[platform/upstream/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 a B<map> space as input.  An old call
C<isl_map_identity(space)> can be rewritten to
C<isl_map_identity(isl_space_map_from_set(space))>.

=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

=head3 Changes since isl-0.06

=over

=item * The format of C<isl_printer_print_qpolynomial>'s
C<ISL_FORMAT_ISL> output has changed.
Use C<ISL_FORMAT_C> to obtain the old output.

=item * The C<*_fast_*> functions have been renamed to C<*_plain_*>.
Some of the old names have been kept for backward compatibility,
but they will be removed in the future.

=back

=head3 Changes since isl-0.07

=over

=item * The function C<isl_pw_aff_max> has been renamed to
C<isl_pw_aff_union_max>.
Similarly, the function C<isl_pw_aff_add> has been renamed to
C<isl_pw_aff_union_add>.

=item * The C<isl_dim> type has been renamed to C<isl_space>
along with the associated functions.
Some of the old names have been kept for backward compatibility,
but they will be removed in the future.

=item * Spaces of maps, sets and parameter domains are now
treated differently.  The distinction between map spaces and set spaces
has always been made on a conceptual level, but proper use of such spaces
was never checked.  Furthermore, up until isl-0.07 there was no way
of explicitly creating a parameter space.  These can now be created
directly using C<isl_space_params_alloc> or from other spaces using
C<isl_space_params>.

=item * The space in which C<isl_aff>, C<isl_pw_aff>, C<isl_qpolynomial>,
C<isl_pw_qpolynomial>, C<isl_qpolynomial_fold> and C<isl_pw_qpolynomial_fold>
objects live is now a map space
instead of a set space.  This means, for example, that the dimensions
of the domain of an C<isl_aff> are now considered to be of type
C<isl_dim_in> instead of C<isl_dim_set>.  Extra functions have been
added to obtain the domain space.  Some of the constructors still
take a domain space and have therefore been renamed.

=item * The functions C<isl_equality_alloc> and C<isl_inequality_alloc>
now take an C<isl_local_space> instead of an C<isl_space>.
An C<isl_local_space> can be created from an C<isl_space>
using C<isl_local_space_from_space>.

=item * The C<isl_div> type has been removed.  Functions that used
to return an C<isl_div> now return an C<isl_aff>.
Note that the space of an C<isl_aff> is that of relation.
When replacing a call to C<isl_div_get_coefficient> by a call to
C<isl_aff_get_coefficient> any C<isl_dim_set> argument needs
to be replaced by C<isl_dim_in>.
A call to C<isl_aff_from_div> can be replaced by a call
to C<isl_aff_floor>.
A call to C<isl_qpolynomial_div(div)> call be replaced by
the nested call

        isl_qpolynomial_from_aff(isl_aff_floor(div))

The function C<isl_constraint_div> has also been renamed
to C<isl_constraint_get_div>.

=item * The C<nparam> argument has been removed from
C<isl_map_read_from_str> and similar functions.
When reading input in the original PolyLib format,
the result will have no parameters.
If parameters are expected, the caller may want to perform
dimension manipulation on the result.

=back

=head3 Changes since isl-0.09

=over

=item * The C<schedule_split_parallel> option has been replaced
by the C<schedule_split_scaled> option.

=item * The first argument of C<isl_pw_aff_cond> is now
an C<isl_pw_aff> instead of an C<isl_set>.
A call C<isl_pw_aff_cond(a, b, c)> can be replaced by

        isl_pw_aff_cond(isl_set_indicator_function(a), b, c)

=back

=head3 Changes since isl-0.10

=over

=item * The functions C<isl_set_dim_has_lower_bound> and
C<isl_set_dim_has_upper_bound> have been renamed to
C<isl_set_dim_has_any_lower_bound> and
C<isl_set_dim_has_any_upper_bound>.
The new C<isl_set_dim_has_lower_bound> and
C<isl_set_dim_has_upper_bound> have slightly different meanings.

=back

=head1 License

C<isl> is released under the MIT license.

=over

Permission is hereby granted, free of charge, to any person obtaining a copy of
this software and associated documentation files (the "Software"), to deal in
the Software without restriction, including without limitation the rights to
use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies
of the Software, and to permit persons to whom the Software is furnished to do
so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.

=back

Note that C<isl> currently requires C<GMP>, which is released
under the GNU Lesser General Public License (LGPL).  This means
that code linked against C<isl> is also linked against LGPL code.

=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 Integer Set 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 Values

An C<isl_val> represents an integer value, a rational value
or one of three special values, infinity, negative infinity and NaN.
Some predefined values can be created using the following functions.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_zero(isl_ctx *ctx);
        __isl_give isl_val *isl_val_one(isl_ctx *ctx);
        __isl_give isl_val *isl_val_nan(isl_ctx *ctx);
        __isl_give isl_val *isl_val_infty(isl_ctx *ctx);
        __isl_give isl_val *isl_val_neginfty(isl_ctx *ctx);

Specific integer values can be created using the following functions.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_int_from_si(isl_ctx *ctx,
                long i);
        __isl_give isl_val *isl_val_int_from_ui(isl_ctx *ctx,
                unsigned long u);

They can be copied and freed using the following functions.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_copy(__isl_keep isl_val *v);
        void *isl_val_free(__isl_take isl_val *v);

They can be inspected using the following functions.

        #include <isl/val.h>
        isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val);
        long isl_val_get_num_si(__isl_keep isl_val *v);
        long isl_val_get_den_si(__isl_keep isl_val *v);
        double isl_val_get_d(__isl_keep isl_val *v);

Note that C<isl_val_get_num_si>, C<isl_val_get_den_si> and
C<isl_val_get_d> can only be applied to rational values.

An C<isl_val> can be modified using the following function.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_set_si(__isl_take isl_val *v,
                long i);

The following unary properties are defined on C<isl_val>s.

        #include <isl/val.h>
        int isl_val_sgn(__isl_keep isl_val *v);
        int isl_val_is_zero(__isl_keep isl_val *v);
        int isl_val_is_one(__isl_keep isl_val *v);
        int isl_val_is_negone(__isl_keep isl_val *v);
        int isl_val_is_nonneg(__isl_keep isl_val *v);
        int isl_val_is_nonpos(__isl_keep isl_val *v);
        int isl_val_is_pos(__isl_keep isl_val *v);
        int isl_val_is_neg(__isl_keep isl_val *v);
        int isl_val_is_int(__isl_keep isl_val *v);
        int isl_val_is_rat(__isl_keep isl_val *v);
        int isl_val_is_nan(__isl_keep isl_val *v);
        int isl_val_is_infty(__isl_keep isl_val *v);
        int isl_val_is_neginfty(__isl_keep isl_val *v);

Note that the sign of NaN is undefined.

The following binary properties are defined on pairs of C<isl_val>s.

        #include <isl/val.h>
        int isl_val_lt(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);
        int isl_val_le(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);
        int isl_val_gt(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);
        int isl_val_ge(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);
        int isl_val_eq(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);
        int isl_val_ne(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);

For integer C<isl_val>s we additionally have the following binary property.

        #include <isl/val.h>
        int isl_val_is_divisible_by(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);

An C<isl_val> can also be compared to an integer using the following
function.  The result is undefined for NaN.

        #include <isl/val.h>
        int isl_val_cmp_si(__isl_keep isl_val *v, long i);

The following unary operations are available on C<isl_val>s.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_abs(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_neg(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_floor(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_ceil(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_trunc(__isl_take isl_val *v);

The following binary operations are available on C<isl_val>s.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_abs(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_neg(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_floor(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_ceil(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_trunc(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_min(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_max(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_add(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_add_ui(__isl_take isl_val *v1,
                unsigned long v2);
        __isl_give isl_val *isl_val_sub(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_sub_ui(__isl_take isl_val *v1,
                unsigned long v2);
        __isl_give isl_val *isl_val_mul(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_mul_ui(__isl_take isl_val *v1,
                unsigned long v2);
        __isl_give isl_val *isl_val_div(__isl_take isl_val *v1,
                __isl_take isl_val *v2);

On integer values, we additionally have the following operations.

        #include <isl/val.h>
        __isl_give isl_val *isl_val_2exp(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_mod(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_gcd(__isl_take isl_val *v1,
                __isl_take isl_val *v2);
        __isl_give isl_val *isl_val_gcdext(__isl_take isl_val *v1,
                __isl_take isl_val *v2, __isl_give isl_val **x,
                __isl_give isl_val **y);

The function C<isl_val_gcdext> returns the greatest common divisor g
of C<v1> and C<v2> as well as two integers C<*x> and C<*y> such
that C<*x> * C<v1> + C<*y> * C<v2> = g.

A value can be read from input using

        #include <isl/val.h>
        __isl_give isl_val *isl_val_read_from_str(isl_ctx *ctx,
                const char *str);

A value can be printed using

        #include <isl/val.h>
        __isl_give isl_printer *isl_printer_print_val(
                __isl_take isl_printer *p, __isl_keep isl_val *v);

=head2 Integers (obsolescent)

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 live in the same space.  C<isl_union_set>s and C<isl_union_map>s
represent unions of C<isl_set>s or C<isl_map>s in I<different> spaces,
where spaces are considered different if they have a different number
of dimensions and/or different names (see L<"Spaces">).
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 documented 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 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 Error Handling

C<isl> supports different ways to react in case a runtime error is triggered.
Runtime errors arise, e.g., if a function such as C<isl_map_intersect> is called
with two maps that have incompatible spaces. There are three possible ways
to react on error: to warn, to continue or to abort.

The default behavior is to warn. In this mode, C<isl> prints a warning, stores
the last error in the corresponding C<isl_ctx> and the function in which the
error was triggered returns C<NULL>. An error does not corrupt internal state,
such that isl can continue to be used. C<isl> also provides functions to
read the last error and to reset the memory that stores the last error. The
last error is only stored for information purposes. Its presence does not
change the behavior of C<isl>. Hence, resetting an error is not required to
continue to use isl, but only to observe new errors.

        #include <isl/ctx.h>
        enum isl_error isl_ctx_last_error(isl_ctx *ctx);
        void isl_ctx_reset_error(isl_ctx *ctx);

Another option is to continue on error. This is similar to warn on error mode,
except that C<isl> does not print any warning. This allows a program to
implement its own error reporting.

The last option is to directly abort the execution of the program from within
the isl library. This makes it obviously impossible to recover from an error,
but it allows to directly spot the error location. By aborting on error,
debuggers break at the location the error occurred and can provide a stack
trace. Other tools that automatically provide stack traces on abort or that do
not want to continue execution after an error was triggered may also prefer to
abort on error.

The on error behavior of isl can be specified by calling
C<isl_options_set_on_error> or by setting the command line option
C<--isl-on-error>. Valid arguments for the function call are
C<ISL_ON_ERROR_WARN>, C<ISL_ON_ERROR_CONTINUE> and C<ISL_ON_ERROR_ABORT>. The
choices for the command line option are C<warn>, C<continue> and C<abort>.
It is also possible to query the current error mode.

        #include <isl/options.h>
        int isl_options_set_on_error(isl_ctx *ctx, int val);
        int isl_options_get_on_error(isl_ctx *ctx);

=head2 Identifiers

Identifiers are used to identify both individual dimensions
and tuples of dimensions.  They consist of an optional name and an optional
user pointer.  The name and the user pointer cannot both be C<NULL>, however.
Identifiers with the same name but different pointer values
are considered to be distinct.
Similarly, identifiers with different names but the same pointer value
are also considered to be distinct.
Equal identifiers are represented using the same object.
Pairs of identifiers can therefore be tested for equality using the
C<==> operator.
Identifiers can be constructed, copied, freed, inspected and printed
using the following functions.

        #include <isl/id.h>
        __isl_give isl_id *isl_id_alloc(isl_ctx *ctx,
                __isl_keep const char *name, void *user);
        __isl_give isl_id *isl_id_set_free_user(
                __isl_take isl_id *id,
                __isl_give void (*free_user)(void *user));
        __isl_give isl_id *isl_id_copy(isl_id *id);
        void *isl_id_free(__isl_take isl_id *id);

        isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id);
        void *isl_id_get_user(__isl_keep isl_id *id);
        __isl_keep const char *isl_id_get_name(__isl_keep isl_id *id);

        __isl_give isl_printer *isl_printer_print_id(
                __isl_take isl_printer *p, __isl_keep isl_id *id);

The callback set by C<isl_id_set_free_user> is called on the user
pointer when the last reference to the C<isl_id> is freed.
Note that C<isl_id_get_name> returns a pointer to some internal
data structure, so the result can only be used while the
corresponding C<isl_id> is alive.

=head2 Spaces

Whenever a new set, relation or similiar object is created from scratch,
the space in which it lives needs to be specified using an C<isl_space>.
Each space involves zero or more parameters and zero, one or two
tuples of set or input/output dimensions.  The parameters and dimensions
are identified by an C<isl_dim_type> and a position.
The type C<isl_dim_param> refers to parameters,
the type C<isl_dim_set> refers to set dimensions (for spaces
with a single tuple of dimensions) and the types C<isl_dim_in>
and C<isl_dim_out> refer to input and output dimensions
(for spaces with two tuples of dimensions).
Local spaces (see L</"Local Spaces">) also contain dimensions
of type C<isl_dim_div>.
Note that parameters are only identified by their position within
a given object.  Across different objects, parameters are (usually)
identified by their names or identifiers.  Only unnamed parameters
are identified by their positions across objects.  The use of unnamed
parameters is discouraged.

        #include <isl/space.h>
        __isl_give isl_space *isl_space_alloc(isl_ctx *ctx,
                unsigned nparam, unsigned n_in, unsigned n_out);
        __isl_give isl_space *isl_space_params_alloc(isl_ctx *ctx,
                unsigned nparam);
        __isl_give isl_space *isl_space_set_alloc(isl_ctx *ctx,
                unsigned nparam, unsigned dim);
        __isl_give isl_space *isl_space_copy(__isl_keep isl_space *space);
        void *isl_space_free(__isl_take isl_space *space);
        unsigned isl_space_dim(__isl_keep isl_space *space,
                enum isl_dim_type type);

The space used for creating a parameter domain
needs to be created using C<isl_space_params_alloc>.
For other sets, the space
needs to be created using C<isl_space_set_alloc>, while
for a relation, the space
needs to be created using C<isl_space_alloc>.
C<isl_space_dim> can be used
to find out the number of dimensions of each type in
a space, 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>.

To check whether a given space is that of a set or a map
or whether it is a parameter space, use these functions:

        #include <isl/space.h>
        int isl_space_is_params(__isl_keep isl_space *space);
        int isl_space_is_set(__isl_keep isl_space *space);
        int isl_space_is_map(__isl_keep isl_space *space);

Spaces can be compared using the following functions:

        #include <isl/space.h>
        int isl_space_is_equal(__isl_keep isl_space *space1,
                __isl_keep isl_space *space2);
        int isl_space_is_domain(__isl_keep isl_space *space1,
                __isl_keep isl_space *space2);
        int isl_space_is_range(__isl_keep isl_space *space1,
                __isl_keep isl_space *space2);

C<isl_space_is_domain> checks whether the first argument is equal
to the domain of the second argument.  This requires in particular that
the first argument is a set space and that the second argument
is a map space.

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 space
of the original object.

