# File cooper_dec.ML

    (*  Title:      HOL/Integ/cooper_dec.ML
        ID:         $Id: cooper_dec.ML,v 1.22 2005/09/20 14:18:15 haftmann Exp $
        Author:     Amine Chaieb and Tobias Nipkow, TU Muenchen

    File containing the implementation of Cooper Algorithm
    decision procedure (intensively inspired from J.Harrison)
    *)

    signature COOPER_DEC =
    sig
      exception COOPER
      val is_arith_rel : term -> bool
      val mk_numeral : IntInf.int -> term
      val dest_numeral : term -> IntInf.int
      val is_numeral : term -> bool
      val zero : term
      val one : term
      val linear_cmul : IntInf.int -> term -> term
      val linear_add : string list -> term -> term -> term
      val linear_sub : string list -> term -> term -> term
      val linear_neg : term -> term
      val lint : string list -> term -> term
      val linform : string list -> term -> term
      val formlcm : term -> term -> IntInf.int
      val adjustcoeff : term -> IntInf.int -> term -> term
      val unitycoeff : term -> term -> term
      val divlcm : term -> term -> IntInf.int
      val bset : term -> term -> term list
      val aset : term -> term -> term list
      val linrep : string list -> term -> term -> term -> term
      val list_disj : term list -> term
      val list_conj : term list -> term
      val simpl : term -> term
      val fv : term -> string list
      val negate : term -> term
      val operations : (string * (IntInf.int * IntInf.int -> bool)) list
      val conjuncts : term -> term list
      val disjuncts : term -> term list
      val has_bound : term -> bool
      val minusinf : term -> term -> term
      val plusinf : term -> term -> term
      val onatoms : (term -> term) -> term -> term
      val evalc : term -> term
      val cooper_w : string list -> term -> (term option * term)
      val integer_qelim : Term.term -> Term.term
    end;

    structure  CooperDec : COOPER_DEC =
    struct

    (* ========================================================================= *)
    (* Cooper's algorithm for Presburger arithmetic.                             *)
    (* ========================================================================= *)
    exception COOPER;

    (* ------------------------------------------------------------------------- *)
    (* Lift operations up to numerals.                                           *)
    (* ------------------------------------------------------------------------- *)

    (*Assumption : The construction of atomar formulas in linearl arithmetic is based on
    relation operations of Type : [IntInf.int,IntInf.int]---> bool *)

    (* ------------------------------------------------------------------------- *)

    (*Function is_arith_rel returns true if and only if the term is an atomar presburger
    formula *)
    fun is_arith_rel tm = case tm of
    	 Const(p,Type ("fun",[Type ("Numeral.bin", []),Type ("fun",[Type ("Numeral.bin",
    	 []),Type ("bool",[])] )])) $ _ $_ => true
    	|Const(p,Type ("fun",[Type ("IntDef.int", []),Type ("fun",[Type ("IntDef.int",
    	 []),Type ("bool",[])] )])) $ _ $_ => true
    	|_ => false;

    (*Function is_arith_rel returns true if and only if the term is an operation of the
    form [int,int]---> int*)

    (*Transform a natural number to a term*)

    fun mk_numeral 0 = Const("0",HOLogic.intT)
       |mk_numeral 1 = Const("1",HOLogic.intT)
       |mk_numeral n = (HOLogic.number_of_const HOLogic.intT) $ (HOLogic.mk_bin n);

    (*Transform an Term to an natural number*)

    fun dest_numeral (Const("0",Type ("IntDef.int", []))) = 0
       |dest_numeral (Const("1",Type ("IntDef.int", []))) = 1
       |dest_numeral (Const ("Numeral.number_of",_) $ n) =
           HOLogic.dest_binum n;
    (*Some terms often used for pattern matching*)

    val zero = mk_numeral 0;
    val one = mk_numeral 1;

    (*Tests if a Term is representing a number*)

    fun is_numeral t = (t = zero) orelse (t = one) orelse (can dest_numeral t);

    (*maps a unary natural function on a term containing an natural number*)

    fun numeral1 f n = mk_numeral (f(dest_numeral n));

    (*maps a binary natural function on 2 term containing  natural numbers*)

    fun numeral2 f m n = mk_numeral(f(dest_numeral m) (dest_numeral n));

    (* ------------------------------------------------------------------------- *)
    (* Operations on canonical linear terms c1 * x1 + ... + cn * xn + k          *)
    (*                                                                           *)
    (* Note that we're quite strict: the ci must be present even if 1            *)
    (* (but if 0 we expect the monomial to be omitted) and k must be there       *)
    (* even if it's zero. Thus, it's a constant iff not an addition term.        *)
    (* ------------------------------------------------------------------------- *)

    fun linear_cmul n tm =  if n = 0 then zero else let fun times n k = n*k in
      ( case tm of
         (Const("op +",T)  $  (Const ("op *",T1 ) $c1 $  x1) $ rest) =>
           Const("op +",T) $ ((Const("op *",T1) $ (numeral1 (times n) c1) $ x1)) $ (linear_cmul n rest)
        |_ =>  numeral1 (times n) tm)
        end ;

