export isl_basic_set_remove_redundancies

[isl.git] / doc / user.pod

=head1 Introduction

C<isl> is a thread-safe C library for manipulating
sets and relations of integer points bounded by affine constraints.
The descriptions of the sets and relations may involve
both parameters and existentially quantified variables.
All computations are performed in exact integer arithmetic
using C<GMP>.
The C<isl> library offers functionality that is similar
to that offered by the C<Omega> and C<Omega+> libraries,
but the underlying algorithms are in most cases completely different.

The library is by no means complete and some fairly basic
functionality is still missing.
Still, even in its current form, the library has been successfully
used as a backend polyhedral library for the polyhedral
scanner C<CLooG> and as part of an equivalence checker of
static affine programs.
For bug reports, feature requests and questions,
visit the the discussion group at
L<http://groups.google.com/group/isl-development>.

=head2 Backward Incompatible Changes

=head3 Changes since isl-0.02

=over

=item * The old printing functions have been deprecated
and replaced by C<isl_printer> functions, see L<Input and Output>.

=item * Most functions related to dependence analysis have acquired
an extra C<must> argument.  To obtain the old behavior, this argument
should be given the value 1.  See L<Dependence Analysis>.

=back

=head3 Changes since isl-0.03

=over

=item * The function C<isl_pw_qpolynomial_fold_add> has been
renamed to C<isl_pw_qpolynomial_fold_fold>.
Similarly, C<isl_union_pw_qpolynomial_fold_add> has been
renamed to C<isl_union_pw_qpolynomial_fold_fold>.

=back

=head3 Changes since isl-0.04

=over

=item * All header files have been renamed from C<isl_header.h>
to C<isl/header.h>.

=back

=head3 Changes since isl-0.05

=over

=item * The functions C<isl_printer_print_basic_set> and
C<isl_printer_print_basic_map> no longer print a newline.

=item * The functions C<isl_flow_get_no_source>
and C<isl_union_map_compute_flow> now return
the accesses for which no source could be found instead of
the iterations where those accesses occur.

=item * The functions C<isl_basic_map_identity> and
C<isl_map_identity> now take the dimension specification
of a B<map> as input.  An old call
C<isl_map_identity(dim)> can be rewritten to
C<isl_map_identity(isl_dim_map_from_set(dim))>.

=item * The function C<isl_map_power> no longer takes
a parameter position as input.  Instead, the exponent
is now expressed as the domain of the resulting relation.

=back

=head1 Installation

The source of C<isl> can be obtained either as a tarball
or from the git repository.  Both are available from
L<http://freshmeat.net/projects/isl/>.
The installation process depends on how you obtained
the source.

=head2 Installation from the git repository

=over

=item 1 Clone or update the repository

The first time the source is obtained, you need to clone
the repository.

        git clone git://repo.or.cz/isl.git

To obtain updates, you need to pull in the latest changes

        git pull

=item 2 Generate C<configure>

        ./autogen.sh

=back

After performing the above steps, continue
with the L<Common installation instructions>.

=head2 Common installation instructions

=over

=item 1 Obtain C<GMP>

Building C<isl> requires C<GMP>, including its headers files.
Your distribution may not provide these header files by default
and you may need to install a package called C<gmp-devel> or something
similar.  Alternatively, C<GMP> can be built from
source, available from L<http://gmplib.org/>.

=item 2 Configure

C<isl> uses the standard C<autoconf> C<configure> script.
To run it, just type

        ./configure

optionally followed by some configure options.
A complete list of options can be obtained by running

        ./configure --help

Below we discuss some of the more common options.

C<isl> can optionally use C<piplib>, but no
C<piplib> functionality is currently used by default.
The C<--with-piplib> option can
be used to specify which C<piplib>
library to use, either an installed version (C<system>),
an externally built version (C<build>)
or no version (C<no>).  The option C<build> is mostly useful
in C<configure> scripts of larger projects that bundle both C<isl>
and C<piplib>.

=over

=item C<--prefix>

Installation prefix for C<isl>

=item C<--with-gmp-prefix>

Installation prefix for C<GMP> (architecture-independent files).

=item C<--with-gmp-exec-prefix>

Installation prefix for C<GMP> (architecture-dependent files).

=item C<--with-piplib>

Which copy of C<piplib> to use, either C<no> (default), C<system> or C<build>.

=item C<--with-piplib-prefix>

Installation prefix for C<system> C<piplib> (architecture-independent files).

=item C<--with-piplib-exec-prefix>

Installation prefix for C<system> C<piplib> (architecture-dependent files).

=item C<--with-piplib-builddir>

Location where C<build> C<piplib> was built.

=back

=item 3 Compile

        make

=item 4 Install (optional)

        make install

=back

=head1 Library

=head2 Initialization

All manipulations of integer sets and relations occur within
the context of an C<isl_ctx>.
A given C<isl_ctx> can only be used within a single thread.
All arguments of a function are required to have been allocated
within the same context.
There are currently no functions available for moving an object
from one C<isl_ctx> to another C<isl_ctx>.  This means that
there is currently no way of safely moving an object from one
thread to another, unless the whole C<isl_ctx> is moved.

An C<isl_ctx> can be allocated using C<isl_ctx_alloc> and
freed using C<isl_ctx_free>.
All objects allocated within an C<isl_ctx> should be freed
before the C<isl_ctx> itself is freed.

        isl_ctx *isl_ctx_alloc();
        void isl_ctx_free(isl_ctx *ctx);

=head2 Integers

All operations on integers, mainly the coefficients
of the constraints describing the sets and relations,
are performed in exact integer arithmetic using C<GMP>.
However, to allow future versions of C<isl> to optionally
support fixed integer arithmetic, all calls to C<GMP>
are wrapped inside C<isl> specific macros.
The basic type is C<isl_int> and the operations below
are available on this type.
The meanings of these operations are essentially the same
as their C<GMP> C<mpz_> counterparts.
As always with C<GMP> types, C<isl_int>s need to be
initialized with C<isl_int_init> before they can be used
and they need to be released with C<isl_int_clear>
after the last use.
The user should not assume that an C<isl_int> is represented
as a C<mpz_t>, but should instead explicitly convert between
C<mpz_t>s and C<isl_int>s using C<isl_int_set_gmp> and
C<isl_int_get_gmp> whenever a C<mpz_t> is required.

=over

=item isl_int_init(i)

=item isl_int_clear(i)

=item isl_int_set(r,i)

=item isl_int_set_si(r,i)

=item isl_int_set_gmp(r,g)

=item isl_int_get_gmp(i,g)

=item isl_int_abs(r,i)

=item isl_int_neg(r,i)

=item isl_int_swap(i,j)

=item isl_int_swap_or_set(i,j)

=item isl_int_add_ui(r,i,j)

=item isl_int_sub_ui(r,i,j)

=item isl_int_add(r,i,j)

=item isl_int_sub(r,i,j)

=item isl_int_mul(r,i,j)

=item isl_int_mul_ui(r,i,j)

=item isl_int_addmul(r,i,j)

=item isl_int_submul(r,i,j)

=item isl_int_gcd(r,i,j)

=item isl_int_lcm(r,i,j)

=item isl_int_divexact(r,i,j)

=item isl_int_cdiv_q(r,i,j)

=item isl_int_fdiv_q(r,i,j)

=item isl_int_fdiv_r(r,i,j)

=item isl_int_fdiv_q_ui(r,i,j)

=item isl_int_read(r,s)

=item isl_int_print(out,i,width)

=item isl_int_sgn(i)

=item isl_int_cmp(i,j)

=item isl_int_cmp_si(i,si)

=item isl_int_eq(i,j)

=item isl_int_ne(i,j)

=item isl_int_lt(i,j)

=item isl_int_le(i,j)

=item isl_int_gt(i,j)

=item isl_int_ge(i,j)

=item isl_int_abs_eq(i,j)

=item isl_int_abs_ne(i,j)

=item isl_int_abs_lt(i,j)

=item isl_int_abs_gt(i,j)

=item isl_int_abs_ge(i,j)

=item isl_int_is_zero(i)

=item isl_int_is_one(i)

=item isl_int_is_negone(i)

=item isl_int_is_pos(i)

=item isl_int_is_neg(i)

=item isl_int_is_nonpos(i)

=item isl_int_is_nonneg(i)

=item isl_int_is_divisible_by(i,j)

=back

=head2 Sets and Relations

C<isl> uses six types of objects for representing sets and relations,
C<isl_basic_set>, C<isl_basic_map>, C<isl_set>, C<isl_map>,
C<isl_union_set> and C<isl_union_map>.
C<isl_basic_set> and C<isl_basic_map> represent sets and relations that
can be described as a conjunction of affine constraints, while
C<isl_set> and C<isl_map> represent unions of
C<isl_basic_set>s and C<isl_basic_map>s, respectively.
However, all C<isl_basic_set>s or C<isl_basic_map>s in the union need
to have the same dimension.  C<isl_union_set>s and C<isl_union_map>s
represent unions of C<isl_set>s or C<isl_map>s of I<different> dimensions,
where dimensions with different space names
(see L<Dimension Specifications>) are considered different as well.
The difference between sets and relations (maps) is that sets have
one set of variables, while relations have two sets of variables,
input variables and output variables.

=head2 Memory Management

Since a high-level operation on sets and/or relations usually involves
several substeps and since the user is usually not interested in
the intermediate results, most functions that return a new object
will also release all the objects passed as arguments.
If the user still wants to use one or more of these arguments
after the function call, she should pass along a copy of the
object rather than the object itself.
The user is then responsible for making sure that the original
object gets used somewhere else or is explicitly freed.

The arguments and return values of all documents functions are
annotated to make clear which arguments are released and which
arguments are preserved.  In particular, the following annotations
are used

=over

=item C<__isl_give>

C<__isl_give> means that a new object is returned.
The user should make sure that the returned pointer is
used exactly once as a value for an C<__isl_take> argument.
In between, it can be used as a value for as many
C<__isl_keep> arguments as the user likes.
There is one exception, and that is the case where the
pointer returned is C<NULL>.  Is this case, the user
is free to use it as an C<__isl_take> argument or not.

=item C<__isl_take>

C<__isl_take> means that the object the argument points to
is taken over by the function and may no longer be used
by the user as an argument to any other function.
The pointer value must be one returned by a function
returning an C<__isl_give> pointer.
If the user passes in a C<NULL> value, then this will
be treated as an error in the sense that the function will
not perform its usual operation.  However, it will still
make sure that all the the other C<__isl_take> arguments
are released.

=item C<__isl_keep>

C<__isl_keep> means that the function will only use the object
temporarily.  After the function has finished, the user
can still use it as an argument to other functions.
A C<NULL> value will be treated in the same way as
a C<NULL> value for an C<__isl_take> argument.

=back

=head2 Dimension Specifications

Whenever a new set or relation is created from scratch,
its dimension needs to be specified using an C<isl_dim>.

