3 C<isl> is a thread-safe C library for manipulating
4 sets and relations of integer points bounded by affine constraints.
5 The descriptions of the sets and relations may involve
6 both parameters and existentially quantified variables.
7 All computations are performed in exact integer arithmetic
9 The C<isl> library offers functionality that is similar
10 to that offered by the C<Omega> and C<Omega+> libraries,
11 but the underlying algorithms are in most cases completely different.
13 The library is by no means complete and some fairly basic
14 functionality is still missing.
15 Still, even in its current form, the library has been successfully
16 used as a backend polyhedral library for the polyhedral
17 scanner C<CLooG> and as part of an equivalence checker of
18 static affine programs.
19 For bug reports, feature requests and questions,
20 visit the the discussion group at
21 L<http://groups.google.com/group/isl-development>.
23 =head2 Backward Incompatible Changes
25 =head3 Changes since isl-0.02
29 =item * The old printing functions have been deprecated
30 and replaced by C<isl_printer> functions, see L<Input and Output>.
32 =item * Most functions related to dependence analysis have acquired
33 an extra C<must> argument. To obtain the old behavior, this argument
34 should be given the value 1. See L<Dependence Analysis>.
38 =head3 Changes since isl-0.03
42 =item * The function C<isl_pw_qpolynomial_fold_add> has been
43 renamed to C<isl_pw_qpolynomial_fold_fold>.
44 Similarly, C<isl_union_pw_qpolynomial_fold_add> has been
45 renamed to C<isl_union_pw_qpolynomial_fold_fold>.
49 =head3 Changes since isl-0.04
53 =item * All header files have been renamed from C<isl_header.h>
58 =head3 Changes since isl-0.05
62 =item * The functions C<isl_printer_print_basic_set> and
63 C<isl_printer_print_basic_map> no longer print a newline.
65 =item * The functions C<isl_flow_get_no_source>
66 and C<isl_union_map_compute_flow> now return
67 the accesses for which no source could be found instead of
68 the iterations where those accesses occur.
70 =item * The functions C<isl_basic_map_identity> and
71 C<isl_map_identity> now take the dimension specification
72 of a B<map> as input. An old call
73 C<isl_map_identity(dim)> can be rewritten to
74 C<isl_map_identity(isl_dim_map_from_set(dim))>.
76 =item * The function C<isl_map_power> no longer takes
77 a parameter position as input. Instead, the exponent
78 is now expressed as the domain of the resulting relation.
82 =head3 Changes since isl-0.06
86 =item * The format of C<isl_printer_print_qpolynomial>'s
87 C<ISL_FORMAT_ISL> output has changed.
88 Use C<ISL_FORMAT_C> to obtain the old output.
94 The source of C<isl> can be obtained either as a tarball
95 or from the git repository. Both are available from
96 L<http://freshmeat.net/projects/isl/>.
97 The installation process depends on how you obtained
100 =head2 Installation from the git repository
104 =item 1 Clone or update the repository
106 The first time the source is obtained, you need to clone
109 git clone git://repo.or.cz/isl.git
111 To obtain updates, you need to pull in the latest changes
115 =item 2 Generate C<configure>
121 After performing the above steps, continue
122 with the L<Common installation instructions>.
124 =head2 Common installation instructions
128 =item 1 Obtain C<GMP>
130 Building C<isl> requires C<GMP>, including its headers files.
131 Your distribution may not provide these header files by default
132 and you may need to install a package called C<gmp-devel> or something
133 similar. Alternatively, C<GMP> can be built from
134 source, available from L<http://gmplib.org/>.
138 C<isl> uses the standard C<autoconf> C<configure> script.
143 optionally followed by some configure options.