    (* Whether the first of two items comes earlier in the list  *)
    fun earlier [] x y = false
    	|earlier (h::t) x y =if h = y then false
                  else if h = x then true
                  	else earlier t x y ;

    fun earlierv vars (Bound i) (Bound j) = i < j
       |earlierv vars (Bound _) _ = true
       |earlierv vars _ (Bound _)  = false
       |earlierv vars (Free (x,_)) (Free (y,_)) = earlier vars x y;

    fun linear_add vars tm1 tm2 =
      let fun addwith x y = x + y in
     (case (tm1,tm2) of
    	((Const ("op +",T1) $ ( Const("op *",T2) $ c1 $  x1) $ rest1),(Const
    	("op +",T3)$( Const("op *",T4) $ c2 $  x2) $ rest2)) =>
             if x1 = x2 then
                  let val c = (numeral2 (addwith) c1 c2)
    	      in
                  if c = zero then (linear_add vars rest1  rest2)
    	      else (Const("op +",T1) $ (Const("op *",T2) $ c $ x1) $ (linear_add vars  rest1 rest2))
                  end
    	   else
    		if earlierv vars x1 x2 then (Const("op +",T1) $
    		(Const("op *",T2)$ c1 $ x1) $ (linear_add vars rest1 tm2))
        	       else (Const("op +",T1) $ (Const("op *",T2) $ c2 $ x2) $ (linear_add vars tm1 rest2))
       	|((Const("op +",T1) $ (Const("op *",T2) $ c1 $ x1) $ rest1) ,_) =>
        	  (Const("op +",T1)$ (Const("op *",T2) $ c1 $ x1) $ (linear_add vars
    	  rest1 tm2))
       	|(_, (Const("op +",T1) $(Const("op *",T2) $ c2 $ x2) $ rest2)) =>
          	  (Const("op +",T1) $ (Const("op *",T2) $ c2 $ x2) $ (linear_add vars tm1
    	  rest2))
       	| (_,_) => numeral2 (addwith) tm1 tm2)

    	end;

    (*To obtain the unary - applyed on a formula*)

    fun linear_neg tm = linear_cmul (0 - 1) tm;

    (*Substraction of two terms *)

    fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);

    (* ------------------------------------------------------------------------- *)
    (* Linearize a term.                                                         *)
    (* ------------------------------------------------------------------------- *)

    (* linearises a term from the point of view of Variable Free (x,T).
    After this fuction the all expressions containig ths variable will have the form
     c*Free(x,T) + t where c is a constant ant t is a Term which is not containing
     Free(x,T)*)

    fun lint vars tm = if is_numeral tm then tm else case tm of
       (Free (x,T)) =>  (HOLogic.mk_binop "op +" ((HOLogic.mk_binop "op *" ((mk_numeral 1),Free (x,T))), zero))
      |(Bound i) =>  (Const("op +",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $
      (Const("op *",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ (mk_numeral 1) $ (Bound i)) $ zero)
      |(Const("uminus",_) $ t ) => (linear_neg (lint vars t))
      |(Const("op +",_) $ s $ t) => (linear_add vars (lint vars s) (lint vars t))
      |(Const("op -",_) $ s $ t) => (linear_sub vars (lint vars s) (lint vars t))
      |(Const ("op *",_) $ s $ t) =>
            let val s' = lint vars s
                val t' = lint vars t
            in
            if is_numeral s' then (linear_cmul (dest_numeral s') t')
            else if is_numeral t' then (linear_cmul (dest_numeral t') s')

             else raise COOPER
             end
      |_ =>  raise COOPER;

    (* ------------------------------------------------------------------------- *)
    (* Linearize the atoms in a formula, and eliminate non-strict inequalities.  *)
    (* ------------------------------------------------------------------------- *)

    fun mkatom vars p t = Const(p,HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ zero $ (lint vars t);

    fun linform vars (Const ("Divides.op dvd",_) $ c $ t) =
        if is_numeral c then
          let val c' = (mk_numeral(abs(dest_numeral c)))
          in (HOLogic.mk_binrel "Divides.op dvd" (c,lint vars t))
          end
        else (warning "Nonlinear term --- Non numeral leftside at dvd"
          ;raise COOPER)
      |linform vars  (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ s $ t ) = (mkatom vars "op =" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s) )
      |linform vars  (Const("op <",_)$ s $t ) = (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s))
      |linform vars  (Const("op >",_) $ s $ t ) = (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ s $ t))
      |linform vars  (Const("op <=",_)$ s $ t ) =
            (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ (Const("op +",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $t $(mk_numeral 1)) $ s))
      |linform vars  (Const("op >=",_)$ s $ t ) =
            (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT -->
    	HOLogic.intT) $ (Const("op +",HOLogic.intT --> HOLogic.intT -->
    	HOLogic.intT) $s $(mk_numeral 1)) $ t))

       |linform vars  fm =  fm;

    (* ------------------------------------------------------------------------- *)
    (* Post-NNF transformation eliminating negated inequalities.                 *)
    (* ------------------------------------------------------------------------- *)

    fun posineq fm = case fm of
     (Const ("Not",_)$(Const("op <",_)$ c $ t)) =>
       (HOLogic.mk_binrel "op <"  (zero , (linear_sub [] (mk_numeral 1) (linear_add [] c t ) )))
      | ( Const ("op &",_) $ p $ q)  => HOLogic.mk_conj (posineq p,posineq q)
      | ( Const ("op |",_) $ p $ q ) => HOLogic.mk_disj (posineq p,posineq q)
      | _ => fm;