        #include <isl/dim.h>
        __isl_give isl_dim *isl_dim_alloc(isl_ctx *ctx,
                unsigned nparam, unsigned n_in, unsigned n_out);
        __isl_give isl_dim *isl_dim_set_alloc(isl_ctx *ctx,
                unsigned nparam, unsigned dim);
        __isl_give isl_dim *isl_dim_copy(__isl_keep isl_dim *dim);
        void isl_dim_free(__isl_take isl_dim *dim);
        unsigned isl_dim_size(__isl_keep isl_dim *dim,
                enum isl_dim_type type);

The dimension specification used for creating a set
needs to be created using C<isl_dim_set_alloc>, while
that for creating a relation
needs to be created using C<isl_dim_alloc>.
C<isl_dim_size> can be used
to find out the number of dimensions of each type in
a dimension specification, where type may be
C<isl_dim_param>, C<isl_dim_in> (only for relations),
C<isl_dim_out> (only for relations), C<isl_dim_set>
(only for sets) or C<isl_dim_all>.

It is often useful to create objects that live in the
same space as some other object.  This can be accomplished
by creating the new objects
(see L<Creating New Sets and Relations> or
L<Creating New (Piecewise) Quasipolynomials>) based on the dimension
specification of the original object.

        #include <isl/set.h>
        __isl_give isl_dim *isl_basic_set_get_dim(
                __isl_keep isl_basic_set *bset);
        __isl_give isl_dim *isl_set_get_dim(__isl_keep isl_set *set);

        #include <isl/union_set.h>
        __isl_give isl_dim *isl_union_set_get_dim(
                __isl_keep isl_union_set *uset);

        #include <isl/map.h>
        __isl_give isl_dim *isl_basic_map_get_dim(
                __isl_keep isl_basic_map *bmap);
        __isl_give isl_dim *isl_map_get_dim(__isl_keep isl_map *map);

        #include <isl/union_map.h>
        __isl_give isl_dim *isl_union_map_get_dim(
                __isl_keep isl_union_map *umap);

        #include <isl/constraint.h>
        __isl_give isl_dim *isl_constraint_get_dim(
                __isl_keep isl_constraint *constraint);

        #include <isl/polynomial.h>
        __isl_give isl_dim *isl_qpolynomial_get_dim(
                __isl_keep isl_qpolynomial *qp);
        __isl_give isl_dim *isl_pw_qpolynomial_get_dim(
                __isl_keep isl_pw_qpolynomial *pwqp);
        __isl_give isl_dim *isl_union_pw_qpolynomial_get_dim(
                __isl_keep isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_dim *isl_union_pw_qpolynomial_fold_get_dim(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);

The names of the individual dimensions may be set or read off
using the following functions.

        #include <isl/dim.h>
        __isl_give isl_dim *isl_dim_set_name(__isl_take isl_dim *dim,
                                 enum isl_dim_type type, unsigned pos,
                                 __isl_keep const char *name);
        __isl_keep const char *isl_dim_get_name(__isl_keep isl_dim *dim,
                                 enum isl_dim_type type, unsigned pos);

Note that C<isl_dim_get_name> returns a pointer to some internal
data structure, so the result can only be used while the
corresponding C<isl_dim> is alive.
Also note that every function that operates on two sets or relations
requires that both arguments have the same parameters.  This also
means that if one of the arguments has named parameters, then the
other needs to have named parameters too and the names need to match.
Pairs of C<isl_union_set> and/or C<isl_union_map> arguments may
have different parameters (as long as they are named), in which case
the result will have as parameters the union of the parameters of
the arguments.

The names of entire spaces may be set or read off
using the following functions.

        #include <isl/dim.h>
        __isl_give isl_dim *isl_dim_set_tuple_name(
                __isl_take isl_dim *dim,
                enum isl_dim_type type, const char *s);
        const char *isl_dim_get_tuple_name(__isl_keep isl_dim *dim,
                enum isl_dim_type type);

The C<dim> argument needs to be one of C<isl_dim_in>, C<isl_dim_out>
or C<isl_dim_set>.  As with C<isl_dim_get_name>,
the C<isl_dim_get_tuple_name> function returns a pointer to some internal
data structure.
Binary operations require the corresponding spaces of their arguments
to have the same name.

Spaces can be nested.  In particular, the domain of a set or
the domain or range of a relation can be a nested relation.
The following functions can be used to construct and deconstruct
such nested dimension specifications.

        #include <isl/dim.h>
        int isl_dim_is_wrapping(__isl_keep isl_dim *dim);
        __isl_give isl_dim *isl_dim_wrap(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_unwrap(__isl_take isl_dim *dim);

The input to C<isl_dim_is_wrapping> and C<isl_dim_unwrap> should
be the dimension specification of a set, while that of
C<isl_dim_wrap> should be the dimension specification of a relation.
Conversely, the output of C<isl_dim_unwrap> is the dimension specification
of a relation, while that of C<isl_dim_wrap> is the dimension specification
of a set.

Dimension specifications can be created from other dimension
specifications using the following functions.

        __isl_give isl_dim *isl_dim_domain(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_from_domain(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_range(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_from_range(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_reverse(__isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_join(__isl_take isl_dim *left,
                __isl_take isl_dim *right);
        __isl_give isl_dim *isl_dim_insert(__isl_take isl_dim *dim,
                enum isl_dim_type type, unsigned pos, unsigned n);
        __isl_give isl_dim *isl_dim_add(__isl_take isl_dim *dim,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_dim *isl_dim_drop(__isl_take isl_dim *dim,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_dim *isl_dim_map_from_set(
                __isl_take isl_dim *dim);
        __isl_give isl_dim *isl_dim_zip(__isl_take isl_dim *dim);

Note that if dimensions are added or removed from a space, then
the name and the internal structure are lost.

=head2 Input and Output

C<isl> supports its own input/output format, which is similar
to the C<Omega> format, but also supports the C<PolyLib> format
in some cases.

=head3 C<isl> format

The C<isl> format is similar to that of C<Omega>, but has a different
syntax for describing the parameters and allows for the definition
of an existentially quantified variable as the integer division
of an affine expression.
For example, the set of integers C<i> between C<0> and C<n>
such that C<i % 10 <= 6> can be described as

        [n] -> { [i] : exists (a = [i/10] : 0 <= i and i <= n and
                                i - 10 a <= 6) }

A set or relation can have several disjuncts, separated
by the keyword C<or>.  Each disjunct is either a conjunction
of constraints or a projection (C<exists>) of a conjunction
of constraints.  The constraints are separated by the keyword
C<and>.

=head3 C<PolyLib> format

If the represented set is a union, then the first line
contains a single number representing the number of disjuncts.
Otherwise, a line containing the number C<1> is optional.

Each disjunct is represented by a matrix of constraints.
The first line contains two numbers representing
the number of rows and columns,
where the number of rows is equal to the number of constraints
and the number of columns is equal to two plus the number of variables.
The following lines contain the actual rows of the constraint matrix.
In each row, the first column indicates whether the constraint
is an equality (C<0>) or inequality (C<1>).  The final column
corresponds to the constant term.

If the set is parametric, then the coefficients of the parameters
appear in the last columns before the constant column.
The coefficients of any existentially quantified variables appear
between those of the set variables and those of the parameters.

=head3 Extended C<PolyLib> format

The extended C<PolyLib> format is nearly identical to the
C<PolyLib> format.  The only difference is that the line
containing the number of rows and columns of a constraint matrix
also contains four additional numbers:
the number of output dimensions, the number of input dimensions,
the number of local dimensions (i.e., the number of existentially
quantified variables) and the number of parameters.
For sets, the number of ``output'' dimensions is equal
to the number of set dimensions, while the number of ``input''
dimensions is zero.

=head3 Input

        #include <isl/set.h>
        __isl_give isl_basic_set *isl_basic_set_read_from_file(
                isl_ctx *ctx, FILE *input, int nparam);
        __isl_give isl_basic_set *isl_basic_set_read_from_str(
                isl_ctx *ctx, const char *str, int nparam);
        __isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx,
                FILE *input, int nparam);
        __isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx,
                const char *str, int nparam);

        #include <isl/map.h>
        __isl_give isl_basic_map *isl_basic_map_read_from_file(
                isl_ctx *ctx, FILE *input, int nparam);
        __isl_give isl_basic_map *isl_basic_map_read_from_str(
                isl_ctx *ctx, const char *str, int nparam);
        __isl_give isl_map *isl_map_read_from_file(
                struct isl_ctx *ctx, FILE *input, int nparam);
        __isl_give isl_map *isl_map_read_from_str(isl_ctx *ctx,
                const char *str, int nparam);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_read_from_file(
                isl_ctx *ctx, FILE *input);
        __isl_give isl_union_set *isl_union_set_read_from_str(
                struct isl_ctx *ctx, const char *str);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_read_from_file(
                isl_ctx *ctx, FILE *input);
        __isl_give isl_union_map *isl_union_map_read_from_str(
                struct isl_ctx *ctx, const char *str);

The input format is autodetected and may be either the C<PolyLib> format
or the C<isl> format.
C<nparam> specifies how many of the final columns in
the C<PolyLib> format correspond to parameters.
If input is given in the C<isl> format, then the number
of parameters needs to be equal to C<nparam>.
If C<nparam> is negative, then any number of parameters
is accepted in the C<isl> format and zero parameters
are assumed in the C<PolyLib> format.

=head3 Output

Before anything can be printed, an C<isl_printer> needs to
be created.

        __isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx,
                FILE *file);
        __isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx);
        void isl_printer_free(__isl_take isl_printer *printer);
        __isl_give char *isl_printer_get_str(
                __isl_keep isl_printer *printer);

The behavior of the printer can be modified in various ways

        __isl_give isl_printer *isl_printer_set_output_format(
                __isl_take isl_printer *p, int output_format);
        __isl_give isl_printer *isl_printer_set_indent(
                __isl_take isl_printer *p, int indent);
        __isl_give isl_printer *isl_printer_set_prefix(
                __isl_take isl_printer *p, const char *prefix);
        __isl_give isl_printer *isl_printer_set_suffix(
                __isl_take isl_printer *p, const char *suffix);

The C<output_format> may be either C<ISL_FORMAT_ISL>, C<ISL_FORMAT_OMEGA>,
C<ISL_FORMAT_POLYLIB>, C<ISL_FORMAT_EXT_POLYLIB> or C<ISL_FORMAT_LATEX>
and defaults to C<ISL_FORMAT_ISL>.
Each line in the output is indented by C<indent> spaces
(default: 0), prefixed by C<prefix> and suffixed by C<suffix>.
In the C<PolyLib> format output,
the coefficients of the existentially quantified variables
appear between those of the set variables and those
of the parameters.

