add isl_schedule_node_get_prefix_schedule_relation

[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> or C<imath>.
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 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

=head3 Changes since isl-0.12

=over

=item * C<isl_int> has been replaced by C<isl_val>.
Some of the old functions are still available in C<isl/deprecated/*.h>
but they will be removed in the future.

=item * The functions C<isl_pw_qpolynomial_eval>,
C<isl_union_pw_qpolynomial_eval>, C<isl_pw_qpolynomial_fold_eval>
and C<isl_union_pw_qpolynomial_fold_eval> have been changed to return
an C<isl_val> instead of an C<isl_qpolynomial>.

=item * The function C<isl_band_member_is_zero_distance>
has been removed.  Essentially the same functionality is available
through C<isl_band_member_is_coincident>, except that is requires
setting up coincidence constraints.
The option C<schedule_outer_zero_distance> has accordingly been
replaced by the option C<schedule_outer_coincidence>.

=item * The function C<isl_vertex_get_expr> has been changed
to return an C<isl_multi_aff> instead of a rational C<isl_basic_set>.
The function C<isl_vertex_get_domain> has been changed to return
a regular basic set, rather than a rational basic set.

=back

=head3 Changes since isl-0.14

=over

=item * Objects of type C<isl_union_pw_multi_aff> can no longer contain
two or more C<isl_pw_multi_aff> objects with the same domain space.

=item * The function C<isl_union_pw_multi_aff_add> now consistently
computes the sum on the shared definition domain.
The function C<isl_union_pw_multi_aff_union_add> has been added
to compute the sum on the union of definition domains.
The original behavior of C<isl_union_pw_multi_aff_add> was
confused and is no longer available.

=item * Band forests have been replaced by schedule trees.

=item * The function C<isl_union_map_compute_flow> has been
replaced by the function C<isl_union_access_info_compute_flow>.
Note that the may dependence relation returned by
C<isl_union_flow_get_may_dependence> is the union of
the two dependence relations returned by
C<isl_union_map_compute_flow>.  Similarly for the no source relations.
The function C<isl_union_map_compute_flow> is still available
for backward compatibility, but it will be removed in the future.

=item * The function C<isl_basic_set_drop_constraint> has been
deprecated.

=item * The function C<isl_ast_build_ast_from_schedule> has been
renamed to C<isl_ast_build_node_from_schedule_map>.
The original name is still available
for backward compatibility, but it will be removed in the future.

=item * The C<separation_class> AST generation option has been
deprecated.

=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 by default C<isl> 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.

When configuring with C<--with-int=imath>, C<isl> will link against C<imath>, a
library for exact integer arithmetic released under the MIT license.

=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 Optionally get C<imath> submodule

To build C<isl> with C<imath>, you need to obtain the C<imath>
submodule by running in the git source tree of C<isl>

       git submodule init
       git submodule update

This will fetch the required version of C<imath> in a subdirectory of C<isl>.

=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>

By default, 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/>.
C<GMP> is not needed if you build C<isl> with C<imath>.

=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.

=over

=item C<--prefix>

Installation prefix for C<isl>

=item C<--with-int=[gmp|imath]>

Select the integer library to be used by C<isl>, the default is C<gmp>.
Note that C<isl> may run significantly slower if you use C<imath>.

=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).

=back

=item 3 Compile

        make

=item 4 Install (optional)

        make install

=back

=head1 Integer Set Library

=head2 Memory Management

Since a high-level operation on isl objects 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.
When applied to a C<char *>, the returned pointer needs to be
freed using C<free>.

=item C<__isl_null>

C<__isl_null> means that a C<NULL> value is returned.

=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.
This annotation may also be used on return values of
type C<const char *>, in which case the returned pointer should
not be freed by the user and is only valid until the object
from which it was derived is updated or freed.

=back

=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);

The user can impose a bound on the number of low-level I<operations>
that can be performed by an C<isl_ctx>.  This bound can be set and
retrieved using the following functions.  A bound of zero means that
no bound is imposed.  The number of operations performed can be
reset using C<isl_ctx_reset_operations>.  Note that the number
of low-level operations needed to perform a high-level computation
may differ significantly across different versions
of C<isl>, but it should be the same across different platforms
for the same version of C<isl>.

Warning: This feature is experimental.  C<isl> has good support to abort and
bail out during the computation, but this feature may exercise error code paths
that are normally not used that much. Consequently, it is not unlikely that
hidden bugs will be exposed.

        void isl_ctx_set_max_operations(isl_ctx *ctx,
                unsigned long max_operations);
        unsigned long isl_ctx_get_max_operations(isl_ctx *ctx);
        void isl_ctx_reset_operations(isl_ctx *ctx);

In order to be able to create an object in the same context
as another object, most object types (described later in
this document) provide a function to obtain the context
in which the object was created.

        #include <isl/val.h>
        isl_ctx *isl_val_get_ctx(__isl_keep isl_val *val);
        isl_ctx *isl_multi_val_get_ctx(
                __isl_keep isl_multi_val *mv);

        #include <isl/id.h>
        isl_ctx *isl_id_get_ctx(__isl_keep isl_id *id);

        #include <isl/local_space.h>
        isl_ctx *isl_local_space_get_ctx(
                __isl_keep isl_local_space *ls);

        #include <isl/set.h>
        isl_ctx *isl_set_list_get_ctx(
                __isl_keep isl_set_list *list);

        #include <isl/aff.h>
        isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
        isl_ctx *isl_multi_aff_get_ctx(
                __isl_keep isl_multi_aff *maff);
        isl_ctx *isl_pw_aff_get_ctx(__isl_keep isl_pw_aff *pa);
        isl_ctx *isl_pw_multi_aff_get_ctx(
                __isl_keep isl_pw_multi_aff *pma);
        isl_ctx *isl_multi_pw_aff_get_ctx(
                __isl_keep isl_multi_pw_aff *mpa);
        isl_ctx *isl_union_pw_aff_get_ctx(
                __isl_keep isl_union_pw_aff *upa);
        isl_ctx *isl_union_pw_multi_aff_get_ctx(
                __isl_keep isl_union_pw_multi_aff *upma);
        isl_ctx *isl_multi_union_pw_aff_get_ctx(
                __isl_keep isl_multi_union_pw_aff *mupa);

        #include <isl/id_to_ast_expr.h>
        isl_ctx *isl_id_to_ast_expr_get_ctx(
                __isl_keep isl_id_to_ast_expr *id2expr);

        #include <isl/point.h>
        isl_ctx *isl_point_get_ctx(__isl_keep isl_point *pnt);

        #include <isl/vec.h>
        isl_ctx *isl_vec_get_ctx(__isl_keep isl_vec *vec);

        #include <isl/mat.h>
        isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);

        #include <isl/vertices.h>
        isl_ctx *isl_vertices_get_ctx(
                __isl_keep isl_vertices *vertices);
        isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
        isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);

        #include <isl/flow.h>
        isl_ctx *isl_restriction_get_ctx(
                __isl_keep isl_restriction *restr);

        #include <isl/schedule.h>
        isl_ctx *isl_schedule_get_ctx(
                __isl_keep isl_schedule *sched);
        isl_ctx *isl_schedule_constraints_get_ctx(
                __isl_keep isl_schedule_constraints *sc);

        #include <isl/schedule_node.h>
        isl_ctx *isl_schedule_node_get_ctx(
                __isl_keep isl_schedule_node *node);

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

        #include <isl/ast_build.h>
        isl_ctx *isl_ast_build_get_ctx(
                __isl_keep isl_ast_build *build);

        #include <isl/ast.h>
        isl_ctx *isl_ast_expr_get_ctx(
                __isl_keep isl_ast_expr *expr);
        isl_ctx *isl_ast_node_get_ctx(
                __isl_keep isl_ast_node *node);

=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_negone(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);
        __isl_give isl_val *isl_val_int_from_chunks(isl_ctx *ctx,
                size_t n, size_t size, const void *chunks);

The function C<isl_val_int_from_chunks> constructs an C<isl_val>
from the C<n> I<digits>, each consisting of C<size> bytes, stored at C<chunks>.
The least significant digit is assumed to be stored first.

Value objects 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);
        __isl_null isl_val *isl_val_free(__isl_take isl_val *v);

They can be inspected using the following functions.

        #include <isl/val.h>
        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);
        size_t isl_val_n_abs_num_chunks(__isl_keep isl_val *v,
                size_t size);
        int isl_val_get_abs_num_chunks(__isl_keep isl_val *v,
                size_t size, void *chunks);

C<isl_val_n_abs_num_chunks> returns the number of I<digits>
of C<size> bytes needed to store the absolute value of the
numerator of C<v>.
C<isl_val_get_abs_num_chunks> stores these digits at C<chunks>,
which is assumed to have been preallocated by the caller.
The least significant digit is stored first.
Note that C<isl_val_get_num_si>, C<isl_val_get_den_si>,
C<isl_val_get_d>, C<isl_val_n_abs_num_chunks>
and C<isl_val_get_abs_num_chunks> 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);
        int isl_val_abs_eq(__isl_keep isl_val *v1,
                __isl_keep isl_val *v2);

The function C<isl_val_abs_eq> checks whether its two arguments
are equal in absolute value.

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);
        __isl_give isl_val *isl_val_inv(__isl_take isl_val *v);
        __isl_give isl_val *isl_val_2exp(__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_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.

=head3 GMP specific functions

These functions are only available if C<isl> has been compiled with C<GMP>
support.

Specific integer and rational values can be created from C<GMP> values using
the following functions.

        #include <isl/val_gmp.h>
        __isl_give isl_val *isl_val_int_from_gmp(isl_ctx *ctx,
                mpz_t z);
        __isl_give isl_val *isl_val_from_gmp(isl_ctx *ctx,
                const mpz_t n, const mpz_t d);

The numerator and denominator of a rational value can be extracted as
C<GMP> values using the following functions.

        #include <isl/val_gmp.h>
        int isl_val_get_num_gmp(__isl_keep isl_val *v, mpz_t z);
        int isl_val_get_den_gmp(__isl_keep isl_val *v, mpz_t z);

=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 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);
        __isl_null isl_id *isl_id_free(__isl_take 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 similar 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);
        __isl_null isl_space *isl_space_free(__isl_take isl_space *space);

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>.

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);
        int isl_space_tuple_is_equal(
                __isl_keep isl_space *space1,
                enum isl_dim_type type1,
                __isl_keep isl_space *space2,
                enum isl_dim_type type2);

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.  C<isl_space_tuple_is_equal> checks whether the given
tuples (C<isl_dim_in>, C<isl_dim_out> or C<isl_dim_set>) of the given
spaces are the same.  That is, it checks if they have the same
identifier (if any), the same dimension and the same internal structure
(if any).

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</"Functions">) 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/val.h>
        __isl_give isl_space *isl_multi_val_get_space(
                __isl_keep isl_multi_val *mv);

        #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_aff_get_space(
                __isl_keep isl_union_pw_aff *upa);
        __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);
        __isl_give isl_space *
        isl_multi_union_pw_aff_get_domain_space(
                __isl_keep isl_multi_union_pw_aff *mupa);
        __isl_give isl_space *
        isl_multi_union_pw_aff_get_space(
                __isl_keep isl_multi_union_pw_aff *mupa);

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

The number of dimensions of a given type of space
may be read off from a space or an object that lives
in a space using the following functions.
In case of C<isl_space_dim>, 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>.

        #include <isl/space.h>
        unsigned isl_space_dim(__isl_keep isl_space *space,
                enum isl_dim_type type);

        #include <isl/local_space.h>
        int isl_local_space_dim(__isl_keep isl_local_space *ls,
                enum isl_dim_type type);

        #include <isl/set.h>
        unsigned isl_basic_set_dim(__isl_keep isl_basic_set *bset,
                enum isl_dim_type type);
        unsigned isl_set_dim(__isl_keep isl_set *set,
                enum isl_dim_type type);

        #include <isl/union_set.h>
        unsigned isl_union_set_dim(__isl_keep isl_union_set *uset,
                enum isl_dim_type type);

        #include <isl/map.h>
        unsigned isl_basic_map_dim(__isl_keep isl_basic_map *bmap,
                enum isl_dim_type type);
        unsigned isl_map_dim(__isl_keep isl_map *map,
                enum isl_dim_type type);

        #include <isl/union_map.h>
        unsigned isl_union_map_dim(__isl_keep isl_union_map *umap,
                enum isl_dim_type type);

        #include <isl/val.h>
        unsigned isl_multi_val_dim(__isl_keep isl_multi_val *mv,
                enum isl_dim_type type);

        #include <isl/aff.h>
        int isl_aff_dim(__isl_keep isl_aff *aff,
                enum isl_dim_type type);
        unsigned isl_multi_aff_dim(__isl_keep isl_multi_aff *maff,
                enum isl_dim_type type);
        unsigned isl_pw_aff_dim(__isl_keep isl_pw_aff *pwaff,
                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);
        unsigned isl_union_pw_aff_dim(
                __isl_keep isl_union_pw_aff *upa,
                enum isl_dim_type type);
        unsigned isl_union_pw_multi_aff_dim(
                __isl_keep isl_union_pw_multi_aff *upma,
                enum isl_dim_type type);
        unsigned isl_multi_union_pw_aff_dim(
                __isl_keep isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type);

        #include <isl/polynomial.h>
        unsigned isl_union_pw_qpolynomial_dim(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                enum isl_dim_type type);
        unsigned isl_union_pw_qpolynomial_fold_dim(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf,
                enum isl_dim_type type);

Note that an C<isl_union_set>, an C<isl_union_map>,
an C<isl_union_pw_multi_aff>,
an C<isl_union_pw_qpolynomial> and
an C<isl_union_pw_qpolynomial_fold>
only have parameters.

The identifiers or names of the individual dimensions of spaces
may be set or read off using the following functions on spaces
or objects that live in spaces.
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.

        #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);

        #include <isl/local_space.h>
        __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);
        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);
        __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);
        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);

        #include <isl/constraint.h>
        const char *isl_constraint_get_dim_name(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned pos);

        #include <isl/set.h>
        __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);
        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);

        #include <isl/map.h>
        __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_basic_map_has_dim_id(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos);
        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);
        __isl_give isl_id *isl_union_map_get_dim_id(
                __isl_keep isl_union_map *umap,
                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);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_set_dim_id(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        __isl_give isl_id *isl_multi_val_get_dim_id(
                __isl_keep isl_multi_val *mv,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_multi_val *isl_multi_val_set_dim_name(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, unsigned pos, const char *s);

        #include <isl/aff.h>
        __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_multi_aff *isl_multi_aff_set_dim_id(
                __isl_take isl_multi_aff *maff,
                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_multi_pw_aff *
        isl_multi_pw_aff_set_dim_id(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_set_dim_id(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_id *id);
        __isl_give isl_id *isl_multi_aff_get_dim_id(
                __isl_keep isl_multi_aff *ma,
                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_multi_aff_get_dim_id(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_multi_pw_aff_get_dim_id(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_id *isl_multi_union_pw_aff_get_dim_id(
                __isl_keep isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, unsigned pos);
        __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_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_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_union_pw_aff *
        isl_union_pw_aff_set_dim_name(
                __isl_take isl_union_pw_aff *upa,
                enum isl_dim_type type, unsigned pos,
                const char *s);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_set_dim_name(
                __isl_take isl_union_pw_multi_aff *upma,
                enum isl_dim_type type, unsigned pos,
                const char *s);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_set_dim_name(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, unsigned pos,
        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);
        const char *isl_pw_multi_aff_get_dim_name(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned pos);

        #include <isl/polynomial.h>
        __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);
        __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);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_set_dim_name(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                enum isl_dim_type type, unsigned pos,
                const char *s);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_set_dim_name(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                enum isl_dim_type type, unsigned pos,
                const char *s);

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 functions.

