--- a/src/HOL/MicroJava/J/WellType.thy Thu Feb 01 20:51:48 2001 +0100
+++ b/src/HOL/MicroJava/J/WellType.thy Thu Feb 01 20:53:13 2001 +0100
@@ -15,41 +15,59 @@
*)
-WellType = Term + WellForm +
+theory WellType = Term + WellForm:
types lenv (* local variables, including method parameters and This *)
- = "vname \\<leadsto> ty"
+ = "vname \<leadsto> ty"
'c env
- = "'c prog \\<times> lenv"
+ = "'c prog \<times> lenv"
syntax
prg :: "'c env => 'c prog"
- localT :: "'c env => (vname \\<leadsto> ty)"
+ localT :: "'c env => (vname \<leadsto> ty)"
translations
"prg" => "fst"
"localT" => "snd"
consts
- more_spec :: "'c prog => (ty \\<times> 'x) \\<times> ty list =>
- (ty \\<times> 'x) \\<times> ty list => bool"
- appl_methds :: "'c prog => cname => sig => ((ty \\<times> ty) \\<times> ty list) set"
- max_spec :: "'c prog => cname => sig => ((ty \\<times> ty) \\<times> ty list) set"
+ more_spec :: "'c prog => (ty \<times> 'x) \<times> ty list =>
+ (ty \<times> 'x) \<times> ty list => bool"
+ appl_methds :: "'c prog => cname => sig => ((ty \<times> ty) \<times> ty list) set"
+ max_spec :: "'c prog => cname => sig => ((ty \<times> ty) \<times> ty list) set"
defs
- more_spec_def "more_spec G == \\<lambda>((d,h),pTs). \\<lambda>((d',h'),pTs'). G\\<turnstile>d\\<preceq>d' \\<and>
- list_all2 (\\<lambda>T T'. G\\<turnstile>T\\<preceq>T') pTs pTs'"
+ more_spec_def: "more_spec G == \<lambda>((d,h),pTs). \<lambda>((d',h'),pTs'). G\<turnstile>d\<preceq>d' \<and>
+ list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'"
(* applicable methods, cf. 15.11.2.1 *)
- appl_methds_def "appl_methds G C == \\<lambda>(mn, pTs).
+ appl_methds_def: "appl_methds G C == \<lambda>(mn, pTs).
{((Class md,rT),pTs') |md rT mb pTs'.
- method (G,C) (mn, pTs') = Some (md,rT,mb) \\<and>
- list_all2 (\\<lambda>T T'. G\\<turnstile>T\\<preceq>T') pTs pTs'}"
+ method (G,C) (mn, pTs') = Some (md,rT,mb) \<and>
+ list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'}"
(* maximally specific methods, cf. 15.11.2.2 *)
- max_spec_def "max_spec G C sig == {m. m \\<in>appl_methds G C sig \\<and>
- (\\<forall>m'\\<in>appl_methds G C sig.
- more_spec G m' m --> m' = m)}"
+ max_spec_def: "max_spec G C sig == {m. m \<in>appl_methds G C sig \<and>
+ (\<forall>m'\<in>appl_methds G C sig.