        #include <isl/set.h>
        __isl_give isl_space *isl_basic_set_get_space(
                __isl_keep isl_basic_set *bset);
        __isl_give isl_space *isl_set_get_space(__isl_keep isl_set *set);

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

        #include <isl/map.h>
        __isl_give isl_space *isl_basic_map_get_space(
                __isl_keep isl_basic_map *bmap);
        __isl_give isl_space *isl_map_get_space(__isl_keep isl_map *map);

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

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

        #include <isl/polynomial.h>
        __isl_give isl_space *isl_qpolynomial_get_domain_space(
                __isl_keep isl_qpolynomial *qp);
        __isl_give isl_space *isl_qpolynomial_get_space(
                __isl_keep isl_qpolynomial *qp);
        __isl_give isl_space *isl_qpolynomial_fold_get_space(
                __isl_keep isl_qpolynomial_fold *fold);
        __isl_give isl_space *isl_pw_qpolynomial_get_domain_space(
                __isl_keep isl_pw_qpolynomial *pwqp);
        __isl_give isl_space *isl_pw_qpolynomial_get_space(
                __isl_keep isl_pw_qpolynomial *pwqp);
        __isl_give isl_space *isl_pw_qpolynomial_fold_get_domain_space(
                __isl_keep isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_space *isl_pw_qpolynomial_fold_get_space(
                __isl_keep isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_space *isl_union_pw_qpolynomial_get_space(
                __isl_keep isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_space *isl_union_pw_qpolynomial_fold_get_space(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);

        #include <isl/aff.h>
        __isl_give isl_space *isl_aff_get_domain_space(
                __isl_keep isl_aff *aff);
        __isl_give isl_space *isl_aff_get_space(
                __isl_keep isl_aff *aff);
        __isl_give isl_space *isl_pw_aff_get_domain_space(
                __isl_keep isl_pw_aff *pwaff);
        __isl_give isl_space *isl_pw_aff_get_space(
                __isl_keep isl_pw_aff *pwaff);
        __isl_give isl_space *isl_multi_aff_get_domain_space(
                __isl_keep isl_multi_aff *maff);
        __isl_give isl_space *isl_multi_aff_get_space(
                __isl_keep isl_multi_aff *maff);
        __isl_give isl_space *isl_pw_multi_aff_get_domain_space(
                __isl_keep isl_pw_multi_aff *pma);
        __isl_give isl_space *isl_pw_multi_aff_get_space(
                __isl_keep isl_pw_multi_aff *pma);
        __isl_give isl_space *isl_union_pw_multi_aff_get_space(
                __isl_keep isl_union_pw_multi_aff *upma);
        __isl_give isl_space *isl_multi_pw_aff_get_domain_space(
                __isl_keep isl_multi_pw_aff *mpa);
        __isl_give isl_space *isl_multi_pw_aff_get_space(
                __isl_keep isl_multi_pw_aff *mpa);

        #include <isl/point.h>
        __isl_give isl_space *isl_point_get_space(
                __isl_keep isl_point *pnt);

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

        #include <isl/space.h>
        __isl_give isl_space *isl_space_set_dim_id(
                __isl_take isl_space *space,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        int isl_space_has_dim_id(__isl_keep isl_space *space,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_space_get_dim_id(
                __isl_keep isl_space *space,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_space *isl_space_set_dim_name(
                __isl_take isl_space *space,
                 enum isl_dim_type type, unsigned pos,
                 __isl_keep const char *name);
        int isl_space_has_dim_name(__isl_keep isl_space *space,
                enum isl_dim_type type, unsigned pos);
        __isl_keep const char *isl_space_get_dim_name(
                __isl_keep isl_space *space,
                enum isl_dim_type type, unsigned pos);

Note that C<isl_space_get_name> returns a pointer to some internal
data structure, so the result can only be used while the
corresponding C<isl_space> 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_set>, C<isl_map>, 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.

Given the identifier or name of a dimension (typically a parameter),
its position can be obtained from the following function.

        #include <isl/space.h>
        int isl_space_find_dim_by_id(__isl_keep isl_space *space,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_space_find_dim_by_name(__isl_keep isl_space *space,
                enum isl_dim_type type, const char *name);

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

        #include <isl/space.h>
        __isl_give isl_space *isl_space_set_tuple_id(
                __isl_take isl_space *space,
                enum isl_dim_type type, __isl_take isl_id *id);
        __isl_give isl_space *isl_space_reset_tuple_id(
                __isl_take isl_space *space, enum isl_dim_type type);
        int isl_space_has_tuple_id(__isl_keep isl_space *space,
                enum isl_dim_type type);
        __isl_give isl_id *isl_space_get_tuple_id(
                __isl_keep isl_space *space, enum isl_dim_type type);
        __isl_give isl_space *isl_space_set_tuple_name(
                __isl_take isl_space *space,
                enum isl_dim_type type, const char *s);
        int isl_space_has_tuple_name(__isl_keep isl_space *space,
                enum isl_dim_type type);
        const char *isl_space_get_tuple_name(__isl_keep isl_space *space,
                enum isl_dim_type type);

The C<type> argument needs to be one of C<isl_dim_in>, C<isl_dim_out>
or C<isl_dim_set>.  As with C<isl_space_get_name>,
the C<isl_space_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 spaces.

        #include <isl/space.h>
        int isl_space_is_wrapping(__isl_keep isl_space *space);
        __isl_give isl_space *isl_space_wrap(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_unwrap(__isl_take isl_space *space);

The input to C<isl_space_is_wrapping> and C<isl_space_unwrap> should
be the space of a set, while that of
C<isl_space_wrap> should be the space of a relation.
Conversely, the output of C<isl_space_unwrap> is the space
of a relation, while that of C<isl_space_wrap> is the space of a set.

Spaces can be created from other spaces
using the following functions.

        __isl_give isl_space *isl_space_domain(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_from_domain(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_range(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_from_range(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_params(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_set_from_params(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_reverse(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_join(__isl_take isl_space *left,
                __isl_take isl_space *right);
        __isl_give isl_space *isl_space_align_params(
                __isl_take isl_space *space1, __isl_take isl_space *space2)
        __isl_give isl_space *isl_space_insert_dims(__isl_take isl_space *space,
                enum isl_dim_type type, unsigned pos, unsigned n);
        __isl_give isl_space *isl_space_add_dims(__isl_take isl_space *space,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_space *isl_space_drop_dims(__isl_take isl_space *space,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_space *isl_space_move_dims(__isl_take isl_space *space,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_space *isl_space_map_from_set(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_map_from_domain_and_range(
                __isl_take isl_space *domain,
                __isl_take isl_space *range);
        __isl_give isl_space *isl_space_zip(__isl_take isl_space *space);
        __isl_give isl_space *isl_space_curry(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_uncurry(
                __isl_take isl_space *space);

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

=head2 Local Spaces

A local space is essentially a space with
zero or more existentially quantified variables.
The local space of a (constraint of a) basic set or relation can be obtained
using the following functions.

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

        #include <isl/set.h>
        __isl_give isl_local_space *isl_basic_set_get_local_space(
                __isl_keep isl_basic_set *bset);

        #include <isl/map.h>
        __isl_give isl_local_space *isl_basic_map_get_local_space(
                __isl_keep isl_basic_map *bmap);

A new local space can be created from a space using

        #include <isl/local_space.h>
        __isl_give isl_local_space *isl_local_space_from_space(
                __isl_take isl_space *space);

They can be inspected, modified, copied and freed using the following functions.

        #include <isl/local_space.h>
        isl_ctx *isl_local_space_get_ctx(
                __isl_keep isl_local_space *ls);
        int isl_local_space_is_set(__isl_keep isl_local_space *ls);
        int isl_local_space_dim(__isl_keep isl_local_space *ls,
                enum isl_dim_type type);
        int isl_local_space_has_dim_id(
                __isl_keep isl_local_space *ls,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_local_space_get_dim_id(
                __isl_keep isl_local_space *ls,
                enum isl_dim_type type, unsigned pos);
        int isl_local_space_has_dim_name(
                __isl_keep isl_local_space *ls,
                enum isl_dim_type type, unsigned pos)
        const char *isl_local_space_get_dim_name(
                __isl_keep isl_local_space *ls,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_local_space *isl_local_space_set_dim_name(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned pos, const char *s);
        __isl_give isl_local_space *isl_local_space_set_dim_id(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        __isl_give isl_space *isl_local_space_get_space(
                __isl_keep isl_local_space *ls);
        __isl_give isl_aff *isl_local_space_get_div(
                __isl_keep isl_local_space *ls, int pos);
        __isl_give isl_local_space *isl_local_space_copy(
                __isl_keep isl_local_space *ls);
        void *isl_local_space_free(__isl_take isl_local_space *ls);

Note that C<isl_local_space_get_div> can only be used on local spaces
of sets.

Two local spaces can be compared using

        int isl_local_space_is_equal(__isl_keep isl_local_space *ls1,
                __isl_keep isl_local_space *ls2);

Local spaces can be created from other local spaces
using the following functions.

        __isl_give isl_local_space *isl_local_space_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_local_space *isl_local_space_range(
                __isl_take isl_local_space *ls);
        __isl_give isl_local_space *isl_local_space_from_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_local_space *isl_local_space_intersect(
                __isl_take isl_local_space *ls1,
                __isl_take isl_local_space *ls2);
        __isl_give isl_local_space *isl_local_space_add_dims(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_local_space *isl_local_space_insert_dims(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_local_space *isl_local_space_drop_dims(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned first, unsigned n);

=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);
        __isl_give isl_basic_set *isl_basic_set_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx,
                FILE *input);
        __isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx,
                const char *str);

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

        #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(
                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(
                isl_ctx *ctx, const char *str);

The input format is autodetected and may be either the C<PolyLib> format
or the C<isl> 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 printer can be inspected using the following functions.

        FILE *isl_printer_get_file(
                __isl_keep isl_printer *printer);
        int isl_printer_get_output_format(
                __isl_keep isl_printer *p);

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_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> (set by
C<isl_printer_set_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.
The function C<isl_printer_indent> increases the indentation
by the specified amount (which may be negative).

To actually print something, use

        #include <isl/printer.h>
        __isl_give isl_printer *isl_printer_print_double(
                __isl_take isl_printer *p, double d);

        #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_space *space);
        __isl_give isl_basic_map *isl_basic_map_empty(
                __isl_take isl_space *space);
        __isl_give isl_set *isl_set_empty(
                __isl_take isl_space *space);
        __isl_give isl_map *isl_map_empty(
                __isl_take isl_space *space);
        __isl_give isl_union_set *isl_union_set_empty(
                __isl_take isl_space *space);
        __isl_give isl_union_map *isl_union_map_empty(
                __isl_take isl_space *space);

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

=item * Universe sets and relations

        __isl_give isl_basic_set *isl_basic_set_universe(
                __isl_take isl_space *space);
        __isl_give isl_basic_map *isl_basic_map_universe(
                __isl_take isl_space *space);
        __isl_give isl_set *isl_set_universe(
                __isl_take isl_space *space);
        __isl_give isl_map *isl_map_universe(
                __isl_take isl_space *space);
        __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_space *space);
        __isl_give isl_basic_map *isl_basic_map_nat_universe(
                __isl_take isl_space *space);
        __isl_give isl_set *isl_set_nat_universe(
                __isl_take isl_space *space);
        __isl_give isl_map *isl_map_nat_universe(
                __isl_take isl_space *space);

=item * Identity relations

        __isl_give isl_basic_map *isl_basic_map_identity(
                __isl_take isl_space *space);
        __isl_give isl_map *isl_map_identity(
                __isl_take isl_space *space);

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

=item * Lexicographic order

        __isl_give isl_map *isl_map_lex_lt(
                __isl_take isl_space *set_space);
        __isl_give isl_map *isl_map_lex_le(
                __isl_take isl_space *set_space);
        __isl_give isl_map *isl_map_lex_gt(
                __isl_take isl_space *set_space);
        __isl_give isl_map *isl_map_lex_ge(
                __isl_take isl_space *set_space);
        __isl_give isl_map *isl_map_lex_lt_first(
                __isl_take isl_space *space, unsigned n);
        __isl_give isl_map *isl_map_lex_le_first(
                __isl_take isl_space *space, unsigned n);
        __isl_give isl_map *isl_map_lex_gt_first(
                __isl_take isl_space *space, unsigned n);
        __isl_give isl_map *isl_map_lex_ge_first(
                __isl_take isl_space *space, unsigned n);

The first four functions take a space 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 space 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_set *isl_union_set_from_basic_set(
                __isl_take isl_basic_set *bset);
        __isl_give isl_union_map *isl_union_map_from_basic_map(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_union_set *isl_union_set_from_set(
                __isl_take isl_set *set);
        __isl_give isl_union_map *isl_union_map_from_map(
                __isl_take isl_map *map);

The inverse conversions below can only be used if the input
union set or relation is known to contain elements in exactly one
space.

        __isl_give isl_set *isl_set_from_union_set(
                __isl_take isl_union_set *uset);
        __isl_give isl_map *isl_map_from_union_map(
                __isl_take isl_union_map *umap);

A zero-dimensional (basic) set can be constructed on a given parameter domain
using the following function.

        __isl_give isl_basic_set *isl_basic_set_from_params(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_set_from_params(
                __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 (or removed from) (basic) sets and relations
using the following functions.

        #include <isl/constraint.h>
        __isl_give isl_constraint *isl_equality_alloc(
                __isl_take isl_local_space *ls);
        __isl_give isl_constraint *isl_inequality_alloc(
                __isl_take isl_local_space *ls);
        __isl_give isl_constraint *isl_constraint_set_constant(
                __isl_take isl_constraint *constraint, isl_int v);
        __isl_give isl_constraint *isl_constraint_set_constant_si(
                __isl_take isl_constraint *constraint, int v);
        __isl_give isl_constraint *isl_constraint_set_coefficient(
                __isl_take isl_constraint *constraint,
                enum isl_dim_type type, int pos, isl_int v);
        __isl_give isl_constraint *isl_constraint_set_coefficient_si(
                __isl_take isl_constraint *constraint,
                enum isl_dim_type type, int pos, 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);
        __isl_give isl_map *isl_map_add_constraint(
                __isl_take isl_map *map,
                __isl_take isl_constraint *constraint);
        __isl_give isl_set *isl_set_add_constraint(
                __isl_take isl_set *set,
                __isl_take isl_constraint *constraint);
        __isl_give isl_basic_set *isl_basic_set_drop_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_space *space;
        isl_local_space *ls;
        isl_constraint *c;
        isl_basic_set *bset;

        space = isl_space_set_alloc(ctx, 0, 2);
        bset = isl_basic_set_universe(isl_space_copy(space));
        ls = isl_local_space_from_space(space);

        c = isl_equality_alloc(isl_local_space_copy(ls));
        c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
        c = isl_constraint_set_coefficient_si(c, isl_dim_set, 1, 2);
        bset = isl_basic_set_add_constraint(bset, c);

        c = isl_inequality_alloc(isl_local_space_copy(ls));
        c = isl_constraint_set_constant_si(c, -10);
        c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, 1);
        bset = isl_basic_set_add_constraint(bset, c);

        c = isl_inequality_alloc(ls);
        c = isl_constraint_set_constant_si(c, 42);
        c = isl_constraint_set_coefficient_si(c, isl_dim_set, 0, -1);
        bset = isl_basic_set_add_constraint(bset, c);

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

Or, alternatively,

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

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_space *space,
                __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_space *space,
                __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.

A (basic or union) set or relation can also be constructed from a
(union) (piecewise) (multiple) affine expression
or a list of affine expressions
(See L<"Piecewise Quasi Affine Expressions"> and
L<"Piecewise Multiple Quasi Affine Expressions">).

        __isl_give isl_basic_map *isl_basic_map_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_map *isl_map_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_set *isl_set_from_pw_aff(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_map *isl_map_from_pw_aff(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_basic_map *isl_basic_map_from_aff_list(
                __isl_take isl_space *domain_space,
                __isl_take isl_aff_list *list);
        __isl_give isl_basic_map *isl_basic_map_from_multi_aff(
                __isl_take isl_multi_aff *maff)
        __isl_give isl_map *isl_map_from_multi_aff(
                __isl_take isl_multi_aff *maff)
        __isl_give isl_set *isl_set_from_pw_multi_aff(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_map *isl_map_from_pw_multi_aff(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_map *
        isl_union_map_from_union_pw_multi_aff(
                __isl_take isl_union_pw_multi_aff *upma);

The C<domain_dim> argument describes the domain of the resulting
basic relation.  It is required because the C<list> may consist
of zero affine expressions.