To actually print something, use

        #include <isl/set.h>
        __isl_give isl_printer *isl_printer_print_basic_set(
                __isl_take isl_printer *printer,
                __isl_keep isl_basic_set *bset);
        __isl_give isl_printer *isl_printer_print_set(
                __isl_take isl_printer *printer,
                __isl_keep isl_set *set);

        #include <isl/map.h>
        __isl_give isl_printer *isl_printer_print_basic_map(
                __isl_take isl_printer *printer,
                __isl_keep isl_basic_map *bmap);
        __isl_give isl_printer *isl_printer_print_map(
                __isl_take isl_printer *printer,
                __isl_keep isl_map *map);

        #include <isl/union_set.h>
        __isl_give isl_printer *isl_printer_print_union_set(
                __isl_take isl_printer *p,
                __isl_keep isl_union_set *uset);

        #include <isl/union_map.h>
        __isl_give isl_printer *isl_printer_print_union_map(
                __isl_take isl_printer *p,
                __isl_keep isl_union_map *umap);

When called on a file printer, the following function flushes
the file.  When called on a string printer, the buffer is cleared.

        __isl_give isl_printer *isl_printer_flush(
                __isl_take isl_printer *p);

=head2 Creating New Sets and Relations

C<isl> has functions for creating some standard sets and relations.

=over

=item * Empty sets and relations

        __isl_give isl_basic_set *isl_basic_set_empty(
                __isl_take isl_dim *dim);
        __isl_give isl_basic_map *isl_basic_map_empty(
                __isl_take isl_dim *dim);
        __isl_give isl_set *isl_set_empty(
                __isl_take isl_dim *dim);
        __isl_give isl_map *isl_map_empty(
                __isl_take isl_dim *dim);
        __isl_give isl_union_set *isl_union_set_empty(
                __isl_take isl_dim *dim);
        __isl_give isl_union_map *isl_union_map_empty(
                __isl_take isl_dim *dim);

For C<isl_union_set>s and C<isl_union_map>s, the dimensions specification
is only used to specify the parameters.

=item * Universe sets and relations

        __isl_give isl_basic_set *isl_basic_set_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_basic_map *isl_basic_map_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_set *isl_set_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_map *isl_map_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_union_set *isl_union_set_universe(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_universe(
                __isl_take isl_union_map *umap);

The sets and relations constructed by the functions above
contain all integer values, while those constructed by the
functions below only contain non-negative values.

        __isl_give isl_basic_set *isl_basic_set_nat_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_basic_map *isl_basic_map_nat_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_set *isl_set_nat_universe(
                __isl_take isl_dim *dim);
        __isl_give isl_map *isl_map_nat_universe(
                __isl_take isl_dim *dim);

=item * Identity relations

        __isl_give isl_basic_map *isl_basic_map_identity(
                __isl_take isl_dim *dim);
        __isl_give isl_map *isl_map_identity(
                __isl_take isl_dim *dim);

The number of input and output dimensions in C<dim> needs
to be the same.

=item * Lexicographic order

        __isl_give isl_map *isl_map_lex_lt(
                __isl_take isl_dim *set_dim);
        __isl_give isl_map *isl_map_lex_le(
                __isl_take isl_dim *set_dim);
        __isl_give isl_map *isl_map_lex_gt(
                __isl_take isl_dim *set_dim);
        __isl_give isl_map *isl_map_lex_ge(
                __isl_take isl_dim *set_dim);
        __isl_give isl_map *isl_map_lex_lt_first(
                __isl_take isl_dim *dim, unsigned n);
        __isl_give isl_map *isl_map_lex_le_first(
                __isl_take isl_dim *dim, unsigned n);
        __isl_give isl_map *isl_map_lex_gt_first(
                __isl_take isl_dim *dim, unsigned n);
        __isl_give isl_map *isl_map_lex_ge_first(
                __isl_take isl_dim *dim, unsigned n);

The first four functions take a dimension specification for a B<set>
and return relations that express that the elements in the domain
are lexicographically less
(C<isl_map_lex_lt>), less or equal (C<isl_map_lex_le>),
greater (C<isl_map_lex_gt>) or greater or equal (C<isl_map_lex_ge>)
than the elements in the range.
The last four functions take a dimension specification for a map
and return relations that express that the first C<n> dimensions
in the domain are lexicographically less
(C<isl_map_lex_lt_first>), less or equal (C<isl_map_lex_le_first>),
greater (C<isl_map_lex_gt_first>) or greater or equal (C<isl_map_lex_ge_first>)
than the first C<n> dimensions in the range.

=back

A basic set or relation can be converted to a set or relation
using the following functions.

        __isl_give isl_set *isl_set_from_basic_set(
                __isl_take isl_basic_set *bset);
        __isl_give isl_map *isl_map_from_basic_map(
                __isl_take isl_basic_map *bmap);

Sets and relations can be converted to union sets and relations
using the following functions.

        __isl_give isl_union_map *isl_union_map_from_map(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_from_set(
                __isl_take isl_set *set);

Sets and relations can be copied and freed again using the following
functions.

        __isl_give isl_basic_set *isl_basic_set_copy(
                __isl_keep isl_basic_set *bset);
        __isl_give isl_set *isl_set_copy(__isl_keep isl_set *set);
        __isl_give isl_union_set *isl_union_set_copy(
                __isl_keep isl_union_set *uset);
        __isl_give isl_basic_map *isl_basic_map_copy(
                __isl_keep isl_basic_map *bmap);
        __isl_give isl_map *isl_map_copy(__isl_keep isl_map *map);
        __isl_give isl_union_map *isl_union_map_copy(
                __isl_keep isl_union_map *umap);
        void isl_basic_set_free(__isl_take isl_basic_set *bset);
        void isl_set_free(__isl_take isl_set *set);
        void isl_union_set_free(__isl_take isl_union_set *uset);
        void isl_basic_map_free(__isl_take isl_basic_map *bmap);
        void isl_map_free(__isl_take isl_map *map);
        void isl_union_map_free(__isl_take isl_union_map *umap);

Other sets and relations can be constructed by starting
from a universe set or relation, adding equality and/or
inequality constraints and then projecting out the
existentially quantified variables, if any.
Constraints can be constructed, manipulated and
added to basic sets and relations using the following functions.

        #include <isl/constraint.h>
        __isl_give isl_constraint *isl_equality_alloc(
                __isl_take isl_dim *dim);
        __isl_give isl_constraint *isl_inequality_alloc(
                __isl_take isl_dim *dim);
        void isl_constraint_set_constant(
                __isl_keep isl_constraint *constraint, isl_int v);
        void isl_constraint_set_coefficient(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, int pos, isl_int v);
        __isl_give isl_basic_map *isl_basic_map_add_constraint(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_constraint *constraint);
        __isl_give isl_basic_set *isl_basic_set_add_constraint(
                __isl_take isl_basic_set *bset,
                __isl_take isl_constraint *constraint);

For example, to create a set containing the even integers
between 10 and 42, you would use the following code.

        isl_int v;
        struct isl_dim *dim;
        struct isl_constraint *c;
        struct isl_basic_set *bset;

        isl_int_init(v);
        dim = isl_dim_set_alloc(ctx, 0, 2);
        bset = isl_basic_set_universe(isl_dim_copy(dim));

        c = isl_equality_alloc(isl_dim_copy(dim));
        isl_int_set_si(v, -1);
        isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
        isl_int_set_si(v, 2);
        isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
        bset = isl_basic_set_add_constraint(bset, c);

        c = isl_inequality_alloc(isl_dim_copy(dim));
        isl_int_set_si(v, -10);
        isl_constraint_set_constant(c, v);
        isl_int_set_si(v, 1);
        isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
        bset = isl_basic_set_add_constraint(bset, c);

        c = isl_inequality_alloc(dim);
        isl_int_set_si(v, 42);
        isl_constraint_set_constant(c, v);
        isl_int_set_si(v, -1);
        isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
        bset = isl_basic_set_add_constraint(bset, c);

        bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 1);

        isl_int_clear(v);

Or, alternatively,

        struct isl_basic_set *bset;
        bset = isl_basic_set_read_from_str(ctx,
                "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}", -1);

A basic set or relation can also be constructed from two matrices
describing the equalities and the inequalities.

        __isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
                __isl_take isl_dim *dim,
                __isl_take isl_mat *eq, __isl_take isl_mat *ineq,
                enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4);
        __isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
                __isl_take isl_dim *dim,
                __isl_take isl_mat *eq, __isl_take isl_mat *ineq,
                enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4, enum isl_dim_type c5);

The C<isl_dim_type> arguments indicate the order in which
different kinds of variables appear in the input matrices
and should be a permutation of C<isl_dim_cst>, C<isl_dim_param>,
C<isl_dim_set> and C<isl_dim_div> for sets and
of C<isl_dim_cst>, C<isl_dim_param>,
C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div> for relations.

=head2 Inspecting Sets and Relations

Usually, the user should not have to care about the actual constraints
of the sets and maps, but should instead apply the abstract operations
explained in the following sections.
Occasionally, however, it may be required to inspect the individual
coefficients of the constraints.  This section explains how to do so.
In these cases, it may also be useful to have C<isl> compute
an explicit representation of the existentially quantified variables.

        __isl_give isl_set *isl_set_compute_divs(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_compute_divs(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_compute_divs(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_compute_divs(
                __isl_take isl_union_map *umap);

This explicit representation defines the existentially quantified
variables as integer divisions of the other variables, possibly
including earlier existentially quantified variables.
An explicitly represented existentially quantified variable therefore
has a unique value when the values of the other variables are known.
If, furthermore, the same existentials, i.e., existentials
with the same explicit representations, should appear in the
same order in each of the disjuncts of a set or map, then the user should call
either of the following functions.

        __isl_give isl_set *isl_set_align_divs(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_align_divs(
                __isl_take isl_map *map);

Alternatively, the existentially quantified variables can be removed
using the following functions, which compute an overapproximation.

        __isl_give isl_basic_set *isl_basic_set_remove_divs(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_map *isl_basic_map_remove_divs(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_set_remove_divs(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_remove_divs(
                __isl_take isl_map *map);

To iterate over all the sets or maps in a union set or map, use

        int isl_union_set_foreach_set(__isl_keep isl_union_set *uset,
                int (*fn)(__isl_take isl_set *set, void *user),
                void *user);
        int isl_union_map_foreach_map(__isl_keep isl_union_map *umap,
                int (*fn)(__isl_take isl_map *map, void *user),
                void *user);

The number of sets or maps in a union set or map can be obtained
from

        int isl_union_set_n_set(__isl_keep isl_union_set *uset);
        int isl_union_map_n_map(__isl_keep isl_union_map *umap);

To extract the set or map from a union with a given dimension
specification, use

        __isl_give isl_set *isl_union_set_extract_set(
                __isl_keep isl_union_set *uset,
                __isl_take isl_dim *dim);
        __isl_give isl_map *isl_union_map_extract_map(
                __isl_keep isl_union_map *umap,
                __isl_take isl_dim *dim);

To iterate over all the basic sets or maps in a set or map, use

        int isl_set_foreach_basic_set(__isl_keep isl_set *set,
                int (*fn)(__isl_take isl_basic_set *bset, void *user),
                void *user);
        int isl_map_foreach_basic_map(__isl_keep isl_map *map,
                int (*fn)(__isl_take isl_basic_map *bmap, void *user),
                void *user);

The callback function C<fn> should return 0 if successful and
-1 if an error occurs.  In the latter case, or if any other error
occurs, the above functions will return -1.

