Theory Example

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theory Example
imports Equivalence

(*  Title:      HOL/NanoJava/Example.thy
    ID:         $Id$
    Author:     David von Oheimb
    Copyright   2001 Technische Universitaet Muenchen
*)

header "Example"

theory Example imports Equivalence begin

text {*

\begin{verbatim}
class Nat {

  Nat pred;

  Nat suc() 
    { Nat n = new Nat(); n.pred = this; return n; }

  Nat eq(Nat n)
    { if (this.pred != null) if (n.pred != null) return this.pred.eq(n.pred);
                             else return n.pred; // false
      else if (n.pred != null) return this.pred; // false
           else return this.suc(); // true
    }

  Nat add(Nat n)
    { if (this.pred != null) return this.pred.add(n.suc()); else return n; }

  public static void main(String[] args) // test x+1=1+x
    {
        Nat one = new Nat().suc();
        Nat x   = new Nat().suc().suc().suc().suc();
        Nat ok = x.suc().eq(x.add(one));
        System.out.println(ok != null);
    }
}
\end{verbatim}

*}

axioms This_neq_Par [simp]: "This ≠ Par"
       Res_neq_This [simp]: "Res  ≠ This"


subsection "Program representation"

consts N    :: cname ("Nat") (* with mixfix because of clash with NatDef.Nat *)
consts pred :: fname
consts suc  :: mname
       add  :: mname
consts any  :: vname
syntax dummy:: expr ("<>")
       one  :: expr
translations 
      "<>"  == "LAcc any"
      "one" == "{Nat}new Nat..suc(<>)"

text {* The following properties could be derived from a more complete
        program model, which we leave out for laziness. *}

axioms Nat_no_subclasses [simp]: "D \<preceq>C Nat = (D=Nat)"

axioms method_Nat_add [simp]: "method Nat add = Some 
  (| par=Class Nat, res=Class Nat, lcl=[], 
   bdy= If((LAcc This..pred)) 
          (Res :== {Nat}(LAcc This..pred)..add({Nat}LAcc Par..suc(<>))) 
        Else Res :== LAcc Par |)),"

axioms method_Nat_suc [simp]: "method Nat suc = Some 
  (| par=NT, res=Class Nat, lcl=[], 
   bdy= Res :== new Nat;; LAcc Res..pred :== LAcc This |)),"

axioms field_Nat [simp]: "field Nat = empty(pred\<mapsto>Class Nat)"

lemma init_locs_Nat_add [simp]: "init_locs Nat add s = s"
by (simp add: init_locs_def init_vars_def)

lemma init_locs_Nat_suc [simp]: "init_locs Nat suc s = s"
by (simp add: init_locs_def init_vars_def)

lemma upd_obj_new_obj_Nat [simp]: 
  "upd_obj a pred v (new_obj a Nat s) = hupd(a\<mapsto>(Nat, empty(pred\<mapsto>v))) s"
by (simp add: new_obj_def init_vars_def upd_obj_def Let_def)


subsection "``atleast'' relation for interpretation of Nat ``values''"

consts Nat_atleast :: "state => val => nat => bool" ("_:_ ≥ _" [51, 51, 51] 50)
primrec "s:x≥0     = (x≠Null)"
        "s:x≥Suc n = (∃a. x=Addr a ∧ heap s a ≠ None ∧ s:get_field s a pred≥n)"

lemma Nat_atleast_lupd [rule_format, simp]: 
 "∀s v::val. lupd(x\<mapsto>y) s:v ≥ n = (s:v ≥ n)"
apply (induct n)
by  auto

lemma Nat_atleast_set_locs [rule_format, simp]: 
 "∀s v::val. set_locs l s:v ≥ n = (s:v ≥ n)"
apply (induct n)
by auto

lemma Nat_atleast_del_locs [rule_format, simp]: 
 "∀s v::val. del_locs s:v ≥ n = (s:v ≥ n)"
apply (induct n)
by auto

lemma Nat_atleast_NullD [rule_format]: "s:Null ≥ n --> False"
apply (induct n)
by auto

lemma Nat_atleast_pred_NullD [rule_format]: 
"Null = get_field s a pred ==> s:Addr a ≥ n --> n = 0"
apply (induct n)
by (auto dest: Nat_atleast_NullD)