144 A complete list of options can be obtained by running
148 Below we discuss some of the more common options.
150 C<isl> can optionally use C<piplib>, but no
151 C<piplib> functionality is currently used by default.
152 The C<--with-piplib> option can
153 be used to specify which C<piplib>
154 library to use, either an installed version (C<system>),
155 an externally built version (C<build>)
156 or no version (C<no>). The option C<build> is mostly useful
157 in C<configure> scripts of larger projects that bundle both C<isl>
164 Installation prefix for C<isl>
166 =item C<--with-gmp-prefix>
168 Installation prefix for C<GMP> (architecture-independent files).
170 =item C<--with-gmp-exec-prefix>
172 Installation prefix for C<GMP> (architecture-dependent files).
174 =item C<--with-piplib>
176 Which copy of C<piplib> to use, either C<no> (default), C<system> or C<build>.
178 =item C<--with-piplib-prefix>
180 Installation prefix for C<system> C<piplib> (architecture-independent files).
182 =item C<--with-piplib-exec-prefix>
184 Installation prefix for C<system> C<piplib> (architecture-dependent files).
186 =item C<--with-piplib-builddir>
188 Location where C<build> C<piplib> was built.
196 =item 4 Install (optional)
204 =head2 Initialization
206 All manipulations of integer sets and relations occur within
207 the context of an C<isl_ctx>.
208 A given C<isl_ctx> can only be used within a single thread.
209 All arguments of a function are required to have been allocated
210 within the same context.
211 There are currently no functions available for moving an object
212 from one C<isl_ctx> to another C<isl_ctx>. This means that
213 there is currently no way of safely moving an object from one
214 thread to another, unless the whole C<isl_ctx> is moved.
216 An C<isl_ctx> can be allocated using C<isl_ctx_alloc> and
217 freed using C<isl_ctx_free>.
218 All objects allocated within an C<isl_ctx> should be freed
219 before the C<isl_ctx> itself is freed.
221 isl_ctx *isl_ctx_alloc();
222 void isl_ctx_free(isl_ctx *ctx);
226 All operations on integers, mainly the coefficients
227 of the constraints describing the sets and relations,
228 are performed in exact integer arithmetic using C<GMP>.
229 However, to allow future versions of C<isl> to optionally
230 support fixed integer arithmetic, all calls to C<GMP>
231 are wrapped inside C<isl> specific macros.
232 The basic type is C<isl_int> and the operations below
233 are available on this type.
234 The meanings of these operations are essentially the same
235 as their C<GMP> C<mpz_> counterparts.
