author | nipkow |
Fri, 25 Jul 2003 17:21:22 +0200 | |
changeset 14134 | 0fdf5708c7a8 |
parent 14045 | a34d89ce6097 |
child 14171 | 0cab06e3bbd0 |
permissions | -rw-r--r-- |
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(* Title: HOL/MicroJava/J/TypeRel.thy |
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ID: $Id$ |
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Author: David von Oheimb |
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Copyright 1999 Technische Universitaet Muenchen |
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*) |
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header {* \isaheader{Relations between Java Types} *} |
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theory TypeRel = Decl: |
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consts |
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subcls1 :: "'c prog => (cname \<times> cname) set" -- "subclass" |
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widen :: "'c prog => (ty \<times> ty ) set" -- "widening" |
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cast :: "'c prog => (ty \<times> ty ) set" -- "casting" |
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syntax (xsymbols) |
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subcls1 :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70) |
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subcls :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70) |
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widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) |
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cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) |
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syntax |
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subcls1 :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C1 _" [71,71,71] 70) |
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subcls :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C _" [71,71,71] 70) |
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widen :: "'c prog => [ty , ty ] => bool" ("_ |- _ <= _" [71,71,71] 70) |
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cast :: "'c prog => [ty , ty ] => bool" ("_ |- _ <=? _" [71,71,71] 70) |
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translations |
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"G\<turnstile>C \<prec>C1 D" == "(C,D) \<in> subcls1 G" |
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"G\<turnstile>C \<preceq>C D" == "(C,D) \<in> (subcls1 G)^*" |
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"G\<turnstile>S \<preceq> T" == "(S,T) \<in> widen G" |
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"G\<turnstile>C \<preceq>? D" == "(C,D) \<in> cast G" |
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-- "direct subclass, cf. 8.1.3" |
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inductive "subcls1 G" intros |
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subcls1I: "\<lbrakk>class G C = Some (D,rest); C \<noteq> Object\<rbrakk> \<Longrightarrow> G\<turnstile>C\<prec>C1D" |
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lemma subcls1D: |
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"G\<turnstile>C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>fs ms. class G C = Some (D,fs,ms))" |
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apply (erule subcls1.elims) |
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apply auto |
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done |
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lemma subcls1_def2: |
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"subcls1 G = (\<Sigma>C\<in>{C. is_class G C} . {D. C\<noteq>Object \<and> fst (the (class G C))=D})" |
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by (auto simp add: is_class_def dest: subcls1D intro: subcls1I) |
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lemma finite_subcls1: "finite (subcls1 G)" |
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apply(subst subcls1_def2) |
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apply(rule finite_SigmaI [OF finite_is_class]) |
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apply(rule_tac B = "{fst (the (class G C))}" in finite_subset) |
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apply auto |
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done |
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lemma subcls_is_class: "(C,D) \<in> (subcls1 G)^+ ==> is_class G C" |
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apply (unfold is_class_def) |
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apply(erule trancl_trans_induct) |
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apply (auto dest!: subcls1D) |
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done |
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lemma subcls_is_class2 [rule_format (no_asm)]: |
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"G\<turnstile>C\<preceq>C D \<Longrightarrow> is_class G D \<longrightarrow> is_class G C" |
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apply (unfold is_class_def) |
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apply (erule rtrancl_induct) |
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apply (drule_tac [2] subcls1D) |
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apply auto |
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done |
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constdefs |
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class_rec :: "'c prog \<Rightarrow> cname \<Rightarrow> 'a \<Rightarrow> |
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(cname \<Rightarrow> fdecl list \<Rightarrow> 'c mdecl list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a" |
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"class_rec G == wfrec ((subcls1 G)^-1) |
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(\<lambda>r C t f. case class G C of |
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None \<Rightarrow> arbitrary |
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| Some (D,fs,ms) \<Rightarrow> |
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f C fs ms (if C = Object then t else r D t f))" |
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lemma class_rec_lemma: "wf ((subcls1 G)^-1) \<Longrightarrow> class G C = Some (D,fs,ms) \<Longrightarrow> |
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class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)" |
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by (simp add: class_rec_def wfrec cut_apply [OF converseI [OF subcls1I]]) |
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consts |
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method :: "'c prog \<times> cname => ( sig \<rightharpoonup> cname \<times> ty \<times> 'c)" (* ###curry *) |
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field :: "'c prog \<times> cname => ( vname \<rightharpoonup> cname \<times> ty )" (* ###curry *) |
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fields :: "'c prog \<times> cname => ((vname \<times> cname) \<times> ty) list" (* ###curry *) |
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-- "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6" |
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defs method_def: "method \<equiv> \<lambda>(G,C). class_rec G C empty (\<lambda>C fs ms ts. |
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ts ++ map_of (map (\<lambda>(s,m). (s,(C,m))) ms))" |
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lemma method_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==> |
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method (G,C) = (if C = Object then empty else method (G,D)) ++ |
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map_of (map (\<lambda>(s,m). (s,(C,m))) ms)" |
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apply (unfold method_def) |
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apply (simp split del: split_if) |
