src/HOL/Hoare/Hoare.thy

Replaced obsolete "makestring" by Int.toString

(*  Title:      HOL/Hoare/Hoare.thy
    ID:         $Id$
    Author:     Norbert Galm & Tobias Nipkow
    Copyright   1995 TUM

Sugared semantic embedding of Hoare logic.
*)

Hoare = Arith +

types
  'a prog                       (* program syntax *)
  pvar = nat                    (* encoding of program variables ( < 26! ) *)
  'a state = pvar => 'a         (* program state *)
  'a exp = 'a state => 'a       (* denotation of expressions *)
  'a bexp = 'a state => bool    (* denotation of boolean expressions *)
  'a com = 'a state => 'a state => bool (* denotation of programs *)

syntax
  "@Skip"       :: 'a prog                              ("SKIP")
  "@Assign"     :: [id, 'a] => 'a prog          ("_ := _")
  "@Seq"        :: ['a prog, 'a prog] => 'a prog        ("_;//_" [0,1000] 999)
  "@If"         :: [bool, 'a prog, 'a prog] => 'a prog
                   ("IF _//THEN//  (_)//ELSE//  (_)//END")
  "@While"      :: [bool, bool, 'a prog] => 'a prog
                   ("WHILE _//DO {_}//  (_)//END")

  "@Spec"       :: [bool, 'a prog, bool] => bool        ("{_}//_//{_}")

constdefs
  (* denotational semantics *)

  Skip          :: 'a com
  "Skip s s' == (s=s')"

  Assign        :: [pvar, 'a exp] => 'a com
  "Assign v e s s' == (s' = (%x.if x=v then e(s) else s(x)))"

  Seq           :: ['a com, 'a com] => 'a com
  "Seq c c' s s' == ? s''. c s s'' & c' s'' s'"

  Cond          :: ['a bexp, 'a com, 'a com] => 'a com
  "Cond b c c' s s' == (b(s) --> c s s') & (~b s --> c' s s')"

  While         :: ['a bexp, 'a bexp, 'a com] => 'a com
  "While b inv c s s' == ? n. Iter n b c s s'"

  Iter          :: [nat, 'a bexp, 'a com] => 'a com
  "Iter n b c == nat_rec (%s s'.~b(s) & (s=s'))
             (%n_rec iter_rec.%s s'.b(s) & Seq c iter_rec s s') n"

  Spec          :: ['a bexp, 'a com, 'a bexp] => bool
  "Spec p c q == !s s'. c s s' --> p s --> q s'"

end

ML


(*** term manipulation ***)

(* name_in_term:bool (name:string,trm:term) bestimmt, ob in trm der Name name als Konstante,
   freie Var., scheme-Variable oder als Name fuer eine Lambda-Abstraktion vorkommt *)

fun name_in_term (name,Const (s,t))     =(name=s)
  | name_in_term (name,Free (s,t))      =(name=s)
  | name_in_term (name,Var ((s,i),t))   =(name=s ^ Int.toString i)
  | name_in_term (name,Abs (s,t,trm))   =(name=s) orelse (name_in_term (name,trm))
  | name_in_term (name,trm1 $ trm2)     =(name_in_term (name,trm1)) orelse (name_in_term (name,trm2))
  | name_in_term (_,_)                  =false;

(* variant_name:string (name:string,trm:term) liefert einen von name abgewandelten Namen (durch Anhaengen
   von genuegend vielen "_"), der nicht in trm vorkommt. Im Gegensatz zu variant_abs beruecktsichtigt es
   auch Namen von scheme-Variablen und von Lambda-Abstraktionen in trm *)

fun variant_name (name,trm)     =if name_in_term (name,trm)
                                        then variant_name (name ^ "_",trm)
                                        else name;

(* subst_term:term (von:term,nach:term,trm:term) liefert den Term, der aus
trm entsteht, wenn alle Teilterme, die gleich von sind, durch nach ersetzt
wurden *)

fun subst_term (von,nach,Abs (s,t,trm)) =if von=Abs (s,t,trm)
                                                then nach
                                                else Abs (s,t,subst_term (von,nach,trm))
  | subst_term (von,nach,trm1 $ trm2)   =if von=trm1 $ trm2
                                                then nach
                                                else subst_term (von,nach,trm1) $ subst_term (von,nach,trm2)
  | subst_term (von,nach,trm)           =if von=trm
                                                then nach
                                                else trm;


