(*  Title:      HOL/Integ/cooper_dec.ML

     ID:         $Id$

     Author:     Amine Chaieb and Tobias Nipkow, TU Muenchen

     License:    GPL (GNU GENERAL PUBLIC LICENSE)

 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 : int -> term

   val dest_numeral : term -> int

   val zero : term

   val one : term

   val linear_cmul : 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 -> int

   val adjustcoeff : term -> int -> term -> term

   val unitycoeff : term -> term -> term

   val divlcm : term -> term -> 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 * (int * 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

 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 : [int,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 (warning "lint: apparent nonlinearity"; raise COOPER)

          end 

   |_ =>   (error "COOPER:lint: unknown term  ")

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

 (* 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) =  

       let val c' = (mk_numeral(abs(dest_numeral c)))  

       in (HOLogic.mk_binrel "Divides.op dvd" (c,lint vars t)) 

       end 

   |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)*) 

 fun gcd a 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) andalso (pm1 = one) andalso (c = zero) then HOLogic.false_const else HOLogic.true_const 

   |(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) andalso (pm1 = one) andalso (c = zero) then HOLogic.true_const else HOLogic.false_const

   |(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.                                       *) 

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

 val operations = 

   [("op =",op=), ("op <",op<), ("op >",op>), ("op <=",op<=) , ("op >=",op>=), 

    ("Divides.op dvd",fn (x,y) =>((y mod x) = 0))]; 

 fun applyoperation (Some f) (a,b) = f (a, b) 

     |applyoperation _ (_, _) = false; 

 (*Evaluation of constant atomic formulas*) 

 fun evalc_atom at = case at of  

       (Const (p,_) $ s $ t) =>(  

          case assoc (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 assoc (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.                                     *) 

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

 (*Applyes a function iteratively on the list*) 

 fun end_itlist f []     = error "end_itlist" 

    |end_itlist f [x]    = x 

    |end_itlist f (h::t) = f h (end_itlist f t); 

 (*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 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 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) 

   | _ => psimpl1 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 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 = 1 upto 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 = 1 upto 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";

 (*Cooper main procedure*) 

 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 ast = aset x p

     val bst = bset x p

     val js = 1 upto divlcm x p

     val (p_inf,f,S ) = 

     if (length bst) < (length ast) 

      then (minusinf x p,linear_add,bst)

      else (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)

    in (list_disj (map stage js))

    end

   | _ => error "cooper: not an existential formula";

 (*Function itlist applys a double parametred function f : 'a->'b->b iteratively to a List l : 'a 

 list With End condition b. ict calculates f(e1,f(f(e2,f(e3,...(...f(en,b))..))))) 

  assuming l = [e1,e2,...,en]*) 

 fun itlist f l b = case l of 

     [] => b 

   | (h::t) => f h (itlist f t b); 

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

 (* 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 qelim x vars p = 

   let val cjs = conjuncts p 

       val (ycjs,ncjs) = 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 

                           (itlist 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 o simpl

   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) ; 

 end;