(* Title: HOL/Bali/State.thy
Author: David von Oheimb
*)
subsection \<open>State for evaluation of Java expressions and statements\<close>
theory State
imports DeclConcepts
begin
text \<open>
design issues:
\begin{itemize}
\item all kinds of objects (class instances, arrays, and class objects)
are handeled via a general object abstraction
\item the heap and the map for class objects are combined into a single table
\<open>(recall (loc, obj) table \<times> (qtname, obj) table ~= (loc + qtname, obj) table)\<close>
\end{itemize}
\<close>
subsubsection "objects"
datatype obj_tag = \<comment>\<open>tag for generic object\<close>
CInst qtname \<comment>\<open>class instance\<close>
| Arr ty int \<comment>\<open>array with component type and length\<close>
\<comment>\<open>| CStat qtname the tag is irrelevant for a class object,
i.e. the static fields of a class,
since its type is given already by the reference to
it (see below)\<close>
type_synonym vn = "fspec + int" \<comment>\<open>variable name\<close>
record obj =
tag :: "obj_tag" \<comment>\<open>generalized object\<close>
"values" :: "(vn, val) table"
translations
(type) "fspec" <= (type) "vname \<times> qtname"
(type) "vn" <= (type) "fspec + int"
(type) "obj" <= (type) "\<lparr>tag::obj_tag, values::vn \<Rightarrow> val option\<rparr>"
(type) "obj" <= (type) "\<lparr>tag::obj_tag, values::vn \<Rightarrow> val option,\<dots>::'a\<rparr>"
definition
the_Arr :: "obj option \<Rightarrow> ty \<times> int \<times> (vn, val) table"
where "the_Arr obj = (SOME (T,k,t). obj = Some \<lparr>tag=Arr T k,values=t\<rparr>)"
lemma the_Arr_Arr [simp]: "the_Arr (Some \<lparr>tag=Arr T k,values=cs\<rparr>) = (T,k,cs)"
apply (auto simp: the_Arr_def)
done
lemma the_Arr_Arr1 [simp,intro,dest]:
"\<lbrakk>tag obj = Arr T k\<rbrakk> \<Longrightarrow> the_Arr (Some obj) = (T,k,values obj)"
apply (auto simp add: the_Arr_def)
done
definition
upd_obj :: "vn \<Rightarrow> val \<Rightarrow> obj \<Rightarrow> obj"
where "upd_obj n v = (\<lambda>obj. obj \<lparr>values:=(values obj)(n\<mapsto>v)\<rparr>)"
lemma upd_obj_def2 [simp]:
"upd_obj n v obj = obj \<lparr>values:=(values obj)(n\<mapsto>v)\<rparr>"
apply (auto simp: upd_obj_def)
done
definition
obj_ty :: "obj \<Rightarrow> ty" where
"obj_ty obj = (case tag obj of
CInst C \<Rightarrow> Class C
| Arr T k \<Rightarrow> T.[])"
lemma obj_ty_eq [intro!]: "obj_ty \<lparr>tag=oi,values=x\<rparr> = obj_ty \<lparr>tag=oi,values=y\<rparr>"
by (simp add: obj_ty_def)
lemma obj_ty_eq1 [intro!,dest]:
"tag obj = tag obj' \<Longrightarrow> obj_ty obj = obj_ty obj'"
by (simp add: obj_ty_def)
lemma obj_ty_cong [simp]:
"obj_ty (obj \<lparr>values:=vs\<rparr>) = obj_ty obj"
by auto
lemma obj_ty_CInst [simp]:
"obj_ty \<lparr>tag=CInst C,values=vs\<rparr> = Class C"
by (simp add: obj_ty_def)
lemma obj_ty_CInst1 [simp,intro!,dest]:
"\<lbrakk>tag obj = CInst C\<rbrakk> \<Longrightarrow> obj_ty obj = Class C"
by (simp add: obj_ty_def)
lemma obj_ty_Arr [simp]:
"obj_ty \<lparr>tag=Arr T i,values=vs\<rparr> = T.[]"
by (simp add: obj_ty_def)
lemma obj_ty_Arr1 [simp,intro!,dest]:
"\<lbrakk>tag obj = Arr T i\<rbrakk> \<Longrightarrow> obj_ty obj = T.[]"
by (simp add: obj_ty_def)
lemma obj_ty_widenD:
"G\<turnstile>obj_ty obj\<preceq>RefT t \<Longrightarrow> (\<exists>C. tag obj = CInst C) \<or> (\<exists>T k. tag obj = Arr T k)"