It should be noted that C<isl> does not guarantee that
the basic sets or maps passed to C<fn> are disjoint.
If this is required, then the user should call one of
the following functions first.

        __isl_give isl_set *isl_set_make_disjoint(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_make_disjoint(
                __isl_take isl_map *map);

The number of basic sets in a set can be obtained
from

        int isl_set_n_basic_set(__isl_keep isl_set *set);

To iterate over the constraints of a basic set or map, use

        #include <isl/constraint.h>

        int isl_basic_map_foreach_constraint(
                __isl_keep isl_basic_map *bmap,
                int (*fn)(__isl_take isl_constraint *c, void *user),
                void *user);
        void isl_constraint_free(struct isl_constraint *c);

Again, the callback function C<fn> should return 0 if successful and
-1 if an error occurs.  In the latter case, or if any other error
occurs, the above functions will return -1.
The constraint C<c> represents either an equality or an inequality.
Use the following function to find out whether a constraint
represents an equality.  If not, it represents an inequality.

        int isl_constraint_is_equality(
                __isl_keep isl_constraint *constraint);

The coefficients of the constraints can be inspected using
the following functions.

        void isl_constraint_get_constant(
                __isl_keep isl_constraint *constraint, isl_int *v);
        void isl_constraint_get_coefficient(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, int pos, isl_int *v);

The explicit representations of the existentially quantified
variables can be inspected using the following functions.
Note that the user is only allowed to use these functions
if the inspected set or map is the result of a call
to C<isl_set_compute_divs> or C<isl_map_compute_divs>.

        __isl_give isl_div *isl_constraint_div(
                __isl_keep isl_constraint *constraint, int pos);
        isl_ctx *isl_div_get_ctx(__isl_keep isl_div *div);
        void isl_div_get_constant(__isl_keep isl_div *div,
                isl_int *v);
        void isl_div_get_denominator(__isl_keep isl_div *div,
                isl_int *v);
        void isl_div_get_coefficient(__isl_keep isl_div *div,
                enum isl_dim_type type, int pos, isl_int *v);

To obtain the constraints of a basic set or map in matrix
form, use the following functions.

        __isl_give isl_mat *isl_basic_set_equalities_matrix(
                __isl_keep isl_basic_set *bset,
                enum isl_dim_type c1, enum isl_dim_type c2,
                enum isl_dim_type c3, enum isl_dim_type c4);
        __isl_give isl_mat *isl_basic_set_inequalities_matrix(
                __isl_keep isl_basic_set *bset,
                enum isl_dim_type c1, enum isl_dim_type c2,
                enum isl_dim_type c3, enum isl_dim_type c4);
        __isl_give isl_mat *isl_basic_map_equalities_matrix(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4, enum isl_dim_type c5);
        __isl_give isl_mat *isl_basic_map_inequalities_matrix(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type c1,
                enum isl_dim_type c2, enum isl_dim_type c3,
                enum isl_dim_type c4, enum isl_dim_type c5);

The C<isl_dim_type> arguments dictate the order in which
different kinds of variables appear in the resulting matrix
and should be a permutation of C<isl_dim_cst>, C<isl_dim_param>,
C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div>.

The names of the domain and range spaces of a set or relation can be
read off using the following functions.

        const char *isl_basic_set_get_tuple_name(
                __isl_keep isl_basic_set *bset);
        const char *isl_set_get_tuple_name(
                __isl_keep isl_set *set);
        const char *isl_basic_map_get_tuple_name(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type);
        const char *isl_map_get_tuple_name(
                __isl_keep isl_map *map,
                enum isl_dim_type type);

As with C<isl_dim_get_tuple_name>, the value returned points to
an internal data structure.
The names of individual dimensions can be read off using
the following functions.

        const char *isl_constraint_get_dim_name(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned pos);
        const char *isl_basic_set_get_dim_name(
                __isl_keep isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos);
        const char *isl_set_get_dim_name(
                __isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);
        const char *isl_basic_map_get_dim_name(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos);
        const char *isl_map_get_dim_name(
                __isl_keep isl_map *map,
                enum isl_dim_type type, unsigned pos);

These functions are mostly useful to obtain the names
of the parameters.

=head2 Properties

=head3 Unary Properties

=over

=item * Emptiness

The following functions test whether the given set or relation
contains any integer points.  The ``plain'' variants do not perform
any computations, but simply check if the given set or relation
is already known to be empty.

        int isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset);
        int isl_basic_set_is_empty(__isl_keep isl_basic_set *bset);
        int isl_set_plain_is_empty(__isl_keep isl_set *set);
        int isl_set_is_empty(__isl_keep isl_set *set);
        int isl_union_set_is_empty(__isl_keep isl_union_set *uset);
        int isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap);
        int isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap);
        int isl_map_plain_is_empty(__isl_keep isl_map *map);
        int isl_map_is_empty(__isl_keep isl_map *map);
        int isl_union_map_is_empty(__isl_keep isl_union_map *umap);

=item * Universality

        int isl_basic_set_is_universe(__isl_keep isl_basic_set *bset);
        int isl_basic_map_is_universe(__isl_keep isl_basic_map *bmap);
        int isl_set_plain_is_universe(__isl_keep isl_set *set);

=item * Single-valuedness

        int isl_map_is_single_valued(__isl_keep isl_map *map);
        int isl_union_map_is_single_valued(__isl_keep isl_union_map *umap);

=item * Injectivity

        int isl_map_plain_is_injective(__isl_keep isl_map *map);
        int isl_map_is_injective(__isl_keep isl_map *map);
        int isl_union_map_plain_is_injective(
                __isl_keep isl_union_map *umap);
        int isl_union_map_is_injective(
                __isl_keep isl_union_map *umap);

=item * Bijectivity

        int isl_map_is_bijective(__isl_keep isl_map *map);
        int isl_union_map_is_bijective(__isl_keep isl_union_map *umap);

=item * Wrapping

The following functions check whether the domain of the given
(basic) set is a wrapped relation.

        int isl_basic_set_is_wrapping(
                __isl_keep isl_basic_set *bset);
        int isl_set_is_wrapping(__isl_keep isl_set *set);

=item * Internal Product

        int isl_basic_map_can_zip(
                __isl_keep isl_basic_map *bmap);
        int isl_map_can_zip(__isl_keep isl_map *map);

Check whether the product of domain and range of the given relation
can be computed,
i.e., whether both domain and range are nested relations.

=back

=head3 Binary Properties

=over

=item * Equality

        int isl_set_plain_is_equal(__isl_keep isl_set *set1,
                __isl_keep isl_set *set2);
        int isl_set_is_equal(__isl_keep isl_set *set1,
                __isl_keep isl_set *set2);
        int isl_union_set_is_equal(
                __isl_keep isl_union_set *uset1,
                __isl_keep isl_union_set *uset2);
        int isl_basic_map_is_equal(
                __isl_keep isl_basic_map *bmap1,
                __isl_keep isl_basic_map *bmap2);
        int isl_map_is_equal(__isl_keep isl_map *map1,
                __isl_keep isl_map *map2);
        int isl_map_plain_is_equal(__isl_keep isl_map *map1,
                __isl_keep isl_map *map2);
        int isl_union_map_is_equal(
                __isl_keep isl_union_map *umap1,
                __isl_keep isl_union_map *umap2);

=item * Disjointness

        int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
                __isl_keep isl_set *set2);

=item * Subset

        int isl_set_is_subset(__isl_keep isl_set *set1,
                __isl_keep isl_set *set2);
        int isl_set_is_strict_subset(
                __isl_keep isl_set *set1,
                __isl_keep isl_set *set2);
        int isl_union_set_is_subset(
                __isl_keep isl_union_set *uset1,
                __isl_keep isl_union_set *uset2);
        int isl_union_set_is_strict_subset(
                __isl_keep isl_union_set *uset1,
                __isl_keep isl_union_set *uset2);
        int isl_basic_map_is_subset(
                __isl_keep isl_basic_map *bmap1,
                __isl_keep isl_basic_map *bmap2);
        int isl_basic_map_is_strict_subset(
                __isl_keep isl_basic_map *bmap1,
                __isl_keep isl_basic_map *bmap2);
        int isl_map_is_subset(
                __isl_keep isl_map *map1,
                __isl_keep isl_map *map2);
        int isl_map_is_strict_subset(
                __isl_keep isl_map *map1,
                __isl_keep isl_map *map2);
        int isl_union_map_is_subset(
                __isl_keep isl_union_map *umap1,
                __isl_keep isl_union_map *umap2);
        int isl_union_map_is_strict_subset(
                __isl_keep isl_union_map *umap1,
                __isl_keep isl_union_map *umap2);

=back

=head2 Unary Operations

=over

=item * Complement

        __isl_give isl_set *isl_set_complement(
                __isl_take isl_set *set);

=item * Inverse map

        __isl_give isl_basic_map *isl_basic_map_reverse(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_reverse(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_reverse(
                __isl_take isl_union_map *umap);

=item * Projection

        __isl_give isl_basic_set *isl_basic_set_project_out(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_basic_map *isl_basic_map_project_out(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_basic_set *isl_basic_map_domain(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_set *isl_basic_map_range(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_map_domain(
                __isl_take isl_map *bmap);
        __isl_give isl_set *isl_map_range(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_map_domain(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_set *isl_union_map_range(
                __isl_take isl_union_map *umap);

        __isl_give isl_basic_map *isl_basic_map_domain_map(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_map *isl_basic_map_range_map(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map);
        __isl_give isl_map *isl_map_range_map(__isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_domain_map(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *isl_union_map_range_map(
                __isl_take isl_union_map *umap);

The functions above construct a (basic, regular or union) relation
that maps (a wrapped version of) the input relation to its domain or range.

=item * Elimination

        __isl_give isl_set *isl_set_eliminate(
                __isl_take isl_set *set, enum isl_dim_type type,
                unsigned first, unsigned n);

Eliminate the coefficients for the given dimensions from the constraints,
without removing the dimensions.

=item * Identity

        __isl_give isl_map *isl_set_identity(
                __isl_take isl_set *set);
        __isl_give isl_union_map *isl_union_set_identity(
                __isl_take isl_union_set *uset);

Construct an identity relation on the given (union) set.

=item * Deltas

        __isl_give isl_basic_set *isl_basic_map_deltas(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_map_deltas(__isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_map_deltas(
                __isl_take isl_union_map *umap);

These functions return a (basic) set containing the differences
between image elements and corresponding domain elements in the input.

        __isl_give isl_basic_map *isl_basic_map_deltas_map(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_deltas_map(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_deltas_map(
                __isl_take isl_union_map *umap);

The functions above construct a (basic, regular or union) relation
that maps (a wrapped version of) the input relation to its delta set.