    (* ------------------------------------------------------------------------- *)
    (* Find the LCM of the coefficients of x.                                    *)
    (* ------------------------------------------------------------------------- *)
    (*gcd calculates gcd (a,b) and helps lcm_num calculating lcm (a,b)*)

    (*BEWARE: replaces Library.gcd!! There is also Library.lcm!*)
    fun gcd (a:IntInf.int) b = if a=0 then b else gcd (b mod a) a ;
    fun lcm_num a b = (abs a*b) div (gcd (abs a) (abs b));

    fun formlcm x fm = case fm of
        (Const (p,_)$ _ $(Const ("op +", _)$(Const ("op *",_)$ c $ y ) $z ) ) =>  if
        (is_arith_rel fm) andalso (x = y) then  (abs(dest_numeral c)) else 1
      | ( Const ("Not", _) $p) => formlcm x p
      | ( Const ("op &",_) $ p $ q) => lcm_num (formlcm x p) (formlcm x q)
      | ( Const ("op |",_) $ p $ q )=> lcm_num (formlcm x p) (formlcm x q)
      |  _ => 1;

    (* ------------------------------------------------------------------------- *)
    (* Adjust all coefficients of x in formula; fold in reduction to +/- 1.      *)
    (* ------------------------------------------------------------------------- *)

    fun adjustcoeff x l fm =
         case fm of
          (Const(p,_) $d $( Const ("op +", _)$(Const ("op *",_) $
          c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
            let val m = l div (dest_numeral c)
                val n = (if p = "op <" then abs(m) else m)
                val xtm = HOLogic.mk_binop "op *" ((mk_numeral (m div n)), x)
    	in
            (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
    	end
    	else fm
      |( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeff x l p)
      |( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeff x l p) $(adjustcoeff x l q)
      |( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeff x l p)$ (adjustcoeff x l q)
      |_ => fm;

    (* ------------------------------------------------------------------------- *)
    (* Hence make coefficient of x one in existential formula.                   *)
    (* ------------------------------------------------------------------------- *)

    fun unitycoeff x fm =
      let val l = formlcm x fm
          val fm' = adjustcoeff x l fm in
          if l = 1 then fm'
    	 else
         let val xp = (HOLogic.mk_binop "op +"
         		((HOLogic.mk_binop "op *" ((mk_numeral 1), x )), zero))
    	in
          HOLogic.conj $(HOLogic.mk_binrel "Divides.op dvd" ((mk_numeral l) , xp )) $ (adjustcoeff x l fm)
          end
      end;

    (* adjustcoeffeq l fm adjusts the coeffitients c_i of x  overall in fm to l*)
    (* Here l must be a multiple of all c_i otherwise the obtained formula is not equivalent*)
    (*
    fun adjustcoeffeq x l fm =
        case fm of
          (Const(p,_) $d $( Const ("op +", _)$(Const ("op *",_) $
          c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
            let val m = l div (dest_numeral c)
                val n = (if p = "op <" then abs(m) else m)
                val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x))
                in (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
    	    end
    	else fm
      |( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeffeq x l p)
      |( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeffeq x l p) $(adjustcoeffeq x l q)
      |( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeffeq x l p)$ (adjustcoeffeq x l q)
      |_ => fm;

    *)

    (* ------------------------------------------------------------------------- *)
    (* The "minus infinity" version.                                             *)
    (* ------------------------------------------------------------------------- *)

    fun minusinf x fm = case fm of
        (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
      	 if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
    	 				 else fm

      |(Const("op <",_) $ c $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z
      )) => if (x = y)
    	then if (pm1 = one) andalso (c = zero) then HOLogic.false_const
    	     else if (dest_numeral pm1 = ~1) andalso (c = zero) then HOLogic.true_const
    	          else error "minusinf : term not in normal form!!!"
    	else fm

      |(Const ("Not", _) $ p) => HOLogic.Not $ (minusinf x p)
      |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (minusinf x p) $ (minusinf x q)
      |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (minusinf x p) $ (minusinf x q)
      |_ => fm;

    (* ------------------------------------------------------------------------- *)
    (* The "Plus infinity" version.                                             *)
    (* ------------------------------------------------------------------------- *)

    fun plusinf x fm = case fm of
        (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
      	 if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
    	 				 else fm

      |(Const("op <",_) $ c $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z
      )) => if (x = y)
    	then if (pm1 = one) andalso (c = zero) then HOLogic.true_const
    	     else if (dest_numeral pm1 = ~1) andalso (c = zero) then HOLogic.false_const
    	     else error "plusinf : term not in normal form!!!"
    	else fm

      |(Const ("Not", _) $ p) => HOLogic.Not $ (plusinf x p)
      |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (plusinf x p) $ (plusinf x q)
      |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (plusinf x p) $ (plusinf x q)
      |_ => fm;