236 As always with C<GMP> types, C<isl_int>s need to be
237 initialized with C<isl_int_init> before they can be used
238 and they need to be released with C<isl_int_clear>
240 The user should not assume that an C<isl_int> is represented
241 as a C<mpz_t>, but should instead explicitly convert between
242 C<mpz_t>s and C<isl_int>s using C<isl_int_set_gmp> and
243 C<isl_int_get_gmp> whenever a C<mpz_t> is required.
247 =item isl_int_init(i)
249 =item isl_int_clear(i)
251 =item isl_int_set(r,i)
253 =item isl_int_set_si(r,i)
255 =item isl_int_set_gmp(r,g)
257 =item isl_int_get_gmp(i,g)
259 =item isl_int_abs(r,i)
261 =item isl_int_neg(r,i)
263 =item isl_int_swap(i,j)
265 =item isl_int_swap_or_set(i,j)
267 =item isl_int_add_ui(r,i,j)
269 =item isl_int_sub_ui(r,i,j)
271 =item isl_int_add(r,i,j)
273 =item isl_int_sub(r,i,j)
275 =item isl_int_mul(r,i,j)
277 =item isl_int_mul_ui(r,i,j)
279 =item isl_int_addmul(r,i,j)
281 =item isl_int_submul(r,i,j)
283 =item isl_int_gcd(r,i,j)
285 =item isl_int_lcm(r,i,j)
287 =item isl_int_divexact(r,i,j)
289 =item isl_int_cdiv_q(r,i,j)
291 =item isl_int_fdiv_q(r,i,j)
293 =item isl_int_fdiv_r(r,i,j)
295 =item isl_int_fdiv_q_ui(r,i,j)
297 =item isl_int_read(r,s)
299 =item isl_int_print(out,i,width)
303 =item isl_int_cmp(i,j)
305 =item isl_int_cmp_si(i,si)
307 =item isl_int_eq(i,j)
309 =item isl_int_ne(i,j)
311 =item isl_int_lt(i,j)
313 =item isl_int_le(i,j)
315 =item isl_int_gt(i,j)
317 =item isl_int_ge(i,j)
319 =item isl_int_abs_eq(i,j)
321 =item isl_int_abs_ne(i,j)
323 =item isl_int_abs_lt(i,j)
325 =item isl_int_abs_gt(i,j)
327 =item isl_int_abs_ge(i,j)
329 =item isl_int_is_zero(i)
331 =item isl_int_is_one(i)
333 =item isl_int_is_negone(i)
335 =item isl_int_is_pos(i)
337 =item isl_int_is_neg(i)
339 =item isl_int_is_nonpos(i)
341 =item isl_int_is_nonneg(i)
343 =item isl_int_is_divisible_by(i,j)
347 =head2 Sets and Relations
349 C<isl> uses six types of objects for representing sets and relations,
350 C<isl_basic_set>, C<isl_basic_map>, C<isl_set>, C<isl_map>,
351 C<isl_union_set> and C<isl_union_map>.
352 C<isl_basic_set> and C<isl_basic_map> represent sets and relations that
353 can be described as a conjunction of affine constraints, while
354 C<isl_set> and C<isl_map> represent unions of
355 C<isl_basic_set>s and C<isl_basic_map>s, respectively.
356 However, all C<isl_basic_set>s or C<isl_basic_map>s in the union need
357 to have the same dimension. C<isl_union_set>s and C<isl_union_map>s
358 represent unions of C<isl_set>s or C<isl_map>s of I<different> dimensions,
359 where dimensions with different space names
360 (see L<Dimension Specifications>) are considered different as well.
361 The difference between sets and relations (maps) is that sets have
362 one set of variables, while relations have two sets of variables,
363 input variables and output variables.
365 =head2 Memory Management
367 Since a high-level operation on sets and/or relations usually involves
368 several substeps and since the user is usually not interested in
369 the intermediate results, most functions that return a new object
370 will also release all the objects passed as arguments.
371 If the user still wants to use one or more of these arguments
372 after the function call, she should pass along a copy of the
373 object rather than the object itself.
374 The user is then responsible for making sure that the original
375 object gets used somewhere else or is explicitly freed.
377 The arguments and return values of all documents functions are
378 annotated to make clear which arguments are released and which
379 arguments are preserved. In particular, the following annotations
386 C<__isl_give> means that a new object is returned.
387 The user should make sure that the returned pointer is
388 used exactly once as a value for an C<__isl_take> argument.
389 In between, it can be used as a value for as many
390 C<__isl_keep> arguments as the user likes.
391 There is one exception, and that is the case where the
392 pointer returned is C<NULL>. Is this case, the user
393 is free to use it as an C<__isl_take> argument or not.
397 C<__isl_take> means that the object the argument points to
398 is taken over by the function and may no longer be used
399 by the user as an argument to any other function.
400 The pointer value must be one returned by a function
401 returning an C<__isl_give> pointer.
402 If the user passes in a C<NULL> value, then this will
403 be treated as an error in the sense that the function will
404 not perform its usual operation. However, it will still
405 make sure that all the the other C<__isl_take> arguments
410 C<__isl_keep> means that the function will only use the object
411 temporarily. After the function has finished, the user
412 can still use it as an argument to other functions.
413 A C<NULL> value will be treated in the same way as
414 a C<NULL> value for an C<__isl_take> argument.
418 =head2 Dimension Specifications
420 Whenever a new set or relation is created from scratch,
421 its dimension needs to be specified using an C<isl_dim>.