=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);

It is also possible to only remove those divs that are defined
in terms of a given range of dimensions or only those for which
no explicit representation is known.

        __isl_give isl_basic_set *
        isl_basic_set_remove_divs_involving_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_basic_map *
        isl_basic_map_remove_divs_involving_dims(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_set *isl_set_remove_divs_involving_dims(
                __isl_take isl_set *set, enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_map *isl_map_remove_divs_involving_dims(
                __isl_take isl_map *map, enum isl_dim_type type,
                unsigned first, unsigned n);

        __isl_give isl_basic_set *
        isl_basic_set_remove_unknown_divs(
                __isl_take isl_basic_set *bset);
        __isl_give isl_set *isl_set_remove_unknown_divs(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_remove_unknown_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 in a given space from a union, use

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

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_set_n_constraint(
                __isl_keep isl_basic_set *bset);
        int isl_basic_set_foreach_constraint(
                __isl_keep isl_basic_set *bset,
                int (*fn)(__isl_take isl_constraint *c, void *user),
                void *user);
        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(__isl_take 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.

        int isl_constraint_is_lower_bound(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned pos);
        int isl_constraint_is_upper_bound(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned pos);
        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);
        int isl_constraint_involves_dims(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned first, unsigned n);

The explicit representations of the existentially quantified
variables can be inspected using the following function.
Note that the user is only allowed to use this function
if the inspected set or map is the result of a call
to C<isl_set_compute_divs> or C<isl_map_compute_divs>.
The existentially quantified variable is equal to the floor
of the returned affine expression.  The affine expression
itself can be inspected using the functions in
L<"Piecewise Quasi Affine Expressions">.

        __isl_give isl_aff *isl_constraint_get_div(
                __isl_keep isl_constraint *constraint, int pos);

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 number of parameters, input, output or set dimensions can
be obtained using the following functions.

        unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
                enum isl_dim_type type);
        unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
                enum isl_dim_type type);
        unsigned isl_set_dim(__isl_keep isl_set *set,
                enum isl_dim_type type);
        unsigned isl_map_dim(__isl_keep isl_map *map,
                enum isl_dim_type type);

To check whether the description of a set or relation depends
on one or more given dimensions, it is not necessary to iterate over all
constraints.  Instead the following functions can be used.

        int isl_basic_set_involves_dims(
                __isl_keep isl_basic_set *bset,
                enum isl_dim_type type, unsigned first, unsigned n);
        int isl_set_involves_dims(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned first, unsigned n);
        int isl_basic_map_involves_dims(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned first, unsigned n);
        int isl_map_involves_dims(__isl_keep isl_map *map,
                enum isl_dim_type type, unsigned first, unsigned n);

Similarly, the following functions can be used to check whether
a given dimension is involved in any lower or upper bound.

        int isl_set_dim_has_any_lower_bound(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);
        int isl_set_dim_has_any_upper_bound(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);

Note that these functions return true even if there is a bound on
the dimension on only some of the basic sets of C<set>.
To check if they have a bound for all of the basic sets in C<set>,
use the following functions instead.

        int isl_set_dim_has_lower_bound(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);
        int isl_set_dim_has_upper_bound(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);

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

        __isl_give isl_set *isl_set_set_tuple_id(
                __isl_take isl_set *set, __isl_take isl_id *id);
        __isl_give isl_set *isl_set_reset_tuple_id(
                __isl_take isl_set *set);
        int isl_set_has_tuple_id(__isl_keep isl_set *set);
        __isl_give isl_id *isl_set_get_tuple_id(
                __isl_keep isl_set *set);
        __isl_give isl_map *isl_map_set_tuple_id(
                __isl_take isl_map *map, enum isl_dim_type type,
                __isl_take isl_id *id);
        __isl_give isl_map *isl_map_reset_tuple_id(
                __isl_take isl_map *map, enum isl_dim_type type);
        int isl_map_has_tuple_id(__isl_keep isl_map *map,
                enum isl_dim_type type);
        __isl_give isl_id *isl_map_get_tuple_id(
                __isl_keep isl_map *map, enum isl_dim_type type);

        const char *isl_basic_set_get_tuple_name(
                __isl_keep isl_basic_set *bset);
        __isl_give isl_basic_set *isl_basic_set_set_tuple_name(
                __isl_take isl_basic_set *set, const char *s);
        int isl_set_has_tuple_name(__isl_keep isl_set *set);
        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);
        __isl_give isl_basic_map *isl_basic_map_set_tuple_name(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, const char *s);
        int isl_map_has_tuple_name(__isl_keep isl_map *map,
                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_space_get_tuple_name>, the value returned points to
an internal data structure.
The identifiers, positions or names of individual dimensions can be
read off using the following functions.

        __isl_give isl_id *isl_basic_set_get_dim_id(
                __isl_keep isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_set *isl_set_set_dim_id(
                __isl_take isl_set *set, enum isl_dim_type type,
                unsigned pos, __isl_take isl_id *id);
        int isl_set_has_dim_id(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_set_get_dim_id(
                __isl_keep isl_set *set, enum isl_dim_type type,
                unsigned pos);
        int isl_basic_map_has_dim_id(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_map *isl_map_set_dim_id(
                __isl_take isl_map *map, enum isl_dim_type type,
                unsigned pos, __isl_take isl_id *id);
        int isl_map_has_dim_id(__isl_keep isl_map *map,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_map_get_dim_id(
                __isl_keep isl_map *map, enum isl_dim_type type,
                unsigned pos);

        int isl_set_find_dim_by_id(__isl_keep isl_set *set,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_map_find_dim_by_id(__isl_keep isl_map *map,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_set_find_dim_by_name(__isl_keep isl_set *set,
                enum isl_dim_type type, const char *name);
        int isl_map_find_dim_by_name(__isl_keep isl_map *map,
                enum isl_dim_type type, const char *name);

        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);
        int isl_set_has_dim_name(__isl_keep isl_set *set,
                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);
        int isl_map_has_dim_name(__isl_keep isl_map *map,
                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 identifiers, positions
or names of the parameters.  Identifiers of individual dimensions are
essentially only useful for printing.  They are ignored by all other
operations and may not be preserved across those operations.

=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_basic_map_is_single_valued(
                __isl_keep isl_basic_map *bmap);
        int isl_map_plain_is_single_valued(
                __isl_keep isl_map *map);
        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 * Position

        int isl_basic_map_plain_is_fixed(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos,
                isl_int *val);
        int isl_set_plain_is_fixed(__isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos,
                isl_int *val);
        int isl_map_plain_is_fixed(__isl_keep isl_map *map,
                enum isl_dim_type type, unsigned pos,
                isl_int *val);

Check if the relation obviously lies on a hyperplane where the given dimension
has a fixed value and if so, return that value in C<*val>.

=item * Space

To check whether a set is a parameter domain, use this function:

        int isl_set_is_params(__isl_keep isl_set *set);
        int isl_union_set_is_params(
                __isl_keep isl_union_set *uset);

=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.

=item * Currying

        int isl_basic_map_can_curry(
                __isl_keep isl_basic_map *bmap);
        int isl_map_can_curry(__isl_keep isl_map *map);

Check whether the domain of the (basic) relation is a wrapped relation.

        int isl_basic_map_can_uncurry(
                __isl_keep isl_basic_map *bmap);
        int isl_map_can_uncurry(__isl_keep isl_map *map);

Check whether the range of the (basic) relation is a wrapped relation.

=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);
        int isl_set_is_disjoint(__isl_keep isl_set *set1,
                __isl_keep isl_set *set2);
        int isl_map_is_disjoint(__isl_keep isl_map *map1,
                __isl_keep isl_map *map2);

=item * Subset

        int isl_basic_set_is_subset(
                __isl_keep isl_basic_set *bset1,
                __isl_keep isl_basic_set *bset2);
        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);

Check whether the first argument is a (strict) subset of the
second argument.

=item * Order

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

This function is useful for sorting C<isl_set>s.
The order depends on the internal representation of the inputs.
The order is fixed over different calls to the function (assuming
the internal representation of the inputs has not changed), but may
change over different versions of C<isl>.

=back

=head2 Unary Operations

=over

=item * Complement

        __isl_give isl_set *isl_set_complement(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_complement(
                __isl_take isl_map *map);

=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_set_params(
                __isl_take isl_basic_set *bset);
        __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_set_params(__isl_take isl_set *set);
        __isl_give isl_set *isl_map_params(__isl_take isl_map *map);
        __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_set *isl_union_set_params(
                __isl_take isl_union_set *uset);
        __isl_give isl_set *isl_union_map_params(
                __isl_take isl_union_map *umap);
        __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_basic_set *isl_basic_set_eliminate(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_set *isl_set_eliminate(
                __isl_take isl_set *set, enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_basic_map *isl_basic_map_eliminate(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_map *isl_map_eliminate(
                __isl_take isl_map *map, enum isl_dim_type type,
                unsigned first, unsigned n);

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

=item * Slicing

        __isl_give isl_basic_set *isl_basic_set_fix(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos,
                isl_int value);
        __isl_give isl_basic_set *isl_basic_set_fix_si(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_set *isl_set_fix(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                isl_int value);
        __isl_give isl_set *isl_set_fix_si(__isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_basic_map *isl_basic_map_fix_si(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_map *isl_map_fix(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos,
                isl_int value);
        __isl_give isl_map *isl_map_fix_si(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, int value);

Intersect the set or relation with the hyperplane where the given
dimension has the fixed given value.

        __isl_give isl_basic_map *isl_basic_map_lower_bound_si(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_basic_map *isl_basic_map_upper_bound_si(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_set *isl_set_lower_bound(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                isl_int value);
        __isl_give isl_set *isl_set_lower_bound_si(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_map *isl_map_lower_bound_si(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_set *isl_set_upper_bound(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                isl_int value);
        __isl_give isl_set *isl_set_upper_bound_si(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_map *isl_map_upper_bound_si(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, int value);

Intersect the set or relation with the half-space where the given
dimension has a value bounded by the fixed given value.

        __isl_give isl_set *isl_set_equate(__isl_take isl_set *set,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);
        __isl_give isl_basic_map *isl_basic_map_equate(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);
        __isl_give isl_map *isl_map_equate(__isl_take isl_map *map,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);

Intersect the set or relation with the hyperplane where the given
dimensions are equal to each other.

        __isl_give isl_map *isl_map_oppose(__isl_take isl_map *map,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);

Intersect the relation with the hyperplane where the given
dimensions have opposite values.

        __isl_give isl_basic_map *isl_basic_map_order_ge(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);
        __isl_give isl_map *isl_map_order_lt(__isl_take isl_map *map,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);
        __isl_give isl_basic_map *isl_basic_map_order_gt(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);
        __isl_give isl_map *isl_map_order_gt(__isl_take isl_map *map,
                enum isl_dim_type type1, int pos1,
                enum isl_dim_type type2, int pos2);

Intersect the relation with the half-space where the given
dimensions satisfy the given ordering.

=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);

One of the methods for combining pairs of basic sets or relations
can result in coefficients that are much larger than those that appear
in the constraints of the input.  By default, the coefficients are
not allowed to grow larger, but this can be changed by unsetting
the following option.

        int isl_options_set_coalesce_bounded_wrapping(
                isl_ctx *ctx, int val);
        int isl_options_get_coalesce_bounded_wrapping(
                isl_ctx *ctx);

=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_set *isl_set_remove_redundancies(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_basic_map_remove_redundancies(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_remove_redundancies(
                __isl_take isl_map *map);

=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_unshifted_simple_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *
        isl_map_unshifted_simple_hull(
                __isl_take isl_map *map);
        __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.
In case of C<isl_set_unshifted_simple_hull>, only the original
constraints are used, without any translation.

=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 * Other approximations

        __isl_give isl_basic_set *
        isl_basic_set_drop_constraints_involving_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_basic_map *
        isl_basic_map_drop_constraints_involving_dims(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_basic_set *
        isl_basic_set_drop_constraints_not_involving_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_set *
        isl_set_drop_constraints_involving_dims(
                __isl_take isl_set *set,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_map *
        isl_map_drop_constraints_involving_dims(
                __isl_take isl_map *map,
                enum isl_dim_type type,
                unsigned first, unsigned n);

These functions drop any constraints (not) involving the specified dimensions.
Note that the result depends on the representation of the input.

=item * Feasibility

        __isl_give isl_basic_set *isl_basic_set_sample(
                __isl_take isl_basic_set *bset);
        __isl_give isl_basic_set *isl_set_sample(
                __isl_take isl_set *set);
        __isl_give isl_basic_map *isl_basic_map_sample(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_map *isl_map_sample(
                __isl_take isl_map *map);

If the input (basic) set or relation is non-empty, then return
a singleton subset of the input.  Otherwise, return an empty set.

=item * Optimization

        #include <isl/ilp.h>
        enum isl_lp_result isl_basic_set_max(
                __isl_keep isl_basic_set *bset,
                __isl_keep isl_aff *obj, isl_int *opt)
        enum isl_lp_result isl_set_min(__isl_keep isl_set *set,
                __isl_keep isl_aff *obj, isl_int *opt);
        enum isl_lp_result isl_set_max(__isl_keep isl_set *set,
                __isl_keep isl_aff *obj, isl_int *opt);

Compute the minimum or maximum of the integer affine expression C<obj>
over the points in C<set>, returning the result in C<opt>.
The return value may be one of C<isl_lp_error>,
C<isl_lp_ok>, C<isl_lp_unbounded> or C<isl_lp_empty>.

=item * Parametric optimization

        __isl_give isl_pw_aff *isl_set_dim_min(
                __isl_take isl_set *set, int pos);
        __isl_give isl_pw_aff *isl_set_dim_max(
                __isl_take isl_set *set, int pos);
        __isl_give isl_pw_aff *isl_map_dim_max(
                __isl_take isl_map *map, int pos);

Compute the minimum or maximum of the given set or output dimension
as a function of the parameters (and input dimensions), but independently
of the other set or output dimensions.
For lexicographic optimization, see L<"Lexicographic Optimization">.

=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_fixed_power(
                __isl_take isl_map *map, isl_int exp);
        __isl_give isl_union_map *isl_union_map_fixed_power(
                __isl_take isl_union_map *umap, isl_int exp);

Compute the given power of C<map>, where C<exp> is assumed to be non-zero.
If the exponent C<exp> is negative, then the -C<exp> th power of the inverse
of C<map> is computed.

        __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_domain(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_map *isl_basic_map_flatten_range(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_flatten_range(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_flatten_domain(
                __isl_take isl_map *map);
        __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);

Given a local space that contains the existentially quantified
variables of a set, a basic relation that, when applied to
a basic set, has essentially the same effect as C<isl_basic_set_lift>,
can be constructed using the following function.

        #include <isl/local_space.h>
        __isl_give isl_basic_map *isl_local_space_lifting(
                __isl_take isl_local_space *ls);

=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 * Currying

        __isl_give isl_basic_map *isl_basic_map_curry(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_map *isl_basic_map_uncurry(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_map *isl_map_curry(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_uncurry(
                __isl_take isl_map *map);
        __isl_give isl_union_map *isl_union_map_curry(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *isl_union_map_uncurry(
                __isl_take isl_union_map *umap);

Given a relation with a nested relation for domain,
the C<curry> functions
move the range of the nested relation out of the domain
and use it as the domain of a nested relation in the range,
with the original range as range of this nested relation.
The C<uncurry> functions perform the inverse operation.

=item * Aligning parameters

        __isl_give isl_basic_set *isl_basic_set_align_params(
                __isl_take isl_basic_set *bset,
                __isl_take isl_space *model);
        __isl_give isl_set *isl_set_align_params(
                __isl_take isl_set *set,
                __isl_take isl_space *model);
        __isl_give isl_basic_map *isl_basic_map_align_params(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_space *model);
        __isl_give isl_map *isl_map_align_params(
                __isl_take isl_map *map,
                __isl_take isl_space *model);

Change the order of the parameters of the given set or relation
such that the first parameters match those of C<model>.
This may involve the introduction of extra parameters.
All parameters need to be named.