+ more_spec G m' m --> m' = m)}"
+
+lemma max_spec2appl_meths:
+ "x \<in> max_spec G C sig ==> x \<in> appl_methds G C sig"
+apply (unfold max_spec_def)
+apply (fast)
+done
+
+lemma appl_methsD:
+"((md,rT),pTs')\<in>appl_methds G C (mn, pTs) ==>
+ \<exists>D b. md = Class D \<and> method (G,C) (mn, pTs') = Some (D,rT,b)
+ \<and> list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') pTs pTs'"
+apply (unfold appl_methds_def)
+apply (fast)
+done
+
+lemmas max_spec2mheads = insertI1 [THEN [2] equalityD2 [THEN subsetD],
+ THEN max_spec2appl_meths, THEN appl_methsD]
+
consts
typeof :: "(loc => ty option) => val => ty option"
@@ -61,19 +79,30 @@
"typeof dt (Intg i) = Some (PrimT Integer)"
"typeof dt (Addr a) = dt a"
+lemma is_type_typeof [rule_format (no_asm), simp]: "(\<forall>a. v \<noteq> Addr a) --> (\<exists>T. typeof t v = Some T \<and> is_type G T)"
+apply (rule val.induct)
+apply auto
+done
+
+lemma typeof_empty_is_type [rule_format (no_asm)]:
+ "typeof (\<lambda>a. None) v = Some T \<longrightarrow> is_type G T"
+apply (rule val.induct)
+apply auto
+done
+
types
- java_mb = "vname list \\<times> (vname \\<times> ty) list \\<times> stmt \\<times> expr"
+ java_mb = "vname list \<times> (vname \<times> ty) list \<times> stmt \<times> expr"
(* method body with parameter names, local variables, block, result expression *)
consts
- ty_expr :: "java_mb env => (expr \\<times> ty ) set"
- ty_exprs:: "java_mb env => (expr list \\<times> ty list) set"
+ ty_expr :: "java_mb env => (expr \<times> ty ) set"
+ ty_exprs:: "java_mb env => (expr list \<times> ty list) set"
wt_stmt :: "java_mb env => stmt set"
syntax
- ty_expr :: "java_mb env => [expr , ty ] => bool" ("_ \\<turnstile> _ :: _" [51,51,51]50)
- ty_exprs:: "java_mb env => [expr list, ty list] => bool" ("_ \\<turnstile> _ [::] _" [51,51,51]50)
- wt_stmt :: "java_mb env => stmt => bool" ("_ \\<turnstile> _ \\<surd>" [51,51 ]50)
+ ty_expr :: "java_mb env => [expr , ty ] => bool" ("_ \<turnstile> _ :: _" [51,51,51]50)
+ ty_exprs:: "java_mb env => [expr list, ty list] => bool" ("_ \<turnstile> _ [::] _" [51,51,51]50)
+ wt_stmt :: "java_mb env => stmt => bool" ("_ \<turnstile> _ \<surd>" [51,51 ]50)
syntax (HTML)
ty_expr :: "java_mb env => [expr , ty ] => bool" ("_ |- _ :: _" [51,51,51]50)
@@ -82,107 +111,123 @@
translations
- "E\\<turnstile>e :: T" == "(e,T) \\<in> ty_expr E"
- "E\\<turnstile>e[::]T" == "(e,T) \\<in> ty_exprs E"
- "E\\<turnstile>c \\<surd>" == "c \\<in> wt_stmt E"
+ "E\<turnstile>e :: T" == "(e,T) \<in> ty_expr E"
+ "E\<turnstile>e[::]T" == "(e,T) \<in> ty_exprs E"
+ "E\<turnstile>c \<surd>" == "c \<in> wt_stmt E"
-inductive "ty_expr E" "ty_exprs E" "wt_stmt E" intrs
