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Arithmetic constraints
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<TABLE CELLPADDING=0 CELLSPACING=0 WIDTH="100%">
<TR><TD BGCOLOR="#66dbff"><DIV ALIGN=center><TABLE>
<TR><TD><FONT SIZE=4><B><A NAME="htoc321">8.6</A></B></FONT></TD>
<TD WIDTH="100%" ALIGN=center><FONT SIZE=4><B>Arithmetic constraints</B></FONT></TD>
</TR></TABLE></DIV></TD>
</TR></TABLE><UL>
<LI><A HREF="manual060.html#toc263"> FD arithmetic expressions</A>
<LI><A HREF="manual060.html#toc264"> Partial AC: <TT>(#=)/2</TT> - constraint equal,
<TT>(#\=)/2</TT> - constraint not equal,<BR>
<TT>(#<)/2</TT> - constraint less than,
<TT>(#=<)/2</TT> - constraint less than or equal,<BR>
<TT>(#>)/2</TT> - constraint greater than,
<TT>(#>=)/2</TT> - constraint greater than or equal</A>
<LI><A HREF="manual060.html#toc265"> Full AC: <TT>(#=#)/2</TT> - constraint equal,
<TT>(#\=#)/2</TT> - constraint not equal,<BR>
<TT>(#<#)/2</TT> - constraint less than,
<TT>(#=<#)/2</TT> - constraint less than or equal,<BR>
<TT>(#>#)/2</TT> - constraint greater than,
<TT>(#>=#)/2</TT> - constraint greater than or equal</A>
<LI><A HREF="manual060.html#toc266"> <TT>fd_prime/1</TT>,
<TT>fd_not_prime/1</TT></A>
</UL>
<BR>
<A NAME="toc263"></A><TABLE CELLPADDING=0 CELLSPACING=0 WIDTH="100%">
<TR><TD BGCOLOR="#98e7ff"><DIV ALIGN=center><TABLE>
<TR><TD><B><A NAME="htoc322">8.6.1</A></B></TD>
<TD WIDTH="100%" ALIGN=center><B>FD arithmetic expressions</B></TD>
</TR></TABLE></DIV></TD>
</TR></TABLE>
<A NAME="FD-arithmetic-expressions"></A>
An FD arithmetic expression is a Prolog term built from integers, variables
(Prolog or FD variables), and functors (or operators) that represent
arithmetic functions. The following table details the components of an FD
arithmetic expression:<BR>
<TABLE BORDER=1 CELLSPACING=0 CELLPADDING=1>
<TR><TD VALIGN=top ALIGN=left NOWRAP>FD Expression</TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>Result</DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP>Prolog variable</TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>domain <TT>0..fd_max_integer</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP>FD variable <TT>X</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>domain of <TT>X</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP>integer number <TT>N</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>domain <TT>N..N</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>+ E</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>same as <TT>E</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>- E</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>opposite of <TT>E</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 + E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>sum of <TT>E1</TT> and <TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 - E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>subtraction of <TT>E2</TT> from <TT>E1</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 * E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>multiplication of <TT>E1</TT> by <TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 / E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>integer division of <TT>E1</TT> by <TT>E2</TT> (only
succeeds if the remainder is 0)</DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 ** E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left><TT>E1</TT> raised to the power of <TT>E2
</TT>(<TT>E1</TT> or <TT>E2</TT> must be an integer)</DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>min(E1,E2)</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>minimum of <TT>E1</TT> and <TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>max(E1,E2)</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>maximum of <TT>E1</TT> and <TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>dist(E1,E2)</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>distance, i.e. |<TT>E1 - E2|</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 // E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>quotient of the integer division of <TT>E1</TT> by
<TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>E1 rem E2</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>remainder of the integer division of <TT>E1</TT> by
