(* 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;