apply (unfold obj_ty_def)
apply (auto split: obj_tag.split_asm)
done
definition
obj_class :: "obj \<Rightarrow> qtname" where
"obj_class obj = (case tag obj of
CInst C \<Rightarrow> C
| Arr T k \<Rightarrow> Object)"
lemma obj_class_CInst [simp]: "obj_class \<lparr>tag=CInst C,values=vs\<rparr> = C"
by (auto simp: obj_class_def)
lemma obj_class_CInst1 [simp,intro!,dest]:
"tag obj = CInst C \<Longrightarrow> obj_class obj = C"
by (auto simp: obj_class_def)
lemma obj_class_Arr [simp]: "obj_class \<lparr>tag=Arr T k,values=vs\<rparr> = Object"
by (auto simp: obj_class_def)
lemma obj_class_Arr1 [simp,intro!,dest]:
"tag obj = Arr T k \<Longrightarrow> obj_class obj = Object"
by (auto simp: obj_class_def)
lemma obj_ty_obj_class: "G\<turnstile>obj_ty obj\<preceq> Class statC = G\<turnstile>obj_class obj \<preceq>\<^sub>C statC"
apply (case_tac "tag obj")
apply (auto simp add: obj_ty_def obj_class_def)
apply (case_tac "statC = Object")
apply (auto dest: widen_Array_Class)
done
subsubsection "object references"
type_synonym oref = "loc + qtname" \<comment>\<open>generalized object reference\<close>
syntax
Heap :: "loc \<Rightarrow> oref"
Stat :: "qtname \<Rightarrow> oref"
translations
"Heap" => "CONST Inl"
"Stat" => "CONST Inr"
(type) "oref" <= (type) "loc + qtname"
definition
fields_table :: "prog \<Rightarrow> qtname \<Rightarrow> (fspec \<Rightarrow> field \<Rightarrow> bool) \<Rightarrow> (fspec, ty) table" where
"fields_table G C P =
map_option type \<circ> table_of (filter (case_prod P) (DeclConcepts.fields G C))"
lemma fields_table_SomeI:
"\<lbrakk>table_of (DeclConcepts.fields G C) n = Some f; P n f\<rbrakk>
\<Longrightarrow> fields_table G C P n = Some (type f)"
apply (unfold fields_table_def)
apply clarsimp
apply (rule exI)
apply (rule conjI)
apply (erule map_of_filter_in)
apply assumption
apply simp
done
(* unused *)
lemma fields_table_SomeD': "fields_table G C P fn = Some T \<Longrightarrow>
\<exists>f. (fn,f)\<in>set(DeclConcepts.fields G C) \<and> type f = T"
apply (unfold fields_table_def)
apply clarsimp
apply (drule map_of_SomeD)
apply auto
done
lemma fields_table_SomeD:
"\<lbrakk>fields_table G C P fn = Some T; unique (DeclConcepts.fields G C)\<rbrakk> \<Longrightarrow>
\<exists>f. table_of (DeclConcepts.fields G C) fn = Some f \<and> type f = T"
apply (unfold fields_table_def)
apply clarsimp
apply (rule exI)
apply (rule conjI)
apply (erule table_of_filter_unique_SomeD)
apply assumption
apply simp
done
definition
in_bounds :: "int \<Rightarrow> int \<Rightarrow> bool" ("(_/ in'_bounds _)" [50, 51] 50)
where "i in_bounds k = (0 \<le> i \<and> i < k)"
definition
arr_comps :: "'a \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a option"
where "arr_comps T k = (\<lambda>i. if i in_bounds k then Some T else None)"
definition
var_tys :: "prog \<Rightarrow> obj_tag \<Rightarrow> oref \<Rightarrow> (vn, ty) table" where
"var_tys G oi r =
(case r of
Heap a \<Rightarrow> (case oi of
CInst C \<Rightarrow> fields_table G C (\<lambda>n f. \<not>static f) (+) empty
| Arr T k \<Rightarrow> empty (+) arr_comps T k)
| Stat C \<Rightarrow> fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f)
(+) empty)"
lemma var_tys_Some_eq:
"var_tys G oi r n = Some T
= (case r of
Inl a \<Rightarrow> (case oi of
CInst C \<Rightarrow> (\<exists>nt. n = Inl nt \<and> fields_table G C (\<lambda>n f.