=item * Dimension manipulation

        __isl_give isl_basic_set *isl_basic_set_add_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned n);
        __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);
        __isl_give isl_basic_set *isl_basic_set_insert_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos,
                unsigned n);
        __isl_give isl_basic_map *isl_basic_map_insert_dims(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos,
                unsigned n);
        __isl_give isl_set *isl_set_insert_dims(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, unsigned n);
        __isl_give isl_map *isl_map_insert_dims(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, unsigned n);
        __isl_give isl_basic_set *isl_basic_set_move_dims(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_basic_map *isl_basic_map_move_dims(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_set *isl_set_move_dims(
                __isl_take isl_set *set,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_map *isl_map_move_dims(
                __isl_take isl_map *map,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                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, assuming
C<isl_set_align_params> and C<isl_map_align_params>
are not sufficient.

=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_params(
                __isl_take isl_basic_set *bset1,
                __isl_take isl_basic_set *bset2);
        __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_params(
                __isl_take isl_set *set,
                __isl_take isl_set *params);
        __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_params(
                __isl_take isl_union_set *uset,
                __isl_take isl_set *set);
        __isl_give isl_union_map *isl_union_map_intersect_params(
                __isl_take isl_union_map *umap,
                __isl_take isl_set *set);
        __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_params(
                __isl_take isl_map *map,
                __isl_take isl_set *params);
        __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);

The second argument to the C<_params> functions needs to be
a parametric (basic) set.  For the other functions, a parametric set
for either argument is only allowed if the other argument is
a parametric set as well.

=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_map *isl_map_subtract_domain(
                __isl_take isl_map *map,
                __isl_take isl_set *dom);
        __isl_give isl_map *isl_map_subtract_range(
                __isl_take isl_map *map,
                __isl_take isl_set *dom);
        __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);
        __isl_give isl_union_map *isl_union_map_subtract_domain(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *dom);
        __isl_give isl_union_map *isl_union_map_subtract_range(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *dom);

=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 * Preimage

        __isl_give isl_basic_set *
        isl_basic_set_preimage_multi_aff(
                __isl_take isl_basic_set *bset,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_set *isl_set_preimage_multi_aff(
                __isl_take isl_set *set,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_set *isl_set_preimage_pw_multi_aff(
                __isl_take isl_set *set,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_map *isl_map_preimage_domain_multi_aff(
                __isl_take isl_map *map,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_union_map *
        isl_union_map_preimage_domain_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_aff *ma);

These functions compute the preimage of the given set or map domain under
the given function.  In other words, the expression is plugged
into the set description or into the domain of the map.
Objects of types C<isl_multi_aff> and C<isl_pw_multi_aff> are described in
L</"Piecewise Multiple Quasi Affine Expressions">.

=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_domain_product(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __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_basic_map *isl_basic_map_product(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_domain_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __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_domain_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);
        __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_range_product(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_flat_domain_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_map *isl_map_flat_range_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);
        __isl_give isl_union_map *isl_union_map_flat_range_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);
        __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_set *isl_set_gist_params(
                __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_union_set *isl_union_set_gist_params(
                __isl_take isl_union_set *uset,
                __isl_take isl_set *set);
        __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_map *isl_map_gist_params(
                __isl_take isl_map *map,
                __isl_take isl_set *context);
        __isl_give isl_map *isl_map_gist_domain(
                __isl_take isl_map *map,
                __isl_take isl_set *context);
        __isl_give isl_map *isl_map_gist_range(
                __isl_take isl_map *map,
                __isl_take isl_set *context);
        __isl_give isl_union_map *isl_union_map_gist(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_map *context);
        __isl_give isl_union_map *isl_union_map_gist_params(
                __isl_take isl_union_map *umap,
                __isl_take isl_set *set);
        __isl_give isl_union_map *isl_union_map_gist_domain(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *uset);
        __isl_give isl_union_map *isl_union_map_gist_range(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_set *uset);

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);

The following functions return their result in the form of
a piecewise multi-affine expression
(See L<"Piecewise Multiple Quasi Affine Expressions">),
but are otherwise equivalent to the corresponding functions
returning a basic set or relation.

        __isl_give isl_pw_multi_aff *
        isl_basic_map_lexmin_pw_multi_aff(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_pw_multi_aff *
        isl_basic_set_partial_lexmin_pw_multi_aff(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_pw_multi_aff *
        isl_basic_set_partial_lexmax_pw_multi_aff(
                __isl_take isl_basic_set *bset,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_pw_multi_aff *
        isl_basic_map_partial_lexmin_pw_multi_aff(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_pw_multi_aff *
        isl_basic_map_partial_lexmax_pw_multi_aff(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *dom,
                __isl_give isl_set **empty);
        __isl_give isl_pw_multi_aff *isl_set_lexmin_pw_multi_aff(
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *isl_set_lexmax_pw_multi_aff(
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *isl_map_lexmin_pw_multi_aff(
                __isl_take isl_map *map);
        __isl_give isl_pw_multi_aff *isl_map_lexmax_pw_multi_aff(
                __isl_take isl_map *map);

=head2 Lists

Lists are defined over several element types, including
C<isl_id>, C<isl_aff>, C<isl_pw_aff>, C<isl_constraint>,
C<isl_basic_set>, C<isl_set>, C<isl_ast_expr> and C<isl_ast_node>.
Here we take lists of C<isl_set>s as an example.
Lists can be created, copied, modified and freed using the following functions.

        #include <isl/list.h>
        __isl_give isl_set_list *isl_set_list_from_set(
                __isl_take isl_set *el);
        __isl_give isl_set_list *isl_set_list_alloc(
                isl_ctx *ctx, int n);
        __isl_give isl_set_list *isl_set_list_copy(
                __isl_keep isl_set_list *list);
        __isl_give isl_set_list *isl_set_list_insert(
                __isl_take isl_set_list *list, unsigned pos,
                __isl_take isl_set *el);
        __isl_give isl_set_list *isl_set_list_add(
                __isl_take isl_set_list *list,
                __isl_take isl_set *el);
        __isl_give isl_set_list *isl_set_list_drop(
                __isl_take isl_set_list *list,
                unsigned first, unsigned n);
        __isl_give isl_set_list *isl_set_list_set_set(
                __isl_take isl_set_list *list, int index,
                __isl_take isl_set *set);
        __isl_give isl_set_list *isl_set_list_concat(
                __isl_take isl_set_list *list1,
                __isl_take isl_set_list *list2);
        __isl_give isl_set_list *isl_set_list_sort(
                __isl_take isl_set_list *list,
                int (*cmp)(__isl_keep isl_set *a,
                        __isl_keep isl_set *b, void *user),
                void *user);
        void *isl_set_list_free(__isl_take isl_set_list *list);

C<isl_set_list_alloc> creates an empty list with a capacity for
C<n> elements.  C<isl_set_list_from_set> creates a list with a single
element.

Lists can be inspected using the following functions.

        #include <isl/list.h>
        isl_ctx *isl_set_list_get_ctx(__isl_keep isl_set_list *list);
        int isl_set_list_n_set(__isl_keep isl_set_list *list);
        __isl_give isl_set *isl_set_list_get_set(
                __isl_keep isl_set_list *list, int index);
        int isl_set_list_foreach(__isl_keep isl_set_list *list,
                int (*fn)(__isl_take isl_set *el, void *user),
                void *user);
        int isl_set_list_foreach_scc(__isl_keep isl_set_list *list,
                int (*follows)(__isl_keep isl_set *a,
                        __isl_keep isl_set *b, void *user),
                void *follows_user
                int (*fn)(__isl_take isl_set *el, void *user),
                void *fn_user);

The function C<isl_set_list_foreach_scc> calls C<fn> on each of the
strongly connected components of the graph with as vertices the elements
of C<list> and a directed edge from vertex C<b> to vertex C<a>
iff C<follows(a, b)> returns C<1>.  The callbacks C<follows> and C<fn>
should return C<-1> on error.

Lists can be printed using

        #include <isl/list.h>
        __isl_give isl_printer *isl_printer_print_set_list(
                __isl_take isl_printer *p,
                __isl_keep isl_set_list *list);

=head2 Vectors

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

        #include <isl/vec.h>
        __isl_give isl_vec *isl_vec_alloc(isl_ctx *ctx,
                unsigned size);
        __isl_give isl_vec *isl_vec_copy(__isl_keep isl_vec *vec);
        void *isl_vec_free(__isl_take isl_vec *vec);

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

        isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec);
        int isl_vec_size(__isl_keep isl_vec *vec);
        int isl_vec_get_element(__isl_keep isl_vec *vec,
                int pos, isl_int *v);
        __isl_give isl_vec *isl_vec_set_element(
                __isl_take isl_vec *vec, int pos, isl_int v);
        __isl_give isl_vec *isl_vec_set_element_si(
                __isl_take isl_vec *vec, int pos, int v);
        __isl_give isl_vec *isl_vec_set(__isl_take isl_vec *vec,
                isl_int v);
        __isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec,
                int v);
        __isl_give isl_vec *isl_vec_fdiv_r(__isl_take isl_vec *vec,
                isl_int m);

C<isl_vec_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 concatenate two vectors.

        __isl_give isl_vec *isl_vec_concat(__isl_take isl_vec *vec1,
                __isl_take isl_vec *vec2);

=head2 Matrices

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

        #include <isl/mat.h>
        __isl_give isl_mat *isl_mat_alloc(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 Piecewise Quasi Affine Expressions

The zero quasi affine expression or the quasi affine expression
that is equal to a specified dimension on a given domain can be created using

        __isl_give isl_aff *isl_aff_zero_on_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_aff *isl_aff_var_on_domain(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_pw_aff *isl_pw_aff_var_on_domain(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, unsigned pos);

Note that the space in which the resulting objects live is a map space
with the given space as domain and a one-dimensional range.

An empty piecewise quasi affine expression (one with no cells)
or a piecewise quasi affine expression with a single cell can
be created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_empty(
                __isl_take isl_space *space);
        __isl_give isl_pw_aff *isl_pw_aff_alloc(
                __isl_take isl_set *set, __isl_take isl_aff *aff);
        __isl_give isl_pw_aff *isl_pw_aff_from_aff(
                __isl_take isl_aff *aff);

A piecewise quasi affine expression that is equal to 1 on a set
and 0 outside the set can be created using the following function.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_set_indicator_function(
                __isl_take isl_set *set);

Quasi affine expressions can be copied and freed using

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_copy(__isl_keep isl_aff *aff);
        void *isl_aff_free(__isl_take isl_aff *aff);

        __isl_give isl_pw_aff *isl_pw_aff_copy(
                __isl_keep isl_pw_aff *pwaff);
        void *isl_pw_aff_free(__isl_take isl_pw_aff *pwaff);

A (rational) bound on a dimension can be extracted from an C<isl_constraint>
using the following function.  The constraint is required to have
a non-zero coefficient for the specified dimension.

        #include <isl/constraint.h>
        __isl_give isl_aff *isl_constraint_get_bound(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, int pos);

The entire affine expression of the constraint can also be extracted
using the following function.

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

Conversely, an equality constraint equating
the affine expression to zero or an inequality constraint enforcing
the affine expression to be non-negative, can be constructed using

        __isl_give isl_constraint *isl_equality_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_constraint *isl_inequality_from_aff(
                __isl_take isl_aff *aff);

The expression can be inspected using

        #include <isl/aff.h>
        isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
        int isl_aff_dim(__isl_keep isl_aff *aff,
                enum isl_dim_type type);
        __isl_give isl_local_space *isl_aff_get_domain_local_space(
                __isl_keep isl_aff *aff);
        __isl_give isl_local_space *isl_aff_get_local_space(
                __isl_keep isl_aff *aff);
        const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
                enum isl_dim_type type, unsigned pos);
        const char *isl_pw_aff_get_dim_name(
                __isl_keep isl_pw_aff *pa,
                enum isl_dim_type type, unsigned pos);
        int isl_pw_aff_has_dim_id(__isl_keep isl_pw_aff *pa,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_pw_aff_get_dim_id(
                __isl_keep isl_pw_aff *pa,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_pw_aff_get_tuple_id(
                __isl_keep isl_pw_aff *pa,
                enum isl_dim_type type);
        int isl_aff_get_constant(__isl_keep isl_aff *aff,
                isl_int *v);
        int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
                enum isl_dim_type type, int pos, isl_int *v);
        int isl_aff_get_denominator(__isl_keep isl_aff *aff,
                isl_int *v);
        __isl_give isl_aff *isl_aff_get_div(
                __isl_keep isl_aff *aff, int pos);

        int isl_pw_aff_n_piece(__isl_keep isl_pw_aff *pwaff);
        int isl_pw_aff_foreach_piece(__isl_keep isl_pw_aff *pwaff,
                int (*fn)(__isl_take isl_set *set,
                          __isl_take isl_aff *aff,
                          void *user), void *user);

        int isl_aff_is_cst(__isl_keep isl_aff *aff);
        int isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff);

        int isl_aff_involves_dims(__isl_keep isl_aff *aff,
                enum isl_dim_type type, unsigned first, unsigned n);
        int isl_pw_aff_involves_dims(__isl_keep isl_pw_aff *pwaff,
                enum isl_dim_type type, unsigned first, unsigned n);

        isl_ctx *isl_pw_aff_get_ctx(__isl_keep isl_pw_aff *pwaff);
        unsigned isl_pw_aff_dim(__isl_keep isl_pw_aff *pwaff,
                enum isl_dim_type type);
        int isl_pw_aff_is_empty(__isl_keep isl_pw_aff *pwaff);

It can be modified using

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_set_tuple_id(
                __isl_take isl_pw_aff *pwaff,
                enum isl_dim_type type, __isl_take isl_id *id);
        __isl_give isl_aff *isl_aff_set_dim_name(
                __isl_take isl_aff *aff, enum isl_dim_type type,
                unsigned pos, const char *s);
        __isl_give isl_aff *isl_aff_set_dim_id(
                __isl_take isl_aff *aff, enum isl_dim_type type,
                unsigned pos, __isl_take isl_id *id);
        __isl_give isl_pw_aff *isl_pw_aff_set_dim_id(
                __isl_take isl_pw_aff *pma,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        __isl_give isl_aff *isl_aff_set_constant(
                __isl_take isl_aff *aff, isl_int v);
        __isl_give isl_aff *isl_aff_set_constant_si(
                __isl_take isl_aff *aff, int v);
        __isl_give isl_aff *isl_aff_set_coefficient(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos, isl_int v);
        __isl_give isl_aff *isl_aff_set_coefficient_si(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos, int v);
        __isl_give isl_aff *isl_aff_set_denominator(
                __isl_take isl_aff *aff, isl_int v);

        __isl_give isl_aff *isl_aff_add_constant(
                __isl_take isl_aff *aff, isl_int v);
        __isl_give isl_aff *isl_aff_add_constant_si(
                __isl_take isl_aff *aff, int v);
        __isl_give isl_aff *isl_aff_add_constant_num(
                __isl_take isl_aff *aff, isl_int v);
        __isl_give isl_aff *isl_aff_add_constant_num_si(
                __isl_take isl_aff *aff, int v);
        __isl_give isl_aff *isl_aff_add_coefficient(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos, isl_int v);
        __isl_give isl_aff *isl_aff_add_coefficient_si(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos, int v);

        __isl_give isl_aff *isl_aff_insert_dims(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_pw_aff *isl_pw_aff_insert_dims(
                __isl_take isl_pw_aff *pwaff,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_aff *isl_aff_add_dims(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_pw_aff *isl_pw_aff_add_dims(
                __isl_take isl_pw_aff *pwaff,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_aff *isl_aff_drop_dims(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_pw_aff *isl_pw_aff_drop_dims(
                __isl_take isl_pw_aff *pwaff,
                enum isl_dim_type type, unsigned first, unsigned n);

Note that the C<set_constant> and C<set_coefficient> functions
set the I<numerator> of the constant or coefficient, while
C<add_constant> and C<add_coefficient> add an integer value to
the possibly rational constant or coefficient.
The C<add_constant_num> functions add an integer value to
the numerator.