    (* ------------------------------------------------------------------------- *)
    (* The LCM of all the divisors that involve x.                               *)
    (* ------------------------------------------------------------------------- *)

    fun divlcm x (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z ) ) =
            if x = y then abs(dest_numeral d) else 1
      |divlcm x ( Const ("Not", _) $ p) = divlcm x p
      |divlcm x ( Const ("op &",_) $ p $ q) = lcm_num (divlcm x p) (divlcm x q)
      |divlcm x ( Const ("op |",_) $ p $ q ) = lcm_num (divlcm x p) (divlcm x q)
      |divlcm x  _ = 1;

    (* ------------------------------------------------------------------------- *)
    (* Construct the B-set.                                                      *)
    (* ------------------------------------------------------------------------- *)

    fun bset x fm = case fm of
       (Const ("Not", _) $ p) => if (is_arith_rel p) then
              (case p of
    	      (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $c2 $y) $a ) )
    	             => if (is_arith_rel p) andalso (x=	y) andalso (c2 = one) andalso (c1 = zero)
    	                then [linear_neg a]
    			else  bset x p
       	  |_ =>[])

    			else bset x p
      |(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +",_) $ (Const ("op *",_) $c2 $ x) $ a)) =>  if (c1 =zero) andalso (c2 = one) then [linear_neg(linear_add [] a (mk_numeral 1))]  else []
      |(Const ("op <",_) $ c1$ (Const ("op +",_) $(Const ("op *",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg a] else []
      |(Const ("op &",_) $ p $ q) => (bset x p) union (bset x q)
      |(Const ("op |",_) $ p $ q) => (bset x p) union (bset x q)
      |_ => [];

    (* ------------------------------------------------------------------------- *)
    (* Construct the A-set.                                                      *)
    (* ------------------------------------------------------------------------- *)

    fun aset x fm = case fm of
       (Const ("Not", _) $ p) => if (is_arith_rel p) then
              (case p of
    	      (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $c2 $y) $a ) )
    	             => if (x=	y) andalso (c2 = one) andalso (c1 = zero)
    	                then [linear_neg a]
    			else  []
       	  |_ =>[])

    			else aset x p
      |(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +",_) $ (Const ("op *",_) $c2 $ x) $ a)) =>  if (c1 =zero) andalso (c2 = one) then [linear_sub [] (mk_numeral 1) a]  else []
      |(Const ("op <",_) $ c1$ (Const ("op +",_) $(Const ("op *",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = (mk_numeral (~1))) then [a] else []
      |(Const ("op &",_) $ p $ q) => (aset x p) union (aset x q)
      |(Const ("op |",_) $ p $ q) => (aset x p) union (aset x q)
      |_ => [];

    (* ------------------------------------------------------------------------- *)
    (* Replace top variable with another linear form, retaining canonicality.    *)
    (* ------------------------------------------------------------------------- *)

    fun linrep vars x t fm = case fm of
       ((Const(p,_)$ d $ (Const("op +",_)$(Const("op *",_)$ c $ y) $ z))) =>
          if (x = y) andalso (is_arith_rel fm)
          then
            let val ct = linear_cmul (dest_numeral c) t
    	in (HOLogic.mk_binrel p (d, linear_add vars ct z))
    	end
    	else fm
      |(Const ("Not", _) $ p) => HOLogic.Not $ (linrep vars x t p)
      |(Const ("op &",_) $ p $ q) => HOLogic.conj $ (linrep vars x t p) $ (linrep vars x t q)
      |(Const ("op |",_) $ p $ q) => HOLogic.disj $ (linrep vars x t p) $ (linrep vars x t q)
      |_ => fm;

    (* ------------------------------------------------------------------------- *)
    (* Evaluation of constant expressions.                                       *)
    (* ------------------------------------------------------------------------- *)

    (* An other implementation of divides, that covers more cases*)

    exception DVD_UNKNOWN

    fun dvd_op (d, t) =
     if not(is_numeral d) then raise DVD_UNKNOWN
     else let
       val dn = dest_numeral d
       fun coeffs_of x = case x of
         Const(p,_) $ tl $ tr =>
           if p = "op +" then (coeffs_of tl) union (coeffs_of tr)
              else if p = "op *"
    	        then if (is_numeral tr)
    		 then [(dest_numeral tr) * (dest_numeral tl)]
    		 else [dest_numeral tl]
    	        else []
        |_ => if (is_numeral t) then [dest_numeral t]  else []
       val ts = coeffs_of t
       in case ts of
         [] => raise DVD_UNKNOWN
        |_  => foldr (fn(k,r) => r andalso (k mod dn = 0)) true ts
       end;

    val operations =
      [("op =",op=), ("op <",IntInf.<), ("op >",IntInf.>), ("op <=",IntInf.<=) ,
       ("op >=",IntInf.>=),
       ("Divides.op dvd",fn (x,y) =>((IntInf.mod(y, x)) = 0))];

    fun applyoperation (SOME f) (a,b) = f (a, b)
        |applyoperation _ (_, _) = false;