424 __isl_give isl_dim *isl_dim_alloc(isl_ctx *ctx,
425 unsigned nparam, unsigned n_in, unsigned n_out);
426 __isl_give isl_dim *isl_dim_set_alloc(isl_ctx *ctx,
427 unsigned nparam, unsigned dim);
428 __isl_give isl_dim *isl_dim_copy(__isl_keep isl_dim *dim);
429 void isl_dim_free(__isl_take isl_dim *dim);
430 unsigned isl_dim_size(__isl_keep isl_dim *dim,
431 enum isl_dim_type type);
433 The dimension specification used for creating a set
434 needs to be created using C<isl_dim_set_alloc>, while
435 that for creating a relation
436 needs to be created using C<isl_dim_alloc>.
437 C<isl_dim_size> can be used
438 to find out the number of dimensions of each type in
439 a dimension specification, where type may be
440 C<isl_dim_param>, C<isl_dim_in> (only for relations),
441 C<isl_dim_out> (only for relations), C<isl_dim_set>
442 (only for sets) or C<isl_dim_all>.
444 It is often useful to create objects that live in the
445 same space as some other object. This can be accomplished
446 by creating the new objects
447 (see L<Creating New Sets and Relations> or
448 L<Creating New (Piecewise) Quasipolynomials>) based on the dimension
449 specification of the original object.
452 __isl_give isl_dim *isl_basic_set_get_dim(
453 __isl_keep isl_basic_set *bset);
454 __isl_give isl_dim *isl_set_get_dim(__isl_keep isl_set *set);
456 #include <isl/union_set.h>
457 __isl_give isl_dim *isl_union_set_get_dim(
458 __isl_keep isl_union_set *uset);
461 __isl_give isl_dim *isl_basic_map_get_dim(
462 __isl_keep isl_basic_map *bmap);
463 __isl_give isl_dim *isl_map_get_dim(__isl_keep isl_map *map);
465 #include <isl/union_map.h>
466 __isl_give isl_dim *isl_union_map_get_dim(
467 __isl_keep isl_union_map *umap);
469 #include <isl/constraint.h>
470 __isl_give isl_dim *isl_constraint_get_dim(
471 __isl_keep isl_constraint *constraint);
473 #include <isl/polynomial.h>
474 __isl_give isl_dim *isl_qpolynomial_get_dim(
475 __isl_keep isl_qpolynomial *qp);
476 __isl_give isl_dim *isl_pw_qpolynomial_get_dim(
477 __isl_keep isl_pw_qpolynomial *pwqp);
478 __isl_give isl_dim *isl_union_pw_qpolynomial_get_dim(
479 __isl_keep isl_union_pw_qpolynomial *upwqp);
480 __isl_give isl_dim *isl_union_pw_qpolynomial_fold_get_dim(
481 __isl_keep isl_union_pw_qpolynomial_fold *upwf);
484 __isl_give isl_dim *isl_aff_get_dim(
485 __isl_keep isl_aff *aff);
487 The names of the individual dimensions may be set or read off
488 using the following functions.
491 __isl_give isl_dim *isl_dim_set_name(__isl_take isl_dim *dim,
492 enum isl_dim_type type, unsigned pos,
493 __isl_keep const char *name);
494 __isl_keep const char *isl_dim_get_name(__isl_keep isl_dim *dim,
495 enum isl_dim_type type, unsigned pos);
497 Note that C<isl_dim_get_name> returns a pointer to some internal
498 data structure, so the result can only be used while the
499 corresponding C<isl_dim> is alive.
500 Also note that every function that operates on two sets or relations
501 requires that both arguments have the same parameters. This also
502 means that if one of the arguments has named parameters, then the
503 other needs to have named parameters too and the names need to match.
504 Pairs of C<isl_union_set> and/or C<isl_union_map> arguments may
505 have different parameters (as long as they are named), in which case
506 the result will have as parameters the union of the parameters of
509 The names of entire spaces may be set or read off
510 using the following functions.
