(* Title: HOL/MicroJava/BV/JType.thy
Author: Tobias Nipkow, Gerwin Klein
Copyright 2000 TUM
*)
section \<open>The Java Type System as Semilattice\<close>
theory JType
imports "../DFA/Semilattices" "../J/WellForm"
begin
definition super :: "'a prog \<Rightarrow> cname \<Rightarrow> cname" where
"super G C == fst (the (class G C))"
lemma superI:
"G \<turnstile> C \<prec>C1 D \<Longrightarrow> super G C = D"
by (unfold super_def) (auto dest: subcls1D)
definition is_ref :: "ty \<Rightarrow> bool" where
"is_ref T == case T of PrimT t \<Rightarrow> False | RefT r \<Rightarrow> True"
definition sup :: "'c prog \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> ty err" where
"sup G T1 T2 ==
case T1 of PrimT P1 \<Rightarrow> (case T2 of PrimT P2 \<Rightarrow>
(if P1 = P2 then OK (PrimT P1) else Err) | RefT R \<Rightarrow> Err)
| RefT R1 \<Rightarrow> (case T2 of PrimT P \<Rightarrow> Err | RefT R2 \<Rightarrow>
(case R1 of NullT \<Rightarrow> (case R2 of NullT \<Rightarrow> OK NT | ClassT C \<Rightarrow> OK (Class C))
| ClassT C \<Rightarrow> (case R2 of NullT \<Rightarrow> OK (Class C)
| ClassT D \<Rightarrow> OK (Class (exec_lub (subcls1 G) (super G) C D)))))"
definition subtype :: "'c prog \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool" where
"subtype G T1 T2 == G \<turnstile> T1 \<preceq> T2"
definition is_ty :: "'c prog \<Rightarrow> ty \<Rightarrow> bool" where
"is_ty G T == case T of PrimT P \<Rightarrow> True | RefT R \<Rightarrow>
(case R of NullT \<Rightarrow> True | ClassT C \<Rightarrow> (C, Object) \<in> (subcls1 G)^*)"
abbreviation "types G == Collect (is_type G)"
definition esl :: "'c prog \<Rightarrow> ty esl" where
"esl G == (types G, subtype G, sup G)"
lemma PrimT_PrimT: "(G \<turnstile> xb \<preceq> PrimT p) = (xb = PrimT p)"
by (auto elim: widen.cases)
lemma PrimT_PrimT2: "(G \<turnstile> PrimT p \<preceq> xb) = (xb = PrimT p)"
by (auto elim: widen.cases)
lemma is_tyI:
"\<lbrakk> is_type G T; ws_prog G \<rbrakk> \<Longrightarrow> is_ty G T"
by (auto simp add: is_ty_def intro: subcls_C_Object
split: ty.splits ref_ty.splits)
lemma is_type_conv:
"ws_prog G \<Longrightarrow> is_type G T = is_ty G T"
proof
assume "is_type G T" "ws_prog G"
thus "is_ty G T"
by (rule is_tyI)
next
assume wf: "ws_prog G" and
ty: "is_ty G T"
show "is_type G T"
proof (cases T)
case PrimT
thus ?thesis by simp
next
fix R assume R: "T = RefT R"
with wf
have "R = ClassT Object \<Longrightarrow> ?thesis" by simp
moreover
from R wf ty
have "R \<noteq> ClassT Object \<Longrightarrow> ?thesis"
by (auto simp add: is_ty_def is_class_def split_tupled_all
elim!: subcls1.cases
elim: converse_rtranclE
split: ref_ty.splits)
ultimately
show ?thesis by blast
qed
qed
lemma order_widen:
"acyclic (subcls1 G) \<Longrightarrow> order (subtype G)"
apply (unfold Semilat.order_def lesub_def subtype_def)
apply (auto intro: widen_trans)
apply (case_tac x)
apply (case_tac y)
apply (auto simp add: PrimT_PrimT)
apply (case_tac y)
apply simp
apply simp
apply (rename_tac ref_ty ref_tya, case_tac ref_ty)
apply (case_tac ref_tya)
apply simp
apply simp
apply (case_tac ref_tya)
apply simp
apply simp
