(* Title: HOL/MicroJava/J/WellForm.thy
ID: $Id$
Author: David von Oheimb
Copyright 1999 Technische Universitaet Muenchen
*)
header {* \isaheader{Well-formedness of Java programs} *}
theory WellForm = TypeRel:
text {*
for static checks on expressions and statements, see WellType.
\begin{description}
\item[improvements over Java Specification 1.0 (cf. 8.4.6.3, 8.4.6.4, 9.4.1):]\ \\
\begin{itemize}
\item a method implementing or overwriting another method may have a result type
that widens to the result type of the other method (instead of identical type)
\end{itemize}
\item[simplifications:]\ \\
\begin{itemize}
\item for uniformity, Object is assumed to be declared like any other class
\end{itemize}
\end{description}
*}
types 'c wf_mb = "'c prog => cname => 'c mdecl => bool"
constdefs
wf_fdecl :: "'c prog => fdecl => bool"
"wf_fdecl G == \<lambda>(fn,ft). is_type G ft"
wf_mhead :: "'c prog => sig => ty => bool"
"wf_mhead G == \<lambda>(mn,pTs) rT. (\<forall>T\<in>set pTs. is_type G T) \<and> is_type G rT"
wf_mdecl :: "'c wf_mb => 'c wf_mb"
"wf_mdecl wf_mb G C == \<lambda>(sig,rT,mb). wf_mhead G sig rT \<and> wf_mb G C (sig,rT,mb)"
wf_cdecl :: "'c wf_mb => 'c prog => 'c cdecl => bool"
"wf_cdecl wf_mb G ==
\<lambda>(C,(D,fs,ms)).
(\<forall>f\<in>set fs. wf_fdecl G f) \<and> unique fs \<and>
(\<forall>m\<in>set ms. wf_mdecl wf_mb G C m) \<and> unique ms \<and>
(C \<noteq> Object \<longrightarrow> is_class G D \<and> \<not>G\<turnstile>D\<preceq>C C \<and>
(\<forall>(sig,rT,b)\<in>set ms. \<forall>D' rT' b'.
method(G,D) sig = Some(D',rT',b') --> G\<turnstile>rT\<preceq>rT'))"
wf_prog :: "'c wf_mb => 'c prog => bool"
"wf_prog wf_mb G ==
let cs = set G in ObjectC \<in> cs \<and> (\<forall>c\<in>cs. wf_cdecl wf_mb G c) \<and> unique G"
lemma class_wf:
"[|class G C = Some c; wf_prog wf_mb G|] ==> wf_cdecl wf_mb G (C,c)"
apply (unfold wf_prog_def class_def)
apply (simp)
apply (fast dest: map_of_SomeD)
done
lemma class_Object [simp]:
"wf_prog wf_mb G ==> class G Object = Some (arbitrary, [], [])"
apply (unfold wf_prog_def ObjectC_def class_def)
apply (auto intro: map_of_SomeI)
done
lemma is_class_Object [simp]: "wf_prog wf_mb G ==> is_class G Object"
apply (unfold is_class_def)
apply (simp (no_asm_simp))
done
lemma subcls1_wfD: "[|G\<turnstile>C\<prec>C1D; wf_prog wf_mb G|] ==> D \<noteq> C \<and> \<not>(D,C)\<in>(subcls1 G)^+"
apply( frule r_into_trancl)
apply( drule subcls1D)
apply(clarify)
apply( drule (1) class_wf)
apply( unfold wf_cdecl_def)
apply(force simp add: reflcl_trancl [THEN sym] simp del: reflcl_trancl)
done
lemma wf_cdecl_supD:
"!!r. \<lbrakk>wf_cdecl wf_mb G (C,D,r); C \<noteq> Object\<rbrakk> \<Longrightarrow> is_class G D"
apply (unfold wf_cdecl_def)
apply (auto split add: option.split_asm)
done
lemma subcls_asym: "[|wf_prog wf_mb G; (C,D)\<in>(subcls1 G)^+|] ==> \<not>(D,C)\<in>(subcls1 G)^+"
apply(erule tranclE)
apply(fast dest!: subcls1_wfD )
apply(fast dest!: subcls1_wfD intro: trancl_trans)
done
lemma subcls_irrefl: "[|wf_prog wf_mb G; (C,D)\<in>(subcls1 G)^+|] ==> C \<noteq> D"
apply (erule trancl_trans_induct)
apply (auto dest: subcls1_wfD subcls_asym)
done
lemma acyclic_subcls1: "wf_prog wf_mb G ==> acyclic (subcls1 G)"