    (*Evaluation of constant atomic formulas*)
     (*FIXME : This is an optimation but still incorrect !! *)
    (*
    fun evalc_atom at = case at of
      (Const (p,_) $ s $ t) =>
       (if p="Divides.op dvd" then
         ((if dvd_op(s,t) then HOLogic.true_const
         else HOLogic.false_const)
          handle _ => at)
        else
      case AList.lookup (op =) operations p of
        SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const else HOLogic.false_const)
        handle _ => at)
          | _ =>  at)
          |Const("Not",_)$(Const (p,_) $ s $ t) =>(
      case AList.lookup (op =) operations p of
        SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then
        HOLogic.false_const else HOLogic.true_const)
        handle _ => at)
          | _ =>  at)
          | _ =>  at;

    *)

    fun evalc_atom at = case at of
      (Const (p,_) $ s $ t) =>
       ( case AList.lookup (op =) operations p of
        SOME f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const
                    else HOLogic.false_const)
        handle _ => at)
          | _ =>  at)
          |Const("Not",_)$(Const (p,_) $ s $ t) =>(
      case AList.lookup (op =) operations p of
        SOME f => ((if (f ((dest_numeral s),(dest_numeral t)))
                   then HOLogic.false_const else HOLogic.true_const)
        handle _ => at)
          | _ =>  at)
          | _ =>  at;

     (*Function onatoms apllys function f on the atomic formulas involved in a.*)

    fun onatoms f a = if (is_arith_rel a) then f a else case a of

      	(Const ("Not",_) $ p) => if is_arith_rel p then HOLogic.Not $ (f p)

    				else HOLogic.Not $ (onatoms f p)
      	|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (onatoms f p) $ (onatoms f q)
      	|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (onatoms f p) $ (onatoms f q)
      	|(Const ("op -->",_) $ p $ q) => HOLogic.imp $ (onatoms f p) $ (onatoms f q)
      	|((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) => (Const ("op =", [HOLogic.boolT, HOLogic.boolT] ---> HOLogic.boolT)) $ (onatoms f p) $ (onatoms f q)
      	|(Const("All",_) $ Abs(x,T,p)) => Const("All", [HOLogic.intT -->
    	HOLogic.boolT] ---> HOLogic.boolT)$ Abs (x ,T, (onatoms f p))
      	|(Const("Ex",_) $ Abs(x,T,p)) => Const("Ex", [HOLogic.intT --> HOLogic.boolT]---> HOLogic.boolT) $ Abs( x ,T, (onatoms f p))
      	|_ => a;

    val evalc = onatoms evalc_atom;

    (* ------------------------------------------------------------------------- *)
    (* Hence overall quantifier elimination.                                     *)
    (* ------------------------------------------------------------------------- *)

    (*list_disj[conj] makes a disj[conj] of a given list. used with conjucts or disjuncts
    it liearises iterated conj[disj]unctions. *)

    fun disj_help p q = HOLogic.disj $ p $ q ;

    fun list_disj l =
      if l = [] then HOLogic.false_const else Utils.end_itlist disj_help l;

    fun conj_help p q = HOLogic.conj $ p $ q ;

    fun list_conj l =
      if l = [] then HOLogic.true_const else Utils.end_itlist conj_help l;

    (*Simplification of Formulas *)

    (*Function q_bnd_chk checks if a quantified Formula makes sens : Means if in
    the body of the existential quantifier there are bound variables to the
    existential quantifier.*)

    fun has_bound fm =let fun has_boundh fm i = case fm of
    		 Bound n => (i = n)
    		 |Abs (_,_,p) => has_boundh p (i+1)
    		 |t1 $ t2 => (has_boundh t1 i) orelse (has_boundh t2 i)
    		 |_ =>false

    in  case fm of
    	Bound _ => true
           |Abs (_,_,p) => has_boundh p 0
           |t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
           |_ =>false
    end;

    (*has_sub_abs checks if in a given Formula there are subformulas which are quantifed
    too. Is no used no more.*)

    fun has_sub_abs fm = case fm of
    		 Abs (_,_,_) => true
    		 |t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
    		 |_ =>false ;

    (*update_bounds called with i=0 udates the numeration of bounded variables because the
    formula will not be quantified any more.*)

    fun update_bounds fm i = case fm of
    		 Bound n => if n >= i then Bound (n-1) else fm
    		 |Abs (x,T,p) => Abs(x,T,(update_bounds p (i+1)))
    		 |t1 $ t2 => (update_bounds t1 i) $ (update_bounds t2 i)
    		 |_ => fm ;

    (*psimpl : Simplification of propositions (general purpose)*)
    fun psimpl1 fm = case fm of
        Const("Not",_) $ Const ("False",_) => HOLogic.true_const
      | Const("Not",_) $ Const ("True",_) => HOLogic.false_const
      | Const("op &",_) $ Const ("False",_) $ q => HOLogic.false_const
      | Const("op &",_) $ p $ Const ("False",_)  => HOLogic.false_const
      | Const("op &",_) $ Const ("True",_) $ q => q
      | Const("op &",_) $ p $ Const ("True",_) => p
      | Const("op |",_) $ Const ("False",_) $ q => q
      | Const("op |",_) $ p $ Const ("False",_)  => p
      | Const("op |",_) $ Const ("True",_) $ q => HOLogic.true_const
      | Const("op |",_) $ p $ Const ("True",_)  => HOLogic.true_const
      | Const("op -->",_) $ Const ("False",_) $ q => HOLogic.true_const
      | Const("op -->",_) $ Const ("True",_) $  q => q
      | Const("op -->",_) $ p $ Const ("True",_)  => HOLogic.true_const
      | Const("op -->",_) $ p $ Const ("False",_)  => HOLogic.Not $  p
      | Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("True",_) $ q => q
      | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("True",_) => p
      | Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("False",_) $ q => HOLogic.Not $  q
      | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("False",_)  => HOLogic.Not $  p
      | _ => fm;