        #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);

        #include <isl/local_space.h>
        int isl_local_space_find_dim_by_name(
                __isl_keep isl_local_space *ls,
                enum isl_dim_type type, const char *name);

        #include <isl/val.h>
        int isl_multi_val_find_dim_by_id(
                __isl_keep isl_multi_val *mv,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_multi_val_find_dim_by_name(
                __isl_keep isl_multi_val *mv,
                enum isl_dim_type type, const char *name);

        #include <isl/set.h>
        int isl_set_find_dim_by_id(__isl_keep isl_set *set,
                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);

        #include <isl/map.h>
        int isl_map_find_dim_by_id(__isl_keep isl_map *map,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_basic_map_find_dim_by_name(
                __isl_keep isl_basic_map *bmap,
                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);
        int isl_union_map_find_dim_by_name(
                __isl_keep isl_union_map *umap,
                enum isl_dim_type type, const char *name);

        #include <isl/aff.h>
        int isl_multi_aff_find_dim_by_id(
                __isl_keep isl_multi_aff *ma,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_multi_pw_aff_find_dim_by_id(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_multi_union_pw_aff_find_dim_by_id(
                __isl_keep isl_union_multi_pw_aff *mupa,
                enum isl_dim_type type, __isl_keep isl_id *id);
        int isl_aff_find_dim_by_name(__isl_keep isl_aff *aff,
                enum isl_dim_type type, const char *name);
        int isl_multi_aff_find_dim_by_name(
                __isl_keep isl_multi_aff *ma,
                enum isl_dim_type type, const char *name);
        int isl_pw_aff_find_dim_by_name(__isl_keep isl_pw_aff *pa,
                enum isl_dim_type type, const char *name);
        int isl_multi_pw_aff_find_dim_by_name(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type, const char *name);
        int isl_pw_multi_aff_find_dim_by_name(
                __isl_keep isl_pw_multi_aff *pma,
                enum isl_dim_type type, const char *name);
        int isl_union_pw_aff_find_dim_by_name(
                __isl_keep isl_union_pw_aff *upa,
                enum isl_dim_type type, const char *name);
        int isl_union_pw_multi_aff_find_dim_by_name(
                __isl_keep isl_union_pw_multi_aff *upma,
                enum isl_dim_type type, const char *name);
        int isl_multi_union_pw_aff_find_dim_by_name(
                __isl_keep isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, const char *name);

        #include <isl/polynomial.h>
        int isl_pw_qpolynomial_find_dim_by_name(
                __isl_keep isl_pw_qpolynomial *pwqp,
                enum isl_dim_type type, const char *name);
        int isl_pw_qpolynomial_fold_find_dim_by_name(
                __isl_keep isl_pw_qpolynomial_fold *pwf,
                enum isl_dim_type type, const char *name);
        int isl_union_pw_qpolynomial_find_dim_by_name(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                enum isl_dim_type type, const char *name);
        int isl_union_pw_qpolynomial_fold_find_dim_by_name(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf,
                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);

        #include <isl/local_space.h>
        __isl_give isl_local_space *isl_local_space_set_tuple_id(
                __isl_take isl_local_space *ls,
                enum isl_dim_type type, __isl_take isl_id *id);

        #include <isl/set.h>
        __isl_give isl_basic_set *isl_basic_set_set_tuple_id(
                __isl_take isl_basic_set *bset,
                __isl_take isl_id *id);
        __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_basic_set *isl_basic_set_set_tuple_name(
                __isl_take isl_basic_set *set, const char *s);
        __isl_give isl_set *isl_set_set_tuple_name(
                __isl_take isl_set *set, const char *s);
        const char *isl_basic_set_get_tuple_name(
                __isl_keep isl_basic_set *bset);
        int isl_set_has_tuple_name(__isl_keep isl_set *set);
        const char *isl_set_get_tuple_name(
                __isl_keep isl_set *set);

        #include <isl/map.h>
        __isl_give isl_basic_map *isl_basic_map_set_tuple_id(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, __isl_take isl_id *id);
        __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);
        __isl_give isl_map *isl_map_set_tuple_name(
                __isl_take isl_map *map,
                enum isl_dim_type type, const char *s);
        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);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_set_tuple_id(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, __isl_take isl_id *id);
        __isl_give isl_multi_val *isl_multi_val_reset_tuple_id(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type);
        int isl_multi_val_has_tuple_id(__isl_keep isl_multi_val *mv,
                enum isl_dim_type type);
        __isl_give isl_id *isl_multi_val_get_tuple_id(
                __isl_keep isl_multi_val *mv,
                enum isl_dim_type type);
        __isl_give isl_multi_val *isl_multi_val_set_tuple_name(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, const char *s);
        const char *isl_multi_val_get_tuple_name(
                __isl_keep isl_multi_val *mv,
                enum isl_dim_type type);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_set_tuple_id(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, __isl_take isl_id *id);
        __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_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_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_union_pw_aff *
        isl_multi_union_pw_aff_set_tuple_id(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, __isl_take isl_id *id);
        __isl_give isl_multi_aff *isl_multi_aff_reset_tuple_id(
                __isl_take isl_multi_aff *ma,
                enum isl_dim_type type);
        __isl_give isl_pw_aff *isl_pw_aff_reset_tuple_id(
                __isl_take isl_pw_aff *pa,
                enum isl_dim_type type);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_reset_tuple_id(
                __isl_take isl_multi_pw_aff *mpa,
                enum isl_dim_type type);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_reset_tuple_id(
                __isl_take isl_pw_multi_aff *pma,
                enum isl_dim_type type);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_reset_tuple_id(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type);
        int isl_multi_aff_has_tuple_id(__isl_keep isl_multi_aff *ma,
                enum isl_dim_type type);
        __isl_give isl_id *isl_multi_aff_get_tuple_id(
                __isl_keep isl_multi_aff *ma,
                enum isl_dim_type type);
        int isl_pw_aff_has_tuple_id(__isl_keep isl_pw_aff *pa,
                enum isl_dim_type type);
        __isl_give isl_id *isl_pw_aff_get_tuple_id(
                __isl_keep isl_pw_aff *pa,
                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_multi_pw_aff_has_tuple_id(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type);
        __isl_give isl_id *isl_multi_pw_aff_get_tuple_id(
                __isl_keep isl_multi_pw_aff *mpa,
                enum isl_dim_type type);
        int isl_multi_union_pw_aff_has_tuple_id(
                __isl_keep isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type);
        __isl_give isl_id *isl_multi_union_pw_aff_get_tuple_id(
                __isl_keep isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type);
        __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_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_union_pw_aff *
        isl_multi_union_pw_aff_set_tuple_name(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, const char *s);
        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);
        const char *isl_multi_union_pw_aff_get_tuple_name(
                __isl_keep isl_multi_union_pw_aff *mupa,
                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.

To keep the names of all parameters and tuples, but reset the user pointers
of all the corresponding identifiers, use the following function.

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

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

        #include <isl/map.h>
        __isl_give isl_map *isl_map_reset_user(
                __isl_take isl_map *map);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_reset_user(
                __isl_take isl_union_set *uset);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_reset_user(
                __isl_take isl_union_map *umap);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_reset_user(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_reset_user(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_reset_user(
                __isl_take isl_pw_aff *pa);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_reset_user(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_reset_user(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_reset_user(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_reset_user(
                __isl_take isl_multi_union_pw_aff *mupa);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_reset_user(
                __isl_take isl_union_pw_multi_aff *upma);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_reset_user(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_reset_user(
                __isl_take isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_reset_user(
                __isl_take isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_reset_user(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);

Spaces can be nested.  In particular, the domain of a set or
the domain or range of a relation can be a nested relation.
This process is also called I<wrapping>.
The functions for detecting, constructing and deconstructing
such nested spaces can be found in the wrapping properties
of L</"Unary Properties">, the wrapping operations
of L</"Unary Operations"> and the Cartesian product operations
of L</"Basic Operations">.

Spaces can be created from other spaces
using the functions described in L</"Unary Operations">
and L</"Binary Operations">.

=head2 Local Spaces

A local space is essentially a space with
zero or more existentially quantified variables.
The local space of various objects 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);

        #include <isl/aff.h>
        __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);

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>
        int isl_local_space_is_params(
                __isl_keep isl_local_space *ls);
        int isl_local_space_is_set(__isl_keep isl_local_space *ls);
        __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);
        __isl_null isl_local_space *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 functions described in L</"Unary Operations">
and L</"Binary Operations">.

=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);

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);
        __isl_null isl_basic_set *isl_basic_set_free(
                __isl_take isl_basic_set *bset);
        __isl_null isl_set *isl_set_free(__isl_take isl_set *set);
        __isl_null isl_union_set *isl_union_set_free(
                __isl_take isl_union_set *uset);
        __isl_null isl_basic_map *isl_basic_map_free(
                __isl_take isl_basic_map *bmap);
        __isl_null isl_map *isl_map_free(__isl_take isl_map *map);
        __isl_null isl_union_map *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_si(
                __isl_take isl_constraint *constraint, int v);
        __isl_give isl_constraint *isl_constraint_set_constant_val(
                __isl_take isl_constraint *constraint,
                __isl_take isl_val *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_constraint *
        isl_constraint_set_coefficient_val(
                __isl_take isl_constraint *constraint,
                enum isl_dim_type type, int pos,
                __isl_take isl_val *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);

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</"Functions">).

        __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_set *isl_set_from_multi_pw_aff(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_map *isl_map_from_multi_pw_aff(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_union_map *isl_union_map_from_union_pw_aff(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_map *
        isl_union_map_from_union_pw_multi_aff(
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_union_map *
        isl_union_map_from_multi_union_pw_aff(
                __isl_take isl_multi_union_pw_aff *mupa);

The C<domain_space> argument describes the domain of the resulting
basic relation.  It is required because the C<list> may consist
of zero affine expressions.
The C<mupa> passed to C<isl_union_map_from_multi_union_pw_aff>
is not allowed to be zero-dimensional.  The domain of the result
is the shared domain of the union piecewise affine elements.

=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
or the number of basic maps in a map can be obtained
from

        #include <isl/set.h>
        int isl_set_n_basic_set(__isl_keep isl_set *set);

        #include <isl/map.h>
        int isl_map_n_basic_map(__isl_keep isl_map *map);

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

It is also possible to obtain a list of constraints from a basic
map or set

        #include <isl/constraint.h>
        __isl_give isl_constraint_list *
        isl_basic_map_get_constraint_list(
                __isl_keep isl_basic_map *bmap);
        __isl_give isl_constraint_list *
        isl_basic_set_get_constraint_list(
                __isl_keep isl_basic_set *bset);

These functions require that all existentially quantified variables
have an explicit representation.
The returned list can be manipulated using the functions in L<"Lists">.

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);
        __isl_give isl_val *isl_constraint_get_constant_val(
                __isl_keep isl_constraint *constraint);
        __isl_give isl_val *isl_constraint_get_coefficient_val(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, int pos);

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</"Functions">.

        __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.
For set inputs, they should be a permutation of
C<isl_dim_cst>, C<isl_dim_param>, C<isl_dim_set> and C<isl_dim_div>.
For map inputs, they 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>.

=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

        __isl_give isl_val *isl_point_get_coordinate_val(
                __isl_keep isl_point *pnt,
                enum isl_dim_type type, int pos);
        __isl_give isl_point *isl_point_set_coordinate_val(
                __isl_take isl_point *pnt,
                enum isl_dim_type type, int pos,
                __isl_take isl_val *v);

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

Points can be copied or freed using

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

A singleton set can be created from a point using

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

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

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

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

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

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

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

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

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

        int isl_point_is_void(__isl_keep isl_point *pnt);

=head2 Functions

Besides sets and relation, C<isl> also supports various types of functions.
Each of these types is derived from the value type (see L</"Values">)
or from one of two primitive function types
through the application of zero or more type constructors.
We first describe the primitive type and then we describe
the types derived from these primitive types.

=head3 Primitive Functions

C<isl> support two primitive function types, quasi-affine
expressions and quasipolynomials.
A quasi-affine expression is defined either over a parameter
space or over a set and is composed of integer constants,
parameters and set variables, addition, subtraction and
integer division by an integer constant.
For example, the quasi-affine expression

        [n] -> { [x] -> [2*floor((4 n + x)/9] }

maps C<x> to C<2*floor((4 n + x)/9>.
A quasipolynomial is a polynomial expression in quasi-affine
expression.  That is, it additionally allows for multiplication.
Note, though, that it is not allowed to construct an integer
division of an expression involving multiplications.
Here is an example of a quasipolynomial that is not
quasi-affine expression

        [n] -> { [x] -> (n*floor((4 n + x)/9) }

Note that the external representations of quasi-affine expressions
and quasipolynomials are different.  Quasi-affine expressions
use a notation with square brackets just like binary relations,
while quasipolynomials do not.  This might change at some point.

If a primitive function is defined over a parameter space,
then the space of the function itself is that of a set.
If it is defined over a set, then the space of the function
is that of a relation.  In both cases, the set space (or
the output space) is single-dimensional, anonymous and unstructured.
To create functions with multiple dimensions or with other kinds
of set or output spaces, use multiple expressions
(see L</"Multiple Expressions">).

=over

=item * Quasi-affine Expressions

Besides the expressions described above, a quasi-affine
expression can also be set to NaN.  Such expressions
typically represent a failure to represent a result
as a quasi-affine expression.

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

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_zero_on_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_aff *isl_aff_val_on_domain(
                __isl_take isl_local_space *ls,
                __isl_take isl_val *val);
        __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_aff *isl_aff_nan_on_domain(
                __isl_take isl_local_space *ls);

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);
        __isl_null isl_aff *isl_aff_free(
                __isl_take isl_aff *aff);

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 coefficients and the integer divisions of an affine expression
can be inspected using the following functions.

        #include <isl/aff.h>
        __isl_give isl_val *isl_aff_get_constant_val(
                __isl_keep isl_aff *aff);
        __isl_give isl_val *isl_aff_get_coefficient_val(
                __isl_keep isl_aff *aff,
                enum isl_dim_type type, int pos);
        int isl_aff_coefficient_sgn(__isl_keep isl_aff *aff,
                enum isl_dim_type type, int pos);
        __isl_give isl_val *isl_aff_get_denominator_val(
                __isl_keep isl_aff *aff);
        __isl_give isl_aff *isl_aff_get_div(
                __isl_keep isl_aff *aff, int pos);

They can be modified using the following functions.

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_set_constant_si(
                __isl_take isl_aff *aff, int v);
        __isl_give isl_aff *isl_aff_set_constant_val(
                __isl_take isl_aff *aff, __isl_take isl_val *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_coefficient_val(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos,
                __isl_take isl_val *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_val(
                __isl_take isl_aff *aff, __isl_take isl_val *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_si(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos, int v);
        __isl_give isl_aff *isl_aff_add_coefficient_val(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, int pos,
                __isl_take isl_val *v);

Note that C<isl_aff_set_constant_si> and C<isl_aff_set_coefficient_si>
set the I<numerator> of the constant or coefficient, while
C<isl_aff_set_constant_val> and C<isl_aff_set_coefficient_val> set
the constant or coefficient as a whole.
The C<add_constant> and C<add_coefficient> functions add an integer
or rational value to
the possibly rational constant or coefficient.
The C<add_constant_num> functions add an integer value to
the numerator.

=item * Quasipolynomials

Some simple quasipolynomials can be created using the following functions.

        #include <isl/polynomial.h>
        __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_val_on_domain(
                __isl_take isl_space *domain,
                __isl_take isl_val *val);
        __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);

Recall 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.

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

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_copy(
                __isl_keep isl_qpolynomial *qp);
        __isl_null isl_qpolynomial *isl_qpolynomial_free(
                __isl_take isl_qpolynomial *qp);

The constant term of a quasipolynomial can be extracted using

        __isl_give isl_val *isl_qpolynomial_get_constant_val(
                __isl_keep isl_qpolynomial *qp);

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);
        __isl_give isl_val *isl_term_get_coefficient_val(
                __isl_keep isl_term *term);
        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.

=back

=head3 Reductions

A reduction represents a maximum or a minimum of its
base expressions.
The only reduction type defined by C<isl> is
C<isl_qpolynomial_fold>.

There are currently no functions to directly create such
objects, but they do appear in the piecewise quasipolynomial
reductions returned by the C<isl_pw_qpolynomial_bound> function.
See
L</"Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions">.

Reductions can be copied and freed using
the following functions.

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial_fold *
        isl_qpolynomial_fold_copy(
                __isl_keep isl_qpolynomial_fold *fold);
        void isl_qpolynomial_fold_free(
                __isl_take isl_qpolynomial_fold *fold);

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 Multiple Expressions

A multiple expression represents a sequence of zero or
more base expressions, all defined on the same domain space.
The domain space of the multiple expression is the same
as that of the base expressions, but the range space
can be any space.  In case the base expressions have
a set space, the corresponding multiple expression
also has a set space.
Objects of the value type do not have an associated space.
The space of a multiple value is therefore always a set space.
Similarly, the space of a multiple union piecewise
affine expression is always a set space.

The multiple expression types defined by C<isl>
are C<isl_multi_val>, C<isl_multi_aff>, C<isl_multi_pw_aff>,
C<isl_multi_union_pw_aff>.

A multiple expression with the value zero for
each output (or set) dimension can be created
using the following functions.

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_zero(
                __isl_take isl_space *space);

        #include <isl/aff.h>
        __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_union_pw_aff *
        isl_multi_union_pw_aff_zero(
                __isl_take isl_space *space);

Since there is no canonical way of representing a zero
value of type C<isl_union_pw_aff>, the space passed
to C<isl_multi_union_pw_aff_zero> needs to be zero-dimensional.

An identity function can be created using the following
functions.  The space needs to be that of a relation
with the same number of input and output dimensions.

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_identity(
                __isl_take isl_space *space);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_identity(
                __isl_take isl_space *space);

A function that performs a projection on a universe
relation or set can be created using the following functions.
See also the corresponding
projection operations in L</"Unary Operations">.

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_domain_map(
                __isl_take isl_space *space);
        __isl_give isl_multi_aff *isl_multi_aff_range_map(
                __isl_take isl_space *space);
        __isl_give isl_multi_aff *isl_multi_aff_project_out_map(
                __isl_take isl_space *space,
                enum isl_dim_type type,
                unsigned first, unsigned n);

A multiple expression can be created from a single
base expression using the following functions.
The space of the created multiple expression is the same
as that of the base expression, except for
C<isl_multi_union_pw_aff_from_union_pw_aff> where the input
lives in a parameter space and the output lives
in a single-dimensional set space.

        #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_union_pw_aff *
        isl_multi_union_pw_aff_from_union_pw_aff(
                __isl_take isl_union_pw_aff *upa);

A multiple expression can be created from a list
of base expression in a specified space.
The domain of this space needs to be the same
as the domains of the base expressions in the list.
If the base expressions have a set space (or no associated space),
then this space also needs to be a set space.

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_from_val_list(
                __isl_take isl_space *space,
                __isl_take isl_val_list *list);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_from_aff_list(
                __isl_take isl_space *space,
                __isl_take isl_aff_list *list);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_union_pw_aff_list(
                __isl_take isl_space *space,
                __isl_take isl_union_pw_aff_list *list);

As a convenience, a multiple piecewise expression can
also be created from a multiple expression.
Each piecewise expression in the result has a single
universe cell.

        #include <isl/aff.h>
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_from_multi_aff(
                __isl_take isl_multi_aff *ma);

Similarly, a multiple union expression can be
created from a multiple expression.

        #include <isl/aff.h>
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_multi_aff(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_multi_pw_aff(
                __isl_take isl_multi_pw_aff *mpa);

A multiple quasi-affine expression can be created from
a multiple value with a given domain space using the following
function.

        #include <isl/aff.h>
        __isl_give isl_multi_aff *
        isl_multi_aff_multi_val_on_space(
                __isl_take isl_space *space,
                __isl_take isl_multi_val *mv);

Similarly,
a multiple union piecewise affine expression can be created from
a multiple value with a given domain or
a multiple affine expression with a given domain
using the following functions.

        #include <isl/aff.h>
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_multi_val_on_domain(
                __isl_take isl_union_set *domain,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_multi_aff_on_domain(
                __isl_take isl_union_set *domain,
                __isl_take isl_multi_aff *ma);

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

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_copy(
                __isl_keep isl_multi_val *mv);
        __isl_null isl_multi_val *isl_multi_val_free(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_copy(
                __isl_keep isl_multi_aff *maff);
        __isl_null isl_multi_aff *isl_multi_aff_free(
                __isl_take isl_multi_aff *maff);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_copy(
                __isl_keep isl_multi_pw_aff *mpa);
        __isl_null isl_multi_pw_aff *isl_multi_pw_aff_free(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_copy(
                __isl_keep isl_multi_union_pw_aff *mupa);
        __isl_null isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_free(
                __isl_take isl_multi_union_pw_aff *mupa);

The base expression at a given position of a multiple
expression can be extracted using the following functions.

