=head1 Introduction C 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. The C library offers functionality that is similar to that offered by the C and C 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 and as part of an equivalence checker of static affine programs. =head1 Installation The source of C can be obtained either as a tarball or from the git repository. Both are available from L. 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 Get submodule (optional) C can optionally use the C library and provides this library as a submodule. If you want to use it, then after you have cloned C, you need to grab the submodules git submodule init git submodule update To obtain updates, you only need git submodule update Note that C currently does not use any C functionality by default. =item 3 Generate C ./autogen.sh =back After performing the above steps, continue with the L. =head2 Common installation instructions =over =item 1 Obtain C Building C requires C, including its headers files. Your distribution may not provide these header files by default and you may need to install a package called C or something similar. Alternatively, C can be built from source, available from L. =item 2 Configure C uses the standard C C script. To run it, just type ./configure optionally followed by some configure options. A complete list of options can be obtained by running ./configure --help Below we discuss some of the more common options. C can optionally use both C and C. C is mainly used to convert between C objects and C objects. No C functionality is currently used by default. The C<--with-polylib> and C<--with-piplib> options can be used to specify which C or C library to use, either an installed version (C), an externally built version (C), a bundled version (C) or no version (C). The option C is mostly useful in C scripts of larger projects that bundle both C and either C or C. =over =item C<--prefix> Installation prefix for C =item C<--with-gmp-prefix> Installation prefix for C (architecture-independent files). =item C<--with-gmp-exec-prefix> Installation prefix for C (architecture-dependent files). =item C<--with-polylib> Which copy of C to use, either C (default), C or C. =item C<--with-polylib-prefix> Installation prefix for C C (architecture-independent files). =item C<--with-polylib-exec-prefix> Installation prefix for C C (architecture-dependent files). =item C<--with-polylib-builddir> Location where C C was built. =item C<--with-piplib> Which copy of C to use, either C (default), C, C or C. Note that C only works if you have obtained C and its submodules from the git repository. =item C<--with-piplib-prefix> Installation prefix for C C (architecture-independent files). =item C<--with-piplib-exec-prefix> Installation prefix for C C (architecture-dependent files). =item C<--with-piplib-builddir> Location where C C was built. =back =item 3 Compile make =item 4 Install (optional) make install =back =head1 Library =head2 Initialization All manipulations of integer sets and relations occur within the context of an C. A given C 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 to another C. This means that there is currently no way of safely moving an object from one thread to another, unless the whole C is moved. An C can be allocated using C and freed using C. All objects allocated within an C should be freed before the C itself is freed. isl_ctx *isl_ctx_alloc(); void isl_ctx_free(isl_ctx *ctx); =head2 Integers All operations on integers, mainly the coefficients of the constraints describing the sets and relations, are performed in exact integer arithmetic using C. However, to allow future versions of C to optionally support fixed integer arithmetic, all calls to C are wrapped inside C specific macros. The basic type is C and the following operations are available on this type. =over =item isl_int_init(i) =item isl_int_clear(i) =item isl_int_set(r,i) =item isl_int_set_si(r,i) =item isl_int_abs(r,i) =item isl_int_neg(r,i) =item isl_int_swap(i,j) =item isl_int_swap_or_set(i,j) =item isl_int_add_ui(r,i,j) =item isl_int_sub_ui(r,i,j) =item isl_int_add(r,i,j) =item isl_int_sub(r,i,j) =item isl_int_mul(r,i,j) =item isl_int_mul_ui(r,i,j) =item isl_int_addmul(r,i,j) =item isl_int_submul(r,i,j) =item isl_int_gcd(r,i,j) =item isl_int_lcm(r,i,j) =item isl_int_divexact(r,i,j) =item isl_int_cdiv_q(r,i,j) =item isl_int_fdiv_q(r,i,j) =item isl_int_fdiv_r(r,i,j) =item