=item * Coalescing

Simplify the representation of a set or relation by trying
to combine pairs of basic sets or relations into a single
basic set or relation.

        __isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set);
        __isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_coalesce(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_coalesce(
                __isl_take isl_union_map *umap);

=item * Detecting equalities

        __isl_give isl_basic_set *isl_basic_set_detect_equalities(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_map *isl_basic_map_detect_equalities(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_set_detect_equalities(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_detect_equalities(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_detect_equalities(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_detect_equalities(
                __isl_take isl_union_map *umap);

Simplify the representation of a set or relation by detecting implicit
equalities.

=item * Removing redundant constraints

        __isl_give isl_basic_set *isl_basic_set_remove_redundancies(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_map *isl_basic_map_remove_redundancies(
                __isl_take isl_basic_map *bmap);

=item * Convex hull

        __isl_give isl_basic_set *isl_set_convex_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_map_convex_hull(
                __isl_take isl_map *map);

If the input set or relation has any existentially quantified
variables, then the result of these operations is currently undefined.

=item * Simple hull

        __isl_give isl_basic_set *isl_set_simple_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_map_simple_hull(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_simple_hull(
                __isl_take isl_union_map *umap);

These functions compute a single basic set or relation
that contains the whole input set or relation.
In particular, the output is described by translates
of the constraints describing the basic sets or relations in the input.

=begin latex

(See \autoref{s:simple hull}.)

=end latex

=item * Affine hull

        __isl_give isl_basic_set *isl_basic_set_affine_hull(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_set *isl_set_affine_hull(
                __isl_take isl_set *set);
        __isl_give isl_union_set *isl_union_set_affine_hull(
                __isl_take isl_union_set *uset);
        __isl_give isl_basic_map *isl_basic_map_affine_hull(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_map *isl_map_affine_hull(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_affine_hull(
                __isl_take isl_union_map *umap);

In case of union sets and relations, the affine hull is computed
per space.

=item * Polyhedral hull

        __isl_give isl_basic_set *isl_set_polyhedral_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_map_polyhedral_hull(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_polyhedral_hull(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_polyhedral_hull(
                __isl_take isl_union_map *umap);

These functions compute a single basic set or relation
not involving any existentially quantified variables
that contains the whole input set or relation.
In case of union sets and relations, the polyhedral hull is computed
per space.

=item * Dual

The following functions compute either the set of (rational) coefficient
values of valid constraints for the given set or the set of (rational)
values satisfying the constraints with coefficients from the given set.
Internally, these two sets of functions perform essentially the
same operations, except that the set of coefficients is assumed to
be a cone, while the set of values may be any polyhedron.
The current implementation is based on the Farkas lemma and
Fourier-Motzkin elimination, but this may change or be made optional
in future.  In particular, future implementations may use different
dualization algorithms or skip the elimination step.

        __isl_give isl_basic_set *isl_basic_set_coefficients(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_set *isl_set_coefficients(
                __isl_take isl_set *set);
        __isl_give isl_union_set *isl_union_set_coefficients(
                __isl_take isl_union_set *bset);
        __isl_give isl_basic_set *isl_basic_set_solutions(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_set *isl_set_solutions(
                __isl_take isl_set *set);
        __isl_give isl_union_set *isl_union_set_solutions(
                __isl_take isl_union_set *bset);

=item * Power

        __isl_give isl_map *isl_map_power(__isl_take isl_map *map,
                int *exact);
        __isl_give isl_union_map *isl_union_map_power(
                __isl_take isl_union_map *umap, int *exact);

Compute a parametric representation for all positive powers I<k> of C<map>.
The result maps I<k> to a nested relation corresponding to the
I<k>th power of C<map>.
The result may be an overapproximation.  If the result is known to be exact,
then C<*exact> is set to C<1>.

=item * Transitive closure

        __isl_give isl_map *isl_map_transitive_closure(
                __isl_take isl_map *map, int *exact);
        __isl_give isl_union_map *isl_union_map_transitive_closure(
                __isl_take isl_union_map *umap, int *exact);

Compute the transitive closure of C<map>.
The result may be an overapproximation.  If the result is known to be exact,
then C<*exact> is set to C<1>.

=item * Reaching path lengths

        __isl_give isl_map *isl_map_reaching_path_lengths(
                __isl_take isl_map *map, int *exact);

Compute a relation that maps each element in the range of C<map>
to the lengths of all paths composed of edges in C<map> that
end up in the given element.
The result may be an overapproximation.  If the result is known to be exact,
then C<*exact> is set to C<1>.
To compute the I<maximal> path length, the resulting relation
should be postprocessed by C<isl_map_lexmax>.
In particular, if the input relation is a dependence relation
(mapping sources to sinks), then the maximal path length corresponds
to the free schedule.
Note, however, that C<isl_map_lexmax> expects the maximum to be
finite, so if the path lengths are unbounded (possibly due to
the overapproximation), then you will get an error message.

=item * Wrapping

        __isl_give isl_basic_set *isl_basic_map_wrap(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_map_wrap(
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_map_wrap(
                __isl_take isl_union_map *umap);
        __isl_give isl_basic_map *isl_basic_set_unwrap(
                __isl_take isl_basic_set *bset);
        __isl_give isl_map *isl_set_unwrap(
                __isl_take isl_set *set);
        __isl_give isl_union_map *isl_union_set_unwrap(
                __isl_take isl_union_set *uset);

=item * Flattening

Remove any internal structure of domain (and range) of the given
set or relation.  If there is any such internal structure in the input,
then the name of the space is also removed.

        __isl_give isl_basic_set *isl_basic_set_flatten(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_set_flatten(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_basic_map_flatten(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_flatten(
                __isl_take isl_map *map);

        __isl_give isl_map *isl_set_flatten_map(
                __isl_take isl_set *set);

The function above constructs a relation
that maps the input set to a flattened version of the set.

=item * Lifting

Lift the input set to a space with extra dimensions corresponding
to the existentially quantified variables in the input.
In particular, the result lives in a wrapped map where the domain
is the original space and the range corresponds to the original
existentially quantified variables.

        __isl_give isl_basic_set *isl_basic_set_lift(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_set_lift(
                __isl_take isl_set *set);
        __isl_give isl_union_set *isl_union_set_lift(
                __isl_take isl_union_set *uset);

=item * Internal Product

        __isl_give isl_basic_map *isl_basic_map_zip(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_zip(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_zip(
                __isl_take isl_union_map *umap);

Given a relation with nested relations for domain and range,
interchange the range of the domain with the domain of the range.

=item * Dimension manipulation

        __isl_give isl_set *isl_set_add_dims(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_map *isl_map_add_dims(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned n);

It is usually not advisable to directly change the (input or output)
space of a set or a relation as this removes the name and the internal
structure of the space.  However, the above functions can be useful
to add new parameters.

=back

=head2 Binary Operations

The two arguments of a binary operation not only need to live
in the same C<isl_ctx>, they currently also need to have
the same (number of) parameters.

=head3 Basic Operations

=over

=item * Intersection

        __isl_give isl_basic_set *isl_basic_set_intersect(
                __isl_take isl_basic_set *bset1,
                __isl_take isl_basic_set *bset2);
        __isl_give isl_set *isl_set_intersect(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);
        __isl_give isl_union_set *isl_union_set_intersect(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);
        __isl_give isl_basic_map *isl_basic_map_intersect_domain(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_map *isl_basic_map_intersect_range(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_map *isl_basic_map_intersect(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_intersect_domain(
                __isl_take isl_map *map,
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_intersect_range(
                __isl_take isl_map *map,
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_intersect(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_intersect_domain(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_intersect_range(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_intersect(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

=item * Union

        __isl_give isl_set *isl_basic_set_union(
                __isl_take isl_basic_set *bset1,
                __isl_take isl_basic_set *bset2);
        __isl_give isl_map *isl_basic_map_union(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_set *isl_set_union(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);
        __isl_give isl_map *isl_map_union(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_set *isl_union_set_union(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);
        __isl_give isl_union_map *isl_union_map_union(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

=item * Set difference

        __isl_give isl_set *isl_set_subtract(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);
        __isl_give isl_map *isl_map_subtract(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_set *isl_union_set_subtract(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);
        __isl_give isl_union_map *isl_union_map_subtract(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

=item * Application

        __isl_give isl_basic_set *isl_basic_set_apply(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_set_apply(
                __isl_take isl_set *set,
                __isl_take isl_map *map);
        __isl_give isl_union_set *isl_union_set_apply(
                __isl_take isl_union_set *uset,
                __isl_take isl_union_map *umap);
        __isl_give isl_basic_map *isl_basic_map_apply_domain(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_basic_map *isl_basic_map_apply_range(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_apply_domain(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_apply_domain(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);
        __isl_give isl_map *isl_map_apply_range(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_apply_range(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

=item * Cartesian Product

        __isl_give isl_set *isl_set_product(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);
        __isl_give isl_union_set *isl_union_set_product(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);
        __isl_give isl_basic_map *isl_basic_map_range_product(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_range_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_range_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);
        __isl_give isl_map *isl_map_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

The above functions compute the cross product of the given
sets or relations.  The domains and ranges of the results
are wrapped maps between domains and ranges of the inputs.
To obtain a ``flat'' product, use the following functions
instead.

        __isl_give isl_basic_set *isl_basic_set_flat_product(
                __isl_take isl_basic_set *bset1,
                __isl_take isl_basic_set *bset2);
        __isl_give isl_set *isl_set_flat_product(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);
        __isl_give isl_basic_map *isl_basic_map_flat_product(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_flat_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);

=item * Simplification

        __isl_give isl_basic_set *isl_basic_set_gist(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_set *context);
        __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
                __isl_take isl_set *context);
        __isl_give isl_union_set *isl_union_set_gist(
                __isl_take isl_union_set *uset,
                __isl_take isl_union_set *context);
        __isl_give isl_basic_map *isl_basic_map_gist(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_map *context);
        __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
                __isl_take isl_map *context);
        __isl_give isl_union_map *isl_union_map_gist(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_map *context);

The gist operation returns a set or relation that has the
same intersection with the context as the input set or relation.
Any implicit equality in the intersection is made explicit in the result,
while all inequalities that are redundant with respect to the intersection
are removed.
In case of union sets and relations, the gist operation is performed
per space.

=back

=head3 Lexicographic Optimization

Given a (basic) set C<set> (or C<bset>) and a zero-dimensional domain C<dom>,
the following functions
compute a set that contains the lexicographic minimum or maximum
of the elements in C<set> (or C<bset>) for those values of the parameters
that satisfy C<dom>.
If C<empty> is not C<NULL>, then C<*empty> is assigned a set
that contains the parameter values in C<dom> for which C<set> (or C<bset>)
has no elements.
In other words, the union of the parameter values
for which the result is non-empty and of C<*empty>
is equal to C<dom>.