To check whether an affine expressions is obviously zero
or obviously equal to some other affine expression, use

        #include <isl/aff.h>
        int isl_aff_plain_is_zero(__isl_keep isl_aff *aff);
        int isl_aff_plain_is_equal(__isl_keep isl_aff *aff1,
                __isl_keep isl_aff *aff2);
        int isl_pw_aff_plain_is_equal(
                __isl_keep isl_pw_aff *pwaff1,
                __isl_keep isl_pw_aff *pwaff2);

Operations include

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_add(__isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __isl_give isl_pw_aff *isl_pw_aff_add(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_pw_aff *isl_pw_aff_min(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_pw_aff *isl_pw_aff_max(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_aff *isl_aff_sub(__isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __isl_give isl_pw_aff *isl_pw_aff_sub(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_aff *isl_aff_neg(__isl_take isl_aff *aff);
        __isl_give isl_pw_aff *isl_pw_aff_neg(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff);
        __isl_give isl_pw_aff *isl_pw_aff_ceil(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_aff *isl_aff_floor(__isl_take isl_aff *aff);
        __isl_give isl_pw_aff *isl_pw_aff_floor(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_aff *isl_aff_mod(__isl_take isl_aff *aff,
                isl_int mod);
        __isl_give isl_pw_aff *isl_pw_aff_mod(
                __isl_take isl_pw_aff *pwaff, isl_int mod);
        __isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff,
                isl_int f);
        __isl_give isl_pw_aff *isl_pw_aff_scale(
                __isl_take isl_pw_aff *pwaff, isl_int f);
        __isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff,
                isl_int f);
        __isl_give isl_aff *isl_aff_scale_down_ui(
                __isl_take isl_aff *aff, unsigned f);
        __isl_give isl_pw_aff *isl_pw_aff_scale_down(
                __isl_take isl_pw_aff *pwaff, isl_int f);

        __isl_give isl_pw_aff *isl_pw_aff_list_min(
                __isl_take isl_pw_aff_list *list);
        __isl_give isl_pw_aff *isl_pw_aff_list_max(
                __isl_take isl_pw_aff_list *list);

        __isl_give isl_pw_aff *isl_pw_aff_coalesce(
                __isl_take isl_pw_aff *pwqp);

        __isl_give isl_aff *isl_aff_align_params(
                __isl_take isl_aff *aff,
                __isl_take isl_space *model);
        __isl_give isl_pw_aff *isl_pw_aff_align_params(
                __isl_take isl_pw_aff *pwaff,
                __isl_take isl_space *model);

        __isl_give isl_aff *isl_aff_project_domain_on_params(
                __isl_take isl_aff *aff);

        __isl_give isl_aff *isl_aff_gist_params(
                __isl_take isl_aff *aff,
                __isl_take isl_set *context);
        __isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
                __isl_take isl_set *context);
        __isl_give isl_pw_aff *isl_pw_aff_gist_params(
                __isl_take isl_pw_aff *pwaff,
                __isl_take isl_set *context);
        __isl_give isl_pw_aff *isl_pw_aff_gist(
                __isl_take isl_pw_aff *pwaff,
                __isl_take isl_set *context);

        __isl_give isl_set *isl_pw_aff_domain(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_pw_aff *isl_pw_aff_intersect_domain(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_set *set);
        __isl_give isl_pw_aff *isl_pw_aff_intersect_params(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_set *set);

        __isl_give isl_aff *isl_aff_mul(__isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __isl_give isl_aff *isl_aff_div(__isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __isl_give isl_pw_aff *isl_pw_aff_mul(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_pw_aff *isl_pw_aff_div(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);
        __isl_give isl_pw_aff *isl_pw_aff_tdiv_q(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);
        __isl_give isl_pw_aff *isl_pw_aff_tdiv_r(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);

When multiplying two affine expressions, at least one of the two needs
to be a constant.  Similarly, when dividing an affine expression by another,
the second expression needs to be a constant.
C<isl_pw_aff_tdiv_q> computes the quotient of an integer division with
rounding towards zero.  C<isl_pw_aff_tdiv_r> computes the corresponding
remainder.

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_pullback_multi_aff(
                __isl_take isl_aff *aff,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_pullback_multi_aff(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_pullback_pw_multi_aff(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_pw_multi_aff *pma);

These functions precompose the input expression by the given
C<isl_multi_aff> or C<isl_pw_multi_aff>.  In other words,
the C<isl_multi_aff> or C<isl_pw_multi_aff> is plugged
into the (piecewise) affine expression.
Objects of type C<isl_multi_aff> are described in
L</"Piecewise Multiple Quasi Affine Expressions">.

        #include <isl/aff.h>
        __isl_give isl_basic_set *isl_aff_zero_basic_set(
                __isl_take isl_aff *aff);
        __isl_give isl_basic_set *isl_aff_neg_basic_set(
                __isl_take isl_aff *aff);
        __isl_give isl_basic_set *isl_aff_le_basic_set(
                __isl_take isl_aff *aff1, __isl_take isl_aff *aff2);
        __isl_give isl_basic_set *isl_aff_ge_basic_set(
                __isl_take isl_aff *aff1, __isl_take isl_aff *aff2);
        __isl_give isl_set *isl_pw_aff_eq_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_set *isl_pw_aff_ne_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_set *isl_pw_aff_le_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_set *isl_pw_aff_lt_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_set *isl_pw_aff_ge_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_set *isl_pw_aff_gt_set(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);

        __isl_give isl_set *isl_pw_aff_list_eq_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);
        __isl_give isl_set *isl_pw_aff_list_ne_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);
        __isl_give isl_set *isl_pw_aff_list_le_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);
        __isl_give isl_set *isl_pw_aff_list_lt_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);
        __isl_give isl_set *isl_pw_aff_list_ge_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);
        __isl_give isl_set *isl_pw_aff_list_gt_set(
                __isl_take isl_pw_aff_list *list1,
                __isl_take isl_pw_aff_list *list2);

The function C<isl_aff_neg_basic_set> returns a basic set
containing those elements in the domain space
of C<aff> where C<aff> is negative.
The function C<isl_aff_ge_basic_set> returns a basic set
containing those elements in the shared space
of C<aff1> and C<aff2> where C<aff1> is greater than or equal to C<aff2>.
The function C<isl_pw_aff_ge_set> returns a set
containing those elements in the shared domain
of C<pwaff1> and C<pwaff2> where C<pwaff1> is greater than or equal to C<pwaff2>.
The functions operating on C<isl_pw_aff_list> apply the corresponding
C<isl_pw_aff> function to each pair of elements in the two lists.

        #include <isl/aff.h>
        __isl_give isl_set *isl_pw_aff_nonneg_set(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_set *isl_pw_aff_zero_set(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_set *isl_pw_aff_non_zero_set(
                __isl_take isl_pw_aff *pwaff);

The function C<isl_pw_aff_nonneg_set> returns a set
containing those elements in the domain
of C<pwaff> where C<pwaff> is non-negative.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_cond(
                __isl_take isl_pw_aff *cond,
                __isl_take isl_pw_aff *pwaff_true,
                __isl_take isl_pw_aff *pwaff_false);

The function C<isl_pw_aff_cond> performs a conditional operator
and returns an expression that is equal to C<pwaff_true>
for elements where C<cond> is non-zero and equal to C<pwaff_false> for elements
where C<cond> is zero.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_union_min(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_pw_aff *isl_pw_aff_union_max(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_pw_aff *isl_pw_aff_union_add(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);

The function C<isl_pw_aff_union_max> computes a piecewise quasi-affine
expression with a domain that is the union of those of C<pwaff1> and
C<pwaff2> and such that on each cell, the quasi-affine expression is
the maximum of those of C<pwaff1> and C<pwaff2>.  If only one of
C<pwaff1> or C<pwaff2> is defined on a given cell, then the
associated expression is the defined one.

An expression can be read from input using

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_pw_aff *isl_pw_aff_read_from_str(
                isl_ctx *ctx, const char *str);

An expression can be printed using

        #include <isl/aff.h>
        __isl_give isl_printer *isl_printer_print_aff(
                __isl_take isl_printer *p, __isl_keep isl_aff *aff);

        __isl_give isl_printer *isl_printer_print_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_pw_aff *pwaff);

=head2 Piecewise Multiple Quasi Affine Expressions

An C<isl_multi_aff> object represents a sequence of
zero or more affine expressions, all defined on the same domain space.
Similarly, an C<isl_multi_pw_aff> object represents a sequence of
zero or more piecewise affine expressions.

An C<isl_multi_aff> can be constructed from a single
C<isl_aff> or an C<isl_aff_list> using the
following functions.  Similarly for C<isl_multi_pw_aff>.

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_pw_aff(
                __isl_take isl_pw_aff *pa);
        __isl_give isl_multi_aff *isl_multi_aff_from_aff_list(
                __isl_take isl_space *space,
                __isl_take isl_aff_list *list);

An empty piecewise multiple quasi affine expression (one with no cells),
the zero piecewise multiple quasi affine expression (with value zero
for each output dimension),
a piecewise multiple quasi affine expression with a single cell (with
either a universe or a specified domain) or
a zero-dimensional piecewise multiple quasi affine expression
on a given domain
can be created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_empty(
                __isl_take isl_space *space);
        __isl_give isl_multi_aff *isl_multi_aff_zero(
                __isl_take isl_space *space);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_zero(
                __isl_take isl_space *space);
        __isl_give isl_multi_aff *isl_multi_aff_identity(
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
                __isl_take isl_space *space);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_identity(
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_from_multi_aff(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_alloc(
                __isl_take isl_set *set,
                __isl_take isl_multi_aff *maff);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
                __isl_take isl_set *set);

        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_empty(
                __isl_take isl_space *space);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_add_pw_multi_aff(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_domain(
                __isl_take isl_union_set *uset);

A piecewise multiple quasi affine expression can also be initialized
from an C<isl_set> or C<isl_map>, provided the C<isl_set> is a singleton
and the C<isl_map> is single-valued.
In case of a conversion from an C<isl_union_set> or an C<isl_union_map>
to an C<isl_union_pw_multi_aff>, these properties need to hold in each space.

        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(
                __isl_take isl_map *map);

        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_union_set(
                __isl_take isl_union_set *uset);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_union_map(
                __isl_take isl_union_map *umap);

Multiple quasi affine expressions can be copied and freed using

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_copy(
                __isl_keep isl_multi_aff *maff);
        void *isl_multi_aff_free(__isl_take isl_multi_aff *maff);

        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_copy(
                __isl_keep isl_pw_multi_aff *pma);
        void *isl_pw_multi_aff_free(
                __isl_take isl_pw_multi_aff *pma);

        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_copy(
                __isl_keep isl_union_pw_multi_aff *upma);
        void *isl_union_pw_multi_aff_free(
                __isl_take isl_union_pw_multi_aff *upma);

        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_copy(
                __isl_keep isl_multi_pw_aff *mpa);
        void *isl_multi_pw_aff_free(
                __isl_take isl_multi_pw_aff *mpa);

The expression can be inspected using

        #include <isl/aff.h>
        isl_ctx *isl_multi_aff_get_ctx(
                __isl_keep isl_multi_aff *maff);
        isl_ctx *isl_pw_multi_aff_get_ctx(
                __isl_keep isl_pw_multi_aff *pma);
        isl_ctx *isl_union_pw_multi_aff_get_ctx(
                __isl_keep isl_union_pw_multi_aff *upma);
        isl_ctx *isl_multi_pw_aff_get_ctx(
                __isl_keep isl_multi_pw_aff *mpa);
        unsigned isl_multi_aff_dim(__isl_keep isl_multi_aff *maff,
                enum isl_dim_type type);
        unsigned isl_pw_multi_aff_dim(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type);
        unsigned isl_multi_pw_aff_dim(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type);
        __isl_give isl_aff *isl_multi_aff_get_aff(
                __isl_keep isl_multi_aff *multi, int pos);
        __isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
                __isl_keep isl_pw_multi_aff *pma, int pos);
        __isl_give isl_pw_aff *isl_multi_pw_aff_get_pw_aff(
                __isl_keep isl_multi_pw_aff *mpa, int pos);
        const char *isl_pw_multi_aff_get_dim_name(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_pw_multi_aff_get_dim_id(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned pos);
        const char *isl_multi_aff_get_tuple_name(
                __isl_keep isl_multi_aff *multi,
                enum isl_dim_type type);
        int isl_pw_multi_aff_has_tuple_name(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type);
        const char *isl_pw_multi_aff_get_tuple_name(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type);
        int isl_pw_multi_aff_has_tuple_id(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type);
        __isl_give isl_id *isl_pw_multi_aff_get_tuple_id(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type);

        int isl_pw_multi_aff_foreach_piece(
                __isl_keep isl_pw_multi_aff *pma,
                int (*fn)(__isl_take isl_set *set,
                            __isl_take isl_multi_aff *maff,
                            void *user), void *user);

        int isl_union_pw_multi_aff_foreach_pw_multi_aff(
                __isl_keep isl_union_pw_multi_aff *upma,
                int (*fn)(__isl_take isl_pw_multi_aff *pma,
                            void *user), void *user);

It can be modified using

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_set_aff(
                __isl_take isl_multi_aff *multi, int pos,
                __isl_take isl_aff *aff);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
                __isl_take isl_pw_multi_aff *pma, unsigned pos,
                __isl_take isl_pw_aff *pa);
        __isl_give isl_multi_aff *isl_multi_aff_set_dim_name(
                __isl_take isl_multi_aff *maff,
                enum isl_dim_type type, unsigned pos, const char *s);
        __isl_give isl_multi_aff *isl_multi_aff_set_tuple_name(
                __isl_take isl_multi_aff *maff,
                enum isl_dim_type type, const char *s);
        __isl_give isl_multi_aff *isl_multi_aff_set_tuple_id(
                __isl_take isl_multi_aff *maff,
                enum isl_dim_type type, __isl_take isl_id *id);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_tuple_id(
                __isl_take isl_pw_multi_aff *pma,
                enum isl_dim_type type, __isl_take isl_id *id);

        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_set_dim_name(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type, unsigned pos, const char *s);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_set_tuple_name(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type, const char *s);

        __isl_give isl_multi_aff *isl_multi_aff_insert_dims(
                __isl_take isl_multi_aff *ma,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_multi_aff *isl_multi_aff_add_dims(
                __isl_take isl_multi_aff *ma,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_multi_aff *isl_multi_aff_drop_dims(
                __isl_take isl_multi_aff *maff,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_drop_dims(
                __isl_take isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned first, unsigned n);

        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_insert_dims(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_add_dims(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type, unsigned n);

To check whether two multiple affine expressions are
obviously equal to each other, use

        int isl_multi_aff_plain_is_equal(__isl_keep isl_multi_aff *maff1,
                __isl_keep isl_multi_aff *maff2);
        int isl_pw_multi_aff_plain_is_equal(
                __isl_keep isl_pw_multi_aff *pma1,
                __isl_keep isl_pw_multi_aff *pma2);

Operations include

        #include <isl/aff.h>
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_multi_aff *isl_multi_aff_add(
                __isl_take isl_multi_aff *maff1,
                __isl_take isl_multi_aff *maff2);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_add(
                __isl_take isl_union_pw_multi_aff *upma1,
                __isl_take isl_union_pw_multi_aff *upma2);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_multi_aff *isl_multi_aff_sub(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_sub(
                __isl_take isl_union_pw_multi_aff *upma1,
                __isl_take isl_union_pw_multi_aff *upma2);

C<isl_multi_aff_sub> subtracts the second argument from the first.