        #include <isl/val.h>
        __isl_give isl_val *isl_multi_val_get_val(
                __isl_keep isl_multi_val *mv, int pos);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_multi_aff_get_aff(
                __isl_keep isl_multi_aff *multi, int pos);
        __isl_give isl_pw_aff *isl_multi_pw_aff_get_pw_aff(
                __isl_keep isl_multi_pw_aff *mpa, int pos);
        __isl_give isl_union_pw_aff *
        isl_multi_union_pw_aff_get_union_pw_aff(
                __isl_keep isl_multi_union_pw_aff *mupa, int pos);

It can be replaced using the following functions.

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_set_val(
                __isl_take isl_multi_val *mv, int pos,
                __isl_take isl_val *val);

        #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_multi_union_pw_aff *
        isl_multi_union_pw_aff_set_union_pw_aff(
                __isl_take isl_multi_union_pw_aff *mupa, int pos,
                __isl_take isl_union_pw_aff *upa);

As a convenience, a sequence of base expressions that have
their domains in a given space can be extracted from a sequence
of union expressions using the following function.

        #include <isl/aff.h>
        __isl_give isl_multi_pw_aff *
        isl_multi_union_pw_aff_extract_multi_pw_aff(
                __isl_keep isl_multi_union_pw_aff *mupa,
                __isl_take isl_space *space);

Note that there is a difference between C<isl_multi_union_pw_aff>
and C<isl_union_pw_multi_aff> objects.  The first is a sequence
of unions of piecewise expressions, while the second is a union
of piecewise sequences.  In particular, multiple affine expressions
in an C<isl_union_pw_multi_aff> may live in different spaces,
while there is only a single multiple expression in
an C<isl_multi_union_pw_aff>, which can therefore only live
in a single space.  This means that not every
C<isl_union_pw_multi_aff> can be converted to
an C<isl_multi_union_pw_aff>.  Conversely, a zero-dimensional
C<isl_multi_union_pw_aff> carries no information
about any possible domain and therefore cannot be converted
to an C<isl_union_pw_multi_aff>.  Moreover, the elements
of an C<isl_multi_union_pw_aff> may be defined over different domains,
while each multiple expression inside an C<isl_union_pw_multi_aff>
has a single domain.  The conversion of an C<isl_union_pw_multi_aff>
of dimension greater than one may therefore not be exact.
The following functions can
be used to perform these conversions when they are possible.

        #include <isl/aff.h>
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_union_pw_multi_aff(
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_multi_union_pw_aff(
                __isl_take isl_multi_union_pw_aff *mupa);

=head3 Piecewise Expressions

A piecewise expression is an expression that is described
using zero or more base expression defined over the same
number of cells in the domain space of the base expressions.
All base expressions are defined over the same
domain space and the cells are disjoint.
The space of a piecewise expression is the same as
that of the base expressions.
If the union of the cells is a strict subset of the domain
space, then the value of the piecewise expression outside
this union is different for types derived from quasi-affine
expressions and those derived from quasipolynomials.
Piecewise expressions derived from quasi-affine expressions
are considered to be undefined outside the union of their cells.
Piecewise expressions derived from quasipolynomials
are considered to be zero outside the union of their cells.

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

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

represents the number of points in the map

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

The piecewise expression types defined by C<isl>
are C<isl_pw_aff>, C<isl_pw_multi_aff>,
C<isl_pw_qpolynomial> and C<isl_pw_qpolynomial_fold>.

A piecewise expression with no cells 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_multi_aff *isl_pw_multi_aff_empty(
                __isl_take isl_space *space);

A piecewise expression with a single universe cell can be
created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_from_multi_aff(
                __isl_take isl_multi_aff *ma);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_from_qpolynomial(
                __isl_take isl_qpolynomial *qp);

A piecewise expression with a single specified cell can be
created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_alloc(
                __isl_take isl_set *set, __isl_take isl_aff *aff);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_alloc(
                __isl_take isl_set *set,
                __isl_take isl_multi_aff *maff);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
                __isl_take isl_set *set,
                __isl_take isl_qpolynomial *qp);

The following convenience functions first create a base expression and
then create a piecewise expression over a universe domain.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_pw_aff *isl_pw_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_nan_on_domain(
                __isl_take isl_local_space *ls);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_zero(
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_range_map(
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_project_out_map(
                __isl_take isl_space *space,
                enum isl_dim_type type,
                unsigned first, unsigned n);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(
                __isl_take isl_space *space);

The following convenience functions first create a base expression and
then create a piecewise expression over a given domain.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_val_on_domain(
                __isl_take isl_set *domain,
                __isl_take isl_val *v);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_multi_val_on_domain(
                __isl_take isl_set *domain,
                __isl_take isl_multi_val *mv);

As a convenience, a piecewise multiple expression can
also be created from a piecewise expression.
Each multiple expression in the result is derived
from the corresponding base expression.

        #include <isl/aff.h>
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_pw_aff(
                __isl_take isl_pw_aff *pa);

Similarly, a piecewise quasipolynomial can be
created from a piecewise quasi-affine expression using
the following function.

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_from_pw_aff(
                __isl_take isl_pw_aff *pwaff);

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

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_copy(
                __isl_keep isl_pw_aff *pwaff);
        __isl_null isl_pw_aff *isl_pw_aff_free(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_copy(
                __isl_keep isl_pw_multi_aff *pma);
        __isl_null isl_pw_multi_aff *isl_pw_multi_aff_free(
                __isl_take isl_pw_multi_aff *pma);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
                __isl_keep isl_pw_qpolynomial *pwqp);
        __isl_null isl_pw_qpolynomial *isl_pw_qpolynomial_free(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_copy(
                __isl_keep isl_pw_qpolynomial_fold *pwf);
        __isl_null isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_free(
                __isl_take isl_pw_qpolynomial_fold *pwf);

To iterate over the different cells of a piecewise expression,
use the following functions.

        #include <isl/aff.h>
        int isl_pw_aff_is_empty(__isl_keep isl_pw_aff *pwaff);
        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_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);

        #include <isl/polynomial.h>
        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);
        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);

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>.
Similarly for C<isl_pw_qpolynomial_fold_foreach_piece>
and C<isl_pw_qpolynomial_fold_foreach_lifted_piece>.

A piecewise expression consisting of the expressions at a given
position of a piecewise multiple expression can be extracted
using the following function.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
                __isl_keep isl_pw_multi_aff *pma, int pos);

These expressions can be replaced using the following function.

        #include <isl/aff.h>
        __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);

Note that there is a difference between C<isl_multi_pw_aff> and
C<isl_pw_multi_aff> objects.  The first is a sequence of piecewise
affine expressions, while the second is a piecewise sequence
of affine expressions.  In particular, each of the piecewise
affine expressions in an C<isl_multi_pw_aff> may have a different
domain, while all multiple expressions associated to a cell
in an C<isl_pw_multi_aff> have the same domain.
It is possible to convert between the two, but when converting
an C<isl_multi_pw_aff> to an C<isl_pw_multi_aff>, the domain
of the result is the intersection of the domains of the input.
The reverse conversion is exact.

        #include <isl/aff.h>
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_from_multi_pw_aff(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_from_pw_multi_aff(
                __isl_take isl_pw_multi_aff *pma);

=head3 Union Expressions

A union expression collects base expressions defined
over different domains.  The space of a union expression
is that of the shared parameter space.

The union expression types defined by C<isl>
are C<isl_union_pw_aff>, C<isl_union_pw_multi_aff>,
C<isl_union_pw_qpolynomial> and C<isl_union_pw_qpolynomial_fold>.

An empty union expression can be created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *isl_union_pw_aff_empty(
                __isl_take isl_space *space);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_empty(
                __isl_take isl_space *space);

        #include <isl/polynomial.h>
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_zero(
                __isl_take isl_space *space);

A union expression containing a single base expression
can be created using the following functions.

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_from_pw_aff(
                __isl_take isl_pw_aff *pa);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_aff(
                __isl_take isl_aff *aff);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_pw_multi_aff(
                __isl_take isl_pw_multi_aff *pma);

        #include <isl/polynomial.h>
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_from_pw_qpolynomial(
                __isl_take isl_pw_qpolynomial *pwqp);

The following functions create a base expression on each
of the sets in the union set and collect the results.

        #include <isl/aff.h>
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_from_union_pw_aff(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_pw_aff *
        isl_union_pw_multi_aff_get_union_pw_aff(
                __isl_keep isl_union_pw_multi_aff *upma, int pos);
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_val_on_domain(
                __isl_take isl_union_set *domain,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_multi_val_on_domain(
                __isl_take isl_union_set *domain,
                __isl_take isl_multi_val *mv);

An C<isl_union_pw_aff> that is equal to a (parametric) affine
expression on a given domain can be created using the following
function.

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_aff_on_domain(
                __isl_take isl_union_set *domain,
                __isl_take isl_aff *aff);

A base expression can be added to a union expression using
the following functions.

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_add_pw_aff(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_pw_aff *pa);
        __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);

        #include <isl/polynomial.h>
        __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);

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

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *isl_union_pw_aff_copy(
                __isl_keep isl_union_pw_aff *upa);
        __isl_null isl_union_pw_aff *isl_union_pw_aff_free(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_copy(
                __isl_keep isl_union_pw_multi_aff *upma);
        __isl_null isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_free(
                __isl_take isl_union_pw_multi_aff *upma);

        #include <isl/polynomial.h>
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_copy(
                __isl_keep isl_union_pw_qpolynomial *upwqp);
        __isl_null isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_free(
                __isl_take isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_copy(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);
        __isl_null isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_free(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);

To iterate over the base expressions in a union expression,
use the following functions.

        #include <isl/aff.h>
        int isl_union_pw_aff_n_pw_aff(
                __isl_keep isl_union_pw_aff *upa);
        int isl_union_pw_aff_foreach_pw_aff(
                __isl_keep isl_union_pw_aff *upa,
                int (*fn)(__isl_take isl_pw_aff *ma, void *user),
                        void *user);
        int isl_union_pw_multi_aff_n_pw_multi_aff(
                __isl_keep isl_union_pw_multi_aff *upma);
        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);

        #include <isl/polynomial.h>
        int isl_union_pw_qpolynomial_n_pw_qpolynomial(
                __isl_keep isl_union_pw_qpolynomial *upwqp);
        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);
        int isl_union_pw_qpolynomial_fold_n_pw_qpolynomial_fold(
                __isl_keep isl_union_pw_qpolynomial_fold *upwf);
        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 extract the base expression in a given space from a union, use
the following functions.

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_union_pw_aff_extract_pw_aff(
                __isl_keep isl_union_pw_aff *upa,
                __isl_take isl_space *space);
        __isl_give isl_pw_multi_aff *
        isl_union_pw_multi_aff_extract_pw_multi_aff(
                __isl_keep isl_union_pw_multi_aff *upma,
                __isl_take isl_space *space);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_union_pw_qpolynomial_extract_pw_qpolynomial(
                __isl_keep isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_space *space);

=head2 Input and Output

For set and relation,
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.
For other object types, typically only an C<isl> format is supported.

=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

Objects can be read from input using the following functions.

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

        #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);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_multi_aff *isl_multi_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);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_read_from_str(
                isl_ctx *ctx, const char *str);
        __isl_give isl_multi_pw_aff *isl_multi_pw_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);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_read_from_str(
                isl_ctx *ctx, const char *str);

        #include <isl/polynomial.h>
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_read_from_str(
                isl_ctx *ctx, const char *str);

For sets and relations,
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);
        __isl_null isl_printer *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);
        int isl_printer_get_yaml_style(__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_set_indent_prefix(
                __isl_take isl_printer *p, const char *prefix);
        __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);
        __isl_give isl_printer *isl_printer_set_yaml_style(
                __isl_take isl_printer *p, int yaml_style);

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 prefixed by C<indent_prefix>,
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).
The YAML style may be either C<ISL_YAML_STYLE_BLOCK> or
C<ISL_YAML_STYLE_FLOW> and when we are printing something
in YAML format.

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

        #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);

        #include <isl/val.h>
        __isl_give isl_printer *isl_printer_print_multi_val(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_val *mv);

        #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_multi_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_aff *maff);
        __isl_give isl_printer *isl_printer_print_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_pw_aff *pwaff);
        __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_multi_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_pw_aff *mpa);
        __isl_give isl_printer *isl_printer_print_union_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_union_pw_aff *upa);
        __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_union_pw_aff(
                __isl_take isl_printer *p,
                __isl_keep isl_multi_union_pw_aff *mupa);

        #include <isl/polynomial.h>
        __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);

        __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_qpolynomial>,
C<isl_printer_print_pw_qpolynomial> and
C<isl_printer_print_pw_qpolynomial_fold>,
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> and
C<isl_printer_print_union_pw_qpolynomial_fold>, 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 first.

C<isl> also provides limited support for printing YAML documents,
just enough for the internal use for printing such documents.

        #include <isl/printer.h>
        __isl_give isl_printer *isl_printer_yaml_start_mapping(
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_printer_yaml_end_mapping(
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_printer_yaml_start_sequence(
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_printer_yaml_end_sequence(
                __isl_take isl_printer *p);
        __isl_give isl_printer *isl_printer_yaml_next(
                __isl_take isl_printer *p);

A document is started by a call to either
C<isl_printer_yaml_start_mapping> or C<isl_printer_yaml_start_sequence>.
Anything printed to the printer after such a call belong to the
first key of the mapping or the first element in the sequence.
The function C<isl_printer_yaml_next> moves to the value if
we are currently printing a mapping key, the next key if we
are printing a value or the next element if we are printing
an element in a sequence.
Nested mappings and sequences are initiated by the same
C<isl_printer_yaml_start_mapping> or C<isl_printer_yaml_start_sequence>.
Each call to these functions needs to have a corresponding call to
C<isl_printer_yaml_end_mapping> or C<isl_printer_yaml_end_sequence>.

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);

Alternatively, a string representation can be obtained
directly using the following functions, which always print
in isl format.

        #include <isl/space.h>
        __isl_give char *isl_space_to_str(
                __isl_keep isl_space *space);

        #include <isl/val.h>
        __isl_give char *isl_val_to_str(__isl_keep isl_val *v);
        __isl_give char *isl_multi_val_to_str(
                __isl_keep isl_multi_val *mv);

        #include <isl/set.h>
        __isl_give char *isl_set_to_str(
                __isl_keep isl_set *set);

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

        #include <isl/map.h>
        __isl_give char *isl_map_to_str(
                __isl_keep isl_map *map);

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

        #include <isl/aff.h>
        __isl_give char *isl_multi_aff_to_str(
                __isl_keep isl_multi_aff *aff);
        __isl_give char *isl_union_pw_aff_to_str(
                __isl_keep isl_union_pw_aff *upa);
        __isl_give char *isl_union_pw_multi_aff_to_str(
                __isl_keep isl_union_pw_multi_aff *upma);
        __isl_give char *isl_multi_union_pw_aff_to_str(
                __isl_keep isl_multi_union_pw_aff *mupa);

=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

        #include <isl/set.h>
        int isl_set_is_singleton(__isl_keep isl_set *set);

        #include <isl/map.h>
        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);

        #include <isl/union_map.h>
        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

        __isl_give isl_val *
        isl_basic_map_plain_get_val_if_fixed(
                __isl_keep isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_val *isl_set_plain_get_val_if_fixed(
                __isl_keep isl_set *set,
                enum isl_dim_type type, unsigned pos);
        __isl_give isl_val *isl_map_plain_get_val_if_fixed(
                __isl_keep isl_map *map,
                enum isl_dim_type type, unsigned pos);

If the set or relation obviously lies on a hyperplane where the given dimension
has a fixed value, then return that value.
Otherwise return NaN.

=item * Stride

        int isl_set_dim_residue_class_val(
                __isl_keep isl_set *set,
                int pos, __isl_give isl_val **modulo,
                __isl_give isl_val **residue);

Check if the values of the given set dimension are equal to a fixed
value modulo some integer value.  If so, assign the modulo to C<*modulo>
and the fixed value to C<*residue>.  If the given dimension attains only
a single value, then assign C<0> to C<*modulo> and the fixed value to
C<*residue>.
If the dimension does not attain only a single value and if no modulo
can be found then assign C<1> to C<*modulo> and C<1> to C<*residue>.

=item * Dependence

To check whether the description of a set, relation or function depends
on one or more given dimensions,
the following functions can be used.

        #include <isl/constraint.h>
        int isl_constraint_involves_dims(
                __isl_keep isl_constraint *constraint,
                enum isl_dim_type type, unsigned first, unsigned n);

        #include <isl/set.h>
        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);

        #include <isl/map.h>
        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);

        #include <isl/union_map.h>
        int isl_union_map_involves_dims(
                __isl_keep isl_union_map *umap,
                enum isl_dim_type type, unsigned first, unsigned n);

        #include <isl/aff.h>
        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);
        int isl_multi_aff_involves_dims(
                __isl_keep isl_multi_aff *ma,
                enum isl_dim_type type, unsigned first, unsigned n);
        int isl_multi_pw_aff_involves_dims(
                __isl_keep isl_multi_pw_aff *mpa,
                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.

        #include <isl/set.h>
        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.

        #include <isl/set.h>
        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);

=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 space of the given
(basic) set or relation range is a wrapped relation.

        #include <isl/space.h>
        int isl_space_is_wrapping(
                __isl_keep isl_space *space);
        int isl_space_domain_is_wrapping(
                __isl_keep isl_space *space);
        int isl_space_range_is_wrapping(
                __isl_keep isl_space *space);

        #include <isl/set.h>
        int isl_basic_set_is_wrapping(
                __isl_keep isl_basic_set *bset);
        int isl_set_is_wrapping(__isl_keep isl_set *set);

        #include <isl/map.h>
        int isl_map_domain_is_wrapping(
                __isl_keep isl_map *map);
        int isl_map_range_is_wrapping(
                __isl_keep isl_map *map);

        #include <isl/val.h>
        int isl_multi_val_range_is_wrapping(
                __isl_keep isl_multi_val *mv);

        #include <isl/aff.h>
        int isl_multi_aff_range_is_wrapping(
                __isl_keep isl_multi_aff *ma);
        int isl_multi_pw_aff_range_is_wrapping(
                __isl_keep isl_multi_pw_aff *mpa);
        int isl_multi_union_pw_aff_range_is_wrapping(
                __isl_keep isl_multi_union_pw_aff *mupa);

The input to C<isl_space_is_wrapping> should
be the space of a set, while that of
C<isl_space_domain_is_wrapping> and
C<isl_space_range_is_wrapping> should be the space of a relation.

=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.

=item * Special Values

        #include <isl/aff.h>
        int isl_aff_is_cst(__isl_keep isl_aff *aff);
        int isl_pw_aff_is_cst(__isl_keep isl_pw_aff *pwaff);

Check whether the given expression is a constant.

