Theory EvenOdd

Up to index of Isabelle/HOL/IMPP

theory EvenOdd
imports Misc
uses [EvenOdd.ML]
begin

(*  Title:      HOL/IMPP/EvenOdd.thy
    ID:         $Id: EvenOdd.thy,v 1.6 2005/09/17 18:14:30 wenzelm Exp $
    Author:     David von Oheimb
    Copyright   1999 TUM
*)

header {* Example of mutually recursive procedures verified with Hoare logic *}

theory EvenOdd
imports Misc
begin

constdefs
  even :: "nat => bool"
  "even n == 2 dvd n"

consts
  Even :: pname
  Odd :: pname
axioms
  Even_neq_Odd: "Even ~= Odd"
  Arg_neq_Res:  "Arg  ~= Res"

constdefs
  evn :: com
 "evn == IF (%s. s<Arg> = 0)
         THEN Loc Res:==(%s. 0)
         ELSE(Loc Res:=CALL Odd(%s. s<Arg> - 1);;
              Loc Arg:=CALL Odd(%s. s<Arg> - 1);;
              Loc Res:==(%s. s<Res> * s<Arg>))"
  odd :: com
 "odd == IF (%s. s<Arg> = 0)
         THEN Loc Res:==(%s. 1)
         ELSE(Loc Res:=CALL Even (%s. s<Arg> - 1))"

defs
  bodies_def: "bodies == [(Even,evn),(Odd,odd)]"

consts
  Z_eq_Arg_plus   :: "nat => nat assn" ("Z=Arg+_" [50]50)
 "even_Z=(Res=0)" ::        "nat assn" ("Res'_ok")
defs
  Z_eq_Arg_plus_def: "Z=Arg+n == %Z s.      Z =  s<Arg>+n"
  Res_ok_def:       "Res_ok   == %Z s. even Z = (s<Res> = 0)"

ML {* use_legacy_bindings (the_context ()) *}

end

even

theorem even_0:

  EvenOdd.even 0

theorem not_even_1:

  EvenOdd.even (Suc 0) = False

theorem even_step:

  EvenOdd.even (Suc (Suc n)) = EvenOdd.even n

Arg, Res

theorem Z_eq_Arg_plus_def2:

  (Z=Arg+n) Z s = (Z = s<Arg> + n)

theorem Res_ok_def2:

  Res_ok Z s = (EvenOdd.even Z = (s<Res> = 0))

theorem body_Odd:

  body Odd = Some odd

theorem body_Even:

  body Even = Some evn

verification

theorem Odd_lemma:

  {{Z=Arg+0}. BODY Even .{Res_ok}}|-{Z=Arg+Suc 0}. odd .{Res_ok}

theorem Even_lemma:

  {{Z=Arg+1}. BODY Odd .{Res_ok}}|-{Z=Arg+0}. evn .{Res_ok}

theorem Even_ok_N:

  {}|-{Z=Arg+0}. BODY Even .{Res_ok}

theorem Even_ok_S:

  {}|-{Z=Arg+0}. BODY Even .{Res_ok}