        __isl_give isl_multi_aff *isl_multi_aff_scale(
                __isl_take isl_multi_aff *maff,
                isl_int f);
        __isl_give isl_multi_aff *isl_multi_aff_scale_vec(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_vec *v);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_vec(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_vec *v);
        __isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_vec(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_vec *v);

C<isl_multi_aff_scale_vec> scales the first elements of C<ma>
by the corresponding elements of C<v>.

        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_params(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_intersect_domain(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_set *set);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_intersect_domain(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_union_set *uset);
        __isl_give isl_multi_aff *isl_multi_aff_lift(
                __isl_take isl_multi_aff *maff,
                __isl_give isl_local_space **ls);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_coalesce(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_multi_aff *isl_multi_aff_align_params(
                __isl_take isl_multi_aff *multi,
                __isl_take isl_space *model);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_align_params(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_space *model);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_project_domain_on_params(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_multi_aff *isl_multi_aff_gist_params(
                __isl_take isl_multi_aff *maff,
                __isl_take isl_set *context);
        __isl_give isl_multi_aff *isl_multi_aff_gist(
                __isl_take isl_multi_aff *maff,
                __isl_take isl_set *context);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist_params(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_gist(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_set *set);
        __isl_give isl_set *isl_pw_multi_aff_domain(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_set *isl_union_pw_multi_aff_domain(
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_multi_aff *isl_multi_aff_range_splice(
                __isl_take isl_multi_aff *ma1, unsigned pos,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_multi_aff *isl_multi_aff_splice(
                __isl_take isl_multi_aff *ma1,
                unsigned in_pos, unsigned out_pos,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_multi_aff *isl_multi_aff_range_product(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_multi_aff *isl_multi_aff_flat_range_product(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_multi_aff *isl_multi_aff_product(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_range_product(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_flat_range_product(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_flat_range_product(
                __isl_take isl_union_pw_multi_aff *upma1,
                __isl_take isl_union_pw_multi_aff *upma2);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_range_splice(
                __isl_take isl_multi_pw_aff *mpa1, unsigned pos,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_splice(
                __isl_take isl_multi_pw_aff *mpa1,
                unsigned in_pos, unsigned out_pos,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_range_product(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_flat_range_product(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);

If the C<ls> argument of C<isl_multi_aff_lift> is not C<NULL>,
then it is assigned the local space that lies at the basis of
the lifting applied.

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_pullback_multi_aff(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_pullback_pw_multi_aff(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);

The function C<isl_multi_aff_pullback_multi_aff> precomposes C<ma1> by C<ma2>.
In other words, C<ma2> is plugged
into C<ma1>.

        __isl_give isl_set *isl_multi_aff_lex_le_set(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_set *isl_multi_aff_lex_ge_set(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);

The function C<isl_multi_aff_lex_le_set> returns a set
containing those elements in the shared domain space
where C<ma1> is lexicographically smaller than or
equal to C<ma2>.

An expression can be read from input using

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_read_from_str(
                isl_ctx *ctx, const char *str);

An expression can be printed using

        #include <isl/aff.h>
        __isl_give isl_printer *isl_printer_print_multi_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_aff *maff);
        __isl_give isl_printer *isl_printer_print_pw_multi_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_pw_multi_aff *pma);
        __isl_give isl_printer *isl_printer_print_union_pw_multi_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_union_pw_multi_aff *upma);
        __isl_give isl_printer *isl_printer_print_multi_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_pw_aff *mpa);

=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_space *space);

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

        int 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);

Other properties can be obtained using

        isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt);

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 Input and Output

Piecewise quasipolynomials can be read from input using

        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_read_from_str(
                isl_ctx *ctx, const char *str);

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_on_domain(
                __isl_take isl_space *domain);
        __isl_give isl_qpolynomial *isl_qpolynomial_one_on_domain(
                __isl_take isl_space *domain);
        __isl_give isl_qpolynomial *isl_qpolynomial_infty_on_domain(
                __isl_take isl_space *domain);
        __isl_give isl_qpolynomial *isl_qpolynomial_neginfty_on_domain(
                __isl_take isl_space *domain);
        __isl_give isl_qpolynomial *isl_qpolynomial_nan_on_domain(
                __isl_take isl_space *domain);
        __isl_give isl_qpolynomial *isl_qpolynomial_rat_cst_on_domain(
                __isl_take isl_space *domain,
                const isl_int n, const isl_int d);
        __isl_give isl_qpolynomial *isl_qpolynomial_var_on_domain(
                __isl_take isl_space *domain,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_qpolynomial *isl_qpolynomial_from_aff(
                __isl_take isl_aff *aff);

Note that the space in which a quasipolynomial lives is a map space
with a one-dimensional range.  The C<domain> argument in some of
the functions above corresponds to the domain of this map space.

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_space *space);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
                __isl_take isl_set *set,
                __isl_take isl_qpolynomial *qp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_qpolynomial(
                __isl_take isl_qpolynomial *qp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_from_pw_aff(
                __isl_take isl_pw_aff *pwaff);

        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_zero(
                __isl_take isl_space *space);
        __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 in a given space from a union, use

        __isl_give isl_pw_qpolynomial *
        isl_union_pw_qpolynomial_extract_pw_qpolynomial(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_space *space);

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_aff *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.

To check whether two union piecewise quasipolynomials are
obviously equal, use

        int isl_union_pw_qpolynomial_plain_is_equal(
                __isl_keep isl_union_pw_qpolynomial *upwqp1,
                __isl_keep isl_union_pw_qpolynomial *upwqp2);

=head3 Operations on (Piecewise) Quasipolynomials

        __isl_give isl_qpolynomial *isl_qpolynomial_scale(
                __isl_take isl_qpolynomial *qp, isl_int v);
        __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_pw_qpolynomial *isl_pw_qpolynomial_pow(
                __isl_take isl_pw_qpolynomial *pwqp, unsigned exponent);

        __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_pw_qpolynomial *isl_pw_qpolynomial_intersect_params(
                __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_intersect_params(
                __isl_take isl_union_pw_qpolynomial *upwpq,
                __isl_take isl_set *set);

        __isl_give isl_qpolynomial *isl_qpolynomial_align_params(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_space *model);

        __isl_give isl_qpolynomial *isl_qpolynomial_project_domain_on_params(
                __isl_take isl_qpolynomial *qp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_project_domain_on_params(
                __isl_take isl_pw_qpolynomial *pwqp);

        __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_params(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_set *context);
        __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_params(
                __isl_take isl_pw_qpolynomial *pwqp,
                __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_params(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __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 Properties of Piecewise Quasipolynomial Reductions

To check whether two union piecewise quasipolynomial reductions are
obviously equal, use

        int isl_union_pw_qpolynomial_fold_plain_is_equal(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf1,
                __isl_keep isl_union_pw_qpolynomial_fold *upwf2);

=head3 Operations on Piecewise Quasipolynomial Reductions

        __isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_scale(
                __isl_take isl_qpolynomial_fold *fold, isl_int v);

        __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_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_intersect_params(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_set *set);

        __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_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_intersect_params(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_set *set);

        __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_project_domain_on_params(
                __isl_take isl_pw_qpolynomial_fold *pwf);

        __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_qpolynomial_fold *isl_qpolynomial_fold_gist_params(
                __isl_take isl_qpolynomial_fold *fold,
                __isl_take isl_set *context);
        __isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_gist(
                __isl_take isl_qpolynomial_fold *fold,
                __isl_take isl_set *context);

        __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_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_gist_params(
                __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);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_gist_params(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_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 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 Polyhedral Compilation Library

This section collects functionality in C<isl> that has been specifically
designed for use during polyhedral compilation.

=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.
The resulting dependence relations map source iterations
to the corresponding sink iterations.
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 from the last
I<must> access B<and> from 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.

=head3 Interaction with Dependence Analysis

During the dependence analysis, we frequently need to perform
the following operation.  Given a relation between sink iterations
and potential source iterations from a particular source domain,
what is the last potential source iteration corresponding to each
sink iteration.  It can sometimes be convenient to adjust
the set of potential source iterations before or after each such operation.
The prototypical example is fuzzy array dataflow analysis,
where we need to analyze if, based on data-dependent constraints,
the sink iteration can ever be executed without one or more of
the corresponding potential source iterations being executed.
If so, we can introduce extra parameters and select an unknown
but fixed source iteration from the potential source iterations.
To be able to perform such manipulations, C<isl> provides the following
function.

        #include <isl/flow.h>

        typedef __isl_give isl_restriction *(*isl_access_restrict)(
                __isl_keep isl_map *source_map,
                __isl_keep isl_set *sink, void *source_user,
                void *user);
        __isl_give isl_access_info *isl_access_info_set_restrict(
                __isl_take isl_access_info *acc,
                isl_access_restrict fn, void *user);

The function C<isl_access_info_set_restrict> should be called
before calling C<isl_access_info_compute_flow> and registers a callback function
that will be called any time C<isl> is about to compute the last
potential source.  The first argument is the (reverse) proto-dependence,
mapping sink iterations to potential source iterations.
The second argument represents the sink iterations for which
we want to compute the last source iteration.
The third argument is the token corresponding to the source
and the final argument is the token passed to C<isl_access_info_set_restrict>.
The callback is expected to return a restriction on either the input or
the output of the operation computing the last potential source.
If the input needs to be restricted then restrictions are needed
for both the source and the sink iterations.  The sink iterations
and the potential source iterations will be intersected with these sets.
If the output needs to be restricted then only a restriction on the source
iterations is required.
If any error occurs, the callback should return C<NULL>.
An C<isl_restriction> object can be created, freed and inspected
using the following functions.

        #include <isl/flow.h>

        __isl_give isl_restriction *isl_restriction_input(
                __isl_take isl_set *source_restr,
                __isl_take isl_set *sink_restr);
        __isl_give isl_restriction *isl_restriction_output(
                __isl_take isl_set *source_restr);
        __isl_give isl_restriction *isl_restriction_none(
                __isl_take isl_map *source_map);
        __isl_give isl_restriction *isl_restriction_empty(
                __isl_take isl_map *source_map);
        void *isl_restriction_free(
                __isl_take isl_restriction *restr);
        isl_ctx *isl_restriction_get_ctx(
                __isl_keep isl_restriction *restr);

C<isl_restriction_none> and C<isl_restriction_empty> are special
cases of C<isl_restriction_input>.  C<isl_restriction_none>
is essentially equivalent to

        isl_restriction_input(isl_set_universe(
            isl_space_range(isl_map_get_space(source_map))),
                            isl_set_universe(
            isl_space_domain(isl_map_get_space(source_map))));

whereas C<isl_restriction_empty> is essentially equivalent to

        isl_restriction_input(isl_set_empty(
            isl_space_range(isl_map_get_space(source_map))),
                            isl_set_universe(
            isl_space_domain(isl_map_get_space(source_map))));

=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.
By default, the algorithm used to construct the schedule is similar
to that of C<Pluto>.
Alternatively, Feautrier's multi-dimensional scheduling algorithm can
be selected.
The generated schedule respects all C<validity> dependences.
That is, all dependence distances over these dependences in the
scheduled space are lexicographically positive.
The default algorithm 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.
When using Feautrier's algorithm, the C<proximity> dependence
distances are only minimized during the extension to a
full-dimensional schedule.

        #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);

A representation of the schedule can be printed using
         
        __isl_give isl_printer *isl_printer_print_schedule(
                __isl_take isl_printer *p,
                __isl_keep isl_schedule *schedule);

A representation of the schedule as a forest of bands can be obtained
using the following function.

        __isl_give isl_band_list *isl_schedule_get_band_forest(
                __isl_keep isl_schedule *schedule);

The individual bands can be visited in depth-first post-order
using the following function.

        #include <isl/schedule.h>
        int isl_schedule_foreach_band(
                __isl_keep isl_schedule *sched,
                int (*fn)(__isl_keep isl_band *band, void *user),
                void *user);

The list can be manipulated as explained in L<"Lists">.
The bands inside the list can be copied and freed using the following
functions.

        #include <isl/band.h>
        __isl_give isl_band *isl_band_copy(
                __isl_keep isl_band *band);
        void *isl_band_free(__isl_take isl_band *band);

Each band contains zero or more scheduling dimensions.
These are referred to as the members of the band.
The section of the schedule that corresponds to the band is
referred to as the partial schedule of the band.
For those nodes that participate in a band, the outer scheduling
dimensions form the prefix schedule, while the inner scheduling
dimensions form the suffix schedule.
That is, if we take a cut of the band forest, then the union of
the concatenations of the prefix, partial and suffix schedules of
each band in the cut is equal to the entire schedule (modulo
some possible padding at the end with zero scheduling dimensions).
The properties of a band can be inspected using the following functions.

        #include <isl/band.h>
        isl_ctx *isl_band_get_ctx(__isl_keep isl_band *band);

        int isl_band_has_children(__isl_keep isl_band *band);
        __isl_give isl_band_list *isl_band_get_children(
                __isl_keep isl_band *band);

        __isl_give isl_union_map *isl_band_get_prefix_schedule(
                __isl_keep isl_band *band);
        __isl_give isl_union_map *isl_band_get_partial_schedule(
                __isl_keep isl_band *band);
        __isl_give isl_union_map *isl_band_get_suffix_schedule(
                __isl_keep isl_band *band);

        int isl_band_n_member(__isl_keep isl_band *band);
        int isl_band_member_is_zero_distance(
                __isl_keep isl_band *band, int pos);

        int isl_band_list_foreach_band(
                __isl_keep isl_band_list *list,
                int (*fn)(__isl_keep isl_band *band, void *user),
                void *user);

Note that a scheduling dimension is considered to be ``zero
distance'' if it does not carry any proximity dependences
within its band.
That is, if the dependence distances of the proximity
dependences are all zero in that direction (for fixed
iterations of outer bands).
Like C<isl_schedule_foreach_band>,
the function C<isl_band_list_foreach_band> calls C<fn> on the bands
in depth-first post-order.

A band can be tiled using the following function.

        #include <isl/band.h>
        int isl_band_tile(__isl_keep isl_band *band,
                __isl_take isl_vec *sizes);

        int isl_options_set_tile_scale_tile_loops(isl_ctx *ctx,
                int val);
        int isl_options_get_tile_scale_tile_loops(isl_ctx *ctx);
        int isl_options_set_tile_shift_point_loops(isl_ctx *ctx,
                int val);
        int isl_options_get_tile_shift_point_loops(isl_ctx *ctx);

The C<isl_band_tile> function tiles the band using the given tile sizes
inside its schedule.
A new child band is created to represent the point loops and it is
inserted between the modified band and its children.
The C<tile_scale_tile_loops> option specifies whether the tile
loops iterators should be scaled by the tile sizes.
If the C<tile_shift_point_loops> option is set, then the point loops
are shifted to start at zero.

A band can be split into two nested bands using the following function.

        int isl_band_split(__isl_keep isl_band *band, int pos);

The resulting outer band contains the first C<pos> dimensions of C<band>
while the inner band contains the remaining dimensions.

A representation of the band can be printed using

        #include <isl/band.h>
        __isl_give isl_printer *isl_printer_print_band(
                __isl_take isl_printer *p,
                __isl_keep isl_band *band);

=head3 Options

        #include <isl/schedule.h>
        int isl_options_set_schedule_max_coefficient(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_max_coefficient(
                isl_ctx *ctx);
        int isl_options_set_schedule_max_constant_term(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_max_constant_term(
                isl_ctx *ctx);
        int isl_options_set_schedule_fuse(isl_ctx *ctx, int val);
        int isl_options_get_schedule_fuse(isl_ctx *ctx);
        int isl_options_set_schedule_maximize_band_depth(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_maximize_band_depth(
                isl_ctx *ctx);
        int isl_options_set_schedule_outer_zero_distance(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_outer_zero_distance(
                isl_ctx *ctx);
        int isl_options_set_schedule_split_scaled(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_split_scaled(
                isl_ctx *ctx);
        int isl_options_set_schedule_algorithm(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_algorithm(
                isl_ctx *ctx);
        int isl_options_set_schedule_separate_components(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_separate_components(
                isl_ctx *ctx);

=over

=item * schedule_max_coefficient

This option enforces that the coefficients for variable and parameter
dimensions in the calculated schedule are not larger than the specified value.
This option can significantly increase the speed of the scheduling calculation
and may also prevent fusing of unrelated dimensions. A value of -1 means that
this option does not introduce bounds on the variable or parameter
coefficients.

