(* Title: HOL/MicroJava/J/TypeRel.thy
ID: $Id$
Author: David von Oheimb
Copyright 1999 Technische Universitaet Muenchen
*)
header "Relations between Java Types"
theory TypeRel = Decl:
consts
subcls1 :: "'c prog => (cname \<times> cname) set" (* subclass *)
widen :: "'c prog => (ty \<times> ty ) set" (* widening *)
cast :: "'c prog => (cname \<times> cname) set" (* casting *)
syntax
subcls1 :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70)
subcls :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70)
widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70)
cast :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70)
syntax (HTML)
subcls1 :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C1 _" [71,71,71] 70)
subcls :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C _" [71,71,71] 70)
widen :: "'c prog => [ty , ty ] => bool" ("_ |- _ <= _" [71,71,71] 70)
cast :: "'c prog => [cname, cname] => bool" ("_ |- _ <=? _" [71,71,71] 70)
translations
"G\<turnstile>C \<prec>C1 D" == "(C,D) \<in> subcls1 G"
"G\<turnstile>C \<preceq>C D" == "(C,D) \<in> (subcls1 G)^*"
"G\<turnstile>S \<preceq> T" == "(S,T) \<in> widen G"
"G\<turnstile>C \<preceq>? D" == "(C,D) \<in> cast G"
defs
(* direct subclass, cf. 8.1.3 *)
subcls1_def: "subcls1 G \<equiv> {(C,D). C\<noteq>Object \<and> (\<exists>c. class G C=Some c \<and> fst c=D)}"
lemma subcls1D:
"G\<turnstile>C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>fs ms. class G C = Some (D,fs,ms))"
apply (unfold subcls1_def)
apply auto
done
lemma subcls1I: "\<lbrakk>class G C = Some (D,rest); C \<noteq> Object\<rbrakk> \<Longrightarrow> G\<turnstile>C\<prec>C1D"
apply (unfold subcls1_def)
apply auto
done
lemma subcls1_def2:
"subcls1 G = (\<Sigma>C\<in>{C. is_class G C} . {D. C\<noteq>Object \<and> fst (the (class G C))=D})"
apply (unfold subcls1_def is_class_def)
apply auto
done
lemma finite_subcls1: "finite (subcls1 G)"
apply(subst subcls1_def2)
apply(rule finite_SigmaI [OF finite_is_class])
apply(rule_tac B = "{fst (the (class G C))}" in finite_subset)
apply auto
done
lemma subcls_is_class: "(C,D) \<in> (subcls1 G)^+ ==> is_class G C"
apply (unfold is_class_def)
apply(erule trancl_trans_induct)
apply (auto dest!: subcls1D)
done
lemma subcls_is_class2 [rule_format (no_asm)]:
"G\<turnstile>C\<preceq>C D \<Longrightarrow> is_class G D \<longrightarrow> is_class G C"
apply (unfold is_class_def)
apply (erule rtrancl_induct)
apply (drule_tac [2] subcls1D)
apply auto
done
declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
consts class_rec ::"'c prog \<times> cname \<Rightarrow>
'a \<Rightarrow> (cname \<Rightarrow> fdecl list \<Rightarrow> 'c mdecl list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a"
recdef class_rec "same_fst (\<lambda>G. wf ((subcls1 G)^-1)) (\<lambda>G. (subcls1 G)^-1)"
"class_rec (G,C) = (\<lambda>t f. case class G C of None \<Rightarrow> arbitrary
| Some (D,fs,ms) \<Rightarrow> if wf ((subcls1 G)^-1) then
f C fs ms (if C = Object then t else class_rec (G,D) t f) else arbitrary)"
(hints intro: subcls1I)
declare class_rec.simps [simp del]
lemma class_rec_lemma: "\<lbrakk> wf ((subcls1 G)^-1); class G C = Some (D,fs,ms)\<rbrakk> \<Longrightarrow>
class_rec (G,C) t f = f C fs ms (if C=Object then t else class_rec (G,D) t f)";
apply (rule class_rec.simps [THEN trans [THEN fun_cong [THEN fun_cong]]])
apply simp
done
consts
method :: "'c prog \<times> cname => ( sig \<leadsto> cname \<times> ty \<times> 'c)" (* ###curry *)
field :: "'c prog \<times> cname => ( vname \<leadsto> cname \<times> ty )" (* ###curry *)
fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" (* ###curry *)
(* methods of a class, with inheritance, overriding and hiding, cf. 8.4.6 *)
defs method_def: "method \<equiv> \<lambda>(G,C). class_rec (G,C) empty (\<lambda>C fs ms ts.
ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))"
lemma method_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
method (G,C) = (if C = Object then empty else method (G,D)) ++
map_of (map (\<lambda>(s,m). (s,(C,m))) ms)"
apply (unfold method_def)
apply (simp split del: split_if)
apply (erule (1) class_rec_lemma [THEN trans]);
apply auto
done
(* list of fields of a class, including inherited and hidden ones *)
defs fields_def: "fields \<equiv> \<lambda>(G,C). class_rec (G,C) [] (\<lambda>C fs ms ts.
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)"
lemma fields_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==>
fields (G,C) =
map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))"
apply (unfold fields_def)
apply (simp split del: split_if)
apply (erule (1) class_rec_lemma [THEN trans]);
apply auto
done
defs field_def: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields"
lemma field_fields:
"field (G,C) fn = Some (fd, fT) \<Longrightarrow> map_of (fields (G,C)) (fn, fd) = Some fT"
apply (unfold field_def)
apply (rule table_of_remap_SomeD)
apply simp
done
inductive "widen G" intros (*widening, viz. method invocation conversion,cf. 5.3
i.e. sort of syntactic subtyping *)
refl [intro!, simp]: "G\<turnstile> T \<preceq> T" (* identity conv., cf. 5.1.1 *)
subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D"
null [intro!]: "G\<turnstile> NT \<preceq> RefT R"
inductive "cast G" intros (* casting conversion, cf. 5.5 / 5.1.5 *)
(* left out casts on primitve types *)
widen: "G\<turnstile>C\<preceq>C D ==> G\<turnstile>C \<preceq>? D"
subcls: "G\<turnstile>D\<preceq>C C ==> G\<turnstile>C \<preceq>? D"
lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False"
apply (rule iffI)
apply (erule widen.elims)
apply auto
done
lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done
lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done
lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D"
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done
lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False"
apply (rule iffI)
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply auto
done
lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)"
apply (rule iffI)
apply (ind_cases "G\<turnstile>S\<preceq>T")
apply (auto elim: widen.subcls)
done
lemma widen_trans [rule_format (no_asm)]: "G\<turnstile>S\<preceq>U ==> \<forall>T. G\<turnstile>U\<preceq>T --> G\<turnstile>S\<preceq>T"
apply (erule widen.induct)
apply safe
apply (frule widen_Class)
apply (frule_tac [2] widen_RefT)
apply safe
apply(erule (1) rtrancl_trans)
done
(*####theorem widen_trans: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T"
proof -
assume "G\<turnstile>S\<preceq>U"
thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" (*(is "PROP ?P S U")*)
proof (induct (*cases*) (open) (*?P S U*) rule: widen.induct [consumes 1])
case refl
fix T' assume "G\<turnstile>T\<preceq>T'" thus "G\<turnstile>T\<preceq>T'".
(* fix T' show "PROP ?P T T".*)
next
case subcls
fix T assume "G\<turnstile>Class D\<preceq>T"
then obtain E where "T = Class E" by (blast dest: widen_Class)
from prems show "G\<turnstile>Class C\<preceq>T" proof (auto elim: rtrancl_trans) qed
next
case null
fix RT assume "G\<turnstile>RefT R\<preceq>RT"
then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT)
thus "G\<turnstile>NT\<preceq>RT" by auto
qed
qed
*)
theorem widen_trans: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T"
proof -
assume "G\<turnstile>S\<preceq>U"
thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" (*(is "PROP ?P S U")*)
proof (induct (*cases*) (open) (*?P S U*)) (* rule: widen.induct *)
case refl
fix T' assume "G\<turnstile>T\<preceq>T'" thus "G\<turnstile>T\<preceq>T'".
(* fix T' show "PROP ?P T T".*)
next
case subcls
fix T assume "G\<turnstile>Class D\<preceq>T"
then obtain E where "T = Class E" by (blast dest: widen_Class)
from prems show "G\<turnstile>Class C\<preceq>T" proof (auto elim: rtrancl_trans) qed
next
case null
fix RT assume "G\<turnstile>RefT R\<preceq>RT"
then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT)
thus "G\<turnstile>NT\<preceq>RT" by auto
qed
qed
end