src/HOL/NanoJava/TypeRel.thy

revised fold1 proofs

(*  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 \<rightharpoonup> methd)"
  field  :: "cname => (fname \<rightharpoonup> ty)"


subsection "Declarations and properties not used in the meta theory"

text{* Direct subclass relation *}
defs
 subcls1_def: "subcls1 \<equiv> {(C,D). C\<noteq>Object \<and> (\<exists>c. class C = Some c \<and> super c=D)}"

text{* Widening, viz. method invocation conversion *}
inductive widen intros 
  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: {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)

(* irrefl_tranclI in Transitive_Closure.thy is more general *)
lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
by(blast elim: tranclE dest: trancl_into_rtrancl)


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 \<rightharpoonup> 'b)"

recdef (permissive) 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 flds"

lemma flds_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
field C = (if C=Object then empty else field (super m)) ++ map_of (flds m)"
apply (unfold field_def)
apply (erule (1) class_rec [THEN trans]);
apply simp
done

end