513 __isl_give isl_dim *isl_dim_set_tuple_name(
514 __isl_take isl_dim *dim,
515 enum isl_dim_type type, const char *s);
516 const char *isl_dim_get_tuple_name(__isl_keep isl_dim *dim,
517 enum isl_dim_type type);
519 The C<dim> argument needs to be one of C<isl_dim_in>, C<isl_dim_out>
520 or C<isl_dim_set>. As with C<isl_dim_get_name>,
521 the C<isl_dim_get_tuple_name> function returns a pointer to some internal
523 Binary operations require the corresponding spaces of their arguments
524 to have the same name.
526 Spaces can be nested. In particular, the domain of a set or
527 the domain or range of a relation can be a nested relation.
528 The following functions can be used to construct and deconstruct
529 such nested dimension specifications.
532 int isl_dim_is_wrapping(__isl_keep isl_dim *dim);
533 __isl_give isl_dim *isl_dim_wrap(__isl_take isl_dim *dim);
534 __isl_give isl_dim *isl_dim_unwrap(__isl_take isl_dim *dim);
536 The input to C<isl_dim_is_wrapping> and C<isl_dim_unwrap> should
537 be the dimension specification of a set, while that of
538 C<isl_dim_wrap> should be the dimension specification of a relation.
539 Conversely, the output of C<isl_dim_unwrap> is the dimension specification
540 of a relation, while that of C<isl_dim_wrap> is the dimension specification
543 Dimension specifications can be created from other dimension
544 specifications using the following functions.
546 __isl_give isl_dim *isl_dim_domain(__isl_take isl_dim *dim);
547 __isl_give isl_dim *isl_dim_from_domain(__isl_take isl_dim *dim);
548 __isl_give isl_dim *isl_dim_range(__isl_take isl_dim *dim);
549 __isl_give isl_dim *isl_dim_from_range(__isl_take isl_dim *dim);
550 __isl_give isl_dim *isl_dim_reverse(__isl_take isl_dim *dim);
551 __isl_give isl_dim *isl_dim_join(__isl_take isl_dim *left,
552 __isl_take isl_dim *right);
553 __isl_give isl_dim *isl_dim_align_params(
554 __isl_take isl_dim *dim1, __isl_take isl_dim *dim2)
555 __isl_give isl_dim *isl_dim_insert(__isl_take isl_dim *dim,
556 enum isl_dim_type type, unsigned pos, unsigned n);
557 __isl_give isl_dim *isl_dim_add(__isl_take isl_dim *dim,
558 enum isl_dim_type type, unsigned n);
559 __isl_give isl_dim *isl_dim_drop(__isl_take isl_dim *dim,
560 enum isl_dim_type type, unsigned first, unsigned n);
561 __isl_give isl_dim *isl_dim_map_from_set(
562 __isl_take isl_dim *dim);
563 __isl_give isl_dim *isl_dim_zip(__isl_take isl_dim *dim);
565 Note that if dimensions are added or removed from a space, then
566 the name and the internal structure are lost.
570 A local space is essentially a dimension specification with
571 zero or more existentially quantified variables.
572 The local space of a basic set or relation can be obtained
573 using the following functions.
576 __isl_give isl_local_space *isl_basic_set_get_local_space(
577 __isl_keep isl_basic_set *bset);
580 __isl_give isl_local_space *isl_basic_map_get_local_space(
581 __isl_keep isl_basic_map *bmap);
583 A new local space can be created from a dimension specification using
585 #include <isl/local_space.h>
586 __isl_give isl_local_space *isl_local_space_from_dim(
587 __isl_take isl_dim *dim);