+inductive "ty_expr E" "ty_exprs E" "wt_stmt E" intros
(* well-typed expressions *)
(* cf. 15.8 *)
- NewC "[| is_class (prg E) C |] ==>
- E\\<turnstile>NewC C::Class C"
+ NewC: "[| is_class (prg E) C |] ==>
+ E\<turnstile>NewC C::Class C"
(* cf. 15.15 *)
- Cast "[| E\\<turnstile>e::Class C; is_class (prg E) D;
- prg E\\<turnstile>C\\<preceq>? D |] ==>
- E\\<turnstile>Cast D e::Class D"
+ Cast: "[| E\<turnstile>e::Class C; is_class (prg E) D;
+ prg E\<turnstile>C\<preceq>? D |] ==>
+ E\<turnstile>Cast D e::Class D"
(* cf. 15.7.1 *)
- Lit "[| typeof (\\<lambda>v. None) x = Some T |] ==>
- E\\<turnstile>Lit x::T"
+ Lit: "[| typeof (\<lambda>v. None) x = Some T |] ==>
+ E\<turnstile>Lit x::T"
(* cf. 15.13.1 *)
- LAcc "[| localT E v = Some T; is_type (prg E) T |] ==>
- E\\<turnstile>LAcc v::T"
+ LAcc: "[| localT E v = Some T; is_type (prg E) T |] ==>
+ E\<turnstile>LAcc v::T"
- BinOp "[| E\\<turnstile>e1::T;
- E\\<turnstile>e2::T;
+ BinOp:"[| E\<turnstile>e1::T;
+ E\<turnstile>e2::T;
if bop = Eq then T' = PrimT Boolean
- else T' = T \\<and> T = PrimT Integer|] ==>
- E\\<turnstile>BinOp bop e1 e2::T'"
+ else T' = T \<and> T = PrimT Integer|] ==>
+ E\<turnstile>BinOp bop e1 e2::T'"
(* cf. 15.25, 15.25.1 *)
- LAss "[| E\\<turnstile>LAcc v::T;
- E\\<turnstile>e::T';
- prg E\\<turnstile>T'\\<preceq>T |] ==>
- E\\<turnstile>v::=e::T'"
+ LAss: "[| E\<turnstile>LAcc v::T;
+ E\<turnstile>e::T';
+ prg E\<turnstile>T'\<preceq>T |] ==>
+ E\<turnstile>v::=e::T'"
(* cf. 15.10.1 *)
- FAcc "[| E\\<turnstile>a::Class C;
+ FAcc: "[| E\<turnstile>a::Class C;
field (prg E,C) fn = Some (fd,fT) |] ==>
- E\\<turnstile>{fd}a..fn::fT"
+ E\<turnstile>{fd}a..fn::fT"
(* cf. 15.25, 15.25.1 *)
- FAss "[| E\\<turnstile>{fd}a..fn::T;
- E\\<turnstile>v ::T';
- prg E\\<turnstile>T'\\<preceq>T |] ==>
- E\\<turnstile>{fd}a..fn:=v::T'"
+ FAss: "[| E\<turnstile>{fd}a..fn::T;
+ E\<turnstile>v ::T';
+ prg E\<turnstile>T'\<preceq>T |] ==>
+ E\<turnstile>{fd}a..fn:=v::T'"
(* cf. 15.11.1, 15.11.2, 15.11.3 *)
- Call "[| E\\<turnstile>a::Class C;
- E\\<turnstile>ps[::]pTs;
+ Call: "[| E\<turnstile>a::Class C;
+ E\<turnstile>ps[::]pTs;
max_spec (prg E) C (mn, pTs) = {((md,rT),pTs')} |] ==>
- E\\<turnstile>{C}a..mn({pTs'}ps)::rT"
+ E\<turnstile>{C}a..mn({pTs'}ps)::rT"
(* well-typed expression lists *)
(* cf. 15.11.??? *)
- Nil "E\\<turnstile>[][::][]"
+ Nil: "E\<turnstile>[][::][]"
(* cf. 15.11.??? *)
- Cons "[| E\\<turnstile>e::T;
- E\\<turnstile>es[::]Ts |] ==>
- E\\<turnstile>e#es[::]T#Ts"
+ Cons:"[| E\<turnstile>e::T;
+ E\<turnstile>es[::]Ts |] ==>
+ E\<turnstile>e#es[::]T#Ts"
(* well-typed statements *)
- Skip "E\\<turnstile>Skip\\<surd>"
+ Skip:"E\<turnstile>Skip\<surd>"