\<not>static f) nt = Some T)
| Arr t k \<Rightarrow> (\<exists> i. n = Inr i \<and> i in_bounds k \<and> t = T))
| Inr C \<Rightarrow> (\<exists>nt. n = Inl nt \<and>
fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f) nt
= Some T))"
apply (unfold var_tys_def arr_comps_def)
apply (force split: sum.split_asm sum.split obj_tag.split)
done
subsubsection "stores"
type_synonym globs \<comment>\<open>global variables: heap and static variables\<close>
= "(oref , obj) table"
type_synonym heap
= "(loc , obj) table"
(* type_synonym locals
= "(lname, val) table" *) (* defined in Value.thy local variables *)
translations
(type) "globs" <= (type) "(oref , obj) table"
(type) "heap" <= (type) "(loc , obj) table"
(* (type) "locals" <= (type) "(lname, val) table" *)
datatype st = (* pure state, i.e. contents of all variables *)
st globs locals
subsection "access"
definition
globs :: "st \<Rightarrow> globs"
where "globs = case_st (\<lambda>g l. g)"
definition
locals :: "st \<Rightarrow> locals"
where "locals = case_st (\<lambda>g l. l)"
definition heap :: "st \<Rightarrow> heap" where
"heap s = globs s \<circ> Heap"
lemma globs_def2 [simp]: " globs (st g l) = g"
by (simp add: globs_def)
lemma locals_def2 [simp]: "locals (st g l) = l"
by (simp add: locals_def)
lemma heap_def2 [simp]: "heap s a=globs s (Heap a)"
by (simp add: heap_def)
abbreviation val_this :: "st \<Rightarrow> val"
where "val_this s == the (locals s This)"
abbreviation lookup_obj :: "st \<Rightarrow> val \<Rightarrow> obj"
where "lookup_obj s a' == the (heap s (the_Addr a'))"
subsection "memory allocation"
definition
new_Addr :: "heap \<Rightarrow> loc option" where
"new_Addr h = (if (\<forall>a. h a \<noteq> None) then None else Some (SOME a. h a = None))"
lemma new_AddrD: "new_Addr h = Some a \<Longrightarrow> h a = None"
apply (auto simp add: new_Addr_def)
apply (erule someI)
done
lemma new_AddrD2: "new_Addr h = Some a \<Longrightarrow> \<forall>b. h b \<noteq> None \<longrightarrow> b \<noteq> a"
apply (drule new_AddrD)
apply auto
done
lemma new_Addr_SomeI: "h a = None \<Longrightarrow> \<exists>b. new_Addr h = Some b \<and> h b = None"
apply (simp add: new_Addr_def)
apply (fast intro: someI2)
done
subsection "initialization"
abbreviation init_vals :: "('a, ty) table \<Rightarrow> ('a, val) table"
where "init_vals vs == map_option default_val \<circ> vs"
lemma init_arr_comps_base [simp]: "init_vals (arr_comps T 0) = empty"
apply (unfold arr_comps_def in_bounds_def)
apply (rule ext)
apply auto
done
lemma init_arr_comps_step [simp]:
"0 < j \<Longrightarrow> init_vals (arr_comps T j ) =
init_vals (arr_comps T (j - 1))(j - 1\<mapsto>default_val T)"
apply (unfold arr_comps_def in_bounds_def)
apply (rule ext)
apply auto
done
subsection "update"
definition
gupd :: "oref \<Rightarrow> obj \<Rightarrow> st \<Rightarrow> st" ("gupd'(_\<mapsto>_')" [10, 10] 1000)
where "gupd r obj = case_st (\<lambda>g l. st (g(r\<mapsto>obj)) l)"
definition
lupd :: "lname \<Rightarrow> val \<Rightarrow> st \<Rightarrow> st" ("lupd'(_\<mapsto>_')" [10, 10] 1000)
where "lupd vn v = case_st (\<lambda>g l. st g (l(vn\<mapsto>v)))"