        __isl_give isl_set *isl_basic_set_partial_lexmin(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_set *isl_basic_set_partial_lexmax(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_set *isl_set_partial_lexmin(
                __isl_take isl_set *set, __isl_take isl_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_set *isl_set_partial_lexmax(
                __isl_take isl_set *set, __isl_take isl_set *dom,
                __isl_give isl_set **empty);

Given a (basic) set C<set> (or C<bset>), the following functions simply
return a set containing the lexicographic minimum or maximum
of the elements in C<set> (or C<bset>).
In case of union sets, the optimum is computed per space.

        __isl_give isl_set *isl_basic_set_lexmin(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_basic_set_lexmax(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_set_lexmin(
                __isl_take isl_set *set);
        __isl_give isl_set *isl_set_lexmax(
                __isl_take isl_set *set);
        __isl_give isl_union_set *isl_union_set_lexmin(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_set *isl_union_set_lexmax(
                __isl_take isl_union_set *uset);

Given a (basic) relation C<map> (or C<bmap>) and a domain C<dom>,
the following functions
compute a relation that maps each element of C<dom>
to the single lexicographic minimum or maximum
of the elements that are associated to that same
element in C<map> (or C<bmap>).
If C<empty> is not C<NULL>, then C<*empty> is assigned a set
that contains the elements in C<dom> that do not map
to any elements in C<map> (or C<bmap>).
In other words, the union of the domain of the result and of C<*empty>
is equal to C<dom>.

        __isl_give isl_map *isl_basic_map_partial_lexmax(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_map *isl_basic_map_partial_lexmin(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_map *isl_map_partial_lexmax(
                __isl_take isl_map *map, __isl_take isl_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_map *isl_map_partial_lexmin(
                __isl_take isl_map *map, __isl_take isl_set *dom,
                __isl_give isl_set **empty);

Given a (basic) map C<map> (or C<bmap>), the following functions simply
return a map mapping each element in the domain of
C<map> (or C<bmap>) to the lexicographic minimum or maximum
of all elements associated to that element.
In case of union relations, the optimum is computed per space.

        __isl_give isl_map *isl_basic_map_lexmin(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_basic_map_lexmax(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_lexmin(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_lexmax(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_lexmin(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *isl_union_map_lexmax(
                __isl_take isl_union_map *umap);

=head2 Matrices

Matrices can be created, copied and freed using the following functions.

        #include <isl/mat.h>
        __isl_give isl_mat *isl_mat_alloc(struct isl_ctx *ctx,
                unsigned n_row, unsigned n_col);
        __isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat);
        void isl_mat_free(__isl_take isl_mat *mat);

Note that the elements of a newly created matrix may have arbitrary values.
The elements can be changed and inspected using the following functions.

        isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);
        int isl_mat_rows(__isl_keep isl_mat *mat);
        int isl_mat_cols(__isl_keep isl_mat *mat);
        int isl_mat_get_element(__isl_keep isl_mat *mat,
                int row, int col, isl_int *v);
        __isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
                int row, int col, isl_int v);
        __isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
                int row, int col, int v);

C<isl_mat_get_element> will return a negative value if anything went wrong.
In that case, the value of C<*v> is undefined.

The following function can be used to compute the (right) inverse
of a matrix, i.e., a matrix such that the product of the original
and the inverse (in that order) is a multiple of the identity matrix.
The input matrix is assumed to be of full row-rank.

        __isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat);

The following function can be used to compute the (right) kernel
(or null space) of a matrix, i.e., a matrix such that the product of
the original and the kernel (in that order) is the zero matrix.

        __isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat);

=head2 Points

Points are elements of a set.  They can be used to construct
simple sets (boxes) or they can be used to represent the
individual elements of a set.
The zero point (the origin) can be created using

        __isl_give isl_point *isl_point_zero(__isl_take isl_dim *dim);

The coordinates of a point can be inspected, set and changed
using

        void isl_point_get_coordinate(__isl_keep isl_point *pnt,
                enum isl_dim_type type, int pos, isl_int *v);
        __isl_give isl_point *isl_point_set_coordinate(
                __isl_take isl_point *pnt,
                enum isl_dim_type type, int pos, isl_int v);

        __isl_give isl_point *isl_point_add_ui(
                __isl_take isl_point *pnt,
                enum isl_dim_type type, int pos, unsigned val);
        __isl_give isl_point *isl_point_sub_ui(
                __isl_take isl_point *pnt,
                enum isl_dim_type type, int pos, unsigned val);

Points can be copied or freed using

        __isl_give isl_point *isl_point_copy(
                __isl_keep isl_point *pnt);
        void isl_point_free(__isl_take isl_point *pnt);

A singleton set can be created from a point using

        __isl_give isl_basic_set *isl_basic_set_from_point(
                __isl_take isl_point *pnt);
        __isl_give isl_set *isl_set_from_point(
                __isl_take isl_point *pnt);

and a box can be created from two opposite extremal points using

        __isl_give isl_basic_set *isl_basic_set_box_from_points(
                __isl_take isl_point *pnt1,
                __isl_take isl_point *pnt2);
        __isl_give isl_set *isl_set_box_from_points(
                __isl_take isl_point *pnt1,
                __isl_take isl_point *pnt2);

All elements of a B<bounded> (union) set can be enumerated using
the following functions.

        int isl_set_foreach_point(__isl_keep isl_set *set,
                int (*fn)(__isl_take isl_point *pnt, void *user),
                void *user);
        int isl_union_set_foreach_point(__isl_keep isl_union_set *uset,
                int (*fn)(__isl_take isl_point *pnt, void *user),
                void *user);

The function C<fn> is called for each integer point in
C<set> with as second argument the last argument of
the C<isl_set_foreach_point> call.  The function C<fn>
should return C<0> on success and C<-1> on failure.
In the latter case, C<isl_set_foreach_point> will stop
enumerating and return C<-1> as well.
If the enumeration is performed successfully and to completion,
then C<isl_set_foreach_point> returns C<0>.

To obtain a single point of a (basic) set, use

        __isl_give isl_point *isl_basic_set_sample_point(
                __isl_take isl_basic_set *bset);
        __isl_give isl_point *isl_set_sample_point(
                __isl_take isl_set *set);

If C<set> does not contain any (integer) points, then the
resulting point will be ``void'', a property that can be
tested using

        int isl_point_is_void(__isl_keep isl_point *pnt);

=head2 Piecewise Quasipolynomials

A piecewise quasipolynomial is a particular kind of function that maps
a parametric point to a rational value.
More specifically, a quasipolynomial is a polynomial expression in greatest
integer parts of affine expressions of parameters and variables.
A piecewise quasipolynomial is a subdivision of a given parametric
domain into disjoint cells with a quasipolynomial associated to
each cell.  The value of the piecewise quasipolynomial at a given
point is the value of the quasipolynomial associated to the cell
that contains the point.  Outside of the union of cells,
the value is assumed to be zero.
For example, the piecewise quasipolynomial

        [n] -> { [x] -> ((1 + n) - x) : x <= n and x >= 0 }

maps C<x> to C<1 + n - x> for values of C<x> between C<0> and C<n>.
A given piecewise quasipolynomial has a fixed domain dimension.
Union piecewise quasipolynomials are used to contain piecewise quasipolynomials
defined over different domains.
Piecewise quasipolynomials are mainly used by the C<barvinok>
library for representing the number of elements in a parametric set or map.
For example, the piecewise quasipolynomial above represents
the number of points in the map

        [n] -> { [x] -> [y] : x,y >= 0 and 0 <= x + y <= n }

=head3 Printing (Piecewise) Quasipolynomials

Quasipolynomials and piecewise quasipolynomials can be printed
using the following functions.

        __isl_give isl_printer *isl_printer_print_qpolynomial(
                __isl_take isl_printer *p,
                __isl_keep isl_qpolynomial *qp);

        __isl_give isl_printer *isl_printer_print_pw_qpolynomial(
                __isl_take isl_printer *p,
                __isl_keep isl_pw_qpolynomial *pwqp);

        __isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
                __isl_take isl_printer *p,
                __isl_keep isl_union_pw_qpolynomial *upwqp);

The output format of the printer
needs to be set to either C<ISL_FORMAT_ISL> or C<ISL_FORMAT_C>.
For C<isl_printer_print_union_pw_qpolynomial>, only C<ISL_FORMAT_ISL>
is supported.
In case of printing in C<ISL_FORMAT_C>, the user may want
to set the names of all dimensions

        __isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
                __isl_take isl_qpolynomial *qp,
                enum isl_dim_type type, unsigned pos,
                const char *s);
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_set_dim_name(
                __isl_take isl_pw_qpolynomial *pwqp,
                enum isl_dim_type type, unsigned pos,
                const char *s);

=head3 Creating New (Piecewise) Quasipolynomials

Some simple quasipolynomials can be created using the following functions.
More complicated quasipolynomials can be created by applying
operations such as addition and multiplication
on the resulting quasipolynomials

        __isl_give isl_qpolynomial *isl_qpolynomial_zero(
                __isl_take isl_dim *dim);
        __isl_give isl_qpolynomial *isl_qpolynomial_one(
                __isl_take isl_dim *dim);
        __isl_give isl_qpolynomial *isl_qpolynomial_infty(
                __isl_take isl_dim *dim);
        __isl_give isl_qpolynomial *isl_qpolynomial_neginfty(
                __isl_take isl_dim *dim);
        __isl_give isl_qpolynomial *isl_qpolynomial_nan(
                __isl_take isl_dim *dim);
        __isl_give isl_qpolynomial *isl_qpolynomial_rat_cst(
                __isl_take isl_dim *dim,
                const isl_int n, const isl_int d);
        __isl_give isl_qpolynomial *isl_qpolynomial_div(
                __isl_take isl_div *div);
        __isl_give isl_qpolynomial *isl_qpolynomial_var(
                __isl_take isl_dim *dim,
                enum isl_dim_type type, unsigned pos);

The zero piecewise quasipolynomial or a piecewise quasipolynomial
with a single cell can be created using the following functions.
Multiple of these single cell piecewise quasipolynomials can
be combined to create more complicated piecewise quasipolynomials.

        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(
                __isl_take isl_dim *dim);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
                __isl_take isl_set *set,
                __isl_take isl_qpolynomial *qp);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_zero(
                __isl_take isl_dim *dim);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_from_pw_qpolynomial(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add_pw_qpolynomial(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_pw_qpolynomial *pwqp);

Quasipolynomials can be copied and freed again using the following
functions.