        #include <isl/aff.h>
        int isl_aff_is_nan(__isl_keep isl_aff *aff);
        int isl_pw_aff_involves_nan(__isl_keep isl_pw_aff *pa);

Check whether the given expression is equal to or involves NaN.

        #include <isl/aff.h>
        int isl_aff_plain_is_zero(__isl_keep isl_aff *aff);

Check whether the affine expression is obviously zero.

=back

=head3 Binary Properties

=over

=item * Equality

The following functions check whether two objects
represent the same set, relation or function.
The C<plain> variants only return true if the objects
are obviously the same.  That is, they may return false
even if the objects are the same, but they will never
return true if the objects are not the same.

        #include <isl/set.h>
        int isl_basic_set_plain_is_equal(
                __isl_keep isl_basic_set *bset1,
                __isl_keep isl_basic_set *bset2);
        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);

        #include <isl/map.h>
        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);

        #include <isl/union_set.h>
        int isl_union_set_is_equal(
                __isl_keep isl_union_set *uset1,
                __isl_keep isl_union_set *uset2);

        #include <isl/union_map.h>
        int isl_union_map_is_equal(
                __isl_keep isl_union_map *umap1,
                __isl_keep isl_union_map *umap2);

        #include <isl/aff.h>
        int isl_aff_plain_is_equal(__isl_keep isl_aff *aff1,
                __isl_keep isl_aff *aff2);
        int isl_multi_aff_plain_is_equal(
                __isl_keep isl_multi_aff *maff1,
                __isl_keep isl_multi_aff *maff2);
        int isl_pw_aff_plain_is_equal(
                __isl_keep isl_pw_aff *pwaff1,
                __isl_keep isl_pw_aff *pwaff2);
        int isl_pw_multi_aff_plain_is_equal(
                __isl_keep isl_pw_multi_aff *pma1,
                __isl_keep isl_pw_multi_aff *pma2);
        int isl_multi_pw_aff_plain_is_equal(
                __isl_keep isl_multi_pw_aff *mpa1,
                __isl_keep isl_multi_pw_aff *mpa2);
        int isl_multi_pw_aff_is_equal(
                __isl_keep isl_multi_pw_aff *mpa1,
                __isl_keep isl_multi_pw_aff *mpa2);
        int isl_union_pw_aff_plain_is_equal(
                __isl_keep isl_union_pw_aff *upa1,
                __isl_keep isl_union_pw_aff *upa2);
        int isl_union_pw_multi_aff_plain_is_equal(
                __isl_keep isl_union_pw_multi_aff *upma1,
                __isl_keep isl_union_pw_multi_aff *upma2);
        int isl_multi_union_pw_aff_plain_is_equal(
                __isl_keep isl_multi_union_pw_aff *mupa1,
                __isl_keep isl_multi_union_pw_aff *mupa2);

        #include <isl/polynomial.h>
        int isl_union_pw_qpolynomial_plain_is_equal(
                __isl_keep isl_union_pw_qpolynomial *upwqp1,
                __isl_keep isl_union_pw_qpolynomial *upwqp2);
        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);

=item * Disjointness

        #include <isl/set.h>
        int isl_basic_set_is_disjoint(
                __isl_keep isl_basic_set *bset1,
                __isl_keep isl_basic_set *bset2);
        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);

        #include <isl/map.h>
        int isl_basic_map_is_disjoint(
                __isl_keep isl_basic_map *bmap1,
                __isl_keep isl_basic_map *bmap2);
        int isl_map_is_disjoint(__isl_keep isl_map *map1,
                __isl_keep isl_map *map2);

        #include <isl/union_set.h>
        int isl_union_set_is_disjoint(
                __isl_keep isl_union_set *uset1,
                __isl_keep isl_union_set *uset2);

        #include <isl/union_map.h>
        int isl_union_map_is_disjoint(
                __isl_keep isl_union_map *umap1,
                __isl_keep isl_union_map *umap2);

=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

Every comparison function returns a negative value if the first
argument is considered smaller than the second, a positive value
if the first argument is considered greater and zero if the two
constraints are considered the same by the comparison criterion.

        #include <isl/constraint.h>
        int isl_constraint_plain_cmp(
                __isl_keep isl_constraint *c1,
                __isl_keep isl_constraint *c2);

This function is useful for sorting C<isl_constraint>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>.

        #include <isl/constraint.h>
        int isl_constraint_cmp_last_non_zero(
                __isl_keep isl_constraint *c1,
                __isl_keep isl_constraint *c2);

This function can be used to sort constraints that live in the same
local space.  Constraints that involve ``earlier'' dimensions or
that have a smaller coefficient for the shared latest dimension
are considered smaller than other constraints.
This function only defines a B<partial> order.

        #include <isl/set.h>
        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>.

        #include <isl/aff.h>
        int isl_pw_aff_plain_cmp(__isl_keep isl_pw_aff *pa1,
                __isl_keep isl_pw_aff *pa2);

The function C<isl_pw_aff_plain_cmp> can be used to sort
C<isl_pw_aff>s.  The order is not strictly defined.
The current order sorts expressions that only involve
earlier dimensions before those that involve later dimensions.

=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

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

        #include <isl/map.h>
        __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);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_reverse(
                __isl_take isl_union_map *umap);

=item * Projection

        #include <isl/space.h>
        __isl_give isl_space *isl_space_domain(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_range(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_params(
                __isl_take isl_space *space);

        #include <isl/local_space.h>
        __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);

        #include <isl/set.h>
        __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_set *isl_set_project_out(__isl_take isl_set *set,
                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_set *isl_set_params(__isl_take isl_set *set);

        #include <isl/map.h>
        __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_map *isl_map_project_out(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_basic_set *isl_basic_map_domain(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_basic_set *isl_basic_map_range(
                __isl_take isl_basic_map *bmap);
        __isl_give isl_set *isl_map_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);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_project_out(
                __isl_take isl_union_set *uset,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_set *isl_union_set_params(
                __isl_take isl_union_set *uset);

The function C<isl_union_set_project_out> can only project out
parameters.

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_project_out(
                __isl_take isl_union_map *umap,
                enum isl_dim_type type, unsigned first, unsigned n);
        __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);

The function C<isl_union_map_project_out> can only project out
parameters.

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_project_domain_on_params(
                __isl_take isl_aff *aff);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_project_domain_on_params(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_set *isl_pw_aff_domain(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_set *isl_pw_multi_aff_domain(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_set *isl_multi_pw_aff_domain(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_union_set *isl_union_pw_aff_domain(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_set *isl_union_pw_multi_aff_domain(
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_union_set *
        isl_multi_union_pw_aff_domain(
                __isl_take isl_multi_union_pw_aff *mupa);
        __isl_give isl_set *isl_pw_aff_params(
                __isl_take isl_pw_aff *pwa);

The function C<isl_multi_union_pw_aff_domain> requires its
input to have at least one set dimension.

        #include <isl/polynomial.h>
        __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_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_project_domain_on_params(
                __isl_take isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_set *isl_pw_qpolynomial_domain(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);
        __isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
                __isl_take isl_union_pw_qpolynomial *upwqp);

        #include <isl/space.h>
        __isl_give isl_space *isl_space_domain_map(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_range_map(
                __isl_take isl_space *space);

        #include <isl/map.h>
        __isl_give isl_map *isl_set_wrapped_domain_map(
                __isl_take isl_set *set);
        __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);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_domain_map(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_pw_multi_aff *
        isl_union_map_domain_map_union_pw_multi_aff(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *isl_union_map_range_map(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *
        isl_union_set_wrapped_domain_map(
                __isl_take isl_union_set *uset);

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

=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 * Constructing a set from a parameter domain

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

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

        #include <isl/set.h>
        __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);

=item * Constructing a relation from a set

Create a relation with the given set as domain or range.
The range or domain of the created relation is a zero-dimensional
flat anonymous space.

        #include <isl/space.h>
        __isl_give isl_space *isl_space_from_domain(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_from_range(
                __isl_take isl_space *space);
        __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);

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

        #include <isl/map.h>
        __isl_give isl_map *isl_map_from_domain(
                __isl_take isl_set *set);
        __isl_give isl_map *isl_map_from_range(
                __isl_take isl_set *set);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_from_range(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_from_range(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_from_range(
                __isl_take isl_pw_aff *pwa);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_from_range(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_range(
                __isl_take isl_multi_union_pw_aff *mupa);
        __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_from_domain(
                __isl_take isl_union_set *uset);

=item * Slicing

        #include <isl/set.h>
        __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_basic_set *isl_basic_set_fix_val(
                __isl_take isl_basic_set *bset,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *v);
        __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_set *isl_set_fix_val(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *v);

        #include <isl/map.h>
        __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_basic_map *isl_basic_map_fix_val(
                __isl_take isl_basic_map *bmap,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *v);
        __isl_give isl_map *isl_map_fix_si(__isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_map *isl_map_fix_val(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *v);

        #include <isl/aff.h>
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_fix_si(
                __isl_take isl_pw_multi_aff *pma,
                enum isl_dim_type type, unsigned pos, int value);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_fix_val(
                __isl_take isl_pw_qpolynomial *pwqp,
                enum isl_dim_type type, unsigned n,
                __isl_take isl_val *v);

Intersect the set, relation or function domain
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_si(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_set *isl_set_lower_bound_val(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *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_si(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos, int value);
        __isl_give isl_set *isl_set_upper_bound_val(
                __isl_take isl_set *set,
                enum isl_dim_type type, unsigned pos,
                __isl_take isl_val *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 integer 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_map *isl_map_order_le(
                __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_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_ge(
                __isl_take isl_map *map,
                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 * Locus

        #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_set *isl_pw_aff_pos_set(
                __isl_take isl_pw_aff *pa);
        __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);
        __isl_give isl_union_set *
        isl_union_pw_aff_zero_union_set(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_set *
        isl_multi_union_pw_aff_zero_union_set(
                __isl_take isl_multi_union_pw_aff *mupa);

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_pw_aff_nonneg_set> returns a set
containing those elements in the domain
of C<pwaff> where C<pwaff> is non-negative.
The function C<isl_multi_union_pw_aff_zero_union_set>
returns a union set containing those elements
in the domains of its elements where they are all zero.

=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);
        __isl_give isl_union_pw_multi_aff *
        isl_union_set_identity_union_pw_multi_aff(
                __isl_take isl_union_set *uset);

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

=item * Function Extraction

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);

A piecewise multiple quasi affine expression can be extracted
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_map>
to an C<isl_union_pw_multi_aff>, these properties need to hold
in each domain space.
A conversion to a C<isl_multi_union_pw_aff> additionally
requires that the input is non-empty and involves only a single
range space.

        #include <isl/aff.h>
        __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);

        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_from_union_map(
                __isl_take isl_union_map *umap);

=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, relation or functions by trying
to combine pairs of basic sets or relations into a single
basic set or relation.

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

        #include <isl/map.h>
        __isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_coalesce(
                __isl_take isl_union_set *uset);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_coalesce(
                __isl_take isl_union_map *umap);

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_coalesce(
                __isl_take isl_pw_aff *pwqp);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_coalesce(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_coalesce(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_coalesce(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_coalesce(
                __isl_take isl_union_pw_multi_aff *upma);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_coalesce(
                __isl_take isl_pw_qpolynomial_fold *pwf);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_coalesce(
                __isl_take isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_coalesce(
                __isl_take isl_union_pw_qpolynomial_fold *upwf);

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

        #include <isl/set.h>
        __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);

        #include <isl/union_set.h>
        __isl_give isl_union_set *
        isl_union_set_remove_redundancies(
                __isl_take isl_union_set *uset);

        #include <isl/map.h>
        __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);

        #include <isl/union_map.h>
        __isl_give isl_union_map *
        isl_union_map_remove_redundancies(
                __isl_take isl_union_map *umap);

=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

        #include <isl/set.h>
        __isl_give isl_basic_set *
        isl_set_unshifted_simple_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_set *isl_set_simple_hull(
                __isl_take isl_set *set);
        __isl_give isl_basic_set *
        isl_set_unshifted_simple_hull_from_set_list(
                __isl_take isl_set *set,
                __isl_take isl_set_list *list);

        #include <isl/map.h>
        __isl_give isl_basic_map *
        isl_map_unshifted_simple_hull(
                __isl_take isl_map *map);
        __isl_give isl_basic_map *isl_map_simple_hull(
                __isl_take isl_map *map);
        __isl_give isl_basic_map *
        isl_map_unshifted_simple_hull_from_map_list(
                __isl_take isl_map *map,
                __isl_take isl_map_list *list);

        #include <isl/union_map.h>
        __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.
In case of C<isl_set_unshifted_simple_hull_from_set_list> and
C<isl_map_unshifted_simple_hull_from_map_list>, the
constraints are taken from the elements of the second argument.

=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

        #include <isl/set.h>
        __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_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);

        #include <isl/map.h>
        __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_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.

        #include <isl/polynomial.h>
        __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.

=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>
        __isl_give isl_val *isl_basic_set_max_val(
                __isl_keep isl_basic_set *bset,
                __isl_keep isl_aff *obj);
        __isl_give isl_val *isl_set_min_val(
                __isl_keep isl_set *set,
                __isl_keep isl_aff *obj);
        __isl_give isl_val *isl_set_max_val(
                __isl_keep isl_set *set,
                __isl_keep isl_aff *obj);

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 result is C<NULL> in case of an error, the optimal value in case
there is one, negative infinity or infinity if the problem is unbounded and
NaN if the problem is 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_val(
                __isl_take isl_map *map,
                __isl_take isl_val *exp);
        __isl_give isl_union_map *
        isl_union_map_fixed_power_val(
                __isl_take isl_union_map *umap,
                __isl_take isl_val *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

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

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

        #include <isl/set.h>
        __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);

        #include <isl/map.h>
        __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);

        #include <isl/union_set.h>
        __isl_give isl_union_map *isl_union_set_unwrap(
                __isl_take isl_union_set *uset);

        #include <isl/union_map.h>
        __isl_give isl_union_set *isl_union_map_wrap(
                __isl_take isl_union_map *umap);

The input to 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.

=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.

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

        #include <isl/set.h>
        __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);

        #include <isl/map.h>
        __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);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_flatten_range(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_flatten_domain(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_aff *isl_multi_aff_flatten_range(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_flatten_range(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_flatten_range(
                __isl_take isl_multi_union_pw_aff *mupa);

        #include <isl/map.h>
        __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.

        #include <isl/set.h>
        __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);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_lift(
                __isl_take isl_multi_aff *maff,
                __isl_give isl_local_space **ls);

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.

=item * Internal Product

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

        #include <isl/map.h>
        __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);

        #include <isl/union_map.h>
        __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

        #include <isl/space.h>
        __isl_give isl_space *isl_space_curry(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_uncurry(
                __isl_take isl_space *space);

        #include <isl/map.h>
        __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);

        #include <isl/union_map.h>
        __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

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

        #include <isl/space.h>
        __isl_give isl_space *isl_space_align_params(
                __isl_take isl_space *space1,
                __isl_take isl_space *space2)

        #include <isl/set.h>
        __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);

        #include <isl/map.h>
        __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);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_align_params(
                __isl_take isl_multi_val *mv,
                __isl_take isl_space *model);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_align_params(
                __isl_take isl_aff *aff,
                __isl_take isl_space *model);
        __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_aff *isl_pw_aff_align_params(
                __isl_take isl_pw_aff *pwaff,
                __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_union_pw_aff *
        isl_union_pw_aff_align_params(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_space *model);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_align_params(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_space *model);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_align_params(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_space *model);

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_align_params(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_space *model);

=item * Unary Arithmethic Operations

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_neg(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_neg(
                __isl_take isl_aff *aff);
        __isl_give isl_multi_aff *isl_multi_aff_neg(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_neg(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_neg(
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_neg(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_neg(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_neg(
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_neg(
                __isl_take isl_multi_union_pw_aff *mupa);
        __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_multi_aff *isl_multi_aff_floor(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_pw_aff *isl_pw_aff_floor(
                __isl_take isl_pw_aff *pwaff);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_floor(
                __isl_take isl_union_pw_aff *upa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_floor(
                __isl_take isl_multi_union_pw_aff *mupa);

        #include <isl/aff.h>
        __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);

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_neg(
                __isl_take isl_qpolynomial *qp);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
                __isl_take isl_pw_qpolynomial *pwqp);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_neg(
                __isl_take isl_union_pw_qpolynomial *upwqp);
        __isl_give isl_qpolynomial *isl_qpolynomial_pow(
                __isl_take isl_qpolynomial *qp,
                unsigned exponent);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_pow(
                __isl_take isl_pw_qpolynomial *pwqp,
                unsigned exponent);

=item * Evaluation

The following functions evaluate a function in a point.

        #include <isl/polynomial.h>
        __isl_give isl_val *isl_pw_qpolynomial_eval(
                __isl_take isl_pw_qpolynomial *pwqp,
                __isl_take isl_point *pnt);
        __isl_give isl_val *isl_pw_qpolynomial_fold_eval(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_point *pnt);
        __isl_give isl_val *isl_union_pw_qpolynomial_eval(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_point *pnt);
        __isl_give isl_val *isl_union_pw_qpolynomial_fold_eval(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_point *pnt);

=item * Dimension manipulation

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 functions below can be useful
to add new parameters, assuming
C<isl_set_align_params> and C<isl_map_align_params>
are not sufficient.