    fun psimpl fm = case fm of
       Const ("Not",_) $ p => psimpl1 (HOLogic.Not $ (psimpl p))
      | Const("op &",_) $ p $ q => psimpl1 (HOLogic.mk_conj (psimpl p,psimpl q))
      | Const("op |",_) $ p $ q => psimpl1 (HOLogic.mk_disj (psimpl p,psimpl q))
      | Const("op -->",_) $ p $ q => psimpl1 (HOLogic.mk_imp(psimpl p,psimpl q))
      | Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q => psimpl1 (HOLogic.mk_eq(psimpl p,psimpl q))
      | _ => fm;

    (*simpl : Simplification of Terms involving quantifiers too.
     This function is able to drop out some quantified expressions where there are no
     bound varaibles.*)

    fun simpl1 fm  =
      case fm of
        Const("All",_) $Abs(x,_,p) => if (has_bound fm ) then fm
        				else (update_bounds p 0)
      | Const("Ex",_) $ Abs (x,_,p) => if has_bound fm then fm
        				else (update_bounds p 0)
      | _ => psimpl fm;

    fun simpl fm = case fm of
        Const ("Not",_) $ p => simpl1 (HOLogic.Not $(simpl p))
      | Const ("op &",_) $ p $ q => simpl1 (HOLogic.mk_conj (simpl p ,simpl q))
      | Const ("op |",_) $ p $ q => simpl1 (HOLogic.mk_disj (simpl p ,simpl q ))
      | Const ("op -->",_) $ p $ q => simpl1 (HOLogic.mk_imp(simpl p ,simpl q ))
      | Const("op =", Type ("fun",[Type ("bool", []),_]))$ p $ q => simpl1
      (HOLogic.mk_eq(simpl p ,simpl q ))
    (*  | Const ("All",Ta) $ Abs(Vn,VT,p) => simpl1(Const("All",Ta) $
      Abs(Vn,VT,simpl p ))
      | Const ("Ex",Ta)  $ Abs(Vn,VT,p) => simpl1(Const("Ex",Ta)  $
      Abs(Vn,VT,simpl p ))
    *)
      | _ => fm;

    (* ------------------------------------------------------------------------- *)

    (* Puts fm into NNF*)

    fun  nnf fm = if (is_arith_rel fm) then fm
    else (case fm of
      ( Const ("op &",_) $ p $ q)  => HOLogic.conj $ (nnf p) $(nnf q)
      | (Const("op |",_) $ p $q) => HOLogic.disj $ (nnf p)$(nnf q)
      | (Const ("op -->",_)  $ p $ q) => HOLogic.disj $ (nnf (HOLogic.Not $ p)) $ (nnf q)
      | ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) =>(HOLogic.disj $ (HOLogic.conj $ (nnf p) $ (nnf q)) $ (HOLogic.conj $ (nnf (HOLogic.Not $ p) ) $ (nnf(HOLogic.Not $ q))))
      | (Const ("Not",_)) $ ((Const ("Not",_)) $ p) => (nnf p)
      | (Const ("Not",_)) $ (( Const ("op &",_)) $ p $ q) =>HOLogic.disj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $q))
      | (Const ("Not",_)) $ (( Const ("op |",_)) $ p $ q) =>HOLogic.conj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $ q))
      | (Const ("Not",_)) $ (( Const ("op -->",_)) $ p $ q ) =>HOLogic.conj $ (nnf p) $(nnf(HOLogic.Not $ q))
      | (Const ("Not",_)) $ ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q ) =>(HOLogic.disj $ (HOLogic.conj $(nnf p) $ (nnf(HOLogic.Not $ q))) $ (HOLogic.conj $(nnf(HOLogic.Not $ p)) $ (nnf q)))
      | _ => fm);

    (* Function remred to remove redundancy in a list while keeping the order of appearance of the
    elements. but VERY INEFFICIENT!! *)

    fun remred1 el [] = []
        |remred1 el (h::t) = if el=h then (remred1 el t) else h::(remred1 el t);

    fun remred [] = []
        |remred (x::l) =  x::(remred1 x (remred l));

    (*Makes sure that all free Variables are of the type integer but this function is only
    used temporarily, this job must be done by the parser later on.*)

    fun mk_uni_vars T  (node $ rest) = (case node of
        Free (name,_) => Free (name,T) $ (mk_uni_vars T rest)
        |_=> (mk_uni_vars T node) $ (mk_uni_vars T rest )  )
        |mk_uni_vars T (Free (v,_)) = Free (v,T)
        |mk_uni_vars T tm = tm;

    fun mk_uni_int T (Const ("0",T2)) = if T = T2 then (mk_numeral 0) else (Const ("0",T2))
        |mk_uni_int T (Const ("1",T2)) = if T = T2 then (mk_numeral 1) else (Const ("1",T2))
        |mk_uni_int T (node $ rest) = (mk_uni_int T node) $ (mk_uni_int T rest )
        |mk_uni_int T (Abs(AV,AT,p)) = Abs(AV,AT,mk_uni_int T p)
        |mk_uni_int T tm = tm;