definition
upd_gobj :: "oref \<Rightarrow> vn \<Rightarrow> val \<Rightarrow> st \<Rightarrow> st"
where "upd_gobj r n v = case_st (\<lambda>g l. st (chg_map (upd_obj n v) r g) l)"
definition
set_locals :: "locals \<Rightarrow> st \<Rightarrow> st"
where "set_locals l = case_st (\<lambda>g l'. st g l)"
definition
init_obj :: "prog \<Rightarrow> obj_tag \<Rightarrow> oref \<Rightarrow> st \<Rightarrow> st"
where "init_obj G oi r = gupd(r\<mapsto>\<lparr>tag=oi, values=init_vals (var_tys G oi r)\<rparr>)"
abbreviation
init_class_obj :: "prog \<Rightarrow> qtname \<Rightarrow> st \<Rightarrow> st"
where "init_class_obj G C == init_obj G undefined (Inr C)"
lemma gupd_def2 [simp]: "gupd(r\<mapsto>obj) (st g l) = st (g(r\<mapsto>obj)) l"
apply (unfold gupd_def)
apply (simp (no_asm))
done
lemma lupd_def2 [simp]: "lupd(vn\<mapsto>v) (st g l) = st g (l(vn\<mapsto>v))"
apply (unfold lupd_def)
apply (simp (no_asm))
done
lemma globs_gupd [simp]: "globs (gupd(r\<mapsto>obj) s) = globs s(r\<mapsto>obj)"
apply (induct "s")
by (simp add: gupd_def)
lemma globs_lupd [simp]: "globs (lupd(vn\<mapsto>v ) s) = globs s"
apply (induct "s")
by (simp add: lupd_def)
lemma locals_gupd [simp]: "locals (gupd(r\<mapsto>obj) s) = locals s"
apply (induct "s")
by (simp add: gupd_def)
lemma locals_lupd [simp]: "locals (lupd(vn\<mapsto>v ) s) = locals s(vn\<mapsto>v )"
apply (induct "s")
by (simp add: lupd_def)
lemma globs_upd_gobj_new [rule_format (no_asm), simp]:
"globs s r = None \<longrightarrow> globs (upd_gobj r n v s) = globs s"
apply (unfold upd_gobj_def)
apply (induct "s")
apply auto
done
lemma globs_upd_gobj_upd [rule_format (no_asm), simp]:
"globs s r=Some obj\<longrightarrow> globs (upd_gobj r n v s) = globs s(r\<mapsto>upd_obj n v obj)"
apply (unfold upd_gobj_def)
apply (induct "s")
apply auto
done
lemma locals_upd_gobj [simp]: "locals (upd_gobj r n v s) = locals s"
apply (induct "s")
by (simp add: upd_gobj_def)
lemma globs_init_obj [simp]: "globs (init_obj G oi r s) t =
(if t=r then Some \<lparr>tag=oi,values=init_vals (var_tys G oi r)\<rparr> else globs s t)"
apply (unfold init_obj_def)
apply (simp (no_asm))
done
lemma locals_init_obj [simp]: "locals (init_obj G oi r s) = locals s"
by (simp add: init_obj_def)
lemma surjective_st [simp]: "st (globs s) (locals s) = s"
apply (induct "s")
by auto
lemma surjective_st_init_obj:
"st (globs (init_obj G oi r s)) (locals s) = init_obj G oi r s"
apply (subst locals_init_obj [THEN sym])
apply (rule surjective_st)
done
lemma heap_heap_upd [simp]:
"heap (st (g(Inl a\<mapsto>obj)) l) = heap (st g l)(a\<mapsto>obj)"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_stat_upd [simp]: "heap (st (g(Inr C\<mapsto>obj)) l) = heap (st g l)"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_local_upd [simp]: "heap (st g (l(vn\<mapsto>v))) = heap (st g l)"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_gupd_Heap [simp]: "heap (gupd(Heap a\<mapsto>obj) s) = heap s(a\<mapsto>obj)"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_gupd_Stat [simp]: "heap (gupd(Stat C\<mapsto>obj) s) = heap s"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_lupd [simp]: "heap (lupd(vn\<mapsto>v) s) = heap s"