=item * schedule_max_constant_term

This option enforces that the constant coefficients in the calculated schedule
are not larger than the maximal constant term. This option can significantly
increase the speed of the scheduling calculation and may also prevent fusing of
unrelated dimensions. A value of -1 means that this option does not introduce
bounds on the constant coefficients.

=item * schedule_fuse

This option controls the level of fusion.
If this option is set to C<ISL_SCHEDULE_FUSE_MIN>, then loops in the
resulting schedule will be distributed as much as possible.
If this option is set to C<ISL_SCHEDULE_FUSE_MAX>, then C<isl> will
try to fuse loops in the resulting schedule.

=item * schedule_maximize_band_depth

If this option is set, we do not split bands at the point
where we detect splitting is necessary. Instead, we
backtrack and split bands as early as possible. This
reduces the number of splits and maximizes the width of
the bands. Wider bands give more possibilities for tiling.
Note that if the C<schedule_fuse> option is set to C<ISL_SCHEDULE_FUSE_MIN>,
then bands will be split as early as possible, even if there is no need.
The C<schedule_maximize_band_depth> option therefore has no effect in this case.

=item * schedule_outer_zero_distance

If this option is set, then we try to construct schedules
where the outermost scheduling dimension in each band
results in a zero dependence distance over the proximity
dependences.

=item * schedule_split_scaled

If this option is set, then we try to construct schedules in which the
constant term is split off from the linear part if the linear parts of
the scheduling rows for all nodes in the graphs have a common non-trivial
divisor.
The constant term is then placed in a separate band and the linear
part is reduced.

=item * schedule_algorithm

Selects the scheduling algorithm to be used.
Available scheduling algorithms are C<ISL_SCHEDULE_ALGORITHM_ISL>
and C<ISL_SCHEDULE_ALGORITHM_FEAUTRIER>.

=item * schedule_separate_components

If at any point the dependence graph contains any (weakly connected) components,
then these components are scheduled separately.
If this option is not set, then some iterations of the domains
in these components may be scheduled together.
If this option is set, then the components are given consecutive
schedules.

=back

=head2 AST Generation

This section describes the C<isl> functionality for generating
ASTs that visit all the elements
in a domain in an order specified by a schedule.
In particular, given a C<isl_union_map>, an AST is generated
that visits all the elements in the domain of the C<isl_union_map>
according to the lexicographic order of the corresponding image
element(s).  If the range of the C<isl_union_map> consists of
elements in more than one space, then each of these spaces is handled
separately in an arbitrary order.
It should be noted that the image elements only specify the I<order>
in which the corresponding domain elements should be visited.
No direct relation between the image elements and the loop iterators
in the generated AST should be assumed.

Each AST is generated within a build.  The initial build
simply specifies the constraints on the parameters (if any)
and can be created, inspected, copied and freed using the following functions.

        #include <isl/ast_build.h>
        __isl_give isl_ast_build *isl_ast_build_from_context(
                __isl_take isl_set *set);
        isl_ctx *isl_ast_build_get_ctx(
                __isl_keep isl_ast_build *build);
        __isl_give isl_ast_build *isl_ast_build_copy(
                __isl_keep isl_ast_build *build);
        void *isl_ast_build_free(
                __isl_take isl_ast_build *build);

The C<set> argument is usually a parameter set with zero or more parameters.
More C<isl_ast_build> functions are described in L</"Nested AST Generation">
and L</"Fine-grained Control over AST Generation">.
Finally, the AST itself can be constructed using the following
function.

        #include <isl/ast_build.h>
        __isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
                __isl_keep isl_ast_build *build,
                __isl_take isl_union_map *schedule);

=head3 Inspecting the AST

The basic properties of an AST node can be obtained as follows.

        #include <isl/ast.h>
        isl_ctx *isl_ast_node_get_ctx(
                __isl_keep isl_ast_node *node);
        enum isl_ast_node_type isl_ast_node_get_type(
                __isl_keep isl_ast_node *node);

The type of an AST node is one of
C<isl_ast_node_for>,
C<isl_ast_node_if>,
C<isl_ast_node_block> or
C<isl_ast_node_user>.
An C<isl_ast_node_for> represents a for node.
An C<isl_ast_node_if> represents an if node.
An C<isl_ast_node_block> represents a compound node.
An C<isl_ast_node_user> represents an expression statement.
An expression statement typically corresponds to a domain element, i.e.,
one of the elements that is visited by the AST.

Each type of node has its own additional properties.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_node_for_get_iterator(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_expr *isl_ast_node_for_get_init(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_expr *isl_ast_node_for_get_cond(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_expr *isl_ast_node_for_get_inc(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_node *isl_ast_node_for_get_body(
                __isl_keep isl_ast_node *node);
        int isl_ast_node_for_is_degenerate(
                __isl_keep isl_ast_node *node);

An C<isl_ast_for> is considered degenerate if it is known to execute
exactly once.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_node_if_get_cond(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_node *isl_ast_node_if_get_then(
                __isl_keep isl_ast_node *node);
        int isl_ast_node_if_has_else(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_node *isl_ast_node_if_get_else(
                __isl_keep isl_ast_node *node);

        __isl_give isl_ast_node_list *
        isl_ast_node_block_get_children(
                __isl_keep isl_ast_node *node);

        __isl_give isl_ast_expr *isl_ast_node_user_get_expr(
                __isl_keep isl_ast_node *node);

Each of the returned C<isl_ast_expr>s can in turn be inspected using
the following functions.

        #include <isl/ast.h>
        isl_ctx *isl_ast_expr_get_ctx(
                __isl_keep isl_ast_expr *expr);
        enum isl_ast_expr_type isl_ast_expr_get_type(
                __isl_keep isl_ast_expr *expr);

The type of an AST expression is one of
C<isl_ast_expr_op>,
C<isl_ast_expr_id> or
C<isl_ast_expr_int>.
An C<isl_ast_expr_op> represents the result of an operation.
An C<isl_ast_expr_id> represents an identifier.
An C<isl_ast_expr_int> represents an integer value.

Each type of expression has its own additional properties.

        #include <isl/ast.h>
        enum isl_ast_op_type isl_ast_expr_get_op_type(
                __isl_keep isl_ast_expr *expr);
        int isl_ast_expr_get_op_n_arg(__isl_keep isl_ast_expr *expr);
        __isl_give isl_ast_expr *isl_ast_expr_get_op_arg(
                __isl_keep isl_ast_expr *expr, int pos);
        int isl_ast_node_foreach_ast_op_type(
                __isl_keep isl_ast_node *node,
                int (*fn)(enum isl_ast_op_type type, void *user),
                void *user);

C<isl_ast_expr_get_op_type> returns the type of the operation
performed.  C<isl_ast_expr_get_op_n_arg> returns the number of
arguments.  C<isl_ast_expr_get_op_arg> returns the specified
argument.
C<isl_ast_node_foreach_ast_op_type> calls C<fn> for each distinct
C<isl_ast_op_type> that appears in C<node>.
The operation type is one of the following.

=over

=item C<isl_ast_op_and>

Logical I<and> of two arguments.
Both arguments can be evaluated.

=item C<isl_ast_op_and_then>

Logical I<and> of two arguments.
The second argument can only be evaluated if the first evaluates to true.

=item C<isl_ast_op_or>

Logical I<or> of two arguments.
Both arguments can be evaluated.

=item C<isl_ast_op_or_else>

Logical I<or> of two arguments.
The second argument can only be evaluated if the first evaluates to false.

=item C<isl_ast_op_max>

Maximum of two or more arguments.

=item C<isl_ast_op_min>

Minimum of two or more arguments.

=item C<isl_ast_op_minus>

Change sign.

=item C<isl_ast_op_add>

Sum of two arguments.

=item C<isl_ast_op_sub>

Difference of two arguments.

=item C<isl_ast_op_mul>

Product of two arguments.

=item C<isl_ast_op_div>

Exact division.  That is, the result is known to be an integer.

=item C<isl_ast_op_fdiv_q>

Result of integer division, rounded towards negative
infinity.

=item C<isl_ast_op_pdiv_q>

Result of integer division, where dividend is known to be non-negative.

=item C<isl_ast_op_pdiv_r>

Remainder of integer division, where dividend is known to be non-negative.

=item C<isl_ast_op_cond>

Conditional operator defined on three arguments.
If the first argument evaluates to true, then the result
is equal to the second argument.  Otherwise, the result
is equal to the third argument.
The second and third argument may only be evaluated if
the first argument evaluates to true and false, respectively.
Corresponds to C<a ? b : c> in C.

=item C<isl_ast_op_select>

Conditional operator defined on three arguments.
If the first argument evaluates to true, then the result
is equal to the second argument.  Otherwise, the result
is equal to the third argument.
The second and third argument may be evaluated independently
of the value of the first argument.
Corresponds to C<a * b + (1 - a) * c> in C.

=item C<isl_ast_op_eq>

Equality relation.

=item C<isl_ast_op_le>

Less than or equal relation.

=item C<isl_ast_op_lt>

Less than relation.

=item C<isl_ast_op_ge>

Greater than or equal relation.

=item C<isl_ast_op_gt>

Greater than relation.

=item C<isl_ast_op_call>

A function call.
The number of arguments of the C<isl_ast_expr> is one more than
the number of arguments in the function call, the first argument
representing the function being called.

=back

        #include <isl/ast.h>
        __isl_give isl_id *isl_ast_expr_get_id(
                __isl_keep isl_ast_expr *expr);

Return the identifier represented by the AST expression.

        #include <isl/ast.h>
        int isl_ast_expr_get_int(__isl_keep isl_ast_expr *expr,
                isl_int *v);

Return the integer represented by the AST expression.
Note that the integer is returned through the C<v> argument.
The return value of the function itself indicates whether the
operation was performed successfully.

=head3 Manipulating and printing the AST

AST nodes can be copied and freed using the following functions.

        #include <isl/ast.h>
        __isl_give isl_ast_node *isl_ast_node_copy(
                __isl_keep isl_ast_node *node);
        void *isl_ast_node_free(__isl_take isl_ast_node *node);

AST expressions can be copied and freed using the following functions.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_expr_copy(
                __isl_keep isl_ast_expr *expr);
        void *isl_ast_expr_free(__isl_take isl_ast_expr *expr);

New AST expressions can be created either directly or within
the context of an C<isl_ast_build>.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_expr_from_id(
                __isl_take isl_id *id);
        __isl_give isl_ast_expr *isl_ast_expr_neg(
                __isl_take isl_ast_expr *expr);
        __isl_give isl_ast_expr *isl_ast_expr_add(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_sub(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_mul(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_div(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_and(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2)
        __isl_give isl_ast_expr *isl_ast_expr_or(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2)

        #include <isl/ast_build.h>
        __isl_give isl_ast_expr *isl_ast_build_expr_from_pw_aff(
                __isl_keep isl_ast_build *build,
                __isl_take isl_pw_aff *pa);
        __isl_give isl_ast_expr *
        isl_ast_build_call_from_pw_multi_aff(
                __isl_keep isl_ast_build *build,
                __isl_take isl_pw_multi_aff *pma);

The domains of C<pa> and C<pma> should correspond
to the schedule space of C<build>.
The tuple id of C<pma> is used as the function being called.

User specified data can be attached to an C<isl_ast_node> and obtained
from the same C<isl_ast_node> using the following functions.

        #include <isl/ast.h>
        __isl_give isl_ast_node *isl_ast_node_set_annotation(
                __isl_take isl_ast_node *node,
                __isl_take isl_id *annotation);
        __isl_give isl_id *isl_ast_node_get_annotation(
                __isl_keep isl_ast_node *node);

Basic printing can be performed using the following functions.

        #include <isl/ast.h>
        __isl_give isl_printer *isl_printer_print_ast_expr(
                __isl_take isl_printer *p,
                __isl_keep isl_ast_expr *expr);
        __isl_give isl_printer *isl_printer_print_ast_node(
                __isl_take isl_printer *p,
                __isl_keep isl_ast_node *node);

More advanced printing can be performed using the following functions.

        #include <isl/ast.h>
        __isl_give isl_printer *isl_ast_op_type_print_macro(
                enum isl_ast_op_type type,
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_ast_node_print_macros(
                __isl_keep isl_ast_node *node,
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_ast_node_print(
                __isl_keep isl_ast_node *node,
                __isl_take isl_printer *p,
                __isl_take isl_ast_print_options *options);
        __isl_give isl_printer *isl_ast_node_for_print(
                __isl_keep isl_ast_node *node,
                __isl_take isl_printer *p,
                __isl_take isl_ast_print_options *options);
        __isl_give isl_printer *isl_ast_node_if_print(
                __isl_keep isl_ast_node *node,
                __isl_take isl_printer *p,
                __isl_take isl_ast_print_options *options);

While printing an C<isl_ast_node> in C<ISL_FORMAT_C>,
C<isl> may print out an AST that makes use of macros such
as C<floord>, C<min> and C<max>.
C<isl_ast_op_type_print_macro> prints out the macro
corresponding to a specific C<isl_ast_op_type>.
C<isl_ast_node_print_macros> scans the C<isl_ast_node>
for expressions where these macros would be used and prints
out the required macro definitions.
Essentially, C<isl_ast_node_print_macros> calls
C<isl_ast_node_foreach_ast_op_type> with C<isl_ast_op_type_print_macro>
as function argument.
C<isl_ast_node_print>, C<isl_ast_node_for_print> and
C<isl_ast_node_if_print> print an C<isl_ast_node>
in C<ISL_FORMAT_C>, but allow for some extra control
through an C<isl_ast_print_options> object.
This object can be created using the following functions.

        #include <isl/ast.h>
        __isl_give isl_ast_print_options *
        isl_ast_print_options_alloc(isl_ctx *ctx);
        __isl_give isl_ast_print_options *
        isl_ast_print_options_copy(
                __isl_keep isl_ast_print_options *options);
        void *isl_ast_print_options_free(
                __isl_take isl_ast_print_options *options);

        __isl_give isl_ast_print_options *
        isl_ast_print_options_set_print_user(
                __isl_take isl_ast_print_options *options,
                __isl_give isl_printer *(*print_user)(
                        __isl_take isl_printer *p,
                        __isl_take isl_ast_print_options *options,
                        __isl_keep isl_ast_node *node, void *user),
                void *user);
        __isl_give isl_ast_print_options *
        isl_ast_print_options_set_print_for(
                __isl_take isl_ast_print_options *options,
                __isl_give isl_printer *(*print_for)(
                        __isl_take isl_printer *p,
                        __isl_take isl_ast_print_options *options,
                        __isl_keep isl_ast_node *node, void *user),
                void *user);

The callback set by C<isl_ast_print_options_set_print_user>
is called whenever a node of type C<isl_ast_node_user> needs to
be printed.
The callback set by C<isl_ast_print_options_set_print_for>
is called whenever a node of type C<isl_ast_node_for> needs to
be printed.
Note that C<isl_ast_node_for_print> will I<not> call the
callback set by C<isl_ast_print_options_set_print_for> on the node
on which C<isl_ast_node_for_print> is called, but only on nested
nodes of type C<isl_ast_node_for>.  It is therefore safe to
call C<isl_ast_node_for_print> from within the callback set by
C<isl_ast_print_options_set_print_for>.