lemma Nat_atleast_mono [rule_format]: 
"∀a. s:get_field s a pred ≥ n --> heap s a ≠ None --> s:Addr a ≥ n"
apply (induct n)
by auto

lemma Nat_atleast_newC [rule_format]: 
  "heap s aa = None ==> ∀v::val. s:v ≥ n --> hupd(aa\<mapsto>obj) s:v ≥ n"
apply (induct n)
apply  auto
apply  (case_tac "aa=a")
apply   auto
apply (tactic "smp_tac 1 1")
apply (case_tac "aa=a")
apply  auto
done


subsection "Proof(s) using the Hoare logic"

theorem add_homomorph_lb: 
  "{} \<turnstile> {λs. s:s<This> ≥ X ∧ s:s<Par> ≥ Y} Meth(Nat,add) {λs. s:s<Res> ≥ X+Y}"
apply (rule hoare_ehoare.Meth) (* 1 *)
apply clarsimp
apply (rule_tac P'= "λZ s. (s:s<This> ≥ fst Z ∧ s:s<Par> ≥ snd Z) ∧ D=Nat" and 
        Q'= "λZ s. s:s<Res> ≥ fst Z+snd Z" in AxSem.Conseq)
prefer 2
apply  (clarsimp simp add: init_locs_def init_vars_def)
apply rule
apply (case_tac "D = Nat", simp_all, rule_tac [2] cFalse)
apply (rule_tac P = "λZ Cm s. s:s<This> ≥ fst Z ∧ s:s<Par> ≥ snd Z" in AxSem.Impl1)
apply (clarsimp simp add: body_def)  (* 4 *)
apply (rename_tac n m)
apply (rule_tac Q = "λv s. (s:s<This> ≥ n ∧ s:s<Par> ≥ m) ∧ 
        (∃a. s<This> = Addr a ∧ v = get_field s a pred)" in hoare_ehoare.Cond)
apply  (rule hoare_ehoare.FAcc)
apply  (rule eConseq1)
apply   (rule hoare_ehoare.LAcc)
apply  fast
apply auto
prefer 2
apply  (rule hoare_ehoare.LAss)
apply  (rule eConseq1)
apply   (rule hoare_ehoare.LAcc)
apply  (auto dest: Nat_atleast_pred_NullD)
apply (rule hoare_ehoare.LAss)
apply (rule_tac 
    Q = "λv   s. (∀m. n = Suc m --> s:v ≥ m) ∧ s:s<Par> ≥ m" and 
    R = "λT P s. (∀m. n = Suc m --> s:T ≥ m) ∧ s:P  ≥ Suc m" 
    in hoare_ehoare.Call) (* 13 *)
apply   (rule hoare_ehoare.FAcc)
apply   (rule eConseq1)
apply    (rule hoare_ehoare.LAcc)
apply   clarify
apply   (drule sym, rotate_tac -1, frule (1) trans)
apply   simp
prefer 2
apply  clarsimp
apply  (rule hoare_ehoare.Meth) (* 17 *)
apply  clarsimp
apply  (case_tac "D = Nat", simp_all, rule_tac [2] cFalse)
apply  (rule AxSem.Conseq)
apply   rule
apply   (rule hoare_ehoare.Asm) (* 20 *)
apply   (rule_tac a = "((case n of 0 => 0 | Suc m => m),m+1)" in UN_I, rule+)
apply  (clarsimp split add: nat.split_asm dest!: Nat_atleast_mono)
apply rule
apply (rule hoare_ehoare.Call) (* 21 *)
apply   (rule hoare_ehoare.LAcc)
apply  rule
apply  (rule hoare_ehoare.LAcc)
apply clarify
apply (rule hoare_ehoare.Meth) (* 24 *)
apply clarsimp
apply  (case_tac "D = Nat", simp_all, rule_tac [2] cFalse)
apply (rule AxSem.Impl1)
apply (clarsimp simp add: body_def)
apply (rule hoare_ehoare.Comp) (* 26 *)
prefer 2
apply  (rule hoare_ehoare.FAss)
prefer 2
apply   rule
apply   (rule hoare_ehoare.LAcc)
apply  (rule hoare_ehoare.LAcc)
apply (rule hoare_ehoare.LAss)
apply (rule eConseq1)
apply  (rule hoare_ehoare.NewC) (* 32 *)
apply (auto dest!: new_AddrD elim: Nat_atleast_newC)
done


end