isl_int_fdiv_q_ui(r,i,j) =item isl_int_read(r,s) =item isl_int_print(out,i,width) =item isl_int_sgn(i) =item isl_int_cmp(i,j) =item isl_int_cmp_si(i,si) =item isl_int_eq(i,j) =item isl_int_ne(i,j) =item isl_int_lt(i,j) =item isl_int_le(i,j) =item isl_int_gt(i,j) =item isl_int_ge(i,j) =item isl_int_abs_eq(i,j) =item isl_int_abs_ne(i,j) =item isl_int_abs_lt(i,j) =item isl_int_abs_gt(i,j) =item isl_int_abs_ge(i,j) =item isl_int_is_zero(i) =item isl_int_is_one(i) =item isl_int_is_negone(i) =item isl_int_is_pos(i) =item isl_int_is_neg(i) =item isl_int_is_nonpos(i) =item isl_int_is_nonneg(i) =item isl_int_is_divisible_by(i,j) =back =head2 Sets and Relations C uses four types of objects for representing sets and relations, C, C, C and C. C and C represent sets and relations that can be described as a conjunction of affine constraints, while C and C represent unions of Cs and Cs, respectively. The difference between sets and relations (maps) is that sets have one set of variables, while relations have two sets of variables, input variables and output variables. =head2 Memory Management Since a high-level operation on sets and/or relations usually involves several substeps and since the user is usually not interested in the intermediate results, most functions that return a new object will also release all the objects passed as arguments. If the user still wants to use one or more of these arguments after the function call, she should pass along a copy of the object rather than the object itself. The user is then responsible for make sure that the original object gets used somewhere else or is explicitly freed. The arguments and return values of all documents functions are annotated to make clear which arguments are released and which arguments are preserved. In particular, the following annotations are used =over =item C<__isl_give> C<__isl_give> means that a new object is returned. The user should make sure that the returned pointer is used exactly once as a value for an C<__isl_take> argument. In between, it can be used as a value for as many C<__isl_keep> arguments as the user likes. There is one exception, and that is the case where the pointer returned is C. Is this case, the user is free to use it as an C<__isl_take> argument or not. =item C<__isl_take> C<__isl_take> means that the object the argument points to is taken over by the function and may no longer be used by the user as an argument to any other function. The pointer value must be one returned by a function returning an C<__isl_give> pointer. If the user passes in a C value, then this will be treated as an error in the sense that the function will not perform its usual operation. However, it will still make sure that all the the other C<__isl_take> arguments are released. =item C<__isl_keep> C<__isl_keep> means that the function will only use the object temporarily. After the function has finished, the user can still use it as an argument to other functions. A C value will be treated in the same way as a C value for an C<__isl_take> argument. =back =head2 Dimension Specifications Whenever a new set or relation is created from scratch, its dimension needs to be specified using an C. #include __isl_give isl_dim *isl_dim_alloc(isl_ctx *ctx, unsigned nparam, unsigned n_in, unsigned n_out); __isl_give isl_dim *isl_dim_set_alloc(isl_ctx *ctx, unsigned nparam, unsigned dim); __isl_give isl_dim *isl_dim_copy(__isl_keep isl_dim *dim); void isl_dim_free(__isl_take isl_dim *dim); unsigned isl_dim_size(__isl_keep isl_dim *dim, enum isl_dim_type type); The dimension specification used for creating a set needs to be created using C, while that for creating a relation needs to be created using C. C can be used to find out the number of dimensions of each type in a dimension specification, where type may be C, C (only for relations), C (only for relations), C (only for sets) or C. =head2 Input and Output Proper input and output functions are still in development. However, some functions are provided to read and write to foreign file formats and to convert between C objects and C objects (if C is available). =head3 Input #include __isl_give isl_basic_set *isl_basic_set_read_from_file( isl_ctx *ctx, FILE *input, unsigned nparam, unsigned input_format); __isl_give isl_basic_set *isl_basic_set_read_from_str( isl_ctx *ctx, const char *str, unsigned nparam, unsigned input_format); __isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx, FILE *input, unsigned nparam, unsigned input_format); #include __isl_give isl_basic_map *isl_basic_map_read_from_file( isl_ctx *ctx, FILE *input, unsigned nparam, unsigned input_format); C may be either C or C. However, not all combination are currently supported. Furthermore, only a very limited subset of the C input format is currently supported. In particular, C and C only support C, while C only supports C. C specifies how many of the final columns in the C format correspond to parameters. It should be zero when C is used. =head3 Output #include void isl_basic_set_print(__isl_keep isl_basic_set *bset, FILE *out, int indent, const char *prefix, const char *suffix, unsigned output_format); void isl_set_print(__isl_keep struct isl_set *set, FILE *out, int indent, unsigned output_format); C must be C. Each line in the output is indented by C spaces, prefixed by C and suffixed by C. The coefficients of the existentially quantified variables appear between those of the set variables and those of the parameters. =head3 Conversion from/to C The following functions are only available if C has been configured to use C. #include __isl_give isl_basic_set *isl_basic_set_new_from_polylib( Polyhedron *P, __isl_take isl_dim *dim); Polyhedron *isl_basic_set_to_polylib( __isl_keep isl_basic_set *bset); __isl_give isl_set *isl_set_new_from_polylib(Polyhedron *D, __isl_take isl_dim *dim); Polyhedron *isl_set_to_polylib(__isl_keep isl_set *set); #include __isl_give isl_basic_map *isl_basic_map_new_from_polylib( Polyhedron *P, __isl_take isl_dim *dim); __isl_give isl_map *isl_map_new_from_polylib(Polyhedron *D, __isl_take isl_dim *dim); Polyhedron *isl_basic_map_to_polylib( __isl_keep isl_basic_map *bmap); Polyhedron *isl_map_to_polylib(__isl_keep isl_map *map); =head3 Dumping the internal state For lack of proper output functions, the following functions can be used to dump the internal state of a set or relation. The user should not depend on the output format of these functions. void isl_basic_set_dump(__isl_keep isl_basic_set *bset, FILE *out, int indent); void isl_basic_map_dump(__isl_keep isl_basic_map *bmap, FILE *out, int indent); void isl_set_dump(__isl_keep isl_set *set, FILE *out, int indent); void isl_map_dump(__isl_keep isl_map *map, FILE *out, int indent); =head2 Creating New Sets and Relations C has functions for creating some standard sets and relations. =over =item * Empty sets and relations __isl_give isl_basic_set *isl_basic_set_empty( __isl_take isl_dim *dim); __isl_give isl_basic_map *isl_basic_map_empty( __isl_take isl_dim *dim); __isl_give isl_set *isl_set_empty( __isl_take isl_dim *dim); __isl_give isl_map *isl_map_empty( __isl_take isl_dim *dim); =item * Universe sets and relations __isl_give isl_basic_set *isl_basic_set_universe( __isl_take isl_dim *dim); __isl_give isl_basic_map *isl_basic_map_universe( __isl_take isl_dim *dim); __isl_give isl_set *isl_set_universe( __isl_take isl_dim *dim); __isl_give isl_map *isl_map_universe( __isl_take isl_dim *dim); =item * Identity relations __isl_give isl_basic_map *isl_basic_map_identity( __isl_take isl_dim *set_dim); __isl_give isl_map *isl_map_identity( __isl_take isl_dim *set_dim); These functions take a dimension specification for a B and return an identity relation between two such sets. =item * Lexicographic order __isl_give isl_map *isl_map_lex_lt( __isl_take isl_dim *set_dim); __isl_give isl_map *isl_map_lex_le( __isl_take isl_dim *set_dim); __isl_give isl_map *isl_map_lex_gt( __isl_take isl_dim *set_dim); __isl_give isl_map *isl_map_lex_ge( __isl_take isl_dim *set_dim); These functions take a dimension specification for a B and return relations that express that the elements in the domain are lexicograhically less (C), less or equal (C), greater (C) or greater or equal (C) than the elements 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 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_basic_map *isl_basic_map_copy( __isl_keep isl_basic_map *bmap); __isl_give isl_map *isl_map_copy(__isl_keep isl_map *map); void isl_basic_set_free(__isl_take isl_basic_set *bset); void isl_set_free(__isl_take isl_set *set); void isl_basic_map_free(__isl_take isl_basic_map *bmap); void isl_map_free(__isl_take isl_map *map); Other sets and relations can be constructed by starting from a universe set or relation, adding equality and/or inequality constraints and then projecting