apply (unfold acyclic_def)
apply (fast dest: subcls_irrefl)
done
lemma wf_subcls1: "wf_prog wf_mb G ==> wf ((subcls1 G)^-1)"
apply (rule finite_acyclic_wf)
apply ( subst finite_converse)
apply ( rule finite_subcls1)
apply (subst acyclic_converse)
apply (erule acyclic_subcls1)
done
lemma subcls_induct:
"[|wf_prog wf_mb G; !!C. \<forall>D. (C,D)\<in>(subcls1 G)^+ --> P D ==> P C|] ==> P C"
(is "?A \<Longrightarrow> PROP ?P \<Longrightarrow> _")
proof -
assume p: "PROP ?P"
assume ?A thus ?thesis apply -
apply(drule wf_subcls1)
apply(drule wf_trancl)
apply(simp only: trancl_converse)
apply(erule_tac a = C in wf_induct)
apply(rule p)
apply(auto)
done
qed
lemma subcls1_induct:
"[|is_class G C; wf_prog wf_mb G; P Object;
!!C D fs ms. [|C \<noteq> Object; is_class G C; class G C = Some (D,fs,ms) \<and>
wf_cdecl wf_mb G (C,D,fs,ms) \<and> G\<turnstile>C\<prec>C1D \<and> is_class G D \<and> P D|] ==> P C
|] ==> P C"
(is "?A \<Longrightarrow> ?B \<Longrightarrow> ?C \<Longrightarrow> PROP ?P \<Longrightarrow> _")
proof -
assume p: "PROP ?P"
assume ?A ?B ?C thus ?thesis apply -
apply(unfold is_class_def)
apply( rule impE)
prefer 2
apply( assumption)
prefer 2
apply( assumption)
apply( erule thin_rl)
apply( rule subcls_induct)
apply( assumption)
apply( rule impI)
apply( case_tac "C = Object")
apply( fast)
apply safe
apply( frule (1) class_wf)
apply( frule (1) wf_cdecl_supD)
apply( subgoal_tac "G\<turnstile>C\<prec>C1a")
apply( erule_tac [2] subcls1I)
apply( rule p)
apply (unfold is_class_def)
apply auto
done
qed
lemmas method_rec = wf_subcls1 [THEN [2] method_rec_lemma];
lemmas fields_rec = wf_subcls1 [THEN [2] fields_rec_lemma];
lemma method_Object [simp]: "wf_prog wf_mb G ==> method (G,Object) = empty"
apply(subst method_rec)
apply auto
done
lemma fields_Object [simp]: "wf_prog wf_mb G ==> fields (G,Object) = []"
apply(subst fields_rec)
apply auto
done
lemma field_Object [simp]: "wf_prog wf_mb G ==> field (G,Object) = empty"
apply (unfold field_def)
apply(simp (no_asm_simp))
done
lemma subcls_C_Object: "[|is_class G C; wf_prog wf_mb G|] ==> G\<turnstile>C\<preceq>C Object"
apply(erule subcls1_induct)
apply( assumption)
apply( fast)
apply(auto dest!: wf_cdecl_supD)
apply(erule (1) converse_rtrancl_into_rtrancl)
done
lemma is_type_rTI: "wf_mhead G sig rT ==> is_type G rT"
apply (unfold wf_mhead_def)
apply auto
done
lemma widen_fields_defpl': "[|is_class G C; wf_prog wf_mb G|] ==>
\<forall>((fn,fd),fT)\<in>set (fields (G,C)). G\<turnstile>C\<preceq>C fd"
apply( erule subcls1_induct)
apply( assumption)
apply( simp (no_asm_simp))
apply( tactic "safe_tac HOL_cs")
apply( subst fields_rec)
apply( assumption)
apply( assumption)
apply( simp (no_asm) split del: split_if)
apply( rule ballI)
apply( simp (no_asm_simp) only: split_tupled_all)
apply( simp (no_asm))
apply( erule UnE)
apply( force)
apply( erule r_into_rtrancl [THEN rtrancl_trans])
apply auto
done
lemma widen_fields_defpl:
"[|((fn,fd),fT) \<in> set (fields (G,C)); wf_prog wf_mb G; is_class G C|] ==>
G\<turnstile>C\<preceq>C fd"
apply( drule (1) widen_fields_defpl')
apply (fast)
done
lemma unique_fields:
"[|is_class G C; wf_prog wf_mb G|] ==> unique (fields (G,C))"