        __isl_give isl_qpolynomial *isl_qpolynomial_copy(
                __isl_keep isl_qpolynomial *qp);
        void isl_qpolynomial_free(__isl_take isl_qpolynomial *qp);

        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
                __isl_keep isl_pw_qpolynomial *pwqp);
        void isl_pw_qpolynomial_free(
                __isl_take isl_pw_qpolynomial *pwqp);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_copy(
                __isl_keep isl_union_pw_qpolynomial *upwqp);
        void isl_union_pw_qpolynomial_free(
                __isl_take isl_union_pw_qpolynomial *upwqp);

=head3 Inspecting (Piecewise) Quasipolynomials

To iterate over all piecewise quasipolynomials in a union
piecewise quasipolynomial, use the following function

        int isl_union_pw_qpolynomial_foreach_pw_qpolynomial(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                int (*fn)(__isl_take isl_pw_qpolynomial *pwqp, void *user),
                void *user);

To extract the piecewise quasipolynomial from a union with a given dimension
specification, use

        __isl_give isl_pw_qpolynomial *
        isl_union_pw_qpolynomial_extract_pw_qpolynomial(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_dim *dim);

To iterate over the cells in a piecewise quasipolynomial,
use either of the following two functions

        int isl_pw_qpolynomial_foreach_piece(
                __isl_keep isl_pw_qpolynomial *pwqp,
                int (*fn)(__isl_take isl_set *set,
                          __isl_take isl_qpolynomial *qp,
                          void *user), void *user);
        int isl_pw_qpolynomial_foreach_lifted_piece(
                __isl_keep isl_pw_qpolynomial *pwqp,
                int (*fn)(__isl_take isl_set *set,
                          __isl_take isl_qpolynomial *qp,
                          void *user), void *user);

As usual, the function C<fn> should return C<0> on success
and C<-1> on failure.  The difference between
C<isl_pw_qpolynomial_foreach_piece> and
C<isl_pw_qpolynomial_foreach_lifted_piece> is that
C<isl_pw_qpolynomial_foreach_lifted_piece> will first
compute unique representations for all existentially quantified
variables and then turn these existentially quantified variables
into extra set variables, adapting the associated quasipolynomial
accordingly.  This means that the C<set> passed to C<fn>
will not have any existentially quantified variables, but that
the dimensions of the sets may be different for different
invocations of C<fn>.

To iterate over all terms in a quasipolynomial,
use

        int isl_qpolynomial_foreach_term(
                __isl_keep isl_qpolynomial *qp,
                int (*fn)(__isl_take isl_term *term,
                          void *user), void *user);

The terms themselves can be inspected and freed using
these functions

        unsigned isl_term_dim(__isl_keep isl_term *term,
                enum isl_dim_type type);
        void isl_term_get_num(__isl_keep isl_term *term,
                isl_int *n);
        void isl_term_get_den(__isl_keep isl_term *term,
                isl_int *d);
        int isl_term_get_exp(__isl_keep isl_term *term,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_div *isl_term_get_div(
                __isl_keep isl_term *term, unsigned pos);
        void isl_term_free(__isl_take isl_term *term);

Each term is a product of parameters, set variables and
integer divisions.  The function C<isl_term_get_exp>
returns the exponent of a given dimensions in the given term.
The C<isl_int>s in the arguments of C<isl_term_get_num>
and C<isl_term_get_den> need to have been initialized
using C<isl_int_init> before calling these functions.

=head3 Properties of (Piecewise) Quasipolynomials

To check whether a quasipolynomial is actually a constant,
use the following function.

        int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
                isl_int *n, isl_int *d);

If C<qp> is a constant and if C<n> and C<d> are not C<NULL>
then the numerator and denominator of the constant
are returned in C<*n> and C<*d>, respectively.

=head3 Operations on (Piecewise) Quasipolynomials

        __isl_give isl_qpolynomial *isl_qpolynomial_neg(
                __isl_take isl_qpolynomial *qp);
        __isl_give isl_qpolynomial *isl_qpolynomial_add(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __isl_give isl_qpolynomial *isl_qpolynomial_sub(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __isl_give isl_qpolynomial *isl_qpolynomial_mul(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __isl_give isl_qpolynomial *isl_qpolynomial_pow(
                __isl_take isl_qpolynomial *qp, unsigned exponent);

        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_disjoint(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add(
                __isl_take isl_union_pw_qpolynomial *upwqp1,
                __isl_take isl_union_pw_qpolynomial *upwqp2);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_sub(
                __isl_take isl_union_pw_qpolynomial *upwqp1,
                __isl_take isl_union_pw_qpolynomial *upwqp2);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
                __isl_take isl_union_pw_qpolynomial *upwqp1,
                __isl_take isl_union_pw_qpolynomial *upwqp2);

        __isl_give isl_qpolynomial *isl_pw_qpolynomial_eval(
                __isl_take isl_pw_qpolynomial *pwqp,
                __isl_take isl_point *pnt);

        __isl_give isl_qpolynomial *isl_union_pw_qpolynomial_eval(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_point *pnt);

        __isl_give isl_set *isl_pw_qpolynomial_domain(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_intersect_domain(
                __isl_take isl_pw_qpolynomial *pwpq,
                __isl_take isl_set *set);

        __isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
                __isl_take isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_intersect_domain(
                __isl_take isl_union_pw_qpolynomial *upwpq,
                __isl_take isl_union_set *uset);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_coalesce(
                __isl_take isl_union_pw_qpolynomial *upwqp);

        __isl_give isl_qpolynomial *isl_qpolynomial_gist(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_set *context);

        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist(
                __isl_take isl_pw_qpolynomial *pwqp,
                __isl_take isl_set *context);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_union_set *context);

The gist operation applies the gist operation to each of
the cells in the domain of the input piecewise quasipolynomial.
The context is also exploited
to simplify the quasipolynomials associated to each cell.

        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
                __isl_take isl_pw_qpolynomial *pwqp, int sign);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_to_polynomial(
                __isl_take isl_union_pw_qpolynomial *upwqp, int sign);

Approximate each quasipolynomial by a polynomial.  If C<sign> is positive,
the polynomial will be an overapproximation.  If C<sign> is negative,
it will be an underapproximation.  If C<sign> is zero, the approximation
will lie somewhere in between.

=head2 Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions

A piecewise quasipolynomial reduction is a piecewise
reduction (or fold) of quasipolynomials.
In particular, the reduction can be maximum or a minimum.
The objects are mainly used to represent the result of
an upper or lower bound on a quasipolynomial over its domain,
i.e., as the result of the following function.

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_bound(
                __isl_take isl_pw_qpolynomial *pwqp,
                enum isl_fold type, int *tight);

        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_bound(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                enum isl_fold type, int *tight);

The C<type> argument may be either C<isl_fold_min> or C<isl_fold_max>.
If C<tight> is not C<NULL>, then C<*tight> is set to C<1>
is the returned bound is known be tight, i.e., for each value
of the parameters there is at least
one element in the domain that reaches the bound.
If the domain of C<pwqp> is not wrapping, then the bound is computed
over all elements in that domain and the result has a purely parametric
domain.  If the domain of C<pwqp> is wrapping, then the bound is
computed over the range of the wrapped relation.  The domain of the
wrapped relation becomes the domain of the result.

A (piecewise) quasipolynomial reduction can be copied or freed using the
following functions.

        __isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_copy(
                __isl_keep isl_qpolynomial_fold *fold);
        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_copy(
                __isl_keep isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_copy(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);
        void isl_qpolynomial_fold_free(
                __isl_take isl_qpolynomial_fold *fold);
        void isl_pw_qpolynomial_fold_free(
                __isl_take isl_pw_qpolynomial_fold *pwf);
        void isl_union_pw_qpolynomial_fold_free(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);

=head3 Printing Piecewise Quasipolynomial Reductions

Piecewise quasipolynomial reductions can be printed
using the following function.

        __isl_give isl_printer *isl_printer_print_pw_qpolynomial_fold(
                __isl_take isl_printer *p,
                __isl_keep isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_printer *isl_printer_print_union_pw_qpolynomial_fold(
                __isl_take isl_printer *p,
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);

For C<isl_printer_print_pw_qpolynomial_fold>,
output format of the printer
needs to be set to either C<ISL_FORMAT_ISL> or C<ISL_FORMAT_C>.
For C<isl_printer_print_union_pw_qpolynomial_fold>,
output format of the printer
needs to be set to C<ISL_FORMAT_ISL>.
In case of printing in C<ISL_FORMAT_C>, the user may want
to set the names of all dimensions

        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_set_dim_name(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                enum isl_dim_type type, unsigned pos,
                const char *s);

=head3 Inspecting (Piecewise) Quasipolynomial Reductions

To iterate over all piecewise quasipolynomial reductions in a union
piecewise quasipolynomial reduction, use the following function

        int isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf,
                int (*fn)(__isl_take isl_pw_qpolynomial_fold *pwf,
                            void *user), void *user);

To iterate over the cells in a piecewise quasipolynomial reduction,
use either of the following two functions

        int isl_pw_qpolynomial_fold_foreach_piece(
                __isl_keep isl_pw_qpolynomial_fold *pwf,
                int (*fn)(__isl_take isl_set *set,
                          __isl_take isl_qpolynomial_fold *fold,
                          void *user), void *user);
        int isl_pw_qpolynomial_fold_foreach_lifted_piece(
                __isl_keep isl_pw_qpolynomial_fold *pwf,
                int (*fn)(__isl_take isl_set *set,
                          __isl_take isl_qpolynomial_fold *fold,
                          void *user), void *user);

See L<Inspecting (Piecewise) Quasipolynomials> for an explanation
of the difference between these two functions.

To iterate over all quasipolynomials in a reduction, use

        int isl_qpolynomial_fold_foreach_qpolynomial(
                __isl_keep isl_qpolynomial_fold *fold,
                int (*fn)(__isl_take isl_qpolynomial *qp,
                          void *user), void *user);

=head3 Operations on Piecewise Quasipolynomial Reductions

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
                __isl_take isl_pw_qpolynomial_fold *pwf1,
                __isl_take isl_pw_qpolynomial_fold *pwf2);

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
                __isl_take isl_pw_qpolynomial_fold *pwf1,
                __isl_take isl_pw_qpolynomial_fold *pwf2);

        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold(
                __isl_take isl_union_pw_qpolynomial_fold *upwf1,
                __isl_take isl_union_pw_qpolynomial_fold *upwf2);

        __isl_give isl_qpolynomial *isl_pw_qpolynomial_fold_eval(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_point *pnt);

        __isl_give isl_qpolynomial *isl_union_pw_qpolynomial_fold_eval(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_point *pnt);

        __isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);
        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_intersect_domain(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_union_set *uset);

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_coalesce(
                __isl_take isl_pw_qpolynomial_fold *pwf);

        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_coalesce(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_gist(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_set *context);

        __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_gist(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_union_set *context);

The gist operation applies the gist operation to each of
the cells in the domain of the input piecewise quasipolynomial reduction.
In future, the operation will also exploit the context
to simplify the quasipolynomial reductions associated to each cell.

