--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/NanoJava/TypeRel.thy Sat Jun 16 20:06:42 2001 +0200
@@ -0,0 +1,145 @@
+(* Title: HOL/NanoJava/TypeRel.thy
+ ID: $Id$
+ Author: David von Oheimb
+ Copyright 2001 Technische Universitaet Muenchen
+*)
+
+header "Type relations"
+
+theory TypeRel = Decl:
+
+consts
+ widen :: "(ty \<times> ty ) set" (* widening *)
+ subcls1 :: "(cname \<times> cname) set" (* subclass *)
+
+syntax (xsymbols)
+ widen :: "[ty , ty ] => bool" ("_ \<preceq> _" [71,71] 70)
+ subcls1 :: "[cname, cname] => bool" ("_ \<prec>C1 _" [71,71] 70)
+ subcls :: "[cname, cname] => bool" ("_ \<preceq>C _" [71,71] 70)
+syntax
+ widen :: "[ty , ty ] => bool" ("_ <= _" [71,71] 70)
+ subcls1 :: "[cname, cname] => bool" ("_ <=C1 _" [71,71] 70)
+ subcls :: "[cname, cname] => bool" ("_ <=C _" [71,71] 70)
+
+translations
+ "C \<prec>C1 D" == "(C,D) \<in> subcls1"
+ "C \<preceq>C D" == "(C,D) \<in> subcls1^*"
+ "S \<preceq> T" == "(S,T) \<in> widen"
+
+consts
+ method :: "cname => (mname \<leadsto> methd)"
+ field :: "cname => (vnam \<leadsto> ty)"
+
+
+text {* The rest of this theory is not actually used. *}
+
+defs
+ (* direct subclass relation *)
+ subcls1_def: "subcls1 \<equiv> {(C,D). C\<noteq>Object \<and> (\<exists>c. class C = Some c \<and> super c=D)}"
+
+inductive widen intros (*widening, viz. method invocation conversion,
+ i.e. sort of syntactic subtyping *)
+ refl [intro!, simp]: "T \<preceq> T"
+ subcls : "C\<preceq>C D \<Longrightarrow> Class C \<preceq> Class D"
+ null [intro!]: "NT \<preceq> R"
+
+lemma subcls1D:
+ "C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>c. class C = Some c \<and> super c=D)"
+apply (unfold subcls1_def)
+apply auto
+done
+
+lemma subcls1I: "\<lbrakk>class C = Some m; super m = D; C \<noteq> Object\<rbrakk> \<Longrightarrow> C\<prec>C1D"
+apply (unfold subcls1_def)
+apply auto
+done
+
+lemma subcls1_def2:
+"subcls1 = (\<Sigma>C\<in>{C. is_class C} . {D. C\<noteq>Object \<and> super (the (class C)) = D})"
+apply (unfold subcls1_def is_class_def)
+apply auto
+done
+
+lemma finite_subcls1: "finite subcls1"
+apply(subst subcls1_def2)
+apply(rule finite_SigmaI [OF finite_is_class])
+apply(rule_tac B = "{super (the (class C))}" in finite_subset)
+apply auto
+done
+
+constdefs
+
+ ws_prog :: "bool"
+ "ws_prog \<equiv> \<forall>(C,c)\<in>set Prog. C\<noteq>Object \<longrightarrow>
+ is_class (super c) \<and> (super c,C)\<notin>subcls1^+"
+
+lemma ws_progD: "\<lbrakk>class C = Some c; C\<noteq>Object; ws_prog\<rbrakk> \<Longrightarrow>
+ is_class (super c) \<and> (super c,C)\<notin>subcls1^+"
+apply (unfold ws_prog_def class_def)
+apply (drule_tac map_of_SomeD)
+apply auto
+done
+
+lemma subcls1_irrefl_lemma1: "ws_prog \<Longrightarrow> subcls1^-1 \<inter> subcls1^+ = {}"
+by (fast dest: subcls1D ws_progD)
+
+(* context (theory "Transitive_Closure") *)
+lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
+apply (rule allI)
+apply (erule irrefl_tranclI)
+done
+
+lemmas subcls1_irrefl_lemma2 = subcls1_irrefl_lemma1 [THEN irrefl_tranclI']
+
+lemma subcls1_irrefl: "\<lbrakk>(x, y) \<in> subcls1; ws_prog\<rbrakk> \<Longrightarrow> x \<noteq> y"
+apply (rule irrefl_trancl_rD)
+apply (rule subcls1_irrefl_lemma2)
+apply auto
+done
+
+lemmas subcls1_acyclic = subcls1_irrefl_lemma2 [THEN acyclicI, standard]
+
+lemma wf_subcls1: "ws_prog \<Longrightarrow> wf (subcls1\<inverse>)"
+by (auto intro: finite_acyclic_wf_converse finite_subcls1 subcls1_acyclic)
+
+
+consts class_rec ::"cname \<Rightarrow> (class \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<leadsto> 'b)"
+
+recdef class_rec "subcls1\<inverse>"
+ "class_rec C = (\<lambda>f. case class C of None \<Rightarrow> arbitrary
+ | Some m \<Rightarrow> if wf (subcls1\<inverse>)
+ then (if C=Object then empty else class_rec (super m) f) ++ map_of (f m)
+ else arbitrary)"
+(hints intro: subcls1I)
+
+lemma class_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
+ class_rec C f = (if C = Object then empty else class_rec (super m) f) ++
+ map_of (f m)";
+apply (drule wf_subcls1)
+apply (rule class_rec.simps [THEN trans [THEN fun_cong]])
+apply assumption
+apply simp
+done
+
+(* methods of a class, with inheritance and hiding *)
+defs method_def: "method C \<equiv> class_rec C methods"
+
+lemma method_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
+method C = (if C=Object then empty else method (super m)) ++ map_of (methods m)"
+apply (unfold method_def)
+apply (erule (1) class_rec [THEN trans]);
+apply simp
+done
+
+
+(* fields of a class, with inheritance and hiding *)
+defs field_def: "field C \<equiv> class_rec C fields"
+
+lemma fields_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
+field C = (if C=Object then empty else field (super m)) ++ map_of (fields m)"
+apply (unfold field_def)
+apply (erule (1) class_rec [THEN trans]);
+apply simp
+done
+
+end