apply (rule ext)
apply (simp (no_asm))
done
lemma heap_upd_gobj_Stat [simp]: "heap (upd_gobj (Stat C) n v s) = heap s"
apply (rule ext)
apply (simp (no_asm))
apply (case_tac "globs s (Stat C)")
apply auto
done
lemma set_locals_def2 [simp]: "set_locals l (st g l') = st g l"
apply (unfold set_locals_def)
apply (simp (no_asm))
done
lemma set_locals_id [simp]: "set_locals (locals s) s = s"
apply (unfold set_locals_def)
apply (induct_tac "s")
apply (simp (no_asm))
done
lemma set_set_locals [simp]: "set_locals l (set_locals l' s) = set_locals l s"
apply (unfold set_locals_def)
apply (induct_tac "s")
apply (simp (no_asm))
done
lemma locals_set_locals [simp]: "locals (set_locals l s) = l"
apply (unfold set_locals_def)
apply (induct_tac "s")
apply (simp (no_asm))
done
lemma globs_set_locals [simp]: "globs (set_locals l s) = globs s"
apply (unfold set_locals_def)
apply (induct_tac "s")
apply (simp (no_asm))
done
lemma heap_set_locals [simp]: "heap (set_locals l s) = heap s"
apply (unfold heap_def)
apply (induct_tac "s")
apply (simp (no_asm))
done
subsubsection "abrupt completion"
primrec the_Xcpt :: "abrupt \<Rightarrow> xcpt"
where "the_Xcpt (Xcpt x) = x"
primrec the_Jump :: "abrupt => jump"
where "the_Jump (Jump j) = j"
primrec the_Loc :: "xcpt \<Rightarrow> loc"
where "the_Loc (Loc a) = a"
primrec the_Std :: "xcpt \<Rightarrow> xname"
where "the_Std (Std x) = x"
definition
abrupt_if :: "bool \<Rightarrow> abopt \<Rightarrow> abopt \<Rightarrow> abopt"
where "abrupt_if c x' x = (if c \<and> (x = None) then x' else x)"
lemma abrupt_if_True_None [simp]: "abrupt_if True x None = x"
by (simp add: abrupt_if_def)
lemma abrupt_if_True_not_None [simp]: "x \<noteq> None \<Longrightarrow> abrupt_if True x y \<noteq> None"
by (simp add: abrupt_if_def)
lemma abrupt_if_False [simp]: "abrupt_if False x y = y"
by (simp add: abrupt_if_def)
lemma abrupt_if_Some [simp]: "abrupt_if c x (Some y) = Some y"
by (simp add: abrupt_if_def)
lemma abrupt_if_not_None [simp]: "y \<noteq> None \<Longrightarrow> abrupt_if c x y = y"
apply (simp add: abrupt_if_def)
by auto
lemma split_abrupt_if:
"P (abrupt_if c x' x) =
((c \<and> x = None \<longrightarrow> P x') \<and> (\<not> (c \<and> x = None) \<longrightarrow> P x))"
apply (unfold abrupt_if_def)
apply (split if_split)
apply auto
done
abbreviation raise_if :: "bool \<Rightarrow> xname \<Rightarrow> abopt \<Rightarrow> abopt"
where "raise_if c xn == abrupt_if c (Some (Xcpt (Std xn)))"
abbreviation np :: "val \<Rightarrow> abopt \<Rightarrow> abopt"
where "np v == raise_if (v = Null) NullPointer"
abbreviation check_neg :: "val \<Rightarrow> abopt \<Rightarrow> abopt"
where "check_neg i' == raise_if (the_Intg i'<0) NegArrSize"
abbreviation error_if :: "bool \<Rightarrow> error \<Rightarrow> abopt \<Rightarrow> abopt"
where "error_if c e == abrupt_if c (Some (Error e))"
lemma raise_if_None [simp]: "(raise_if c x y = None) = (\<not>c \<and> y = None)"
apply (simp add: abrupt_if_def)
by auto
declare raise_if_None [THEN iffD1, dest!]