589 They can be inspected, copied and freed using the following functions.
591 #include <isl/local_space.h>
592 isl_ctx *isl_local_space_get_ctx(
593 __isl_keep isl_local_space *ls);
594 int isl_local_space_dim(__isl_keep isl_local_space *ls,
595 enum isl_dim_type type);
596 const char *isl_local_space_get_dim_name(
597 __isl_keep isl_local_space *ls,
598 enum isl_dim_type type, unsigned pos);
599 __isl_give isl_dim *isl_local_space_get_dim(
600 __isl_keep isl_local_space *ls);
601 __isl_give isl_div *isl_local_space_get_div(
602 __isl_keep isl_local_space *ls, int pos);
603 __isl_give isl_local_space *isl_local_space_copy(
604 __isl_keep isl_local_space *ls);
605 void *isl_local_space_free(__isl_take isl_local_space *ls);
607 =head2 Input and Output
609 C<isl> supports its own input/output format, which is similar
610 to the C<Omega> format, but also supports the C<PolyLib> format
615 The C<isl> format is similar to that of C<Omega>, but has a different
616 syntax for describing the parameters and allows for the definition
617 of an existentially quantified variable as the integer division
618 of an affine expression.
619 For example, the set of integers C<i> between C<0> and C<n>
620 such that C<i % 10 <= 6> can be described as
622 [n] -> { [i] : exists (a = [i/10] : 0 <= i and i <= n and
625 A set or relation can have several disjuncts, separated
626 by the keyword C<or>. Each disjunct is either a conjunction
627 of constraints or a projection (C<exists>) of a conjunction
628 of constraints. The constraints are separated by the keyword
631 =head3 C<PolyLib> format
633 If the represented set is a union, then the first line
634 contains a single number representing the number of disjuncts.
635 Otherwise, a line containing the number C<1> is optional.
637 Each disjunct is represented by a matrix of constraints.
638 The first line contains two numbers representing
639 the number of rows and columns,
640 where the number of rows is equal to the number of constraints
641 and the number of columns is equal to two plus the number of variables.
642 The following lines contain the actual rows of the constraint matrix.
643 In each row, the first column indicates whether the constraint
644 is an equality (C<0>) or inequality (C<1>). The final column
645 corresponds to the constant term.
647 If the set is parametric, then the coefficients of the parameters
648 appear in the last columns before the constant column.
649 The coefficients of any existentially quantified variables appear
650 between those of the set variables and those of the parameters.
652 =head3 Extended C<PolyLib> format
654 The extended C<PolyLib> format is nearly identical to the
655 C<PolyLib> format. The only difference is that the line
656 containing the number of rows and columns of a constraint matrix
657 also contains four additional numbers:
658 the number of output dimensions, the number of input dimensions,
659 the number of local dimensions (i.e., the number of existentially
660 quantified variables) and the number of parameters.
661 For sets, the number of ``output'' dimensions is equal
662 to the number of set dimensions, while the number of ``input''
668 __isl_give isl_basic_set *isl_basic_set_read_from_file(
669 isl_ctx *ctx, FILE *input, int nparam);
670 __isl_give isl_basic_set *isl_basic_set_read_from_str(
671 isl_ctx *ctx, const char *str, int nparam);
672 __isl_give isl_set *isl_set_read_from_file(isl_ctx *ctx,
673 FILE *input, int nparam);
674 __isl_give isl_set *isl_set_read_from_str(isl_ctx *ctx,
675 const char *str, int nparam);
678 __isl_give isl_basic_map *isl_basic_map_read_from_file(
679 isl_ctx *ctx, FILE *input, int nparam);
680 __isl_give isl_basic_map *isl_basic_map_read_from_str(
681 isl_ctx *ctx, const char *str, int nparam);
682 __isl_give isl_map *isl_map_read_from_file(
683 struct isl_ctx *ctx, FILE *input, int nparam);
684 __isl_give isl_map *isl_map_read_from_str(isl_ctx *ctx,
685 const char *str, int nparam);
687 #include <isl/union_set.h>
688 __isl_give isl_union_set *isl_union_set_read_from_file(
689 isl_ctx *ctx, FILE *input);
690 __isl_give isl_union_set *isl_union_set_read_from_str(
691 struct isl_ctx *ctx, const char *str);
693 #include <isl/union_map.h>
694 __isl_give isl_union_map *isl_union_map_read_from_file(
695 isl_ctx *ctx, FILE *input);
696 __isl_give isl_union_map *isl_union_map_read_from_str(
697 struct isl_ctx *ctx, const char *str);
699 The input format is autodetected and may be either the C<PolyLib> format
700 or the C<isl> format.