out the existentially quantified variables, if any. Constraints can be constructed, manipulated and added to basic sets and relations using the following functions. #include __isl_give isl_constraint *isl_equality_alloc( __isl_take isl_dim *dim); __isl_give isl_constraint *isl_inequality_alloc( __isl_take isl_dim *dim); void isl_constraint_set_constant( __isl_keep isl_constraint *constraint, isl_int v); void isl_constraint_set_coefficient( __isl_keep isl_constraint *constraint, enum isl_dim_type type, int pos, isl_int v); __isl_give isl_basic_map *isl_basic_map_add_constraint( __isl_take isl_basic_map *bmap, __isl_take isl_constraint *constraint); __isl_give isl_basic_set *isl_basic_set_add_constraint( __isl_take isl_basic_set *bset, __isl_take isl_constraint *constraint); For example, to create a set containing the even integers between 10 and 42, you would use the following code. isl_int v; struct isl_dim *dim; struct isl_constraint *c; struct isl_basic_set *bset; isl_int_init(v); dim = isl_dim_set_alloc(ctx, 0, 2); bset = isl_basic_set_universe(isl_dim_copy(dim)); c = isl_equality_alloc(isl_dim_copy(dim)); isl_int_set_si(v, -1); isl_constraint_set_coefficient(c, isl_dim_set, 0, v); isl_int_set_si(v, 2); isl_constraint_set_coefficient(c, isl_dim_set, 1, v); bset = isl_basic_set_add_constraint(bset, c); c = isl_inequality_alloc(isl_dim_copy(dim)); isl_int_set_si(v, -10); isl_constraint_set_constant(c, v); isl_int_set_si(v, 1); isl_constraint_set_coefficient(c, isl_dim_set, 0, v); bset = isl_basic_set_add_constraint(bset, c); c = isl_inequality_alloc(dim); isl_int_set_si(v, 42); isl_constraint_set_constant(c, v); isl_int_set_si(v, -1); isl_constraint_set_coefficient(c, isl_dim_set, 0, v); bset = isl_basic_set_add_constraint(bset, c); bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 1); isl_int_clear(v); =head2 Properties =head3 Unary Properties =over =item Emptiness The following functions test whether the given set or relation contains any integer points. The ``fast'' 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_fast_is_empty(__isl_keep isl_basic_set *bset); int isl_basic_set_is_empty(__isl_keep isl_basic_set *bset); int isl_set_is_empty(__isl_keep isl_set *set); int isl_basic_map_fast_is_empty(__isl_keep isl_basic_map *bmap); int isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap); int isl_map_fast_is_empty(__isl_keep isl_map *map); int isl_map_is_empty(__isl_keep isl_map *map); =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); =back =head3 Binary Properties =over =item * Equality int isl_set_fast_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2); int isl_set_is_equal(__isl_keep isl_set *set1, __isl_keep isl_set *set2); int isl_map_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2); int isl_map_fast_is_equal(__isl_keep isl_map *map1, __isl_keep isl_map *map2); int isl_basic_map_is_equal( __isl_keep isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2); =item * Disjointness int isl_set_fast_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2); =item * Subset int isl_set_is_subset(__isl_keep isl_set *set1, __isl_keep isl_set *set2); int isl_set_is_strict_subset( __isl_keep isl_set *set1, __isl_keep isl_set *set2); int isl_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); =back =head2 Unary Operations =over =item * Projection __isl_give isl_basic_set *isl_basic_set_project_out( __isl_take isl_basic_set *bset, enum isl_dim_type type, unsigned first, unsigned n); __isl_give isl_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_map_domain( __isl_take isl_basic_map *bmap); __isl_give isl_basic_set *isl_basic_map_range( __isl_take isl_basic_map *bmap); __isl_give isl_set *isl_map_domain( __isl_take isl_map *bmap); __isl_give isl_set *isl_map_range( __isl_take isl_map *map); C currently only supports projecting out the final C dimensions. =item * Coalescing Simplify the representation of a set or relation by trying to combine pairs of basic sets or relations into a single basic set or relation. __isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set); __isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map); =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 * 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_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); =back =head2 Binary Operations The two arguments of a binary operation not only need to live in the same C, they currently also