lemma if_raise_if_None [simp]:
"((if b then y else raise_if c x y) = None) = ((c \<longrightarrow> b) \<and> y = None)"
apply (simp add: abrupt_if_def)
apply auto
done
lemma raise_if_SomeD [dest!]:
"raise_if c x y = Some z \<Longrightarrow> c \<and> z=(Xcpt (Std x)) \<and> y=None \<or> (y=Some z)"
apply (case_tac y)
apply (case_tac c)
apply (simp add: abrupt_if_def)
apply (simp add: abrupt_if_def)
apply auto
done
lemma error_if_None [simp]: "(error_if c e y = None) = (\<not>c \<and> y = None)"
apply (simp add: abrupt_if_def)
by auto
declare error_if_None [THEN iffD1, dest!]
lemma if_error_if_None [simp]:
"((if b then y else error_if c e y) = None) = ((c \<longrightarrow> b) \<and> y = None)"
apply (simp add: abrupt_if_def)
apply auto
done
lemma error_if_SomeD [dest!]:
"error_if c e y = Some z \<Longrightarrow> c \<and> z=(Error e) \<and> y=None \<or> (y=Some z)"
apply (case_tac y)
apply (case_tac c)
apply (simp add: abrupt_if_def)
apply (simp add: abrupt_if_def)
apply auto
done
definition
absorb :: "jump \<Rightarrow> abopt \<Rightarrow> abopt"
where "absorb j a = (if a=Some (Jump j) then None else a)"
lemma absorb_SomeD [dest!]: "absorb j a = Some x \<Longrightarrow> a = Some x"
by (auto simp add: absorb_def)
lemma absorb_same [simp]: "absorb j (Some (Jump j)) = None"
by (auto simp add: absorb_def)
lemma absorb_other [simp]: "a \<noteq> Some (Jump j) \<Longrightarrow> absorb j a = a"
by (auto simp add: absorb_def)
lemma absorb_Some_NoneD: "absorb j (Some abr) = None \<Longrightarrow> abr = Jump j"
by (simp add: absorb_def)
lemma absorb_Some_JumpD: "absorb j s = Some (Jump j') \<Longrightarrow> j'\<noteq>j"
by (simp add: absorb_def)
subsubsection "full program state"
type_synonym
state = "abopt \<times> st" \<comment>\<open>state including abruption information\<close>
translations
(type) "abopt" <= (type) "abrupt option"
(type) "state" <= (type) "abopt \<times> st"
abbreviation
Norm :: "st \<Rightarrow> state"
where "Norm s == (None, s)"
abbreviation (input)
abrupt :: "state \<Rightarrow> abopt"
where "abrupt == fst"
abbreviation (input)
store :: "state \<Rightarrow> st"
where "store == snd"
lemma single_stateE: "\<forall>Z. Z = (s::state) \<Longrightarrow> False"
apply (erule_tac x = "(Some k,y)" for k y in all_dupE)
apply (erule_tac x = "(None,y)" for y in allE)
apply clarify
done
lemma state_not_single: "All ((=) (x::state)) \<Longrightarrow> R"
apply (drule_tac x = "(if abrupt x = None then Some x' else None, y)" for x' y in spec)
apply clarsimp
done
definition
normal :: "state \<Rightarrow> bool"
where "normal = (\<lambda>s. abrupt s = None)"
lemma normal_def2 [simp]: "normal s = (abrupt s = None)"
apply (unfold normal_def)
apply (simp (no_asm))
done
definition
heap_free :: "nat \<Rightarrow> state \<Rightarrow> bool"
where "heap_free n = (\<lambda>s. atleast_free (heap (store s)) n)"