    (* Minusinfinity Version*)
    fun myupto (m:IntInf.int) n = if m > n then [] else m::(myupto (m+1) n)

    fun coopermi vars1 fm =
      case fm of
       Const ("Ex",_) $ Abs(x0,T,p0) =>
       let
        val (xn,p1) = variant_abs (x0,T,p0)
        val x = Free (xn,T)
        val vars = (xn::vars1)
        val p = unitycoeff x  (posineq (simpl p1))
        val p_inf = simpl (minusinf x p)
        val bset = bset x p
        val js = myupto 1 (divlcm x p)
        fun p_element j b = linrep vars x (linear_add vars b (mk_numeral j)) p
        fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) bset)
       in (list_disj (map stage js))
        end
      | _ => error "cooper: not an existential formula";

    (* The plusinfinity version of cooper*)
    fun cooperpi vars1 fm =
      case fm of
       Const ("Ex",_) $ Abs(x0,T,p0) => let
        val (xn,p1) = variant_abs (x0,T,p0)
        val x = Free (xn,T)
        val vars = (xn::vars1)
        val p = unitycoeff x  (posineq (simpl p1))
        val p_inf = simpl (plusinf x p)
        val aset = aset x p
        val js = myupto 1 (divlcm x p)
        fun p_element j a = linrep vars x (linear_sub vars a (mk_numeral j)) p
        fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) aset)
       in (list_disj (map stage js))
       end
      | _ => error "cooper: not an existential formula";

    (* Try to find a withness for the formula *)

    fun inf_w mi d vars x p =
      let val f = if mi then minusinf else plusinf in
       case (simpl (minusinf x p)) of
       Const("True",_)  => (SOME (mk_numeral 1), HOLogic.true_const)
      |Const("False",_) => (NONE,HOLogic.false_const)
      |F =>
          let
          fun h n =
           case ((simpl o evalc) (linrep vars x (mk_numeral n) F)) of
    	Const("True",_) => (SOME (mk_numeral n),HOLogic.true_const)
           |F' => if n=1 then (NONE,F')
    	     else let val (rw,rf) = h (n-1) in
    	       (rw,HOLogic.mk_disj(F',rf))
    	     end

          in (h d)
          end
      end;

    fun set_w d b st vars x p = let
        fun h ns = case ns of
        [] => (NONE,HOLogic.false_const)
       |n::nl => ( case ((simpl o evalc) (linrep vars x n p)) of
          Const("True",_) => (SOME n,HOLogic.true_const)
          |F' => let val (rw,rf) = h nl
                 in (rw,HOLogic.mk_disj(F',rf))
    	     end)
        val f = if b then linear_add else linear_sub
        val p_elements = foldr (fn (i,l) => l union (map (fn e => f [] e (mk_numeral i)) st)) [] (myupto 1 d)
        in h p_elements
        end;

    fun withness d b st vars x p = case (inf_w b d vars x p) of
       (SOME n,_) => (SOME n,HOLogic.true_const)
      |(NONE,Pinf) => (case (set_w d b st vars x p) of
        (SOME n,_) => (SOME n,HOLogic.true_const)
        |(_,Pst) => (NONE,HOLogic.mk_disj(Pinf,Pst)));

    (*Cooper main procedure*)

    exception STAGE_TRUE;

    fun cooper vars1 fm =
      case fm of
       Const ("Ex",_) $ Abs(x0,T,p0) => let
        val (xn,p1) = variant_abs (x0,T,p0)
        val x = Free (xn,T)
        val vars = (xn::vars1)
    (*     val p = unitycoeff x  (posineq (simpl p1)) *)
        val p = unitycoeff x  p1
        val ast = aset x p
        val bst = bset x p
        val js = myupto 1 (divlcm x p)
        val (p_inf,f,S ) =
        if (length bst) <= (length ast)
         then (simpl (minusinf x p),linear_add,bst)
         else (simpl (plusinf x p), linear_sub,ast)
        fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p
        fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S)
        fun stageh n = ((if n = 0 then []
    	else
    	let
    	val nth_stage = simpl (evalc (stage n))
    	in
    	if (nth_stage = HOLogic.true_const)
    	  then raise STAGE_TRUE
    	  else if (nth_stage = HOLogic.false_const) then stageh (n-1)
    	    else nth_stage::(stageh (n-1))
    	end )
            handle STAGE_TRUE => [HOLogic.true_const])
        val slist = stageh (divlcm x p)
       in (list_disj slist)
       end
      | _ => error "cooper: not an existential formula";