apply( erule subcls1_induct)
apply( assumption)
apply( safe dest!: wf_cdecl_supD)
apply( simp (no_asm_simp))
apply( drule subcls1_wfD)
apply( assumption)
apply( subst fields_rec)
apply auto
apply( rotate_tac -1)
apply( frule class_wf)
apply auto
apply( simp add: wf_cdecl_def)
apply( erule unique_append)
apply( rule unique_map_inj)
apply( clarsimp)
apply (rule injI)
apply( simp)
apply(auto dest!: widen_fields_defpl)
done
lemma fields_mono_lemma [rule_format (no_asm)]:
"[|wf_prog wf_mb G; (C',C)\<in>(subcls1 G)^*|] ==>
x \<in> set (fields (G,C)) --> x \<in> set (fields (G,C'))"
apply(erule converse_rtrancl_induct)
apply( safe dest!: subcls1D)
apply(subst fields_rec)
apply( auto)
done
lemma fields_mono:
"\<lbrakk>map_of (fields (G,C)) fn = Some f; G\<turnstile>D\<preceq>C C; is_class G D; wf_prog wf_mb G\<rbrakk>
\<Longrightarrow> map_of (fields (G,D)) fn = Some f"
apply (rule map_of_SomeI)
apply (erule (1) unique_fields)
apply (erule (1) fields_mono_lemma)
apply (erule map_of_SomeD)
done
lemma widen_cfs_fields:
"[|field (G,C) fn = Some (fd, fT); G\<turnstile>D\<preceq>C C; wf_prog wf_mb G|]==>
map_of (fields (G,D)) (fn, fd) = Some fT"
apply (drule field_fields)
apply (drule rtranclD)
apply safe
apply (frule subcls_is_class)
apply (drule trancl_into_rtrancl)
apply (fast dest: fields_mono)
done
lemma method_wf_mdecl [rule_format (no_asm)]:
"wf_prog wf_mb G ==> is_class G C \<Longrightarrow>
method (G,C) sig = Some (md,mh,m)
--> G\<turnstile>C\<preceq>C md \<and> wf_mdecl wf_mb G md (sig,(mh,m))"
apply( erule subcls1_induct)
apply( assumption)
apply( force)
apply( clarify)
apply( frule_tac C = C in method_rec)
apply( assumption)
apply( rotate_tac -1)
apply( simp)
apply( drule override_SomeD)
apply( erule disjE)
apply( erule_tac V = "?P --> ?Q" in thin_rl)
apply (frule map_of_SomeD)
apply (clarsimp simp add: wf_cdecl_def)
apply( clarify)
apply( rule rtrancl_trans)
prefer 2
apply( assumption)
apply( rule r_into_rtrancl)
apply( fast intro: subcls1I)
done
lemma subcls_widen_methd [rule_format (no_asm)]:
"[|G\<turnstile>T\<preceq>C T'; wf_prog wf_mb G|] ==>
\<forall>D rT b. method (G,T') sig = Some (D,rT ,b) -->
(\<exists>D' rT' b'. method (G,T) sig = Some (D',rT',b') \<and> G\<turnstile>rT'\<preceq>rT)"
apply( drule rtranclD)
apply( erule disjE)
apply( fast)
apply( erule conjE)
apply( erule trancl_trans_induct)
prefer 2
apply( clarify)
apply( drule spec, drule spec, drule spec, erule (1) impE)
apply( fast elim: widen_trans)
apply( clarify)
apply( drule subcls1D)
apply( clarify)
apply( subst method_rec)
apply( assumption)
apply( unfold override_def)
apply( simp (no_asm_simp) del: split_paired_Ex)
apply( case_tac "\<exists>z. map_of(map (\<lambda>(s,m). (s, ?C, m)) ms) sig = Some z")
apply( erule exE)
apply( rotate_tac -1, frule ssubst, erule_tac [2] asm_rl)
prefer 2
apply( rotate_tac -1, frule ssubst, erule_tac [2] asm_rl)
apply( tactic "asm_full_simp_tac (HOL_ss addsimps [not_None_eq RS sym]) 1")
apply( simp_all (no_asm_simp) del: split_paired_Ex)
apply( drule (1) class_wf)
apply( simp (no_asm_simp) only: split_tupled_all)
apply( unfold wf_cdecl_def)
apply( drule map_of_SomeD)
apply auto
done
lemma subtype_widen_methd:
"[| G\<turnstile> C\<preceq>C D; wf_prog wf_mb G;
method (G,D) sig = Some (md, rT, b) |]
==> \<exists>mD' rT' b'. method (G,C) sig= Some(mD',rT',b') \<and> G\<turnstile>rT'\<preceq>rT"
apply(auto dest: subcls_widen_methd method_wf_mdecl
simp add: wf_mdecl_def wf_mhead_def split_def)
done
lemma method_in_md [rule_format (no_asm)]:
"wf_prog wf_mb G ==> is_class G C \<Longrightarrow> \<forall>D. method (G,C) sig = Some(D,mh,code)
--> is_class G D \<and> method (G,D) sig = Some(D,mh,code)"
apply (erule (1) subcls1_induct)
apply (simp)
apply (erule conjE)
apply (subst method_rec)
apply (assumption)
apply (assumption)
apply (clarify)
apply (erule_tac "x" = "Da" in allE)
apply (clarsimp)
apply (simp add: map_of_map)
apply (clarify)
apply (subst method_rec)
apply (assumption)
apply (assumption)
apply (simp add: override_def map_of_map split add: option.split)
done
lemma widen_methd:
"[| method (G,C) sig = Some (md,rT,b); wf_prog wf_mb G; G\<turnstile>T''\<preceq>C C|]
==> \<exists>md' rT' b'. method (G,T'') sig = Some (md',rT',b') \<and> G\<turnstile>rT'\<preceq>rT"
apply( drule subcls_widen_methd)
apply auto
done
lemma Call_lemma:
"[|method (G,C) sig = Some (md,rT,b); G\<turnstile>T''\<preceq>C C; wf_prog wf_mb G;
class G C = Some y|] ==> \<exists>T' rT' b. method (G,T'') sig = Some (T',rT',b) \<and>
G\<turnstile>rT'\<preceq>rT \<and> G\<turnstile>T''\<preceq>C T' \<and> wf_mhead G sig rT' \<and> wf_mb G T' (sig,rT',b)"
apply( drule (2) widen_methd)
apply( clarify)
apply( frule subcls_is_class2)
apply (unfold is_class_def)
apply (simp (no_asm_simp))
apply( drule method_wf_mdecl)
apply( unfold wf_mdecl_def)
apply( unfold is_class_def)
apply auto
done
lemma fields_is_type_lemma [rule_format (no_asm)]:
"[|is_class G C; wf_prog wf_mb G|] ==>
\<forall>f\<in>set (fields (G,C)). is_type G (snd f)"
apply( erule (1) subcls1_induct)
apply( simp (no_asm_simp))
apply( subst fields_rec)
apply( fast)
apply( assumption)
apply( clarsimp)
apply( safe)
prefer 2
apply( force)
apply( drule (1) class_wf)
apply( unfold wf_cdecl_def)
apply( clarsimp)
apply( drule (1) bspec)
apply( unfold wf_fdecl_def)
apply auto
done
lemma fields_is_type:
"[|map_of (fields (G,C)) fn = Some f; wf_prog wf_mb G; is_class G C|] ==>
is_type G f"
apply(drule map_of_SomeD)
apply(drule (2) fields_is_type_lemma)
apply(auto)
done
lemma methd:
"[| wf_prog wf_mb G; (C,S,fs,mdecls) \<in> set G; (sig,rT,code) \<in> set mdecls |]
==> method (G,C) sig = Some(C,rT,code) \<and> is_class G C"
proof -
assume wf: "wf_prog wf_mb G"
assume C: "(C,S,fs,mdecls) \<in> set G"
assume m: "(sig,rT,code) \<in> set mdecls"
moreover
from wf have "class G Object = Some (arbitrary, [], [])" by simp
moreover
from wf C have c: "class G C = Some (S,fs,mdecls)"
by (auto simp add: wf_prog_def class_def is_class_def intro: map_of_SomeI)
ultimately
have O: "C \<noteq> Object" by auto
from wf C have "unique mdecls" by (unfold wf_prog_def wf_cdecl_def) auto
hence "unique (map (\<lambda>(s,m). (s,C,m)) mdecls)" by (induct mdecls, auto)
with m have "map_of (map (\<lambda>(s,m). (s,C,m)) mdecls) sig = Some (C,rT,code)"
by (force intro: map_of_SomeI)
with wf C m c O
show ?thesis by (auto simp add: is_class_def dest: method_rec)
qed
end