apply (auto dest: acyclic_impl_antisym_rtrancl antisymD)
done
lemma wf_converse_subcls1_impl_acc_subtype:
"wf ((subcls1 G)^-1) \<Longrightarrow> acc (subtype G)"
apply (unfold Semilat.acc_def lesssub_def)
apply (drule_tac p = "((subcls1 G)^-1) - Id" in wf_subset)
apply auto
apply (drule wf_trancl)
apply (simp add: wf_eq_minimal)
apply clarify
apply (unfold lesub_def subtype_def)
apply (rename_tac M T)
apply (case_tac "EX C. Class C : M")
prefer 2
apply (case_tac T)
apply (fastforce simp add: PrimT_PrimT2)
apply simp
apply (rename_tac ref_ty)
apply (subgoal_tac "ref_ty = NullT")
apply simp
apply (rule_tac x = NT in bexI)
apply (rule allI)
apply (rule impI, erule conjE)
apply (drule widen_RefT)
apply clarsimp
apply (case_tac t)
apply simp
apply simp
apply simp
apply (case_tac ref_ty)
apply simp
apply simp
apply (erule_tac x = "{C. Class C : M}" in allE)
apply auto
apply (rename_tac D)
apply (rule_tac x = "Class D" in bexI)
prefer 2
apply assumption
apply clarify
apply (frule widen_RefT)
apply (erule exE)
apply (case_tac t)
apply simp
apply simp
apply (insert rtrancl_r_diff_Id [symmetric, of "subcls1 G"])
apply simp
apply (erule rtrancl.cases)
apply blast
apply (drule rtrancl_converseI)
apply (subgoal_tac "(subcls1 G - Id)^-1 = (subcls1 G)^-1 - Id")
prefer 2
apply (simp add: converse_Int) apply safe[1]
apply simp
apply (blast intro: rtrancl_into_trancl2)
done
lemma closed_err_types:
"\<lbrakk> ws_prog G; single_valued (subcls1 G); acyclic (subcls1 G) \<rbrakk>
\<Longrightarrow> closed (err (types G)) (lift2 (sup G))"
apply (unfold closed_def plussub_def lift2_def sup_def)
apply (auto split: err.split)
apply (drule is_tyI, assumption)
apply (auto simp add: is_ty_def is_type_conv simp del: is_type.simps
split: ty.split ref_ty.split)
apply (blast dest!: is_lub_exec_lub is_lubD is_ubD intro!: is_ubI superI)
done
lemma sup_subtype_greater:
"\<lbrakk> ws_prog G; single_valued (subcls1 G); acyclic (subcls1 G);
is_type G t1; is_type G t2; sup G t1 t2 = OK s \<rbrakk>
\<Longrightarrow> subtype G t1 s \<and> subtype G t2 s"
proof -
assume ws_prog: "ws_prog G"
assume single_valued: "single_valued (subcls1 G)"
assume acyclic: "acyclic (subcls1 G)"
{ fix c1 c2
assume is_class: "is_class G c1" "is_class G c2"
with ws_prog
obtain
"G \<turnstile> c1 \<preceq>C Object"
"G \<turnstile> c2 \<preceq>C Object"
by (blast intro: subcls_C_Object)
with ws_prog single_valued
obtain u where
"is_lub ((subcls1 G)^* ) c1 c2 u"
by (blast dest: single_valued_has_lubs)
moreover
note acyclic
moreover
have "\<forall>x y. G \<turnstile> x \<prec>C1 y \<longrightarrow> super G x = y"
by (blast intro: superI)
ultimately
have "G \<turnstile> c1 \<preceq>C exec_lub (subcls1 G) (super G) c1 c2 \<and>
G \<turnstile> c2 \<preceq>C exec_lub (subcls1 G) (super G) c1 c2"
by (simp add: exec_lub_conv) (blast dest: is_lubD is_ubD)
} note this [simp]
assume "is_type G t1" "is_type G t2" "sup G t1 t2 = OK s"
thus ?thesis
apply (unfold sup_def subtype_def)
apply (cases s)
apply (auto split: ty.split_asm ref_ty.split_asm if_split_asm)
done
qed
lemma sup_subtype_smallest:
"\<lbrakk> ws_prog G; single_valued (subcls1 G); acyclic (subcls1 G);
is_type G a; is_type G b; is_type G c;