    (* A Version of cooper that returns a withness *)
    fun cooper_w vars1 fm =
      case fm of
       Const ("Ex",_) $ Abs(x0,T,p0) => let
        val (xn,p1) = variant_abs (x0,T,p0)
        val x = Free (xn,T)
        val vars = (xn::vars1)
    (*     val p = unitycoeff x  (posineq (simpl p1)) *)
        val p = unitycoeff x  p1
        val ast = aset x p
        val bst = bset x p
        val d = divlcm x p
        val (p_inf,S ) =
        if (length bst) <= (length ast)
         then (true,bst)
         else (false,ast)
        in withness d p_inf S vars x p
    (*    fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p
        fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S)
       in (list_disj (map stage js))
    *)
       end
      | _ => error "cooper: not an existential formula";

    (* ------------------------------------------------------------------------- *)
    (* Free variables in terms and formulas.	                             *)
    (* ------------------------------------------------------------------------- *)

    fun fvt tml = case tml of
        [] => []
      | Free(x,_)::r => x::(fvt r)

    fun fv fm = fvt (term_frees fm);

    (* ========================================================================= *)
    (* Quantifier elimination.                                                   *)
    (* ========================================================================= *)
    (*conj[/disj]uncts lists iterated conj[disj]unctions*)

    fun disjuncts fm = case fm of
        Const ("op |",_) $ p $ q => (disjuncts p) @ (disjuncts q)
      | _ => [fm];

    fun conjuncts fm = case fm of
        Const ("op &",_) $p $ q => (conjuncts p) @ (conjuncts q)
      | _ => [fm];

    (* ------------------------------------------------------------------------- *)
    (* Lift procedure given literal modifier, formula normalizer & basic quelim. *)
    (* ------------------------------------------------------------------------- *)

    fun lift_qelim afn nfn qfn isat =
    let
    fun qelift vars fm = if (isat fm) then afn vars fm
    else
    case fm of
      Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
      | Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
      | Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
      | Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
      | Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
      | Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
      | (e as Const ("Ex",_)) $ Abs (x,T,p)  =>  qfn vars (e$Abs (x,T,(nfn(qelift (x::vars) p))))
      | _ => fm

    in (fn fm => qelift (fv fm) fm)
    end;

    (*
    fun lift_qelim afn nfn qfn isat =
     let   fun qelim x vars p =
      let val cjs = conjuncts p
          val (ycjs,ncjs) = List.partition (has_bound) cjs in
          (if ycjs = [] then p else
                              let val q = (qfn vars ((HOLogic.exists_const HOLogic.intT
    			  ) $ Abs(x,HOLogic.intT,(list_conj ycjs)))) in
                              (fold_rev conj_help ncjs q)
    			  end)
           end

      fun qelift vars fm = if (isat fm) then afn vars fm
        else
        case fm of
          Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
        | Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
        | Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
        | Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
        | Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
        | Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
        | Const ("Ex",_) $ Abs (x,T,p)  => let  val djs = disjuncts(nfn(qelift (x::vars) p)) in
        			list_disj(map (qelim x vars) djs) end
        | _ => fm

      in (fn fm => simpl(qelift (fv fm) fm))
      end;
    *)

    (* ------------------------------------------------------------------------- *)
    (* Cleverer (proposisional) NNF with conditional and literal modification.   *)
    (* ------------------------------------------------------------------------- *)

    (*Function Negate used by cnnf, negates a formula p*)

    fun negate (Const ("Not",_) $ p) = p
        |negate p = (HOLogic.Not $ p);

    fun cnnf lfn =
      let fun cnnfh fm = case  fm of
          (Const ("op &",_) $ p $ q) => HOLogic.mk_conj(cnnfh p,cnnfh q)
        | (Const ("op |",_) $ p $ q) => HOLogic.mk_disj(cnnfh p,cnnfh q)
        | (Const ("op -->",_) $ p $q) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh q)
        | (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q) => HOLogic.mk_disj(
        		HOLogic.mk_conj(cnnfh p,cnnfh q),
    		HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $q)))

        | (Const ("Not",_) $ (Const("Not",_) $ p)) => cnnfh p
        | (Const ("Not",_) $ (Const ("op &",_) $ p $ q)) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
        | (Const ("Not",_) $(Const ("op |",_) $ (Const ("op &",_) $ p $ q) $
        			(Const ("op &",_) $ p1 $ r))) => if p1 = negate p then
    		         HOLogic.mk_disj(
    			   cnnfh (HOLogic.mk_conj(p,cnnfh(HOLogic.Not $ q))),
    			   cnnfh (HOLogic.mk_conj(p1,cnnfh(HOLogic.Not $ r))))
    			 else  HOLogic.mk_conj(
    			  cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))),
    			   cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p1),cnnfh(HOLogic.Not $ r)))
    			 )
        | (Const ("Not",_) $ (Const ("op |",_) $ p $ q)) => HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
        | (Const ("Not",_) $ (Const ("op -->",_) $ p $q)) => HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q))
        | (Const ("Not",_) $ (Const ("op =",Type ("fun",[Type ("bool", []),_]))  $ p $ q)) => HOLogic.mk_disj(HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q)),HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh q))
        | _ => lfn fm
    in cnnfh
     end;

    (*End- function the quantifierelimination an decion procedure of presburger formulas.*)

    (*
    val integer_qelim = simpl o evalc o (lift_qelim linform (simpl o (cnnf posineq o evalc)) cooper is_arith_rel) ;
    *)

    val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ;

    end;