(* Translation between program vars ("a" - "z") and their encoding as
   natural numbers: "a" <==> 0, "b" <==> Suc(0), ..., "z" <==> 25 *)

fun is_pvarID s = size s=1 andalso "a"<=s andalso s<="z";

fun pvarID2pvar s =
  let fun rest2pvar (i,arg) =
            if i=0 then arg else rest2pvar(i-1, Syntax.const "Suc" $ arg)
  in rest2pvar(ord s - ord "a", Syntax.const "0") end;

fun pvar2pvarID trm     =
        let
                fun rest2pvarID (Const("0",_),i)                =chr (i + ord "a")
                  | rest2pvarID (Const("Suc",_) $ trm,i)        =rest2pvarID (trm,i+1)
        in
                rest2pvarID (trm,0)
        end;


(*** parse translations: syntax -> semantics ***)

(* term_tr:term (name:string,trm:term) ersetzt in trm alle freien Variablen, die eine pvarID
   darstellen, durch name $ pvarID2pvar(pvarID). Beispiel:
   bei name="s" und dem Term "a=b & a=X" wird der Term "s(0)=s(Suc(0)) & s(0)=X" geliefert *)

fun term_tr (name,Free (s,t)) = if is_pvarID s
                                then Syntax.free name $ pvarID2pvar s
                                else Free (s,t)
  | term_tr (name,Abs (s,t,trm)) = Abs (s,t,term_tr (name,trm))
  | term_tr (name,trm1 $ trm2)  = term_tr (name,trm1) $ term_tr (name,trm2)
  | term_tr (name,trm) = trm;

fun bool_tr B =
  let val name = variant_name("s",B)
  in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,B))) end;

fun exp_tr E =
  let val name = variant_name("s",E)
  in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,E))) end;

fun prog_tr (Const ("@Skip",_)) = Syntax.const "Skip"
  | prog_tr (Const ("@Assign",_) $ Free (V,_) $ E) =
      if is_pvarID V
      then Syntax.const "Assign" $ pvarID2pvar V $ exp_tr E
      else error("Not a valid program variable: " ^ quote V)
  | prog_tr (Const ("@Seq",_) $ C $ C') =
      Syntax.const "Seq" $ prog_tr C $ prog_tr C'
  | prog_tr (Const ("@If",_) $ B $ C $ C') =
      Syntax.const "Cond" $ bool_tr B $ prog_tr C $ prog_tr C'
  | prog_tr (Const ("@While",_) $ B $ INV $ C) =
      Syntax.const "While" $ bool_tr B $ bool_tr INV $ prog_tr C;

fun spec_tr [P,C,Q] = Syntax.const "Spec" $ bool_tr P $ prog_tr C $ bool_tr Q;

val parse_translation = [("@Spec",spec_tr)];


(*** print translations: semantics -> syntax ***)

(* term_tr':term (name:string,trm:term) ersetzt in trm alle Vorkommen von name $ pvar durch
   entsprechende freie Variablen, welche die pvarID zu pvar darstellen. Beispiel:
        bei name="s" und dem Term "s(0)=s(Suc(0)) & s(0)=X" wird der Term "a=b & a=X" geliefert *)

fun term_tr' (name,Free (s,t) $ trm)    =if name=s
                                                then Syntax.free (pvar2pvarID trm)
                                                else Free (s,t) $ term_tr' (name,trm)
  | term_tr' (name,Abs (s,t,trm))       =Abs (s,t,term_tr' (name,trm))
  | term_tr' (name,trm1 $ trm2)         =term_tr' (name,trm1) $ term_tr' (name,trm2)
  | term_tr' (name,trm)                 =trm;

fun bexp_tr' (Abs(_,_,b))       =term_tr' (variant_abs ("s",dummyT,b));

fun exp_tr' (Abs(_,_,e))        =term_tr' (variant_abs ("s",dummyT,e));

fun com_tr' (Const ("Skip",_))                  =Syntax.const "@Skip"
  | com_tr' (Const ("Assign",_) $ v $ e)                =Syntax.const "@Assign" $ Syntax.free (pvar2pvarID v) $ exp_tr' e
  | com_tr' (Const ("Seq",_) $ c $ c')          =Syntax.const "@Seq" $ com_tr' c $ com_tr' c'
  | com_tr' (Const ("Cond",_) $ b $ c $ c')             =Syntax.const "@If" $ bexp_tr' b $ com_tr' c $ com_tr' c'
  | com_tr' (Const ("While",_) $ b $ inv $ c)   =Syntax.const "@While" $ bexp_tr' b $ bexp_tr' inv $ com_tr' c;

fun spec_tr' [p,c,q]            =Syntax.const "@Spec" $ bexp_tr' p $ com_tr' c $ bexp_tr' q;

val print_translation   =[("Spec",spec_tr')];