        #include <isl/space.h>
        __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_insert_dims(
                __isl_take isl_space *space,
                enum isl_dim_type type, unsigned pos, 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);

        #include <isl/local_space.h>
        __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);

        #include <isl/set.h>
        __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_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_set *isl_set_insert_dims(
                __isl_take isl_set *set,
                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_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);

        #include <isl/map.h>
        __isl_give isl_map *isl_map_add_dims(
                __isl_take isl_map *map,
                enum isl_dim_type type, 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_map *isl_map_insert_dims(
                __isl_take isl_map *map,
                enum isl_dim_type type, unsigned 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_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);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_insert_dims(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_multi_val *isl_multi_val_add_dims(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, unsigned n);
        __isl_give isl_multi_val *isl_multi_val_drop_dims(
                __isl_take isl_multi_val *mv,
                enum isl_dim_type type, unsigned first, unsigned n);

        #include <isl/aff.h>
        __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_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_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_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_aff *isl_aff_add_dims(
                __isl_take isl_aff *aff,
                enum isl_dim_type type, 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_pw_aff *isl_pw_aff_add_dims(
                __isl_take isl_pw_aff *pwaff,
                enum isl_dim_type type, 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);
        __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_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_aff *isl_pw_aff_drop_dims(
                __isl_take isl_pw_aff *pwaff,
                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_union_pw_aff *isl_union_pw_aff_drop_dims(
                __isl_take isl_union_pw_aff *upa,
                enum isl_dim_type type, unsigned first, unsigned n);
        __isl_give isl_union_pw_multi_aff *
                isl_union_pw_multi_aff_drop_dims(
                __isl_take isl_union_pw_multi_aff *upma,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_drop_dims(
                __isl_take isl_multi_union_pw_aff *mupa,
                enum isl_dim_type type, unsigned first,
                unsigned n);
        __isl_give isl_aff *isl_aff_move_dims(
                __isl_take isl_aff *aff,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_multi_aff *isl_multi_aff_move_dims(
                __isl_take isl_multi_aff *ma,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_pw_aff *isl_pw_aff_move_dims(
                __isl_take isl_pw_aff *pa,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_move_dims(
                __isl_take isl_multi_pw_aff *pma,
                enum isl_dim_type dst_type, unsigned dst_pos,
                enum isl_dim_type src_type, unsigned src_pos,
                unsigned n);

        #include <isl/polynomial.h>
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_drop_dims(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                enum isl_dim_type type,
                unsigned first, unsigned n);
        __isl_give isl_union_pw_qpolynomial_fold *
                isl_union_pw_qpolynomial_fold_drop_dims(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                enum isl_dim_type type,
                unsigned first, unsigned n);

The operations on union expressions can only manipulate parameters.

=back

=head2 Binary Operations

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

=head3 Basic Operations

=over

=item * Intersection

        #include <isl/local_space.h>
        __isl_give isl_local_space *isl_local_space_intersect(
                __isl_take isl_local_space *ls1,
                __isl_take isl_local_space *ls2);

        #include <isl/set.h>
        __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_basic_set *isl_basic_set_list_intersect(
                __isl_take struct isl_basic_set_list *list);
        __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);

        #include <isl/map.h>
        __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_basic_map *isl_basic_map_list_intersect(
                __isl_take isl_basic_map_list *list);
        __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);

        #include <isl/union_set.h>
        __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_set *isl_union_set_intersect(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);

        #include <isl/union_map.h>
        __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_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);

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_intersect_domain(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_set *set);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_intersect_domain(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_set *domain);
        __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_aff *isl_union_pw_aff_intersect_domain(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_union_set *uset);
        __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_union_pw_aff *
        isl_multi_union_pw_aff_intersect_domain(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_union_set *uset);
        __isl_give isl_pw_aff *isl_pw_aff_intersect_params(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_set *set);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_intersect_params(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_set *set);
        __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_union_pw_aff *
        isl_union_pw_aff_intersect_params(
                __isl_take isl_union_pw_aff *upa,
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_intersect_params(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_set *set);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_intersect_params(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_set *params);
        isl_multi_union_pw_aff_intersect_range(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_set *set);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_intersect_domain(
                __isl_take isl_pw_qpolynomial *pwpq,
                __isl_take isl_set *set);
        __isl_give isl_union_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_fold *
        isl_union_pw_qpolynomial_fold_intersect_domain(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_union_set *uset);
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_intersect_params(
                __isl_take isl_pw_qpolynomial *pwpq,
                __isl_take isl_set *set);
        __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_pw_qpolynomial *
        isl_union_pw_qpolynomial_intersect_params(
                __isl_take isl_union_pw_qpolynomial *upwpq,
                __isl_take isl_set *set);
        __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);

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.
The list passed to C<isl_basic_set_list_intersect> needs to have
at least one element and all elements need to live in the same space.
The function C<isl_multi_union_pw_aff_intersect_range>
restricts the input function to those shared domain elements
that map to the specified range.

=item * Union

        #include <isl/set.h>
        __isl_give isl_set *isl_basic_set_union(
                __isl_take isl_basic_set *bset1,
                __isl_take isl_basic_set *bset2);
        __isl_give isl_set *isl_set_union(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);

        #include <isl/map.h>
        __isl_give isl_map *isl_basic_map_union(
                __isl_take isl_basic_map *bmap1,
                __isl_take isl_basic_map *bmap2);
        __isl_give isl_map *isl_map_union(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);

        #include <isl/union_set.h>
        __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_set *isl_union_set_list_union(
                __isl_take isl_union_set_list *list);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_union(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

=item * Set difference

        #include <isl/set.h>
        __isl_give isl_set *isl_set_subtract(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);

        #include <isl/map.h>
        __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);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_subtract(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);

        #include <isl/union_map.h>
        __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);

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_subtract_domain(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_set *set);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_subtract_domain(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_set *set);
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_subtract_domain(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_union_set *uset);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_subtract_domain(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_set *set);

        #include <isl/polynomial.h>
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_subtract_domain(
                __isl_take isl_pw_qpolynomial *pwpq,
                __isl_take isl_set *set);
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_subtract_domain(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_set *set);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_subtract_domain(
                __isl_take isl_union_pw_qpolynomial *upwpq,
                __isl_take isl_union_set *uset);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_subtract_domain(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_union_set *uset);

=item * Application

        #include <isl/space.h>
        __isl_give isl_space *isl_space_join(
                __isl_take isl_space *left,
                __isl_take isl_space *right);

        #include <isl/map.h>
        __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_map *isl_map_apply_range(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);

        #include <isl/union_map.h>
        __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_union_map *isl_union_map_apply_range(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

        #include <isl/aff.h>
        __isl_give isl_union_pw_aff *
        isl_multi_union_pw_aff_apply_aff(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_aff *aff);
        __isl_give isl_union_pw_aff *
        isl_multi_union_pw_aff_apply_pw_aff(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_pw_aff *pa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_apply_multi_aff(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_apply_pw_multi_aff(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_pw_multi_aff *pma);

The result of C<isl_multi_union_pw_aff_apply_aff> is defined
over the shared domain of the elements of the input.  The dimension is
required to be greater than zero.
The C<isl_multi_union_pw_aff> argument of
C<isl_multi_union_pw_aff_apply_multi_aff> is allowed to be zero-dimensional,
but only if the range of the C<isl_multi_aff> argument
is also zero-dimensional.
Similarly for C<isl_multi_union_pw_aff_apply_pw_multi_aff>.

        #include <isl/polynomial.h>
        __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.

=item * Preimage

        #include <isl/set.h>
        __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_set *isl_set_preimage_multi_pw_aff(
                __isl_take isl_set *set,
                __isl_take isl_multi_pw_aff *mpa);

        #include <isl/union_set.h>
        __isl_give isl_union_set *
        isl_union_set_preimage_multi_aff(
                __isl_take isl_union_set *uset,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_union_set *
        isl_union_set_preimage_pw_multi_aff(
                __isl_take isl_union_set *uset,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_set *
        isl_union_set_preimage_union_pw_multi_aff(
                __isl_take isl_union_set *uset,
                __isl_take isl_union_pw_multi_aff *upma);

        #include <isl/map.h>
        __isl_give isl_basic_map *
        isl_basic_map_preimage_domain_multi_aff(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_map *isl_map_preimage_domain_multi_aff(
                __isl_take isl_map *map,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_map *isl_map_preimage_range_multi_aff(
                __isl_take isl_map *map,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_map *
        isl_map_preimage_domain_pw_multi_aff(
                __isl_take isl_map *map,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_map *
        isl_map_preimage_range_pw_multi_aff(
                __isl_take isl_map *map,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_map *
        isl_map_preimage_domain_multi_pw_aff(
                __isl_take isl_map *map,
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_basic_map *
        isl_basic_map_preimage_range_multi_aff(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_multi_aff *ma);

        #include <isl/union_map.h>
        __isl_give isl_union_map *
        isl_union_map_preimage_domain_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_union_map *
        isl_union_map_preimage_range_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_aff *ma);
        __isl_give isl_union_map *
        isl_union_map_preimage_domain_pw_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_map *
        isl_union_map_preimage_range_pw_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_union_map *
        isl_union_map_preimage_domain_union_pw_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_union_map *
        isl_union_map_preimage_range_union_pw_multi_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_union_pw_multi_aff *upma);

These functions compute the preimage of the given set or map domain/range under
the given function.  In other words, the expression is plugged
into the set description or into the domain/range of the map.

=item * Pullback

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_pullback_aff(
                __isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __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);
        __isl_give isl_pw_aff *isl_pw_aff_pullback_multi_pw_aff(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_multi_pw_aff *mpa);
        __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_multi_pw_aff *
        isl_multi_pw_aff_pullback_multi_aff(
                __isl_take isl_multi_pw_aff *mpa,
                __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);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_pullback_pw_multi_aff(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_pullback_multi_pw_aff(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_union_pw_aff *
        isl_union_pw_aff_pullback_union_pw_multi_aff(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_union_pw_multi_aff *upma);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_pullback_union_pw_multi_aff(
                __isl_take isl_union_pw_multi_aff *upma1,
                __isl_take isl_union_pw_multi_aff *upma2);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_pullback_union_pw_multi_aff(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_union_pw_multi_aff *upma);

These functions precompose the first expression by the second function.
In other words, the second function is plugged
into the first expression.

=item * Locus

        #include <isl/aff.h>
        __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_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);

        __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_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 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>.
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_map *isl_pw_aff_eq_map(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);
        __isl_give isl_map *isl_pw_aff_lt_map(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);
        __isl_give isl_map *isl_pw_aff_gt_map(
                __isl_take isl_pw_aff *pa1,
                __isl_take isl_pw_aff *pa2);

        __isl_give isl_map *isl_multi_pw_aff_eq_map(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_map *isl_multi_pw_aff_lex_lt_map(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __isl_give isl_map *isl_multi_pw_aff_lex_gt_map(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);

These functions return a map between domain elements of the arguments
where the function values satisfy the given relation.

        #include <isl/union_map.h>
        __isl_give isl_union_map *
        isl_union_map_eq_at_multi_union_pw_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_union_pw_aff *mupa);
        __isl_give isl_union_map *
        isl_union_map_lex_lt_at_multi_union_pw_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_union_pw_aff *mupa);
        __isl_give isl_union_map *
        isl_union_map_lex_gt_at_multi_union_pw_aff(
                __isl_take isl_union_map *umap,
                __isl_take isl_multi_union_pw_aff *mupa);

These functions select the subset of elements in the union map
that have an equal or lexicographically smaller function value.

=item * Cartesian Product

        #include <isl/space.h>
        __isl_give isl_space *isl_space_product(
                __isl_take isl_space *space1,
                __isl_take isl_space *space2);
        __isl_give isl_space *isl_space_domain_product(
                __isl_take isl_space *space1,
                __isl_take isl_space *space2);
        __isl_give isl_space *isl_space_range_product(
                __isl_take isl_space *space1,
                __isl_take isl_space *space2);

The functions
C<isl_space_product>, C<isl_space_domain_product>
and C<isl_space_range_product> take pairs or relation spaces and
produce a single relations space, where either the domain, the range
or both domain and range are wrapped spaces of relations between
the domains and/or ranges of the input spaces.
If the product is only constructed over the domain or the range
then the ranges or the domains of the inputs should be the same.
The function C<isl_space_product> also accepts a pair of set spaces,
in which case it returns a wrapped space of a relation between the
two input spaces.

        #include <isl/set.h>
        __isl_give isl_set *isl_set_product(
                __isl_take isl_set *set1,
                __isl_take isl_set *set2);

        #include <isl/map.h>
        __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_map *isl_map_product(
                __isl_take isl_map *map1,
                __isl_take isl_map *map2);

        #include <isl/union_set.h>
        __isl_give isl_union_set *isl_union_set_product(
                __isl_take isl_union_set *uset1,
                __isl_take isl_union_set *uset2);

        #include <isl/union_map.h>
        __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_union_map *isl_union_map_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_range_product(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);
        __isl_give isl_multi_val *isl_multi_val_product(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);

        #include <isl/aff.h>
        __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_product(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __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_product(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __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_product(
                __isl_take isl_pw_multi_aff *pma1,
                __isl_take isl_pw_multi_aff *pma2);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_range_product(
                __isl_take isl_multi_union_pw_aff *mupa1,
                __isl_take isl_multi_union_pw_aff *mupa2);

The above functions compute the cross product of the given
sets, relations or functions.  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.

        #include <isl/set.h>
        __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);

        #include <isl/map.h>
        __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_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);

        #include <isl/union_map.h>
        __isl_give isl_union_map *
        isl_union_map_flat_domain_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);
        __isl_give isl_union_map *
        isl_union_map_flat_range_product(
                __isl_take isl_union_map *umap1,
                __isl_take isl_union_map *umap2);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_flat_range_product(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_aff *mv2);

        #include <isl/aff.h>
        __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_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_multi_pw_aff *
        isl_multi_pw_aff_flat_range_product(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __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_union_pw_aff *
        isl_multi_union_pw_aff_flat_range_product(
                __isl_take isl_multi_union_pw_aff *mupa1,
                __isl_take isl_multi_union_pw_aff *mupa2);

        #include <isl/space.h>
        __isl_give isl_space *isl_space_factor_domain(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_factor_range(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_domain_factor_domain(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_domain_factor_range(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_range_factor_domain(
                __isl_take isl_space *space);
        __isl_give isl_space *isl_space_range_factor_range(
                __isl_take isl_space *space);

The functions C<isl_space_range_factor_domain> and
C<isl_space_range_factor_range> extract the two arguments from
the result of a call to C<isl_space_range_product>.

The arguments of a call to C<isl_map_range_product> can be extracted
from the result using the following functions.

        #include <isl/map.h>
        __isl_give isl_map *isl_map_factor_domain(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_factor_range(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_domain_factor_domain(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_domain_factor_range(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_range_factor_domain(
                __isl_take isl_map *map);
        __isl_give isl_map *isl_map_range_factor_range(
                __isl_take isl_map *map);

        #include <isl/union_map.h>
        __isl_give isl_union_map *isl_union_map_factor_domain(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *isl_union_map_factor_range(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *
        isl_union_map_domain_factor_domain(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *
        isl_union_map_domain_factor_range(
                __isl_take isl_union_map *umap);
        __isl_give isl_union_map *
        isl_union_map_range_factor_range(
                __isl_take isl_union_map *umap);

        #include <isl/val.h>
        __isl_give isl_multi_val *
        isl_multi_val_range_factor_domain(
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_val *
        isl_multi_val_range_factor_range(
                __isl_take isl_multi_val *mv);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *
        isl_multi_aff_range_factor_domain(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_aff *
        isl_multi_aff_range_factor_range(
                __isl_take isl_multi_aff *ma);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_range_factor_domain(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_range_factor_range(
                __isl_take isl_multi_pw_aff *mpa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_range_factor_domain(
                __isl_take isl_multi_union_pw_aff *mupa);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_range_factor_range(
                __isl_take isl_multi_union_pw_aff *mupa);

The splice functions are a generalization of the flat product functions,
where the second argument may be inserted at any position inside
the first argument rather than being placed at the end.

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_range_splice(
                __isl_take isl_multi_val *mv1, unsigned pos,
                __isl_take isl_multi_val *mv2);

        #include <isl/aff.h>
        __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_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_union_pw_aff *
        isl_multi_union_pw_aff_range_splice(
                __isl_take isl_multi_union_pw_aff *mupa1,
                unsigned pos,
                __isl_take isl_multi_union_pw_aff *mupa2);

=item * Simplification

When applied to a set or relation,
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.

When applied to a function,
the gist operation applies the set gist operation to each of
the cells in the domain of the input piecewise expression.
The context is also exploited
to simplify the expression associated to each cell.

        #include <isl/set.h>
        __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);

        #include <isl/map.h>
        __isl_give isl_basic_map *isl_basic_map_gist(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_map *context);
        __isl_give isl_basic_map *isl_basic_map_gist_domain(
                __isl_take isl_basic_map *bmap,
                __isl_take isl_basic_set *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);

        #include <isl/union_set.h>
        __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);

        #include <isl/union_map.h>
        __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);

        #include <isl/aff.h>
        __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_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_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_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_multi_pw_aff *isl_multi_pw_aff_gist_params(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_set *set);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_gist(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_set *set);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_gist(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_union_set *context);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_gist_params(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_set *context);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_gist_params(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_set *context);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_gist(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_union_set *context);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_gist_params(
                __isl_take isl_multi_union_pw_aff *aff,
                __isl_take isl_set *context);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_gist(
                __isl_take isl_multi_union_pw_aff *aff,
                __isl_take isl_union_set *context);

        #include <isl/polynomial.h>
        __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_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 *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_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 *
        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);
        __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);

=item * Binary Arithmethic Operations

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_sub(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_add(
                __isl_take isl_aff *aff1,
                __isl_take isl_aff *aff2);
        __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_aff *isl_pw_aff_add(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __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_aff *isl_union_pw_aff_add(
                __isl_take isl_union_pw_aff *upa1,
                __isl_take isl_union_pw_aff *upa2);
        __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_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_multi_aff *isl_multi_aff_sub(
                __isl_take isl_multi_aff *ma1,
                __isl_take isl_multi_aff *ma2);
        __isl_give isl_pw_aff *isl_pw_aff_sub(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_sub(
                __isl_take isl_multi_pw_aff *mpa1,
                __isl_take isl_multi_pw_aff *mpa2);
        __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_aff *isl_union_pw_aff_sub(
                __isl_take isl_union_pw_aff *upa1,
                __isl_take isl_union_pw_aff *upa2);
        __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);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_sub(
                __isl_take isl_multi_union_pw_aff *mupa1,
                __isl_take isl_multi_union_pw_aff *mupa2);

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

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_add(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __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_add_disjoint(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __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_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_qpolynomial *isl_qpolynomial_sub(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __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_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);

        #include <isl/aff.h>
        __isl_give isl_pw_aff *isl_pw_aff_union_add(
                __isl_take isl_pw_aff *pwaff1,
                __isl_take isl_pw_aff *pwaff2);
        __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_union_pw_aff *isl_union_pw_aff_union_add(
                __isl_take isl_union_pw_aff *upa1,
                __isl_take isl_union_pw_aff *upa2);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_union_add(
                __isl_take isl_union_pw_multi_aff *upma1,
                __isl_take isl_union_pw_multi_aff *upma2);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_union_add(
                __isl_take isl_multi_union_pw_aff *mupa1,
                __isl_take isl_multi_union_pw_aff *mupa2);
        __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);

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.
This in contrast to the C<isl_pw_aff_max> function, which is
only defined on the shared definition domain of the arguments.