<TT>E2</TT></DIV></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left NOWRAP><TT>quot_rem(E1,E2,R)</TT></TD>
<TD VALIGN=top ALIGN=left><DIV ALIGN=left>quotient of the integer division of
<TT>E1</TT> by <TT>E2</TT>
<BR>
(<TT>R</TT> is the remainder of the integer division of <TT>E1</TT> by
<TT>E2</TT>)</DIV></TD>
</TR></TABLE><BR>
FD expressions are not restricted to be linear. However non-linear
constraints usually yield less constraint propagation than linear
constraints.<BR>
<BR>
<TT>+</TT>, <TT>-</TT>, <TT>*</TT>, <TT>/</TT>, <TT>//</TT>, <TT>rem</TT>
and <TT>**</TT> are predefined infix operators. <TT>+</TT> and <TT>-</TT>
are predefined prefix operators (section <A HREF="manual037.html#op/3:(Term-input/output)">7.14.10</A>).<BR>
<BR>
<B>Errors</B><BR>
<TABLE CELLSPACING=2 CELLPADDING=0>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>a sub-expression is of the form <TT>_ ** E</TT> and <TT>E</TT> is
a variable</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>instantiation_error</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>a sub-expression <TT>E</TT> is neither a variable nor an integer
nor an FD arithmetic functor</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(fd_evaluable, E)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left>an expression is too complex</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>resource_error(too_big_fd_constraint)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR></TABLE><BR>
<A NAME="toc264"></A><TABLE CELLPADDING=0 CELLSPACING=0 WIDTH="100%">
<TR><TD BGCOLOR="#98e7ff"><DIV ALIGN=center><TABLE>
<TR><TD><B><A NAME="htoc323">8.6.2</A></B></TD>
<TD WIDTH="100%" ALIGN=center><B>Partial AC: <TT>(#=)/2</TT> - constraint equal,
<TT>(#\=)/2</TT> - constraint not equal,<BR>
<TT>(#<)/2</TT> - constraint less than,
<TT>(#=<)/2</TT> - constraint less than or equal,<BR>
<TT>(#>)/2</TT> - constraint greater than,
<TT>(#>=)/2</TT> - constraint greater than or equal</B></TD>
</TR></TABLE></DIV></TD>
</TR></TABLE>
<A NAME="Partial-AC:-(:=)/2"></A>
<BR>
<B>Templates</B>
<DL COMPACT=compact><DT><DD><TT>
#=(?fd_evaluable, ?fd_evaluable)<BR>
#\=(?fd_evaluable, ?fd_evaluable)<BR>
#<(?fd_evaluable, ?fd_evaluable)<BR>
#=<(?fd_evaluable, ?fd_evaluable)<BR>
#>(?fd_evaluable, ?fd_evaluable)<BR>
#>=(?fd_evaluable, ?fd_evaluable)</TT></DL>
<B>Description</B><BR>
<BR>
<TT>FdExpr1 #= FdExpr2</TT> constrains <TT>FdExpr1</TT> to be
equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #\= FdExpr2</TT> constrains <TT>FdExpr1</TT>
to be different from <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #< FdExpr2</TT> constrains <TT>FdExpr1</TT> to
be less than <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #=< FdExpr2</TT> constrains <TT>FdExpr1</TT>
to be less than or equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #> FdExpr2</TT> constrains <TT>FdExpr1</TT> to
be greater than <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #>= FdExpr2</TT> constrains <TT>FdExpr1</TT>
to be greater than or equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1</TT> and <TT>FdExpr2</TT> are arithmetic FD expressions
(section <A HREF="#FD-arithmetic-expressions">8.6.1</A>).<BR>
<BR>
<TT>#=</TT>, <TT>#\=</TT>, <TT>#<</TT>, <TT>#=<</TT>,
<TT>#></TT> and <TT>#>=</TT> are predefined infix operators
(section <A HREF="manual037.html#op/3:(Term-input/output)">7.14.10</A>).<BR>
<BR>
These predicates post arithmetic constraints that are managed by the solver
using a partial arc-consistency algorithm to reduce the domain of involved
variables. In this scheme only the bounds of the domain of variables are
updated. This leads to less propagation than full arc-consistency techniques
(section <A HREF="#Full-AC:-(:=:)/2">8.6.3</A>) but is generally more efficient for
arithmetic. These arithmetic constraints can be reified (section <A HREF="manual061.html#Boolean-and-reified-constraints">8.7</A>).<BR>
<BR>
<B>Errors</B><BR>
<BR>
Refer to the syntax of arithmetic FD expressions for possible errors
(section <A HREF="#FD-arithmetic-expressions">8.6.1</A>).<BR>
<BR>
<B>Portability</B><BR>
<BR>
GNU Prolog predicates.<BR>