lemma heap_free_def2 [simp]: "heap_free n s = atleast_free (heap (store s)) n"
apply (unfold heap_free_def)
apply simp
done
subsection "update"
definition
abupd :: "(abopt \<Rightarrow> abopt) \<Rightarrow> state \<Rightarrow> state"
where "abupd f = map_prod f id"
definition
supd :: "(st \<Rightarrow> st) \<Rightarrow> state \<Rightarrow> state"
where "supd = map_prod id"
lemma abupd_def2 [simp]: "abupd f (x,s) = (f x,s)"
by (simp add: abupd_def)
lemma abupd_abrupt_if_False [simp]: "\<And> s. abupd (abrupt_if False xo) s = s"
by simp
lemma supd_def2 [simp]: "supd f (x,s) = (x,f s)"
by (simp add: supd_def)
lemma supd_lupd [simp]:
"\<And> s. supd (lupd vn v ) s = (abrupt s,lupd vn v (store s))"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp (no_asm))
done
lemma supd_gupd [simp]:
"\<And> s. supd (gupd r obj) s = (abrupt s,gupd r obj (store s))"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp (no_asm))
done
lemma supd_init_obj [simp]:
"supd (init_obj G oi r) s = (abrupt s,init_obj G oi r (store s))"
apply (unfold init_obj_def)
apply (simp (no_asm))
done
lemma abupd_store_invariant [simp]: "store (abupd f s) = store s"
by (cases s) simp
lemma supd_abrupt_invariant [simp]: "abrupt (supd f s) = abrupt s"
by (cases s) simp
abbreviation set_lvars :: "locals \<Rightarrow> state \<Rightarrow> state"
where "set_lvars l == supd (set_locals l)"
abbreviation restore_lvars :: "state \<Rightarrow> state \<Rightarrow> state"
where "restore_lvars s' s == set_lvars (locals (store s')) s"
lemma set_set_lvars [simp]: "\<And> s. set_lvars l (set_lvars l' s) = set_lvars l s"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp (no_asm))
done
lemma set_lvars_id [simp]: "\<And> s. set_lvars (locals (store s)) s = s"
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp (no_asm))
done
subsubsection "initialisation test"
definition
inited :: "qtname \<Rightarrow> globs \<Rightarrow> bool"
where "inited C g = (g (Stat C) \<noteq> None)"
definition
initd :: "qtname \<Rightarrow> state \<Rightarrow> bool"
where "initd C = inited C \<circ> globs \<circ> store"
lemma not_inited_empty [simp]: "\<not>inited C empty"
apply (unfold inited_def)
apply (simp (no_asm))
done
lemma inited_gupdate [simp]: "inited C (g(r\<mapsto>obj)) = (inited C g \<or> r = Stat C)"
apply (unfold inited_def)
apply (auto split: st.split)
done
lemma inited_init_class_obj [intro!]: "inited C (globs (init_class_obj G C s))"
apply (unfold inited_def)
apply (simp (no_asm))
done
lemma not_initedD: "\<not> inited C g \<Longrightarrow> g (Stat C) = None"
apply (unfold inited_def)
apply (erule notnotD)
done
lemma initedD: "inited C g \<Longrightarrow> \<exists> obj. g (Stat C) = Some obj"
apply (unfold inited_def)
apply auto
done
lemma initd_def2 [simp]: "initd C s = inited C (globs (store s))"
apply (unfold initd_def)
apply (simp (no_asm))
done
subsubsection \<open>\<open>error_free\<close>\<close>
definition