subtype G a c; subtype G b c; sup G a b = OK d \<rbrakk>
\<Longrightarrow> subtype G d c"
proof -
assume ws_prog: "ws_prog G"
assume single_valued: "single_valued (subcls1 G)"
assume acyclic: "acyclic (subcls1 G)"
{ fix c1 c2 D
assume is_class: "is_class G c1" "is_class G c2"
assume le: "G \<turnstile> c1 \<preceq>C D" "G \<turnstile> c2 \<preceq>C D"
from ws_prog is_class
obtain
"G \<turnstile> c1 \<preceq>C Object"
"G \<turnstile> c2 \<preceq>C Object"
by (blast intro: subcls_C_Object)
with ws_prog single_valued
obtain u where
lub: "is_lub ((subcls1 G)^*) c1 c2 u"
by (blast dest: single_valued_has_lubs)
with acyclic
have "exec_lub (subcls1 G) (super G) c1 c2 = u"
by (blast intro: superI exec_lub_conv)
moreover
from lub le
have "G \<turnstile> u \<preceq>C D"
by (simp add: is_lub_def is_ub_def)
ultimately
have "G \<turnstile> exec_lub (subcls1 G) (super G) c1 c2 \<preceq>C D"
by blast
} note this [intro]
have [dest!]:
"\<And>C T. G \<turnstile> Class C \<preceq> T \<Longrightarrow> \<exists>D. T=Class D \<and> G \<turnstile> C \<preceq>C D"
by (frule widen_Class, auto)
assume "is_type G a" "is_type G b" "is_type G c"
"subtype G a c" "subtype G b c" "sup G a b = OK d"
thus ?thesis
by (auto simp add: subtype_def sup_def
split: ty.split_asm ref_ty.split_asm if_split_asm)
qed
lemma sup_exists:
"\<lbrakk> subtype G a c; subtype G b c; sup G a b = Err \<rbrakk> \<Longrightarrow> False"
by (auto simp add: PrimT_PrimT PrimT_PrimT2 sup_def subtype_def
split: ty.splits ref_ty.splits)
lemma err_semilat_JType_esl_lemma:
"\<lbrakk> ws_prog G; single_valued (subcls1 G); acyclic (subcls1 G) \<rbrakk>
\<Longrightarrow> err_semilat (esl G)"
proof -
assume ws_prog: "ws_prog G"
assume single_valued: "single_valued (subcls1 G)"
assume acyclic: "acyclic (subcls1 G)"
hence "order (subtype G)"
by (rule order_widen)
moreover
from ws_prog single_valued acyclic
have "closed (err (types G)) (lift2 (sup G))"
by (rule closed_err_types)
moreover
from ws_prog single_valued acyclic
have
"(\<forall>x\<in>err (types G). \<forall>y\<in>err (types G). x <=_(Err.le (subtype G)) x +_(lift2 (sup G)) y) \<and>
(\<forall>x\<in>err (types G). \<forall>y\<in>err (types G). y <=_(Err.le (subtype G)) x +_(lift2 (sup G)) y)"
by (auto simp add: lesub_def plussub_def Err.le_def lift2_def sup_subtype_greater split: err.split)
moreover
from ws_prog single_valued acyclic
have
"\<forall>x\<in>err (types G). \<forall>y\<in>err (types G). \<forall>z\<in>err (types G).
x <=_(Err.le (subtype G)) z \<and> y <=_(Err.le (subtype G)) z \<longrightarrow> x +_(lift2 (sup G)) y <=_(Err.le (subtype G)) z"
by (unfold lift2_def plussub_def lesub_def Err.le_def)
(auto intro: sup_subtype_smallest sup_exists split: err.split)
ultimately
show ?thesis
by (unfold esl_def semilat_def Err.sl_def) auto
qed
lemma single_valued_subcls1:
"ws_prog G \<Longrightarrow> single_valued (subcls1 G)"
by (auto simp add: ws_prog_def unique_def single_valued_def
intro: subcls1I elim!: subcls1.cases)
theorem err_semilat_JType_esl:
"ws_prog G \<Longrightarrow> err_semilat (esl G)"
by (frule acyclic_subcls1, frule single_valued_subcls1, rule err_semilat_JType_esl_lemma)
end