        __isl_give isl_pw_qpolynomial_fold *
        isl_set_apply_pw_qpolynomial_fold(
                __isl_take isl_set *set,
                __isl_take isl_pw_qpolynomial_fold *pwf,
                int *tight);
        __isl_give isl_pw_qpolynomial_fold *
        isl_map_apply_pw_qpolynomial_fold(
                __isl_take isl_map *map,
                __isl_take isl_pw_qpolynomial_fold *pwf,
                int *tight);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_set_apply_union_pw_qpolynomial_fold(
                __isl_take isl_union_set *uset,
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                int *tight);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_map_apply_union_pw_qpolynomial_fold(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                int *tight);

The functions taking a map
compose the given map with the given piecewise quasipolynomial reduction.
That is, compute a bound (of the same type as C<pwf> or C<upwf> itself)
over all elements in the intersection of the range of the map
and the domain of the piecewise quasipolynomial reduction
as a function of an element in the domain of the map.
The functions taking a set compute a bound over all elements in the
intersection of the set and the domain of the
piecewise quasipolynomial reduction.

=head2 Dependence Analysis

C<isl> contains specialized functionality for performing
array dataflow analysis.  That is, given a I<sink> access relation
and a collection of possible I<source> access relations,
C<isl> can compute relations that describe
for each iteration of the sink access, which iteration
of which of the source access relations was the last
to access the same data element before the given iteration
of the sink access.
To compute standard flow dependences, the sink should be
a read, while the sources should be writes.
If any of the source accesses are marked as being I<may>
accesses, then there will be a dependence to the last
I<must> access B<and> to any I<may> access that follows
this last I<must> access.
In particular, if I<all> sources are I<may> accesses,
then memory based dependence analysis is performed.
If, on the other hand, all sources are I<must> accesses,
then value based dependence analysis is performed.

        #include <isl/flow.h>

        typedef int (*isl_access_level_before)(void *first, void *second);

        __isl_give isl_access_info *isl_access_info_alloc(
                __isl_take isl_map *sink,
                void *sink_user, isl_access_level_before fn,
                int max_source);
        __isl_give isl_access_info *isl_access_info_add_source(
                __isl_take isl_access_info *acc,
                __isl_take isl_map *source, int must,
                void *source_user);
        void isl_access_info_free(__isl_take isl_access_info *acc);

        __isl_give isl_flow *isl_access_info_compute_flow(
                __isl_take isl_access_info *acc);

        int isl_flow_foreach(__isl_keep isl_flow *deps,
                int (*fn)(__isl_take isl_map *dep, int must,
                          void *dep_user, void *user),
                void *user);
        __isl_give isl_map *isl_flow_get_no_source(
                __isl_keep isl_flow *deps, int must);
        void isl_flow_free(__isl_take isl_flow *deps);

The function C<isl_access_info_compute_flow> performs the actual
dependence analysis.  The other functions are used to construct
the input for this function or to read off the output.

The input is collected in an C<isl_access_info>, which can
be created through a call to C<isl_access_info_alloc>.
The arguments to this functions are the sink access relation
C<sink>, a token C<sink_user> used to identify the sink
access to the user, a callback function for specifying the
relative order of source and sink accesses, and the number
of source access relations that will be added.
The callback function has type C<int (*)(void *first, void *second)>.
The function is called with two user supplied tokens identifying
either a source or the sink and it should return the shared nesting
level and the relative order of the two accesses.
In particular, let I<n> be the number of loops shared by
the two accesses.  If C<first> precedes C<second> textually,
then the function should return I<2 * n + 1>; otherwise,
it should return I<2 * n>.
The sources can be added to the C<isl_access_info> by performing
(at most) C<max_source> calls to C<isl_access_info_add_source>.
C<must> indicates whether the source is a I<must> access
or a I<may> access.  Note that a multi-valued access relation
should only be marked I<must> if every iteration in the domain
of the relation accesses I<all> elements in its image.
The C<source_user> token is again used to identify
the source access.  The range of the source access relation
C<source> should have the same dimension as the range
of the sink access relation.
The C<isl_access_info_free> function should usually not be
called explicitly, because it is called implicitly by
C<isl_access_info_compute_flow>.

The result of the dependence analysis is collected in an
C<isl_flow>.  There may be elements of
the sink access for which no preceding source access could be
found or for which all preceding sources are I<may> accesses.
The relations containing these elements can be obtained through
calls to C<isl_flow_get_no_source>, the first with C<must> set
and the second with C<must> unset.
In the case of standard flow dependence analysis,
with the sink a read and the sources I<must> writes,
the first relation corresponds to the reads from uninitialized
array elements and the second relation is empty.
The actual flow dependences can be extracted using
C<isl_flow_foreach>.  This function will call the user-specified
callback function C<fn> for each B<non-empty> dependence between
a source and the sink.  The callback function is called
with four arguments, the actual flow dependence relation
mapping source iterations to sink iterations, a boolean that
indicates whether it is a I<must> or I<may> dependence, a token
identifying the source and an additional C<void *> with value
equal to the third argument of the C<isl_flow_foreach> call.
A dependence is marked I<must> if it originates from a I<must>
source and if it is not followed by any I<may> sources.

After finishing with an C<isl_flow>, the user should call
C<isl_flow_free> to free all associated memory.

A higher-level interface to dependence analysis is provided
by the following function.

        #include <isl/flow.h>

        int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
                __isl_take isl_union_map *must_source,
                __isl_take isl_union_map *may_source,
                __isl_take isl_union_map *schedule,
                __isl_give isl_union_map **must_dep,
                __isl_give isl_union_map **may_dep,
                __isl_give isl_union_map **must_no_source,
                __isl_give isl_union_map **may_no_source);

The arrays are identified by the tuple names of the ranges
of the accesses.  The iteration domains by the tuple names
of the domains of the accesses and of the schedule.
The relative order of the iteration domains is given by the
schedule.  The relations returned through C<must_no_source>
and C<may_no_source> are subsets of C<sink>.
Any of C<must_dep>, C<may_dep>, C<must_no_source>
or C<may_no_source> may be C<NULL>, but a C<NULL> value for
any of the other arguments is treated as an error.

=head2 Scheduling

B<The functionality described in this section is fairly new
and may be subject to change.>

The following function can be used to compute a schedule
for a union of domains.  The generated schedule respects
all C<validity> dependences.  That is, all dependence distances
over these dependences in the scheduled space are lexicographically
positive.  The generated schedule schedule also tries to minimize
the dependence distances over C<proximity> dependences.
Moreover, it tries to obtain sequences (bands) of schedule dimensions
for groups of domains where the dependence distances have only
non-negative values.
The algorithm used to construct the schedule is similar to that
of C<Pluto>.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_union_set_compute_schedule(
                __isl_take isl_union_set *domain,
                __isl_take isl_union_map *validity,
                __isl_take isl_union_map *proximity);
        void *isl_schedule_free(__isl_take isl_schedule *sched);

A mapping from the domains to the scheduled space can be obtained
from an C<isl_schedule> using the following function.

        __isl_give isl_union_map *isl_schedule_get_map(
                __isl_keep isl_schedule *sched);

This mapping can also be obtained in pieces using the following functions.

        int isl_schedule_n_band(__isl_keep isl_schedule *sched);
        __isl_give isl_union_map *isl_schedule_get_band(
                __isl_keep isl_schedule *sched, unsigned band);

C<isl_schedule_n_band> returns the maximal number of bands.
C<isl_schedule_get_band> returns a union of mappings from a domain to
the band of consecutive schedule dimensions with the given sequence
number for that domain.  Bands with the same sequence number but for
different domains may be completely unrelated.
Within a band, the corresponding coordinates of the distance vectors
are all non-negative, assuming that the coordinates for all previous
bands are all zero.

=head2 Parametric Vertex Enumeration

The parametric vertex enumeration described in this section
is mainly intended to be used internally and by the C<barvinok>
library.

        #include <isl/vertices.h>
        __isl_give isl_vertices *isl_basic_set_compute_vertices(
                __isl_keep isl_basic_set *bset);

The function C<isl_basic_set_compute_vertices> performs the
actual computation of the parametric vertices and the chamber
decomposition and store the result in an C<isl_vertices> object.
This information can be queried by either iterating over all
the vertices or iterating over all the chambers or cells
and then iterating over all vertices that are active on the chamber.

        int isl_vertices_foreach_vertex(
                __isl_keep isl_vertices *vertices,
                int (*fn)(__isl_take isl_vertex *vertex, void *user),
                void *user);

        int isl_vertices_foreach_cell(
                __isl_keep isl_vertices *vertices,
                int (*fn)(__isl_take isl_cell *cell, void *user),
                void *user);
        int isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
                int (*fn)(__isl_take isl_vertex *vertex, void *user),
                void *user);

Other operations that can be performed on an C<isl_vertices> object are
the following.

        isl_ctx *isl_vertices_get_ctx(
                __isl_keep isl_vertices *vertices);
        int isl_vertices_get_n_vertices(
                __isl_keep isl_vertices *vertices);
        void isl_vertices_free(__isl_take isl_vertices *vertices);

Vertices can be inspected and destroyed using the following functions.

        isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
        int isl_vertex_get_id(__isl_keep isl_vertex *vertex);
        __isl_give isl_basic_set *isl_vertex_get_domain(
                __isl_keep isl_vertex *vertex);
        __isl_give isl_basic_set *isl_vertex_get_expr(
                __isl_keep isl_vertex *vertex);
        void isl_vertex_free(__isl_take isl_vertex *vertex);

C<isl_vertex_get_expr> returns a singleton parametric set describing
the vertex, while C<isl_vertex_get_domain> returns the activity domain
of the vertex.
Note that C<isl_vertex_get_domain> and C<isl_vertex_get_expr> return
B<rational> basic sets, so they should mainly be used for inspection
and should not be mixed with integer sets.

Chambers can be inspected and destroyed using the following functions.

        isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);
        __isl_give isl_basic_set *isl_cell_get_domain(
                __isl_keep isl_cell *cell);
        void isl_cell_free(__isl_take isl_cell *cell);

=head1 Applications

Although C<isl> is mainly meant to be used as a library,
it also contains some basic applications that use some
of the functionality of C<isl>.
The input may be specified in either the L<isl format>
or the L<PolyLib format>.

=head2 C<isl_polyhedron_sample>

C<isl_polyhedron_sample> takes a polyhedron as input and prints
an integer element of the polyhedron, if there is any.
The first column in the output is the denominator and is always
equal to 1.  If the polyhedron contains no integer points,
then a vector of length zero is printed.

=head2 C<isl_pip>

C<isl_pip> takes the same input as the C<example> program
from the C<piplib> distribution, i.e., a set of constraints
on the parameters, a line containing only -1 and finally a set
of constraints on a parametric polyhedron.
The coefficients of the parameters appear in the last columns
(but before the final constant column).
The output is the lexicographic minimum of the parametric polyhedron.
As C<isl> currently does not have its own output format, the output
is just a dump of the internal state.

=head2 C<isl_polyhedron_minimize>

C<isl_polyhedron_minimize> computes the minimum of some linear
or affine objective function over the integer points in a polyhedron.
If an affine objective function
is given, then the constant should appear in the last column.

=head2 C<isl_polytope_scan>

Given a polytope, C<isl_polytope_scan> prints
all integer points in the polytope.