C \<equiv> inited C \<circ> globs \<circ> store"
|
|
705 |
|
|
706 |
lemma not_inited_empty [simp]: "\<not>inited C empty"
|
|
707 |
apply (unfold inited_def)
|
|
708 |
apply (simp (no_asm))
|
|
709 |
done
|
|
710 |
|
|
711 |
lemma inited_gupdate [simp]: "inited C (g(r\<mapsto>obj)) = (inited C g \<or> r = Stat C)"
|
|
712 |
apply (unfold inited_def)
|
|
713 |
apply (auto split add: st.split)
|
|
714 |
done
|
|
715 |
|
|
716 |
lemma inited_init_class_obj [intro!]: "inited C (globs (init_class_obj G C s))"
|
|
717 |
apply (unfold inited_def)
|
|
718 |
apply (simp (no_asm))
|
|
719 |
done
|
|
720 |
|
|
721 |
lemma not_initedD: "\<not> inited C g \<Longrightarrow> g (Stat C) = None"
|
|
722 |
apply (unfold inited_def)
|
|
723 |
apply (erule notnotD)
|
|
724 |
done
|
|
725 |
|
|
726 |
lemma initedD: "inited C g \<Longrightarrow> \<exists> obj. g (Stat C) = Some obj"
|
|
727 |
apply (unfold inited_def)
|
|
728 |
apply auto
|
|
729 |
done
|
|
730 |
|
|
731 |
lemma initd_def2 [simp]: "initd C s = inited C (globs (store s))"
|
|
732 |
apply (unfold initd_def)
|
|
733 |
apply (simp (no_asm))
|
|
734 |
done
|
|
735 |
|
|
736 |
|
|
737 |
|
|
738 |
end
|
|
739 |
|