The following option determines the type to be used for iterators
while printing the AST.

        int isl_options_set_ast_iterator_type(
                isl_ctx *ctx, const char *val);
        const char *isl_options_get_ast_iterator_type(
                isl_ctx *ctx);

=head3 Options

        #include <isl/ast_build.h>
        int isl_options_set_ast_build_atomic_upper_bound(
                isl_ctx *ctx, int val);
        int isl_options_get_ast_build_atomic_upper_bound(
                isl_ctx *ctx);
        int isl_options_set_ast_build_prefer_pdiv(isl_ctx *ctx,
                int val);
        int isl_options_get_ast_build_prefer_pdiv(isl_ctx *ctx);
        int isl_options_set_ast_build_exploit_nested_bounds(
                isl_ctx *ctx, int val);
        int isl_options_get_ast_build_exploit_nested_bounds(
                isl_ctx *ctx);
        int isl_options_set_ast_build_group_coscheduled(
                isl_ctx *ctx, int val);
        int isl_options_get_ast_build_group_coscheduled(
                isl_ctx *ctx);
        int isl_options_set_ast_build_scale_strides(
                isl_ctx *ctx, int val);
        int isl_options_get_ast_build_scale_strides(
                isl_ctx *ctx);
        int isl_options_set_ast_build_allow_else(isl_ctx *ctx,
                int val);
        int isl_options_get_ast_build_allow_else(isl_ctx *ctx);
        int isl_options_set_ast_build_allow_or(isl_ctx *ctx,
                int val);
        int isl_options_get_ast_build_allow_or(isl_ctx *ctx);

=over

=item * ast_build_atomic_upper_bound

Generate loop upper bounds that consist of the current loop iterator,
an operator and an expression not involving the iterator.
If this option is not set, then the current loop iterator may appear
several times in the upper bound.
For example, when this option is turned off, AST generation
for the schedule

        [n] -> { A[i] -> [i] : 0 <= i <= 100, n }

produces

        for (int c0 = 0; c0 <= 100 && n >= c0; c0 += 1)
          A(c0);

When the option is turned on, the following AST is generated

        for (int c0 = 0; c0 <= min(100, n); c0 += 1)
          A(c0);

=item * ast_build_prefer_pdiv

If this option is turned off, then the AST generation will
produce ASTs that may only contain C<isl_ast_op_fdiv_q>
operators, but no C<isl_ast_op_pdiv_q> or
C<isl_ast_op_pdiv_r> operators.
If this options is turned on, then C<isl> will try to convert
some of the C<isl_ast_op_fdiv_q> operators to (expressions containing)
C<isl_ast_op_pdiv_q> or C<isl_ast_op_pdiv_r> operators.

=item * ast_build_exploit_nested_bounds

Simplify conditions based on bounds of nested for loops.
In particular, remove conditions that are implied by the fact
that one or more nested loops have at least one iteration,
meaning that the upper bound is at least as large as the lower bound.
For example, when this option is turned off, AST generation
for the schedule

        [N,M] -> { A[i,j] -> [i,j] : 0 <= i <= N and
                                        0 <= j <= M }

produces

        if (M >= 0)
          for (int c0 = 0; c0 <= N; c0 += 1)
            for (int c1 = 0; c1 <= M; c1 += 1)
              A(c0, c1);

When the option is turned on, the following AST is generated

        for (int c0 = 0; c0 <= N; c0 += 1)
          for (int c1 = 0; c1 <= M; c1 += 1)
            A(c0, c1);

=item * ast_build_group_coscheduled

If two domain elements are assigned the same schedule point, then
they may be executed in any order and they may even appear in different
loops.  If this options is set, then the AST generator will make
sure that coscheduled domain elements do not appear in separate parts
of the AST.  This is useful in case of nested AST generation
if the outer AST generation is given only part of a schedule
and the inner AST generation should handle the domains that are
coscheduled by this initial part of the schedule together.
For example if an AST is generated for a schedule

        { A[i] -> [0]; B[i] -> [0] }

then the C<isl_ast_build_set_create_leaf> callback described
below may get called twice, once for each domain.
Setting this option ensures that the callback is only called once
on both domains together.

=item * ast_build_separation_bounds

This option specifies which bounds to use during separation.
If this option is set to C<ISL_AST_BUILD_SEPARATION_BOUNDS_IMPLICIT>
then all (possibly implicit) bounds on the current dimension will
be used during separation.
If this option is set to C<ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT>
then only those bounds that are explicitly available will
be used during separation.

=item * ast_build_scale_strides

This option specifies whether the AST generator is allowed
to scale down iterators of strided loops.

=item * ast_build_allow_else

This option specifies whether the AST generator is allowed
to construct if statements with else branches.

=item * ast_build_allow_or

This option specifies whether the AST generator is allowed
to construct if conditions with disjunctions.

=back

=head3 Fine-grained Control over AST Generation

Besides specifying the constraints on the parameters,
an C<isl_ast_build> object can be used to control
various aspects of the AST generation process.
The most prominent way of control is through ``options'',
which can be set using the following function.

        #include <isl/ast_build.h>
        __isl_give isl_ast_build *
        isl_ast_build_set_options(
                __isl_take isl_ast_build *control,
                __isl_take isl_union_map *options);

The options are encoded in an <isl_union_map>.
The domain of this union relation refers to the schedule domain,
i.e., the range of the schedule passed to C<isl_ast_build_ast_from_schedule>.
In the case of nested AST generation (see L</"Nested AST Generation">),
the domain of C<options> should refer to the extra piece of the schedule.
That is, it should be equal to the range of the wrapped relation in the
range of the schedule.
The range of the options can consist of elements in one or more spaces,
the names of which determine the effect of the option.
The values of the range typically also refer to the schedule dimension
to which the option applies.  In case of nested AST generation
(see L</"Nested AST Generation">), these values refer to the position
of the schedule dimension within the innermost AST generation.
The constraints on the domain elements of
the option should only refer to this dimension and earlier dimensions.
We consider the following spaces.

=over

=item C<separation_class>

This space is a wrapped relation between two one dimensional spaces.
The input space represents the schedule dimension to which the option
applies and the output space represents the separation class.
While constructing a loop corresponding to the specified schedule
dimension(s), the AST generator will try to generate separate loops
for domain elements that are assigned different classes.
If only some of the elements are assigned a class, then those elements
that are not assigned any class will be treated as belonging to a class
that is separate from the explicitly assigned classes.
The typical use case for this option is to separate full tiles from
partial tiles.
The other options, described below, are applied after the separation
into classes.

As an example, consider the separation into full and partial tiles
of a tiling of a triangular domain.
Take, for example, the domain

        { A[i,j] : 0 <= i,j and i + j <= 100 }

and a tiling into tiles of 10 by 10.  The input to the AST generator
is then the schedule

        { A[i,j] -> [([i/10]),[j/10],i,j] : 0 <= i,j and
                                                i + j <= 100 }

Without any options, the following AST is generated

        for (int c0 = 0; c0 <= 10; c0 += 1)
          for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
            for (int c2 = 10 * c0;
                 c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
                 c2 += 1)
              for (int c3 = 10 * c1;
                   c3 <= min(10 * c1 + 9, -c2 + 100);
                   c3 += 1)
                A(c2, c3);

Separation into full and partial tiles can be obtained by assigning
a class, say C<0>, to the full tiles.  The full tiles are represented by those
values of the first and second schedule dimensions for which there are
values of the third and fourth dimensions to cover an entire tile.
That is, we need to specify the following option

        { [a,b,c,d] -> separation_class[[0]->[0]] :
                exists b': 0 <= 10a,10b' and
                           10a+9+10b'+9 <= 100;
          [a,b,c,d] -> separation_class[[1]->[0]] :
                0 <= 10a,10b and 10a+9+10b+9 <= 100 }

which simplifies to

        { [a, b, c, d] -> separation_class[[1] -> [0]] :
                a >= 0 and b >= 0 and b <= 8 - a;
          [a, b, c, d] -> separation_class[[0] -> [0]] :
                a >= 0 and a <= 8 }

With this option, the generated AST is as follows

        {
          for (int c0 = 0; c0 <= 8; c0 += 1) {
            for (int c1 = 0; c1 <= -c0 + 8; c1 += 1)
              for (int c2 = 10 * c0;
                   c2 <= 10 * c0 + 9; c2 += 1)
                for (int c3 = 10 * c1;
                     c3 <= 10 * c1 + 9; c3 += 1)
                  A(c2, c3);
            for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
              for (int c2 = 10 * c0;
                   c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
                   c2 += 1)
                for (int c3 = 10 * c1;
                     c3 <= min(-c2 + 100, 10 * c1 + 9);
                     c3 += 1)
                  A(c2, c3);
          }
          for (int c0 = 9; c0 <= 10; c0 += 1)
            for (int c1 = 0; c1 <= -c0 + 10; c1 += 1)
              for (int c2 = 10 * c0;
                   c2 <= min(-10 * c1 + 100, 10 * c0 + 9);
                   c2 += 1)
                for (int c3 = 10 * c1;
                     c3 <= min(10 * c1 + 9, -c2 + 100);
                     c3 += 1)
                  A(c2, c3);
        }

=item C<separate>

This is a single-dimensional space representing the schedule dimension(s)
to which ``separation'' should be applied.  Separation tries to split
a loop into several pieces if this can avoid the generation of guards
inside the loop.
See also the C<atomic> option.

=item C<atomic>

This is a single-dimensional space representing the schedule dimension(s)
for which the domains should be considered ``atomic''.  That is, the
AST generator will make sure that any given domain space will only appear
in a single loop at the specified level.

Consider the following schedule

        { a[i] -> [i] : 0 <= i < 10;
          b[i] -> [i+1] : 0 <= i < 10 }

If the following option is specified

        { [i] -> separate[x] }

then the following AST will be generated

        {
          a(0);
          for (int c0 = 1; c0 <= 9; c0 += 1) {
            a(c0);
            b(c0 - 1);
          }
          b(9);
        }

If, on the other hand, the following option is specified

        { [i] -> atomic[x] }

then the following AST will be generated

        for (int c0 = 0; c0 <= 10; c0 += 1) {
          if (c0 <= 9)
            a(c0);
          if (c0 >= 1)
            b(c0 - 1);
        }

If neither C<atomic> nor C<separate> is specified, then the AST generator
may produce either of these two results or some intermediate form.

=item C<unroll>

This is a single-dimensional space representing the schedule dimension(s)
that should be I<completely> unrolled.
To obtain a partial unrolling, the user should apply an additional
strip-mining to the schedule and fully unroll the inner loop.

=back

Additional control is available through the following functions.

        #include <isl/ast_build.h>
        __isl_give isl_ast_build *
        isl_ast_build_set_iterators(
                __isl_take isl_ast_build *control,
                __isl_take isl_id_list *iterators);

The function C<isl_ast_build_set_iterators> allows the user to
specify a list of iterator C<isl_id>s to be used as iterators.
If the input schedule is injective, then
the number of elements in this list should be as large as the dimension
of the schedule space, but no direct correspondence should be assumed
between dimensions and elements.
If the input schedule is not injective, then an additional number
of C<isl_id>s equal to the largest dimension of the input domains
may be required.
If the number of provided C<isl_id>s is insufficient, then additional
names are automatically generated.

        #include <isl/ast_build.h>
        __isl_give isl_ast_build *
        isl_ast_build_set_create_leaf(
                __isl_take isl_ast_build *control,
                __isl_give isl_ast_node *(*fn)(
                        __isl_take isl_ast_build *build,
                        void *user), void *user);

The
C<isl_ast_build_set_create_leaf> function allows for the
specification of a callback that should be called whenever the AST
generator arrives at an element of the schedule domain.
The callback should return an AST node that should be inserted
at the corresponding position of the AST.  The default action (when
the callback is not set) is to continue generating parts of the AST to scan
all the domain elements associated to the schedule domain element
and to insert user nodes, ``calling'' the domain element, for each of them.
The C<build> argument contains the current state of the C<isl_ast_build>.
To ease nested AST generation (see L</"Nested AST Generation">),
all control information that is
specific to the current AST generation such as the options and
the callbacks has been removed from this C<isl_ast_build>.
The callback would typically return the result of a nested
AST generation or a
user defined node created using the following function.

        #include <isl/ast.h>
        __isl_give isl_ast_node *isl_ast_node_alloc_user(
                __isl_take isl_ast_expr *expr);

        #include <isl/ast_build.h>
        __isl_give isl_ast_build *
        isl_ast_build_set_at_each_domain(
                __isl_take isl_ast_build *build,
                __isl_give isl_ast_node *(*fn)(
                        __isl_take isl_ast_node *node,
                        __isl_keep isl_ast_build *build,
                        void *user), void *user);
        __isl_give isl_ast_build *
        isl_ast_build_set_before_each_for(
                __isl_take isl_ast_build *build,
                __isl_give isl_id *(*fn)(
                        __isl_keep isl_ast_build *build,
                        void *user), void *user);
        __isl_give isl_ast_build *
        isl_ast_build_set_after_each_for(
                __isl_take isl_ast_build *build,
                __isl_give isl_ast_node *(*fn)(
                        __isl_take isl_ast_node *node,
                        __isl_keep isl_ast_build *build,
                        void *user), void *user);

The callback set by C<isl_ast_build_set_at_each_domain> will
be called for each domain AST node.
The callbacks set by C<isl_ast_build_set_before_each_for>
and C<isl_ast_build_set_after_each_for> will be called
for each for AST node.  The first will be called in depth-first
pre-order, while the second will be called in depth-first post-order.
Since C<isl_ast_build_set_before_each_for> is called before the for
node is actually constructed, it is only passed an C<isl_ast_build>.
The returned C<isl_id> will be added as an annotation (using
C<isl_ast_node_set_annotation>) to the constructed for node.
In particular, if the user has also specified an C<after_each_for>
callback, then the annotation can be retrieved from the node passed to
that callback using C<isl_ast_node_get_annotation>.
All callbacks should C<NULL> on failure.
The given C<isl_ast_build> can be used to create new
C<isl_ast_expr> objects using C<isl_ast_build_expr_from_pw_aff>
or C<isl_ast_build_call_from_pw_multi_aff>.

=head3 Nested AST Generation

C<isl> allows the user to create an AST within the context
of another AST.  These nested ASTs are created using the
same C<isl_ast_build_ast_from_schedule> function that is used to create the
outer AST.  The C<build> argument should be an C<isl_ast_build>
passed to a callback set by
C<isl_ast_build_set_create_leaf>.
The space of the range of the C<schedule> argument should refer
to this build.  In particular, the space should be a wrapped
relation and the domain of this wrapped relation should be the
same as that of the range of the schedule returned by
C<isl_ast_build_get_schedule> below.
In practice, the new schedule is typically
created by calling C<isl_union_map_range_product> on the old schedule
and some extra piece of the schedule.
The space of the schedule domain is also available from
the C<isl_ast_build>.

        #include <isl/ast_build.h>
        __isl_give isl_union_map *isl_ast_build_get_schedule(
                __isl_keep isl_ast_build *build);
        __isl_give isl_space *isl_ast_build_get_schedule_space(
                __isl_keep isl_ast_build *build);
        __isl_give isl_ast_build *isl_ast_build_restrict(
                __isl_take isl_ast_build *build,
                __isl_take isl_set *set);

The C<isl_ast_build_get_schedule> function returns a (partial)
schedule for the domains elements for which part of the AST still needs to
be generated in the current build.
In particular, the domain elements are mapped to those iterations of the loops
enclosing the current point of the AST generation inside which
the domain elements are executed.
No direct correspondence between
the input schedule and this schedule should be assumed.
The space obtained from C<isl_ast_build_get_schedule_space> can be used
to create a set for C<isl_ast_build_restrict> to intersect
with the current build.  In particular, the set passed to
C<isl_ast_build_restrict> can have additional parameters.
The ids of the set dimensions in the space returned by
C<isl_ast_build_get_schedule_space> correspond to the
iterators of the already generated loops.
The user should not rely on the ids of the output dimensions
of the relations in the union relation returned by
C<isl_ast_build_get_schedule> having any particular value.

=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.

=head2 C<isl_codegen>

Given a schedule, a context set and an options relation,
C<isl_codegen> prints out an AST that scans the domain elements
of the schedule in the order of their image(s) taking into account
the constraints in the context set.