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apply (erule (1) class_rec_lemma [THEN trans]); |
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apply auto |
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done |
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-- "list of fields of a class, including inherited and hidden ones" |
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defs fields_def: "fields \<equiv> \<lambda>(G,C). class_rec G C [] (\<lambda>C fs ms ts. |
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map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ ts)" |
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lemma fields_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==> |
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fields (G,C) = |
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map (\<lambda>(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))" |
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apply (unfold fields_def) |
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apply (simp split del: split_if) |
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apply (erule (1) class_rec_lemma [THEN trans]); |
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apply auto |
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done |
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defs field_def: "field == map_of o (map (\<lambda>((fn,fd),ft). (fn,(fd,ft)))) o fields" |
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lemma field_fields: |
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"field (G,C) fn = Some (fd, fT) \<Longrightarrow> map_of (fields (G,C)) (fn, fd) = Some fT" |
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apply (unfold field_def) |
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apply (rule table_of_remap_SomeD) |
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apply simp |
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done |
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-- "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" |
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inductive "widen G" intros |
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refl [intro!, simp]: "G\<turnstile> T \<preceq> T" -- "identity conv., cf. 5.1.1" |
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subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" |
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null [intro!]: "G\<turnstile> NT \<preceq> RefT R" |
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-- "casting conversion, cf. 5.5 / 5.1.5" |
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-- "left out casts on primitve types" |
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inductive "cast G" intros |
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widen: "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D" |
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subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D" |
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lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False" |
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apply (rule iffI) |
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apply (erule widen.elims) |
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apply auto |
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done |
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lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> \<exists>t. T=RefT t" |
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apply (ind_cases "G\<turnstile>S\<preceq>T") |
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apply auto |
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done |
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lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> \<exists>t. S=RefT t" |
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apply (ind_cases "G\<turnstile>S\<preceq>T") |
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apply auto |
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done |
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153 |
|
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lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> \<exists>D. T=Class D" |
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155 |
apply (ind_cases "G\<turnstile>S\<preceq>T") |
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156 |
apply auto |
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157 |
done |
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158 |
|
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lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False" |
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160 |
apply (rule iffI) |
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161 |
apply (ind_cases "G\<turnstile>S\<preceq>T") |
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162 |
apply auto |
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163 |
done |
8011 | 164 |
|
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lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)" |
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166 |
apply (rule iffI) |
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167 |
apply (ind_cases "G\<turnstile>S\<preceq>T") |
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168 |
apply (auto elim: widen.subcls) |
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169 |
done |
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170 |
|
14045 | 171 |
lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT \<Longrightarrow> G \<turnstile> T \<preceq> Class D" |
172 |
by (ind_cases "G \<turnstile> T \<preceq> NT", auto) |
|
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||
174 |
lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" |
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175 |
apply (rule iffI) |
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176 |
apply (erule cast.elims) |
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177 |
apply auto |
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178 |
done |
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179 |
||
180 |
lemma cast_RefT: "G \<turnstile> C \<preceq>? Class D \<Longrightarrow> \<exists> rT. C = RefT rT" |
|
181 |
apply (erule cast.cases) |
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182 |
apply simp apply (erule widen.cases) |
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183 |
apply auto |
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184 |
done |
|
185 |
||
12517 | 186 |
theorem widen_trans[trans]: "\<lbrakk>G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T\<rbrakk> \<Longrightarrow> G\<turnstile>S\<preceq>T" |
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187 |
proof - |
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assume "G\<turnstile>S\<preceq>U" thus "\<And>T. G\<turnstile>U\<preceq>T \<Longrightarrow> G\<turnstile>S\<preceq>T" |
11987 | 189 |
proof induct |
12517 | 190 |
case (refl T T') thus "G\<turnstile>T\<preceq>T'" . |
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next |
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case (subcls C D T) |
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then obtain E where "T = Class E" by (blast dest: widen_Class) |
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with subcls show "G\<turnstile>Class C\<preceq>T" by (auto elim: rtrancl_trans) |
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next |
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case (null R RT) |
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then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT) |
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198 |
thus "G\<turnstile>NT\<preceq>RT" by auto |
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qed |
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200 |
qed |
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201 |
|
8011 | 202 |
end |