<BR>
<A NAME="toc265"></A><TABLE CELLPADDING=0 CELLSPACING=0 WIDTH="100%">
<TR><TD BGCOLOR="#98e7ff"><DIV ALIGN=center><TABLE>
<TR><TD><B><A NAME="htoc324">8.6.3</A></B></TD>
<TD WIDTH="100%" ALIGN=center><B>Full AC: <TT>(#=#)/2</TT> - constraint equal,
<TT>(#\=#)/2</TT> - constraint not equal,<BR>
<TT>(#<#)/2</TT> - constraint less than,
<TT>(#=<#)/2</TT> - constraint less than or equal,<BR>
<TT>(#>#)/2</TT> - constraint greater than,
<TT>(#>=#)/2</TT> - constraint greater than or equal</B></TD>
</TR></TABLE></DIV></TD>
</TR></TABLE>
<A NAME="Full-AC:-(:=:)/2"></A>
<BR>
<B>Templates</B>
<DL COMPACT=compact><DT><DD><TT>
#=#(?fd_evaluable, ?fd_evaluable)<BR>
#\=#(?fd_evaluable, ?fd_evaluable)<BR>
#<#(?fd_evaluable, ?fd_evaluable)<BR>
#=<#(?fd_evaluable, ?fd_evaluable)<BR>
#>#(?fd_evaluable, ?fd_evaluable)<BR>
#>=#(?fd_evaluable, ?fd_evaluable)</TT></DL>
<B>Description</B><BR>
<BR>
<TT>FdExpr1 #=# FdExpr2</TT> constrains <TT>FdExpr1</TT> to
be equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #\=# FdExpr2</TT> constrains <TT>FdExpr1</TT> to be
different from <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #<# FdExpr2</TT> constrains <TT>FdExpr1</TT>
to be less than <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #=<# FdExpr2</TT> constrains <TT>FdExpr1</TT> to be
less than or equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #># FdExpr2</TT> constrains <TT>FdExpr1</TT>
to be greater than <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1 #>=# FdExpr2</TT> constrains <TT>FdExpr1</TT> to be
greater than or equal to <TT>FdExpr2</TT>.<BR>
<BR>
<TT>FdExpr1</TT> and <TT>FdExpr2</TT> are arithmetic FD expressions
(section <A HREF="#FD-arithmetic-expressions">8.6.1</A>).<BR>
<BR>
<TT>#=#</TT>, <TT>#\=#</TT>, <TT>#<#</TT>,
<TT>#=<#</TT>, <TT>#>#</TT> and <TT>#>=#</TT> are
predefined infix operators (section <A HREF="manual037.html#op/3:(Term-input/output)">7.14.10</A>).<BR>
<BR>
These predicates post arithmetic constraints that are managed by the solver
using a full arc-consistency algorithm to reduce the domain of involved
variables. In this scheme the full domain of variables is updated. This
leads to more propagation than partial arc-consistency techniques (section <A HREF="#FD-arithmetic-expressions">8.6.1</A>) but is generally less efficient for arithmetic.
These arithmetic constraints can be reified (section <A HREF="manual061.html#Boolean-FD-expressions">8.7.1</A>).<BR>
<BR>
<B>Errors</B><BR>
<BR>
Refer to the syntax of arithmetic FD expressions for possible errors
(section <A HREF="#FD-arithmetic-expressions">8.6.1</A>).<BR>
<BR>
<B>Portability</B><BR>
<BR>
GNU Prolog predicates.<BR>
<BR>
<A NAME="toc266"></A><TABLE CELLPADDING=0 CELLSPACING=0 WIDTH="100%">
<TR><TD BGCOLOR="#98e7ff"><DIV ALIGN=center><TABLE>
<TR><TD><B><A NAME="htoc325">8.6.4</A></B></TD>
<TD WIDTH="100%" ALIGN=center><B><TT>fd_prime/1</TT>,
<TT>fd_not_prime/1</TT></B></TD>
</TR></TABLE></DIV></TD>
</TR></TABLE>
<BR>
<B>Templates</B>
<DL COMPACT=compact><DT><DD><TT>
fd_prime(?fd_variable)<BR>
fd_not_prime(?fd_variable)</TT></DL>
<B>Description</B><BR>
<BR>
<TT>fd_prime(X)</TT> constraints <TT>X</TT> to be a prime number between
<TT>0..vector_max</TT>.
This constraint enforces a sparse representation
for the domain of <TT>X</TT> (section <A HREF="manual055.html#Intro-FD">8.1</A>).<BR>
<BR>
<TT>fd_not_prime(X)</TT> constraints <TT>X</TT> to be a non prime number
between <TT>0..vector_max</TT>. This constraint enforces a sparse
representation for the domain of <TT>X</TT> (section <A HREF="manual055.html#Intro-FD">8.1</A>).<BR>
<BR>
<B>Errors</B><BR>
<TABLE CELLSPACING=2 CELLPADDING=0>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR>
<TR><TD VALIGN=top ALIGN=left><TT>X</TT> is neither an FD variable nor an integer</TD>
<TD VALIGN=top ALIGN=center NOWRAP> </TD>
<TD VALIGN=top ALIGN=left><TT>type_error(fd_variable, X)</TT></TD>
</TR>
<TR><TD BGCOLOR=black COLSPAN=3><TABLE BORDER=0 WIDTH="100%" CELLSPACING=0 CELLPADDING=1><TR><TD></TD></TR></TABLE></TD>
</TR></TABLE><BR>
<B>Portability</B><BR>
<BR>
GNU Prolog predicates.<BR>
<BR>
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<BR>
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