need to have the same (number of) parameters. =head3 Basic Operations =over =item * Intersection __isl_give isl_basic_set *isl_basic_set_intersect( __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2); __isl_give isl_set *isl_set_intersect( __isl_take isl_set *set1, __isl_take isl_set *set2); __isl_give isl_basic_map *isl_basic_map_intersect_domain( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *bset); __isl_give isl_basic_map *isl_basic_map_intersect_range( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *bset); __isl_give isl_basic_map *isl_basic_map_intersect( __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2); __isl_give isl_map *isl_map_intersect_domain( __isl_take isl_map *map, __isl_take isl_set *set); __isl_give isl_map *isl_map_intersect_range( __isl_take isl_map *map, __isl_take isl_set *set); __isl_give isl_map *isl_map_intersect( __isl_take isl_map *map1, __isl_take isl_map *map2); =item * Union __isl_give isl_set *isl_basic_set_union( __isl_take isl_basic_set *bset1, __isl_take isl_basic_set *bset2); __isl_give isl_map *isl_basic_map_union( __isl_take isl_basic_map *bmap1, __isl_take isl_basic_map *bmap2); __isl_give isl_set *isl_set_union( __isl_take isl_set *set1, __isl_take isl_set *set2); __isl_give isl_map *isl_map_union( __isl_take isl_map *map1, __isl_take isl_map *map2); =item * Set difference __isl_give isl_set *isl_set_subtract( __isl_take isl_set *set1, __isl_take isl_set *set2); __isl_give isl_map *isl_map_subtract( __isl_take isl_map *map1, __isl_take isl_map *map2); =item * Application __isl_give isl_basic_set *isl_basic_set_apply( __isl_take isl_basic_set *bset, __isl_take isl_basic_map *bmap); __isl_give isl_set *isl_set_apply( __isl_take isl_set *set, __isl_take isl_map *map); __isl_give isl_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); =back =head3 Lexicographic Optimization Given a basic set C and a zero-dimensional domain C, the following functions compute a set that contains the lexicographic minimum or maximum of the elements in C for those values of the parameters that satisfy C. If C is not C, then C<*empty> is assigned a set that contains the parameter values in C for which C 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. __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); Given a basic set C, the following function simply returns a set containing the lexicographic minimum of the elements in C. __isl_give isl_set *isl_basic_set_lexmin( __isl_take isl_basic_set *bset); Given a basic relation C and a domain C, the following functions compute a relation that maps each element of C to the single lexicographic minimum or maximum of the elements that are associated to that same element in C. If C is not C, then C<*empty> is assigned a set that contains the elements in C that do not map to any elements in C. In other words, the union of the domain of the result and of C<*empty> is equal to C. __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); Given a basic map C, the following function simply returns a map mapping each element in the domain of C to the lexicographic minimum of all elements associated to that element. __isl_give isl_map *isl_basic_map_lexmin( __isl_take isl_basic_map *bmap); =head1 Applications Although C is mainly meant to be used as a library, it also contains some basic applications that use some of the functionality of C. Since C does not have its own input format yet, these applications currently take input in C style. That is, a line with 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, followed by the actual rows. 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. =head2 C C takes a polyhedron in C format 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 C takes the same input as the C program from the C distribution, i.e., a set of constraints on the parameters in C format, a line contains only -1 and finally a set of constraints on a parametric polyhedron, again in C format. 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 currently does not have its own output format, the output is just a dump of the internal state. =head2 C C computes the minimum of some linear or affine objective function over the integer points in a polyhedron. The input is in C format. If an affine objective function is given, then the constant should appear in the last column. =head2 C Given a polytope in C format, C prints all integer points in the polytope.