- Expr "[| E\\<turnstile>e::T |] ==>
- E\\<turnstile>Expr e\\<surd>"
+ Expr:"[| E\<turnstile>e::T |] ==>
+ E\<turnstile>Expr e\<surd>"
- Comp "[| E\\<turnstile>s1\\<surd>;
- E\\<turnstile>s2\\<surd> |] ==>
- E\\<turnstile>s1;; s2\\<surd>"
+ Comp:"[| E\<turnstile>s1\<surd>;
+ E\<turnstile>s2\<surd> |] ==>
+ E\<turnstile>s1;; s2\<surd>"
(* cf. 14.8 *)
- Cond "[| E\\<turnstile>e::PrimT Boolean;
- E\\<turnstile>s1\\<surd>;
- E\\<turnstile>s2\\<surd> |] ==>
- E\\<turnstile>If(e) s1 Else s2\\<surd>"
+ Cond:"[| E\<turnstile>e::PrimT Boolean;
+ E\<turnstile>s1\<surd>;
+ E\<turnstile>s2\<surd> |] ==>
+ E\<turnstile>If(e) s1 Else s2\<surd>"
(* cf. 14.10 *)
- Loop "[| E\\<turnstile>e::PrimT Boolean;
- E\\<turnstile>s\\<surd> |] ==>
- E\\<turnstile>While(e) s\\<surd>"
+ Loop:"[| E\<turnstile>e::PrimT Boolean;
+ E\<turnstile>s\<surd> |] ==>
+ E\<turnstile>While(e) s\<surd>"
constdefs
- wf_java_mdecl :: java_mb prog => cname => java_mb mdecl => bool
-"wf_java_mdecl G C == \\<lambda>((mn,pTs),rT,(pns,lvars,blk,res)).
- length pTs = length pns \\<and>
- nodups pns \\<and>
- unique lvars \\<and>
- (\\<forall>pn\\<in>set pns. map_of lvars pn = None) \\<and>
- (\\<forall>(vn,T)\\<in>set lvars. is_type G T) &
- (let E = (G,map_of lvars(pns[\\<mapsto>]pTs)(This\\<mapsto>Class C)) in
- E\\<turnstile>blk\\<surd> \\<and> (\\<exists>T. E\\<turnstile>res::T \\<and> G\\<turnstile>T\\<preceq>rT))"
+ wf_java_mdecl :: "java_mb prog => cname => java_mb mdecl => bool"
+"wf_java_mdecl G C == \<lambda>((mn,pTs),rT,(pns,lvars,blk,res)).
+ length pTs = length pns \<and>
+ nodups pns \<and>
+ unique lvars \<and>
+ (\<forall>pn\<in>set pns. map_of lvars pn = None) \<and>
+ (\<forall>(vn,T)\<in>set lvars. is_type G T) &
+ (let E = (G,map_of lvars(pns[\<mapsto>]pTs)(This\<mapsto>Class C)) in
+ E\<turnstile>blk\<surd> \<and> (\<exists>T. E\<turnstile>res::T \<and> G\<turnstile>T\<preceq>rT))"
- wf_java_prog :: java_mb prog => bool
+ wf_java_prog :: "java_mb prog => bool"
"wf_java_prog G == wf_prog wf_java_mdecl G"
+
+lemma wt_is_type: "wf_prog wf_mb G \<Longrightarrow> ((G,L)\<turnstile>e::T \<longrightarrow> is_type G T) \<and>
+ ((G,L)\<turnstile>es[::]Ts \<longrightarrow> Ball (set Ts) (is_type G)) \<and> ((G,L)\<turnstile>c \<surd> \<longrightarrow> True)"
+apply (rule ty_expr_ty_exprs_wt_stmt.induct)
+apply auto
+apply ( erule typeof_empty_is_type)
+apply ( simp split add: split_if_asm)
+apply ( drule field_fields)
+apply ( drule (1) fields_is_type)
+apply ( simp (no_asm_simp))
+apply (assumption)
+apply (auto dest!: max_spec2mheads method_wf_mdecl is_type_rTI simp add: wf_mdecl_def)
+done
+
+lemmas ty_expr_is_type = wt_is_type [THEN conjunct1,THEN mp, COMP swap_prems_rl]
+
end