renamed HOL + - * etc. to HOL.plus HOL.minus HOL.times etc.
(* Title: HOL/Integ/cooper_dec.ML
ID: $Id$
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("HOL.plus",T) $ (Const ("HOL.times",T1 ) $c1 $ x1) $ rest) =>
Const("HOL.plus",T) $ ((Const("HOL.times",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 ("HOL.plus",T1) $ ( Const("HOL.times",T2) $ c1 $ x1) $ rest1),(Const
("HOL.plus",T3)$( Const("HOL.times",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("HOL.plus",T1) $ (Const("HOL.times",T2) $ c $ x1) $ (linear_add vars rest1 rest2))
end
else
if earlierv vars x1 x2 then (Const("HOL.plus",T1) $
(Const("HOL.times",T2)$ c1 $ x1) $ (linear_add vars rest1 tm2))
else (Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c2 $ x2) $ (linear_add vars tm1 rest2))
|((Const("HOL.plus",T1) $ (Const("HOL.times",T2) $ c1 $ x1) $ rest1) ,_) =>
(Const("HOL.plus",T1)$ (Const("HOL.times",T2) $ c1 $ x1) $ (linear_add vars
rest1 tm2))
|(_, (Const("HOL.plus",T1) $(Const("HOL.times",T2) $ c2 $ x2) $ rest2)) =>
(Const("HOL.plus",T1) $ (Const("HOL.times",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 "HOL.plus" ((HOLogic.mk_binop "HOL.times" ((mk_numeral 1),Free (x,T))), zero))
|(Bound i) => (Const("HOL.plus",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $
(Const("HOL.times",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ (mk_numeral 1) $ (Bound i)) $ zero)
|(Const("HOL.uminus",_) $ t ) => (linear_neg (lint vars t))
|(Const("HOL.plus",_) $ s $ t) => (linear_add vars (lint vars s) (lint vars t))
|(Const("HOL.minus",_) $ s $ t) => (linear_sub vars (lint vars s) (lint vars t))
|(Const ("HOL.times",_) $ 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 ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s) )
|linform vars (Const("op <",_)$ s $t ) = (mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s))
|linform vars (Const("op >",_) $ s $ t ) = (mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ s $ t))
|linform vars (Const("op <=",_)$ s $ t ) =
(mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ (Const("HOL.plus",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $t $(mk_numeral 1)) $ s))
|linform vars (Const("op >=",_)$ s $ t ) =
(mkatom vars "op <" (Const ("HOL.minus",HOLogic.intT --> HOLogic.intT -->
HOLogic.intT) $ (Const("HOL.plus",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,_)$ _ $(Const ("HOL.plus", _)$(Const ("HOL.times",_)$ 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 $( Const ("HOL.plus", _)$(Const ("HOL.times",_) $
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 "HOL.times" ((mk_numeral (m div n)), x)
in
(HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "HOL.plus" ( 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 "HOL.plus"
((HOLogic.mk_binop "HOL.times" ((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 $( Const ("HOL.plus", _)$(Const ("HOL.times",_) $
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 "HOL.times" ((mk_numeral ((m div n)*l) ), x))
in (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "HOL.plus" ( 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 ("HOL.plus", _) $(Const ("HOL.times",_) $ 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 ("HOL.plus", _) $(Const ("HOL.times",_) $ 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 ("HOL.plus", _) $(Const ("HOL.times",_) $ 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 ("HOL.plus", _) $(Const ("HOL.times",_) $ 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 ("HOL.plus",_) $ (Const ("HOL.times",_) $ 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 ("HOL.plus", _) $(Const ("HOL.times",_) $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 ("HOL.plus",_) $ (Const ("HOL.times",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg(linear_add [] a (mk_numeral 1))] else []
|(Const ("op <",_) $ c1$ (Const ("HOL.plus",_) $(Const ("HOL.times",_)$ 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 ("HOL.plus", _) $(Const ("HOL.times",_) $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 ("HOL.plus",_) $ (Const ("HOL.times",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_sub [] (mk_numeral 1) a] else []
|(Const ("op <",_) $ c1$ (Const ("HOL.plus",_) $(Const ("HOL.times",_)$ 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("HOL.plus",_)$(Const("HOL.times",_)$ 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 = "HOL.plus" then (coeffs_of tl) union (coeffs_of tr)
else if p = "HOL.times"
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;