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_add_val(
                __isl_take isl_multi_val *mv,
                __isl_take isl_val *v);
        __isl_give isl_multi_val *isl_multi_val_mod_val(
                __isl_take isl_multi_val *mv,
                __isl_take isl_val *v);
        __isl_give isl_multi_val *isl_multi_val_scale_val(
                __isl_take isl_multi_val *mv,
                __isl_take isl_val *v);
        __isl_give isl_multi_val *isl_multi_val_scale_down_val(
                __isl_take isl_multi_val *mv,
                __isl_take isl_val *v);

        #include <isl/aff.h>
        __isl_give isl_aff *isl_aff_mod_val(__isl_take isl_aff *aff,
                __isl_take isl_val *mod);
        __isl_give isl_pw_aff *isl_pw_aff_mod_val(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_val *mod);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_mod_val(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_val *f);
        __isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
                __isl_take isl_val *v);
        __isl_give isl_multi_aff *isl_multi_aff_scale_val(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_val *v);
        __isl_give isl_pw_aff *isl_pw_aff_scale_val(
                __isl_take isl_pw_aff *pa, __isl_take isl_val *v);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_val(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_val *v);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_val(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_multi_aff *
        __isl_give isl_union_pw_aff *isl_union_pw_aff_scale_val(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_val *f);
        isl_union_pw_multi_aff_scale_val(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_val *val);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_scale_val(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_val *v);
        __isl_give isl_aff *isl_aff_scale_down_ui(
                __isl_take isl_aff *aff, unsigned f);
        __isl_give isl_aff *isl_aff_scale_down_val(
                __isl_take isl_aff *aff, __isl_take isl_val *v);
        __isl_give isl_multi_aff *isl_multi_aff_scale_down_val(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_val *v);
        __isl_give isl_pw_aff *isl_pw_aff_scale_down_val(
                __isl_take isl_pw_aff *pa,
                __isl_take isl_val *f);
        __isl_give isl_multi_pw_aff *isl_multi_pw_aff_scale_down_val(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_val *v);
        __isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_down_val(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_aff *isl_union_pw_aff_scale_down_val(
                __isl_take isl_union_pw_aff *upa,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_scale_down_val(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_val *val);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_scale_down_val(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_val *v);

        #include <isl/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_scale_val(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_val *v);
        __isl_give isl_qpolynomial_fold *
        isl_qpolynomial_fold_scale_val(
                __isl_take isl_qpolynomial_fold *fold,
                __isl_take isl_val *v);
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_scale_val(
                __isl_take isl_pw_qpolynomial *pwqp,
                __isl_take isl_val *v);
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_scale_val(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_scale_val(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_scale_val(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_val *v);
        __isl_give isl_qpolynomial *
        isl_qpolynomial_scale_down_val(
                __isl_take isl_qpolynomial *qp,
                __isl_take isl_val *v);
        __isl_give isl_qpolynomial_fold *
        isl_qpolynomial_fold_scale_down_val(
                __isl_take isl_qpolynomial_fold *fold,
                __isl_take isl_val *v);
        __isl_give isl_pw_qpolynomial *
        isl_pw_qpolynomial_scale_down_val(
                __isl_take isl_pw_qpolynomial *pwqp,
                __isl_take isl_val *v);
        __isl_give isl_pw_qpolynomial_fold *
        isl_pw_qpolynomial_fold_scale_down_val(
                __isl_take isl_pw_qpolynomial_fold *pwf,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_qpolynomial *
        isl_union_pw_qpolynomial_scale_down_val(
                __isl_take isl_union_pw_qpolynomial *upwqp,
                __isl_take isl_val *v);
        __isl_give isl_union_pw_qpolynomial_fold *
        isl_union_pw_qpolynomial_fold_scale_down_val(
                __isl_take isl_union_pw_qpolynomial_fold *upwf,
                __isl_take isl_val *v);

        #include <isl/val.h>
        __isl_give isl_multi_val *isl_multi_val_mod_multi_val(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);
        __isl_give isl_multi_val *isl_multi_val_scale_multi_val(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);
        __isl_give isl_multi_val *
        isl_multi_val_scale_down_multi_val(
                __isl_take isl_multi_val *mv1,
                __isl_take isl_multi_val *mv2);

        #include <isl/aff.h>
        __isl_give isl_multi_aff *isl_multi_aff_mod_multi_val(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_mod_multi_val(
                __isl_take isl_multi_union_pw_aff *upma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_mod_multi_val(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_aff *isl_multi_aff_scale_multi_val(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_pw_multi_aff *
        isl_pw_multi_aff_scale_multi_val(
                __isl_take isl_pw_multi_aff *pma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_scale_multi_val(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_scale_multi_val(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_multi_val *mv);
        __isl_give isl_union_pw_multi_aff *
        isl_union_pw_multi_aff_scale_multi_val(
                __isl_take isl_union_pw_multi_aff *upma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_aff *
        isl_multi_aff_scale_down_multi_val(
                __isl_take isl_multi_aff *ma,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_pw_aff *
        isl_multi_pw_aff_scale_down_multi_val(
                __isl_take isl_multi_pw_aff *mpa,
                __isl_take isl_multi_val *mv);
        __isl_give isl_multi_union_pw_aff *
        isl_multi_union_pw_aff_scale_down_multi_val(
                __isl_take isl_multi_union_pw_aff *mupa,
                __isl_take isl_multi_val *mv);

C<isl_multi_aff_scale_multi_val> scales the elements of C<ma>
by the corresponding elements of C<mv>.

        #include <isl/aff.h>
        __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/polynomial.h>
        __isl_give isl_qpolynomial *isl_qpolynomial_mul(
                __isl_take isl_qpolynomial *qp1,
                __isl_take isl_qpolynomial *qp2);
        __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
                __isl_take isl_pw_qpolynomial *pwqp1,
                __isl_take isl_pw_qpolynomial *pwqp2);
        __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
                __isl_take isl_union_pw_qpolynomial *upwqp1,
                __isl_take isl_union_pw_qpolynomial *upwqp2);

=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>.

        #include <isl/set.h>
        __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.

        #include <isl/set.h>
        __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>.

        #include <isl/map.h>
        __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.

        #include <isl/map.h>
        __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,
but are otherwise equivalent to the corresponding functions
returning a basic set or relation.

        #include <isl/set.h>
        __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_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);

        #include <isl/map.h>
        __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_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_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);

The following functions return the lexicographic minimum or maximum
on the shared domain of the inputs and the single defined function
on those parts of the domain where only a single function is defined.

        #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);

=head2 Ternary Operations

        #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.

=head2 Lists

Lists are defined over several element types, including
C<isl_val>, C<isl_id>, C<isl_aff>, C<isl_pw_aff>, C<isl_union_pw_aff>,
C<isl_union_pw_multi_aff>, C<isl_constraint>,
C<isl_basic_set>, C<isl_set>, C<isl_basic_map>, C<isl_map>, C<isl_union_set>,
C<isl_union_map>, 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/set.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);
        __isl_null isl_set_list *isl_set_list_free(
                __isl_take isl_set_list *list);

C<isl_set_list_alloc> creates an empty list with an initial capacity
for C<n> elements.  C<isl_set_list_insert> and C<isl_set_list_add>
add elements to a list, increasing its capacity as needed.
C<isl_set_list_from_set> creates a list with a single element.

Lists can be inspected using the following functions.

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

=head2 Associative arrays

Associative arrays map isl objects of a specific type to isl objects
of some (other) specific type.  They are defined for several pairs
of types, including (C<isl_map>, C<isl_basic_set>),
(C<isl_id>, C<isl_ast_expr>) and.
(C<isl_id>, C<isl_pw_aff>).
Here, we take associative arrays that map C<isl_id>s to C<isl_ast_expr>s
as an example.

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

        #include <isl/id_to_ast_expr.h>
        __isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_alloc(
                isl_ctx *ctx, int min_size);
        __isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_copy(
                __isl_keep isl_id_to_ast_expr *id2expr);
        __isl_null isl_id_to_ast_expr *isl_id_to_ast_expr_free(
                __isl_take isl_id_to_ast_expr *id2expr);

The C<min_size> argument to C<isl_id_to_ast_expr_alloc> can be used
to specify the expected size of the associative array.
The associative array will be grown automatically as needed.

Associative arrays can be inspected using the following functions.

        #include <isl/id_to_ast_expr.h>
        int isl_id_to_ast_expr_has(
                __isl_keep isl_id_to_ast_expr *id2expr,
                __isl_keep isl_id *key);
        __isl_give isl_ast_expr *isl_id_to_ast_expr_get(
                __isl_keep isl_id_to_ast_expr *id2expr,
                __isl_take isl_id *key);
        int isl_id_to_ast_expr_foreach(
                __isl_keep isl_id_to_ast_expr *id2expr,
                int (*fn)(__isl_take isl_id *key,
                        __isl_take isl_ast_expr *val, void *user),
                void *user);

They can be modified using the following function.

        #include <isl/id_to_ast_expr.h>
        __isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_set(
                __isl_take isl_id_to_ast_expr *id2expr,
                __isl_take isl_id *key,
                __isl_take isl_ast_expr *val);
        __isl_give isl_id_to_ast_expr *isl_id_to_ast_expr_drop(
                __isl_take isl_id_to_ast_expr *id2expr,
                __isl_take isl_id *key);

Associative arrays can be printed using the following function.

        #include <isl/id_to_ast_expr.h>
        __isl_give isl_printer *isl_printer_print_id_to_ast_expr(
                __isl_take isl_printer *p,
                __isl_keep isl_id_to_ast_expr *id2expr);

=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);
        __isl_null isl_vec *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.

        int isl_vec_size(__isl_keep isl_vec *vec);
        __isl_give isl_val *isl_vec_get_element_val(
                __isl_keep isl_vec *vec, int pos);
        __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_element_val(
                __isl_take isl_vec *vec, int pos,
                __isl_take isl_val *v);
        __isl_give isl_vec *isl_vec_set_si(__isl_take isl_vec *vec,
                int v);
        __isl_give isl_vec *isl_vec_set_val(
                __isl_take isl_vec *vec, __isl_take isl_val *v);
        int isl_vec_cmp_element(__isl_keep isl_vec *vec1,
                __isl_keep isl_vec *vec2, int pos);

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);
        __isl_null isl_mat *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.

        int isl_mat_rows(__isl_keep isl_mat *mat);
        int isl_mat_cols(__isl_keep isl_mat *mat);
        __isl_give isl_val *isl_mat_get_element_val(
                __isl_keep isl_mat *mat, int row, int col);
        __isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
                int row, int col, int v);
        __isl_give isl_mat *isl_mat_set_element_val(
                __isl_take isl_mat *mat, int row, int col,
                __isl_take isl_val *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 Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions

The following functions determine
an upper or lower bound on a quasipolynomial over its domain.

        __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.

=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.

        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.

        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_multi_aff *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 multiple quasi-affine expression
describing the vertex in terms of the parameters,
while C<isl_vertex_get_domain> returns the activity domain
of the vertex.

Chambers can be inspected and destroyed using the following functions.

        __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 Schedule Trees

A schedule tree is a structured representation of a schedule,
assigning a relative order to a set of domain elements.
The relative order expressed by the schedule tree is
defined recursively.  In particular, the order between
two domain elements is determined by the node that is closest
to the root that refers to both elements and that orders them apart.
Each node in the tree is of one of several types.
The root node is always of type C<isl_schedule_node_domain>
and it describes the domain elements to which the schedule applies.
The other types of nodes are as follows.

=over

=item C<isl_schedule_node_band>

A band of schedule dimensions.  Each schedule dimension is represented
by a union piecewise quasi-affine expression.  If this expression
assigns a different value to two domain elements, while all previous
schedule dimensions in the same band assign them the same value,
then the two domain elements are ordered according to these two
different values.

=item C<isl_schedule_node_expansion>

An expansion node maps each of the domain elements that reach the node
to one or more domain elements.  The image of this mapping forms
the set of domain elements that reach the child of the expansion node.
The function that maps each of the expanded domain elements
to the original domain element from which it was expanded
is called the contraction.

=item C<isl_schedule_node_filter>

A filter node does not impose any ordering, but rather intersects
the set of domain elements that the current subtree refers to
with a given union set.  The subtree of the filter node only
refers to domain elements in the intersection.
A filter node is typically only used a child of a sequence or
set node.

=item C<isl_schedule_node_leaf>

A leaf of the schedule tree.  Leaf nodes do not impose any ordering.

=item C<isl_schedule_node_mark>

A mark node can be used to attach any kind of information to a subtree
of the schedule tree.

=item C<isl_schedule_node_sequence>

A sequence node has one or more children, each of which is a filter node.
The filters on these filter nodes form a partition of
the domain elements that the current subtree refers to.
If two domain elements appear in distinct filters then the sequence
node orders them according to the child positions of the corresponding
filter nodes.

=item C<isl_schedule_node_set>

A set node is similar to a sequence node, except that
it expresses that domain elements appearing in distinct filters
may have any order.  The order of the children of a set node
is therefore also immaterial.

=back

The following node type is only supported by the AST generator.

=over

=item C<isl_schedule_node_context>

The context describes constraints on the parameters and
the schedule dimensions of outer
bands that the AST generator may assume to hold.  It is also the only
kind of node that may introduce additional parameters.
The space of the context is that of the flat product of the outer
band nodes.  In particular, if there are no outer band nodes, then
this space is the unnamed zero-dimensional space.
Since a context node references the outer band nodes, any tree
containing a context node is considered to be anchored.

=back

Except for the C<isl_schedule_node_context> nodes,
none of the nodes may introduce any parameters that were not
already present in the root domain node.

A schedule tree is encapsulated in an C<isl_schedule> object.
The simplest such objects, those with a tree consisting of single domain node,
can be created using the following functions with either an empty
domain or a given domain.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_empty(
                __isl_take isl_space *space);
        __isl_give isl_schedule *isl_schedule_from_domain(
                __isl_take isl_union_set *domain);

The function C<isl_schedule_constraints_compute_schedule> described
in L</"Scheduling"> can also be used to construct schedules.

C<isl_schedule> objects may be copied and freed using the following functions.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_copy(
                __isl_keep isl_schedule *sched);
        __isl_null isl_schedule *isl_schedule_free(
                __isl_take isl_schedule *sched);

The following functions checks whether two C<isl_schedule> objects
are obviously the same.

        #include <isl/schedule.h>
        int isl_schedule_plain_is_equal(
                __isl_keep isl_schedule *schedule1,
                __isl_keep isl_schedule *schedule2);

The domain of the schedule, i.e., the domain described by the root node,
can be obtained using the following function.

        #include <isl/schedule.h>
        __isl_give isl_union_set *isl_schedule_get_domain(
                __isl_keep isl_schedule *schedule);

An extra top-level band node (right underneath the domain node) can
be introduced into the schedule using the following function.
The schedule tree is assumed not to have any anchored nodes.

        #include <isl/schedule.h>
        __isl_give isl_schedule *
        isl_schedule_insert_partial_schedule(
                __isl_take isl_schedule *schedule,
                __isl_take isl_multi_union_pw_aff *partial);

A top-level context node (right underneath the domain node) can
be introduced into the schedule using the following function.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_insert_context(
                __isl_take isl_schedule *schedule,
                __isl_take isl_set *context)

A schedule that combines two schedules either in the given
order or in an arbitrary order, i.e., with an C<isl_schedule_node_sequence>
or an C<isl_schedule_node_set> node,
can be created using the following functions.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_sequence(
                __isl_take isl_schedule *schedule1,
                __isl_take isl_schedule *schedule2);
        __isl_give isl_schedule *isl_schedule_set(
                __isl_take isl_schedule *schedule1,
                __isl_take isl_schedule *schedule2);

The domains of the two input schedules need to be disjoint.

The following function can be used to restrict the domain
of a schedule to be a subset of the given union set.
This operation may remove nodes in the tree that have become
redundant.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_intersect_domain(
                __isl_take isl_schedule *schedule,
                __isl_take isl_union_set *domain);

The following function resets the user pointers on all parameter
and tuple identifiers referenced by the nodes of the given schedule.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_reset_user(
                __isl_take isl_schedule *schedule);

The following function aligns the parameters of all nodes
in the given schedule to the given space.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_align_params(
                __isl_take isl_schedule *schedule,
                __isl_take isl_space *space);

The following function allows the user to plug in a given function
in the iteration domains.  The input schedule is not allowed to contain
any expansion nodes.

        #include <isl/schedule.h>
        __isl_give isl_schedule *
        isl_schedule_pullback_union_pw_multi_aff(
                __isl_take isl_schedule *schedule,
                __isl_take isl_union_pw_multi_aff *upma);

An C<isl_union_map> representation of the schedule can be obtained
from an C<isl_schedule> using the following function.

        #include <isl/schedule.h>
        __isl_give isl_union_map *isl_schedule_get_map(
                __isl_keep isl_schedule *sched);

The resulting relation encodes the same relative ordering as
the schedule by mapping the domain elements to a common schedule space.
If the schedule_separate_components option is set, then the order
of the children of a set node is explicitly encoded in the result.
If the tree contains any expansion nodes, then the relation
is formulated in terms of the expanded domain elements.

Schedules can be read from input using the following functions.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_read_from_file(
                isl_ctx *ctx, FILE *input);
        __isl_give isl_schedule *isl_schedule_read_from_str(
                isl_ctx *ctx, const char *str);

A representation of the schedule can be printed using

        #include <isl/schedule.h>
        __isl_give isl_printer *isl_printer_print_schedule(
                __isl_take isl_printer *p,
                __isl_keep isl_schedule *schedule);

The schedule tree can be traversed through the use of
C<isl_schedule_node> objects that point to a particular
position in the schedule tree.  Whenever a C<isl_schedule_node>
is use to modify a node in the schedule tree, the original schedule
tree is left untouched and the modifications are performed to a copy
of the tree.  The returned C<isl_schedule_node> then points to
this modified copy of the tree.

The root of the schedule tree can be obtained using the following function.

        #include <isl/schedule.h>
        __isl_give isl_schedule_node *isl_schedule_get_root(
                __isl_keep isl_schedule *schedule);

A pointer to a newly created schedule tree with a single domain
node can be created using the following function.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_from_domain(
                __isl_take isl_union_set *domain);

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

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_copy(
                __isl_keep isl_schedule_node *node);
        __isl_null isl_schedule_node *isl_schedule_node_free(
                __isl_take isl_schedule_node *node);

The following functions can be used to check if two schedule
nodes point to the same position in the same schedule.