error_free :: "state \<Rightarrow> bool"
where "error_free s = (\<not> (\<exists> err. abrupt s = Some (Error err)))"
lemma error_free_Norm [simp,intro]: "error_free (Norm s)"
by (simp add: error_free_def)
lemma error_free_normal [simp,intro]: "normal s \<Longrightarrow> error_free s"
by (simp add: error_free_def)
lemma error_free_Xcpt [simp]: "error_free (Some (Xcpt x),s)"
by (simp add: error_free_def)
lemma error_free_Jump [simp,intro]: "error_free (Some (Jump j),s)"
by (simp add: error_free_def)
lemma error_free_Error [simp]: "error_free (Some (Error e),s) = False"
by (simp add: error_free_def)
lemma error_free_Some [simp,intro]:
"\<not> (\<exists> err. x=Error err) \<Longrightarrow> error_free ((Some x),s)"
by (auto simp add: error_free_def)
lemma error_free_abupd_absorb [simp,intro]:
"error_free s \<Longrightarrow> error_free (abupd (absorb j) s)"
by (cases s)
(auto simp add: error_free_def absorb_def
split: if_split_asm)
lemma error_free_absorb [simp,intro]:
"error_free (a,s) \<Longrightarrow> error_free (absorb j a, s)"
by (auto simp add: error_free_def absorb_def
split: if_split_asm)
lemma error_free_abrupt_if [simp,intro]:
"\<lbrakk>error_free s; \<not> (\<exists> err. x=Error err)\<rbrakk>
\<Longrightarrow> error_free (abupd (abrupt_if p (Some x)) s)"
by (cases s)
(auto simp add: abrupt_if_def
split: if_split)
lemma error_free_abrupt_if1 [simp,intro]:
"\<lbrakk>error_free (a,s); \<not> (\<exists> err. x=Error err)\<rbrakk>
\<Longrightarrow> error_free (abrupt_if p (Some x) a, s)"
by (auto simp add: abrupt_if_def
split: if_split)
lemma error_free_abrupt_if_Xcpt [simp,intro]:
"error_free s
\<Longrightarrow> error_free (abupd (abrupt_if p (Some (Xcpt x))) s)"
by simp
lemma error_free_abrupt_if_Xcpt1 [simp,intro]:
"error_free (a,s)
\<Longrightarrow> error_free (abrupt_if p (Some (Xcpt x)) a, s)"
by simp
lemma error_free_abrupt_if_Jump [simp,intro]:
"error_free s
\<Longrightarrow> error_free (abupd (abrupt_if p (Some (Jump j))) s)"
by simp
lemma error_free_abrupt_if_Jump1 [simp,intro]:
"error_free (a,s)
\<Longrightarrow> error_free (abrupt_if p (Some (Jump j)) a, s)"
by simp
lemma error_free_raise_if [simp,intro]:
"error_free s \<Longrightarrow> error_free (abupd (raise_if p x) s)"
by simp
lemma error_free_raise_if1 [simp,intro]:
"error_free (a,s) \<Longrightarrow> error_free ((raise_if p x a), s)"
by simp
lemma error_free_supd [simp,intro]:
"error_free s \<Longrightarrow> error_free (supd f s)"
by (cases s) (simp add: error_free_def)
lemma error_free_supd1 [simp,intro]:
"error_free (a,s) \<Longrightarrow> error_free (a,f s)"
by (simp add: error_free_def)
lemma error_free_set_lvars [simp,intro]:
"error_free s \<Longrightarrow> error_free ((set_lvars l) s)"
by (cases s) simp
lemma error_free_set_locals [simp,intro]:
"error_free (x, s)
\<Longrightarrow> error_free (x, set_locals l s')"
by (simp add: error_free_def)
end