        #include <isl/schedule_node.h>
        int isl_schedule_node_is_equal(
                __isl_keep isl_schedule_node *node1,
                __isl_keep isl_schedule_node *node2);

The following properties can be obtained from a schedule node.

        #include <isl/schedule_node.h>
        enum isl_schedule_node_type isl_schedule_node_get_type(
                __isl_keep isl_schedule_node *node);
        enum isl_schedule_node_type
        isl_schedule_node_get_parent_type(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule *isl_schedule_node_get_schedule(
                __isl_keep isl_schedule_node *node);

The function C<isl_schedule_node_get_type> returns the type of
the node, while C<isl_schedule_node_get_parent_type> returns
type of the parent of the node, which is required to exist.
The function C<isl_schedule_node_get_schedule> returns a copy
to the schedule to which the node belongs.

The following functions can be used to move the schedule node
to a different position in the tree or to check if such a position
exists.

        #include <isl/schedule_node.h>
        int isl_schedule_node_has_parent(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *isl_schedule_node_parent(
                __isl_take isl_schedule_node *node);
        __isl_give isl_schedule_node *isl_schedule_node_root(
                __isl_take isl_schedule_node *node);
        __isl_give isl_schedule_node *isl_schedule_node_ancestor(
                __isl_take isl_schedule_node *node,
                int generation);
        int isl_schedule_node_n_children(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *isl_schedule_node_child(
                __isl_take isl_schedule_node *node, int pos);
        int isl_schedule_node_has_children(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *isl_schedule_node_first_child(
                __isl_take isl_schedule_node *node);
        int isl_schedule_node_has_previous_sibling(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *
        isl_schedule_node_previous_sibling(
                __isl_take isl_schedule_node *node);
        int isl_schedule_node_has_next_sibling(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *
        isl_schedule_node_next_sibling(
                __isl_take isl_schedule_node *node);

For C<isl_schedule_node_ancestor>, the ancestor of generation 0
is the node itself, the ancestor of generation 1 is its parent and so on.

It is also possible to query the number of ancestors of a node,
the position of the current node
within the children of its parent, the position of the subtree
containing a node within the children of an ancestor
or to obtain a copy of a given
child without destroying the current node.
Given two nodes that point to the same schedule, their closest
shared ancestor can be obtained using
C<isl_schedule_node_get_shared_ancestor>.

        #include <isl/schedule_node.h>
        int isl_schedule_node_get_tree_depth(
                __isl_keep isl_schedule_node *node);
        int isl_schedule_node_get_child_position(
                __isl_keep isl_schedule_node *node);
        int isl_schedule_node_get_ancestor_child_position(
                __isl_keep isl_schedule_node *node,
                __isl_keep isl_schedule_node *ancestor);
        __isl_give isl_schedule_node *isl_schedule_node_get_child(
                __isl_keep isl_schedule_node *node, int pos);
        __isl_give isl_schedule_node *
        isl_schedule_node_get_shared_ancestor(
                __isl_keep isl_schedule_node *node1,
                __isl_keep isl_schedule_node *node2);

All nodes in a schedule tree or
all descendants of a specific node (including the node) can be visited
in depth-first pre-order using the following functions.

        #include <isl/schedule.h>
        int isl_schedule_foreach_schedule_node(
                __isl_keep isl_schedule *sched,
                int (*fn)(__isl_keep isl_schedule_node *node,
                        void *user), void *user);

        #include <isl/schedule_node.h>
        int isl_schedule_node_foreach_descendant(
                __isl_keep isl_schedule_node *node,
                int (*fn)(__isl_keep isl_schedule_node *node,
                        void *user), void *user);

The callback function is slightly different from the usual
callbacks in that it not only indicates success (non-negative result)
or failure (negative result), but also indicates whether the children
of the given node should be visited.  In particular, if the callback
returns a positive value, then the children are visited, but if
the callback returns zero, then the children are not visited.

The ancestors of a node in a schedule tree can be visited from
the root down to and including the parent of the node using
the following function.

        #include <isl/schedule_node.h>
        int isl_schedule_node_foreach_ancestor_top_down(
                __isl_keep isl_schedule_node *node,
                int (*fn)(__isl_keep isl_schedule_node *node,
                        void *user), void *user);

The following functions allows for a depth-first post-order
traversal of the nodes in a schedule tree or
of the descendants of a specific node (including the node
itself), where the user callback is allowed to modify the
visited node.

        #include <isl/schedule.h>
        __isl_give isl_schedule *isl_schedule_map_schedule_node(
                __isl_take isl_schedule *schedule,
                __isl_give isl_schedule_node *(*fn)(
                        __isl_take isl_schedule_node *node,
                        void *user), void *user);

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_map_descendant(
                __isl_take isl_schedule_node *node,
                __isl_give isl_schedule_node *(*fn)(
                        __isl_take isl_schedule_node *node,
                        void *user), void *user);

The traversal continues from the node returned by the callback function.
It is the responsibility of the user to ensure that this does not
lead to an infinite loop.  It is safest to always return a pointer
to the same position (same ancestors and child positions) as the input node.

The following function removes a node (along with its descendants)
from a schedule tree and returns a pointer to the leaf at the
same position in the updated tree.
It is not allowed to remove the root of a schedule tree or
a child of a set or sequence node.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_cut(
                __isl_take isl_schedule_node *node);

The following function removes a single node
from a schedule tree and returns a pointer to the child
of the node, now located at the position of the original node
or to a leaf node at that position if there was no child.
It is not allowed to remove the root of a schedule tree,
a set or sequence node, a child of a set or sequence node or
a band node with an anchored subtree.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_delete(
                __isl_take isl_schedule_node *node);

Most nodes in a schedule tree only contain local information.
In some cases, however, a node may also refer to outer band nodes.
This means that the position of the node within the tree should
not be changed, or at least that no changes are performed to the
outer band nodes.  The following function can be used to test
whether the subtree rooted at a given node contains any such nodes.

        #include <isl/schedule_node.h>
        int isl_schedule_node_is_subtree_anchored(
                __isl_keep isl_schedule_node *node);

The following function resets the user pointers on all parameter
and tuple identifiers referenced by the given schedule node.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_reset_user(
                __isl_take isl_schedule_node *node);

The following function aligns the parameters of the given schedule
node to the given space.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_align_params(
                __isl_take isl_schedule_node *node,
                __isl_take isl_space *space);

Several node types have their own functions for querying
(and in some cases setting) some node type specific properties.

        #include <isl/schedule_node.h>
        __isl_give isl_space *isl_schedule_node_band_get_space(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_multi_union_pw_aff *
        isl_schedule_node_band_get_partial_schedule(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_map *
        isl_schedule_node_band_get_partial_schedule_union_map(
                __isl_keep isl_schedule_node *node);
        unsigned isl_schedule_node_band_n_member(
                __isl_keep isl_schedule_node *node);
        int isl_schedule_node_band_member_get_coincident(
                __isl_keep isl_schedule_node *node, int pos);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_member_set_coincident(
                __isl_take isl_schedule_node *node, int pos,
                int coincident);
        int isl_schedule_node_band_get_permutable(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_set_permutable(
                __isl_take isl_schedule_node *node, int permutable);
        enum isl_ast_loop_type
        isl_schedule_node_band_member_get_ast_loop_type(
                __isl_keep isl_schedule_node *node, int pos);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_member_set_ast_loop_type(
                __isl_take isl_schedule_node *node, int pos,
                enum isl_ast_loop_type type);
        __isl_give isl_union_set *
        enum isl_ast_loop_type
        isl_schedule_node_band_member_get_isolate_ast_loop_type(
                __isl_keep isl_schedule_node *node, int pos);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_member_set_isolate_ast_loop_type(
                __isl_take isl_schedule_node *node, int pos,
                enum isl_ast_loop_type type);
        isl_schedule_node_band_get_ast_build_options(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_set_ast_build_options(
                __isl_take isl_schedule_node *node,
                __isl_take isl_union_set *options);

The function C<isl_schedule_node_band_get_space> returns the space
of the partial schedule of the band.
The function C<isl_schedule_node_band_get_partial_schedule_union_map>
returns a representation of the partial schedule of the band node
in the form of an C<isl_union_map>.
The coincident and permutable properties are set by
C<isl_schedule_constraints_compute_schedule> on the schedule tree
it produces.
A scheduling dimension is considered to be ``coincident''
if it satisfies the coincidence constraints within its band.
That is, if the dependence distances of the coincidence
constraints are all zero in that direction (for fixed
iterations of outer bands).
A band is marked permutable if it was produced using the Pluto-like scheduler.
Note that the scheduler may have to resort to a Feautrier style scheduling
step even if the default scheduler is used.
An C<isl_ast_loop_type> is one of C<isl_ast_loop_default>,
C<isl_ast_loop_atomic>, C<isl_ast_loop_unroll> or C<isl_ast_loop_separate>.
For the meaning of these loop AST generation types and the difference
between the regular loop AST generation type and the isolate
loop AST generation type, see L</"AST Generation Options (Schedule Tree)">.
The functions C<isl_schedule_node_band_member_get_ast_loop_type>
and C<isl_schedule_node_band_member_get_isolate_ast_loop_type>
may return C<isl_ast_loop_error> if an error occurs.
The AST build options govern how an AST is generated for
the individual schedule dimensions during AST generation.
See L</"AST Generation Options (Schedule Tree)">.

        #include <isl/schedule_node.h>
        __isl_give isl_set *
        isl_schedule_node_context_get_context(
                __isl_keep isl_schedule_node *node);

        #include <isl/schedule_node.h>
        __isl_give isl_union_set *
        isl_schedule_node_domain_get_domain(
                __isl_keep isl_schedule_node *node);

        #include <isl/schedule_node.h>
        __isl_give isl_union_map *
        isl_schedule_node_expansion_get_expansion(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_pw_multi_aff *
        isl_schedule_node_expansion_get_contraction(
                __isl_keep isl_schedule_node *node);

        #include <isl/schedule_node.h>
        __isl_give isl_union_set *
        isl_schedule_node_filter_get_filter(
                __isl_keep isl_schedule_node *node);

        #include <isl/schedule_node.h>
        __isl_give isl_id *isl_schedule_node_mark_get_id(
                __isl_keep isl_schedule_node *node);

The following functions can be used to obtain an C<isl_multi_union_pw_aff>,
an C<isl_union_pw_multi_aff> or C<isl_union_map> representation of
partial schedules related to the node.

        #include <isl/schedule_node.h>
        __isl_give isl_multi_union_pw_aff *
        isl_schedule_node_get_prefix_schedule_multi_union_pw_aff(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_pw_multi_aff *
        isl_schedule_node_get_prefix_schedule_union_pw_multi_aff(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_map *
        isl_schedule_node_get_prefix_schedule_union_map(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_map *
        isl_schedule_node_get_prefix_schedule_relation(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_map *
        isl_schedule_node_get_subtree_schedule_union_map(
                __isl_keep isl_schedule_node *node);

In particular, the functions
C<isl_schedule_node_get_prefix_schedule_multi_union_pw_aff>,
C<isl_schedule_node_get_prefix_schedule_union_pw_multi_aff>
and C<isl_schedule_node_get_prefix_schedule_union_map>
return a relative ordering on the domain elements that reach the given
node determined by its ancestors.
The function C<isl_schedule_node_get_prefix_schedule_relation>
additionally includes the domain constraints in the result.
The function C<isl_schedule_node_get_subtree_schedule_union_map>
returns a representation of the partial schedule defined by the
subtree rooted at the given node.
If the tree contains any expansion nodes, then the subtree schedule
is formulated in terms of the expanded domain elements.

The expansion/contraction defined by an entire subtree, combining
the expansions/contractions
on the expansion nodes in the subtree, can be obtained using
the following functions.

        #include <isl/schedule_node.h>
        __isl_give isl_union_map *
        isl_schedule_node_get_subtree_expansion(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_pw_multi_aff *
        isl_schedule_node_get_subtree_contraction(
                __isl_keep isl_schedule_node *node);

The total number of outer band members of given node, i.e.,
the shared output dimension of the maps in the result
of C<isl_schedule_node_get_prefix_schedule_union_map> can be obtained
using the following function.

        #include <isl/schedule_node.h>
        int isl_schedule_node_get_schedule_depth(
                __isl_keep isl_schedule_node *node);

The following functions return the elements that reach the given node
or the union of universes in the spaces that contain these elements.

        #include <isl/schedule_node.h>
        __isl_give isl_union_set *
        isl_schedule_node_get_domain(
                __isl_keep isl_schedule_node *node);
        __isl_give isl_union_set *
        isl_schedule_node_get_universe_domain(
                __isl_keep isl_schedule_node *node);

The following functions can be used to introduce additional nodes
in the schedule tree.  The new node is introduced at the point
in the tree where the C<isl_schedule_node> points to and
the results points to the new node.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_partial_schedule(
                __isl_take isl_schedule_node *node,
                __isl_take isl_multi_union_pw_aff *schedule);

This function inserts a new band node with (the greatest integer
part of) the given partial schedule.
The subtree rooted at the given node is assumed not to have
any anchored nodes.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_context(
                __isl_take isl_schedule_node *node,
                __isl_take isl_set *context);

This function inserts a new context node with the given context constraints.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_filter(
                __isl_take isl_schedule_node *node,
                __isl_take isl_union_set *filter);

This function inserts a new filter node with the given filter.
If the original node already pointed to a filter node, then the
two filter nodes are merged into one.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_mark(
                __isl_take isl_schedule_node *node,
                __isl_take isl_id *mark);

This function inserts a new mark node with the give mark identifier.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_sequence(
                __isl_take isl_schedule_node *node,
                __isl_take isl_union_set_list *filters);
        __isl_give isl_schedule_node *
        isl_schedule_node_insert_set(
                __isl_take isl_schedule_node *node,
                __isl_take isl_union_set_list *filters);

These functions insert a new sequence or set node with the given
filters as children.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_group(
                __isl_take isl_schedule_node *node,
                __isl_take isl_id *group_id);

This function introduces an expansion node in between the current
node and its parent that expands instances of a space with tuple
identifier C<group_id> to the original domain elements that reach
the node.  The group instances are identified by the prefix schedule
of those domain elements.  The ancestors of the node are adjusted
to refer to the group instances instead of the original domain
elements.  The return value points to the same node in the updated
schedule tree as the input node, i.e., to the child of the newly
introduced expansion node.  Grouping instances of different statements
ensures that they will be treated as a single statement by the
AST generator up to the point of the expansion node.

The partial schedule of a band node can be scaled (down) using
the following functions.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *
        isl_schedule_node_band_scale(
                __isl_take isl_schedule_node *node,
                __isl_take isl_multi_val *mv);
        __isl_give isl_schedule_node *
        isl_schedule_node_band_scale_down(
                __isl_take isl_schedule_node *node,
                __isl_take isl_multi_val *mv);

The spaces of the two arguments need to match.
After scaling, the partial schedule is replaced by its greatest
integer part to ensure that the schedule remains integral.

A band node can be tiled using the following function.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_band_tile(
                __isl_take isl_schedule_node *node,
                __isl_take isl_multi_val *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_schedule_node_band_tile> function tiles
the band using the given tile sizes inside its schedule.
A new child band node is created to represent the point loops and it is
inserted between the modified band and its children.
The subtree rooted at the given node is assumed not to have
any anchored nodes.
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 node can be split into two nested band nodes
using the following function.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_band_split(
                __isl_take isl_schedule_node *node, int pos);

The resulting outer band node contains the first C<pos> dimensions of
the schedule of C<node> while the inner band contains the remaining dimensions.
The schedules of the two band nodes live in anonymous spaces.

A band node can be moved down to the leaves of the subtree rooted
at the band node using the following function.

        #include <isl/schedule_node.h>
        __isl_give isl_schedule_node *isl_schedule_node_band_sink(
                __isl_take isl_schedule_node *node);

The subtree rooted at the given node is assumed not to have
any anchored nodes.
The result points to the node in the resulting tree that is in the same
position as the node pointed to by C<node> in the original tree.

A representation of the schedule node can be printed using

        #include <isl/schedule_node.h>
        __isl_give isl_printer *isl_printer_print_schedule_node(
                __isl_take isl_printer *p,
                __isl_keep isl_schedule_node *node);

=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.

=head3 High-level Interface

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

        #include <isl/flow.h>
        __isl_give isl_union_flow *
        isl_union_access_info_compute_flow(
                __isl_take isl_union_access_info *access);

The input C<isl_union_access_info> object describes the sink
access relations, the source access relations and a schedule,
while the output C<isl_union_flow> object describes
the resulting dependence relations and the subsets of the
sink relations for which no source was found.

An C<isl_union_access_info> is created, modified and freed using
the following functions.

        #include <isl/flow.h>
        __isl_give isl_union_access_info *
        isl_union_access_info_from_sink(
                __isl_take isl_union_map *sink);
        __isl_give isl_union_access_info *
        isl_union_access_info_set_must_source(
                __isl_take isl_union_access_info *access,
                __isl_take isl_union_map *must_source);
        __isl_give isl_union_access_info *
        isl_union_access_info_set_may_source(
                __isl_take isl_union_access_info *access,
                __isl_take isl_union_map *may_source);
        __isl_give isl_union_access_info *
        isl_union_access_info_set_schedule(
                __isl_take isl_union_access_info *access,
                __isl_take isl_schedule *schedule);
        __isl_give isl_union_access_info *
        isl_union_access_info_set_schedule_map(
                __isl_take isl_union_access_info *access,
                __isl_take isl_union_map *schedule_map);
        __isl_null isl_union_access_info *
        isl_union_access_info_free(
                __isl_take isl_union_access_info *access);

The may sources set by C<isl_union_access_info_set_may_source>
do not need to include the must sources set by
C<isl_union_access_info_set_must_source> as a subset.
The user is free not to call one (or both) of these functions,
in which case the corresponding set is kept to its empty default.
Similarly, the default schedule initialized by
C<isl_union_access_info_from_sink> is empty.
The current schedule is determined by the last call to either
C<isl_union_access_info_set_schedule> or
C<isl_union_access_info_set_schedule_map>.
The domain of the schedule corresponds to the domains of
the access relations.  In particular, the domains of the access
relations are effectively intersected with the domain of the schedule
and only the resulting accesses are considered by the dependence analysis.

The output of C<isl_union_access_info_compute_flow> can be examined
and freed using the following functions.

        #include <isl/flow.h>
        __isl_give isl_union_map *isl_union_flow_get_must_dependence(
                __isl_keep isl_union_flow *flow);
        __isl_give isl_union_map *isl_union_flow_get_may_dependence(
                __isl_keep isl_union_flow *flow);
        __isl_give isl_union_map *isl_union_flow_get_must_no_source(
                __isl_keep isl_union_flow *flow);
        __isl_give isl_union_map *isl_union_flow_get_may_no_source(
                __isl_keep isl_union_flow *flow);
        __isl_null isl_union_flow *isl_union_flow_free(
                __isl_take isl_union_flow *flow);

The relation returned by C<isl_union_flow_get_must_dependence>
relates domain elements of must sources to domain elements of the sink.
The relation returned by C<isl_union_flow_get_may_dependence>
relates domain elements of must or may sources to domain elements of the sink
and includes the previous relation as a subset.
The relation returned by C<isl_union_flow_get_must_no_source> is the subset
of the sink relation for which no dependences have been found.
The relation returned by C<isl_union_flow_get_may_no_source> is the subset
of the sink relation for which no definite dependences have been found.
That is, it contains those sink access that do not contribute to any
of the elements in the relation returned
by C<isl_union_flow_get_must_dependence>.

=head3 Low-level Interface

A lower-level interface is provided by the following functions.

        #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);
        __isl_null isl_access_info *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.

=head3 Interaction with the Low-level Interface

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);
        __isl_null isl_restriction *isl_restriction_free(
                __isl_take 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.>

        #include <isl/schedule.h>
        __isl_give isl_schedule *
        isl_schedule_constraints_compute_schedule(
                __isl_take isl_schedule_constraints *sc);

The function C<isl_schedule_constraints_compute_schedule> can be
used to compute a schedule that satisfies the given schedule constraints.
These schedule constraints include the iteration domain for which
a schedule should be computed and dependences between pairs of
iterations.  In particular, these dependences include
I<validity> dependences and I<proximity> dependences.
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 validity dependences.
That is, all dependence distances over these dependences in the
scheduled space are lexicographically positive.
The default algorithm tries to ensure that the dependence distances
over coincidence constraints are zero and to minimize the
dependence distances over proximity dependences.
Moreover, it tries to obtain sequences (bands) of schedule dimensions
for groups of domains where the dependence distances over validity
dependences have only non-negative values.
When using Feautrier's algorithm, the coincidence and proximity constraints
are only taken into account during the extension to a
full-dimensional schedule.

An C<isl_schedule_constraints> object can be constructed
and manipulated using the following functions.

        #include <isl/schedule.h>
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_copy(
                __isl_keep isl_schedule_constraints *sc);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_on_domain(
                __isl_take isl_union_set *domain);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_set_context(
                __isl_take isl_schedule_constraints *sc,
                __isl_take isl_set *context);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_set_validity(
                __isl_take isl_schedule_constraints *sc,
                __isl_take isl_union_map *validity);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_set_coincidence(
                __isl_take isl_schedule_constraints *sc,
                __isl_take isl_union_map *coincidence);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_set_proximity(
                __isl_take isl_schedule_constraints *sc,
                __isl_take isl_union_map *proximity);
        __isl_give isl_schedule_constraints *
        isl_schedule_constraints_set_conditional_validity(
                __isl_take isl_schedule_constraints *sc,
                __isl_take isl_union_map *condition,
                __isl_take isl_union_map *validity);
        __isl_null isl_schedule_constraints *
        isl_schedule_constraints_free(
                __isl_take isl_schedule_constraints *sc);

The initial C<isl_schedule_constraints> object created by
C<isl_schedule_constraints_on_domain> does not impose any constraints.
That is, it has an empty set of dependences.
The function C<isl_schedule_constraints_set_context> allows the user
to specify additional constraints on the parameters that may
be assumed to hold during the construction of the schedule.
The function C<isl_schedule_constraints_set_validity> replaces the
validity dependences, mapping domain elements I<i> to domain
elements that should be scheduled after I<i>.
The function C<isl_schedule_constraints_set_coincidence> replaces the
coincidence dependences, mapping domain elements I<i> to domain
elements that should be scheduled together with I<I>, if possible.
The function C<isl_schedule_constraints_set_proximity> replaces the
proximity dependences, mapping domain elements I<i> to domain
elements that should be scheduled either before I<I>
or as early as possible after I<i>.

The function C<isl_schedule_constraints_set_conditional_validity>
replaces the conditional validity constraints.
A conditional validity constraint is only imposed when any of the corresponding
conditions is satisfied, i.e., when any of them is non-zero.
That is, the scheduler ensures that within each band if the dependence
distances over the condition constraints are not all zero
then all corresponding conditional validity constraints are respected.
A conditional validity constraint corresponds to a condition
if the two are adjacent, i.e., if the domain of one relation intersect
the range of the other relation.
The typical use case of conditional validity constraints is
to allow order constraints between live ranges to be violated
as long as the live ranges themselves are local to the band.
To allow more fine-grained control over which conditions correspond
to which conditional validity constraints, the domains and ranges
of these relations may include I<tags>.  That is, the domains and
ranges of those relation may themselves be wrapped relations
where the iteration domain appears in the domain of those wrapped relations
and the range of the wrapped relations can be arbitrarily chosen
by the user.  Conditions and conditional validity constraints are only
considered adjacent to each other if the entire wrapped relation matches.
In particular, a relation with a tag will never be considered adjacent
to a relation without a tag.

The following function computes a schedule directly from
an iteration domain and validity and proximity dependences
and is implemented in terms of the functions described above.
The use of C<isl_union_set_compute_schedule> is discouraged.

        #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);

The generated schedule represents a schedule tree.
For more information on schedule trees, see
L</"Schedule Trees">.

=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_coincidence(
                isl_ctx *ctx, int val);
        int isl_options_get_schedule_outer_coincidence(
                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_coincidence

If this option is set, then we try to construct schedules
where the outermost scheduling dimension in each band
satisfies the coincidence constraints.

=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 this option is set then the function C<isl_schedule_get_map>
will treat set nodes in the same way as sequence nodes.

=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 tree or
a schedule map.
In case the schedule given as 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 schedule tree or the image elements
in a schedule map only specify the I<order>
in which the corresponding domain elements should be visited.
No direct relation between the partial schedule values
or the image elements on the one hand and the loop iterators
in the generated AST on the other hand 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_alloc(
                isl_ctx *ctx);
        __isl_give isl_ast_build *isl_ast_build_from_context(
                __isl_take isl_set *set);
        __isl_give isl_ast_build *isl_ast_build_copy(
                __isl_keep isl_ast_build *build);
        __isl_null isl_ast_build *isl_ast_build_free(
                __isl_take isl_ast_build *build);

The C<set> argument is usually a parameter set with zero or more parameters.
In fact, when creating an AST using C<isl_ast_build_node_from_schedule>,
this set is required to be a parameter set.
An C<isl_ast_build> created using C<isl_ast_build_alloc> does not
specify any parameter constraints.
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 one of the following
functions.

        #include <isl/ast_build.h>
        __isl_give isl_ast_node *isl_ast_build_node_from_schedule(
                __isl_keep isl_ast_build *build,
                __isl_take isl_schedule *schedule);
        __isl_give isl_ast_node *
        isl_ast_build_node_from_schedule_map(
                __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>
        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>,
C<isl_ast_node_mark> 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_mark> introduces a mark in the AST.
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_id *isl_ast_node_mark_get_id(
                __isl_keep isl_ast_node *node);
        __isl_give isl_ast_node *isl_ast_node_mark_get_node(
                __isl_keep isl_ast_node *node);

C<isl_ast_node_mark_get_id> returns the identifier of the mark.
C<isl_ast_node_mark_get_node> returns the child node that is being marked.

        #include <isl/ast.h>
        __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>
        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_zdiv_r>

Equal to zero iff the remainder on integer division is zero.

=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.

=item C<isl_ast_op_access>

An array access.
The number of arguments of the C<isl_ast_expr> is one more than
the number of index expressions in the array access, the first argument
representing the array being accessed.

=item C<isl_ast_op_member>

A member access.
This operation has two arguments, a structure and the name of
the member of the structure being accessed.

=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>
        __isl_give isl_val *isl_ast_expr_get_val(
                __isl_keep isl_ast_expr *expr);

Return the integer represented by the AST expression.

=head3 Properties of ASTs

        #include <isl/ast.h>
        int isl_ast_expr_is_equal(__isl_keep isl_ast_expr *expr1,
                __isl_keep isl_ast_expr *expr2);

Check if two C<isl_ast_expr>s are equal to each other.

=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);
        __isl_null isl_ast_node *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);
        __isl_null isl_ast_expr *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_val(
                __isl_take isl_val *v);
        __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_address_of(
                __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_pdiv_q(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_pdiv_r(
                __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_and_then(
                __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)
        __isl_give isl_ast_expr *isl_ast_expr_or_else(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2)
        __isl_give isl_ast_expr *isl_ast_expr_eq(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_le(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_lt(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_ge(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_gt(
                __isl_take isl_ast_expr *expr1,
                __isl_take isl_ast_expr *expr2);
        __isl_give isl_ast_expr *isl_ast_expr_access(
                __isl_take isl_ast_expr *array,
                __isl_take isl_ast_expr_list *indices);
        __isl_give isl_ast_expr *isl_ast_expr_call(
                __isl_take isl_ast_expr *function,
                __isl_take isl_ast_expr_list *arguments);

The function C<isl_ast_expr_address_of> can be applied to an
C<isl_ast_expr> of type C<isl_ast_op_access> only. It is meant
to represent the address of the C<isl_ast_expr_access>. The function
C<isl_ast_expr_and_then> as well as C<isl_ast_expr_or_else> are short-circuit
versions of C<isl_ast_expr_and> and C<isl_ast_expr_or>, respectively.

        #include <isl/ast_build.h>
        __isl_give isl_ast_expr *isl_ast_build_expr_from_set(
                __isl_keep isl_ast_build *build,
                __isl_take isl_set *set);
        __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_access_from_pw_multi_aff(
                __isl_keep isl_ast_build *build,
                __isl_take isl_pw_multi_aff *pma);
        __isl_give isl_ast_expr *
        isl_ast_build_access_from_multi_pw_aff(
                __isl_keep isl_ast_build *build,
                __isl_take isl_multi_pw_aff *mpa);
        __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);
        __isl_give isl_ast_expr *
        isl_ast_build_call_from_multi_pw_aff(
                __isl_keep isl_ast_build *build,
                __isl_take isl_multi_pw_aff *mpa);

The set <set> and
the domains of C<pa>, C<mpa> and C<pma> should correspond
to the schedule space of C<build>.
The tuple id of C<mpa> or C<pma> is used as the array being accessed or
the function being called.
If the accessed space is a nested relation, then it is taken
to represent an access of the member specified by the range
of this nested relation of the structure specified by the domain
of the nested relation.

The following functions can be used to modify an C<isl_ast_expr>.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_expr_set_op_arg(
                __isl_take isl_ast_expr *expr, int pos,
                __isl_take isl_ast_expr *arg);

Replace the argument of C<expr> at position C<pos> by C<arg>.

        #include <isl/ast.h>
        __isl_give isl_ast_expr *isl_ast_expr_substitute_ids(
                __isl_take isl_ast_expr *expr,
                __isl_take isl_id_to_ast_expr *id2expr);

The function C<isl_ast_expr_substitute_ids> replaces the
subexpressions of C<expr> of type C<isl_ast_expr_id>
by the corresponding expression in C<id2expr>, if there is any.


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);
        __isl_give char *isl_ast_expr_to_str(
                __isl_keep isl_ast_expr *expr);

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);
        __isl_null isl_ast_print_options *
        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);

The AST printer only prints body nodes as blocks if these
blocks cannot be safely omitted.
For example, a C<for> node with one body node will not be
surrounded with braces in C<ISL_FORMAT_C>.
A block will always be printed by setting the following option.

        int isl_options_set_ast_always_print_block(isl_ctx *ctx,
                int val);
        int isl_options_get_ast_always_print_block(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 AST Generation Options (Schedule Tree)

In case of AST construction from a schedule tree, the options
that control how an AST is created from the individual schedule
dimensions are stored in the band nodes of the tree
(see L</"Schedule Trees">).

In particular, a schedule dimension can be handled in four
different ways, atomic, separate, unroll or the default.
This loop AST generation type can be set using
C<isl_schedule_node_band_member_set_ast_loop_type>.
Alternatively,
the first three can be selected by including a one-dimensional
element with as value the position of the schedule dimension
within the band and as name one of C<atomic>, C<separate>
or C<unroll> in the options
set by C<isl_schedule_node_band_set_ast_build_options>.
Only one of these three may be specified for
any given schedule dimension within a band node.
If none of these is specified, then the default
is used.  The meaning of the options is as follows.

=over

=item C<atomic>

When this option is specified, the AST generator will make
sure that a given domains space only appears in a single
loop at the specified level.

For example, for the schedule tree

        domain: "{ a[i] : 0 <= i < 10; b[i] : 0 <= i < 10 }"
        child:
          schedule: "[{ a[i] -> [i]; b[i] -> [i+1] }]"
          options: "{ atomic[x] }"

the following AST will be generated

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

On the other hand, for the schedule tree

        domain: "{ a[i] : 0 <= i < 10; b[i] : 0 <= i < 10 }"
        child:
          schedule: "[{ a[i] -> [i]; b[i] -> [i+1] }]"
          options: "{ separate[x] }"

the following AST will be generated

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

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<separate>

When this option is specified, the AST generator will
split the domain of the specified schedule dimension
into pieces with a fixed set of statements for which
instances need to be executed by the iterations in
the schedule domain part.  This option tends to avoid
the generation of guards inside the corresponding loops.
See also the C<atomic> option.

=item C<unroll>

When this option is specified, the AST generator will
I<completely> unroll the corresponding schedule dimension.
It is the responsibility of the user to ensure that such
unrolling is possible.
To obtain a partial unrolling, the user should apply an additional
strip-mining to the schedule and fully unroll the inner schedule
dimension.

=back

The C<isolate> option is a bit more involved.  It allows the user
to isolate a range of schedule dimension values from smaller and
greater values.  Additionally, the user may specify a different
atomic/separate/unroll choice for the isolated part and the remaining
parts.  The typical use case of the C<isolate> option is to isolate
full tiles from partial tiles.
The part that needs to be isolated may depend on outer schedule dimensions.
The option therefore needs to be able to reference those outer schedule
dimensions.  In particular, the space of the C<isolate> option is that
of a wrapped map with as domain the flat product of all outer band nodes
and as range the space of the current band node.
The atomic/separate/unroll choice for the isolated part is determined
by an option that lives in an unnamed wrapped space with as domain
a zero-dimensional C<isolate> space and as range the regular
C<atomic>, C<separate> or C<unroll> space.
This option may also be set directly using
C<isl_schedule_node_band_member_set_isolate_ast_loop_type>.
The atomic/separate/unroll choice for the remaining part is determined
by the regular C<atomic>, C<separate> or C<unroll> option.
The use of the C<isolate> option causes any tree containing the node
to be considered anchored.

As an example, consider the isolation of full tiles from partial tiles
in a tiling of a triangular domain.  The original schedule is as follows.

        domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
        child:
          schedule: "[{ A[i,j] -> [floor(i/10)] }, \
                { A[i,j] -> [floor(j/10)] }, \
                { A[i,j] -> [i] }, { A[i,j] -> [j] }]"

The output is

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

Isolating the full tiles, we have the following input

        domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
        child:
          schedule: "[{ A[i,j] -> [floor(i/10)] }, \
                { A[i,j] -> [floor(j/10)] }, \
                { A[i,j] -> [i] }, { A[i,j] -> [j] }]"
          options: "{ isolate[[] -> [a,b,c,d]] : 0 <= 10a,10b and \
                10a+9+10b+9 <= 100 }"

and output

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

We may then additionally unroll the innermost loop of the isolated part

        domain: "{ A[i,j] : 0 <= i,j and i + j <= 100 }"
        child:
          schedule: "[{ A[i,j] -> [floor(i/10)] }, \
                { A[i,j] -> [floor(j/10)] }, \
                { A[i,j] -> [i] }, { A[i,j] -> [j] }]"
          options: "{ isolate[[] -> [a,b,c,d]] : 0 <= 10a,10b and \
                10a+9+10b+9 <= 100; [isolate[] -> unroll[3]] }"

to obtain

        {
          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) {
                A(c2, 10 * c1);
                A(c2, 10 * c1 + 1);
                A(c2, 10 * c1 + 2);
                A(c2, 10 * c1 + 3);
                A(c2, 10 * c1 + 4);
                A(c2, 10 * c1 + 5);
                A(c2, 10 * c1 + 6);
                A(c2, 10 * c1 + 7);
                A(c2, 10 * c1 + 8);
                A(c2, 10 * c1 + 9);
              }
            for (int c1 = -c0 + 9; c1 <= -c0 + 10; c1 += 1)
              for (int c2 = 10 * c0;
                   c2 <= min(10 * c0 + 9, -10 * c1 + 100); c2 += 1)
                for (int c3 = 10 * c1;
                     c3 <= min(10 * c1 + 9, -c2 + 100); 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 * c0 + 9, -10 * c1 + 100); c2 += 1)
                for (int c3 = 10 * c1;
                     c3 <= min(10 * c1 + 9, -c2 + 100); c3 += 1)
                  A(c2, c3);
        }


=head3 AST Generation Options (Schedule Map)

In case of AST construction using
C<isl_ast_build_node_from_schedule_map>, the options
that control how an AST is created from the individual schedule
dimensions are stored in the C<isl_ast_build>.
They 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 C<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_node_from_schedule_map>.
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>

B<This option has been deprecated.  Use the isolate option on
schedule trees instead.>

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

=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.
In case of AST construction using
C<isl_ast_build_node_from_schedule_map>,
the most prominent way of control is through ``options'',
as explained above.

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);
        __isl_give isl_ast_build *
        isl_ast_build_set_before_each_mark(
                __isl_take isl_ast_build *build,
                int (*fn)(__isl_keep isl_id *mark,
                        __isl_keep isl_ast_build *build,
                        void *user), void *user);
        __isl_give isl_ast_build *
        isl_ast_build_set_after_each_mark(
                __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>.
The callbacks set by C<isl_ast_build_set_before_each_mark>
and C<isl_ast_build_set_after_each_mark> will be called for each
mark AST node that is created, i.e., for each mark schedule node
in the input schedule tree.  The first will be called in depth-first
pre-order, while the second will be called in depth-first post-order.
Since the callback set by C<isl_ast_build_set_before_each_mark>
is called before the mark AST node is actually constructed, it is passed
the identifier of the mark node.
All callbacks should C<NULL> (or -1) 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_node_from_schedule_map> 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.