701 C<nparam> specifies how many of the final columns in
702 the C<PolyLib> format correspond to parameters.
703 If input is given in the C<isl> format, then the number
704 of parameters needs to be equal to C<nparam>.
705 If C<nparam> is negative, then any number of parameters
706 is accepted in the C<isl> format and zero parameters
707 are assumed in the C<PolyLib> format.
711 Before anything can be printed, an C<isl_printer> needs to
714 __isl_give isl_printer *isl_printer_to_file(isl_ctx *ctx,
716 __isl_give isl_printer *isl_printer_to_str(isl_ctx *ctx);
717 void isl_printer_free(__isl_take isl_printer *printer);
718 __isl_give char *isl_printer_get_str(
719 __isl_keep isl_printer *printer);
721 The behavior of the printer can be modified in various ways
723 __isl_give isl_printer *isl_printer_set_output_format(
724 __isl_take isl_printer *p, int output_format);
725 __isl_give isl_printer *isl_printer_set_indent(
726 __isl_take isl_printer *p, int indent);
727 __isl_give isl_printer *isl_printer_set_prefix(
728 __isl_take isl_printer *p, const char *prefix);
729 __isl_give isl_printer *isl_printer_set_suffix(
730 __isl_take isl_printer *p, const char *suffix);
732 The C<output_format> may be either C<ISL_FORMAT_ISL>, C<ISL_FORMAT_OMEGA>,
733 C<ISL_FORMAT_POLYLIB>, C<ISL_FORMAT_EXT_POLYLIB> or C<ISL_FORMAT_LATEX>
734 and defaults to C<ISL_FORMAT_ISL>.
735 Each line in the output is indented by C<indent> spaces
736 (default: 0), prefixed by C<prefix> and suffixed by C<suffix>.
737 In the C<PolyLib> format output,
738 the coefficients of the existentially quantified variables
739 appear between those of the set variables and those
742 To actually print something, use
745 __isl_give isl_printer *isl_printer_print_basic_set(
746 __isl_take isl_printer *printer,
747 __isl_keep isl_basic_set *bset);
748 __isl_give isl_printer *isl_printer_print_set(
749 __isl_take isl_printer *printer,
750 __isl_keep isl_set *set);
753 __isl_give isl_printer *isl_printer_print_basic_map(
754 __isl_take isl_printer *printer,
755 __isl_keep isl_basic_map *bmap);
756 __isl_give isl_printer *isl_printer_print_map(
757 __isl_take isl_printer *printer,
758 __isl_keep isl_map *map);
760 #include <isl/union_set.h>
761 __isl_give isl_printer *isl_printer_print_union_set(
762 __isl_take isl_printer *p,
763 __isl_keep isl_union_set *uset);
765 #include <isl/union_map.h>
766 __isl_give isl_printer *isl_printer_print_union_map(
767 __isl_take isl_printer *p,
768 __isl_keep isl_union_map *umap);
770 When called on a file printer, the following function flushes
771 the file. When called on a string printer, the buffer is cleared.
773 __isl_give isl_printer *isl_printer_flush(
774 __isl_take isl_printer *p);
776 =head2 Creating New Sets and Relations
778 C<isl> has functions for creating some standard sets and relations.
782 =item * Empty sets and relations
784 __isl_give isl_basic_set *isl_basic_set_empty(
785 __isl_take isl_dim *dim);
786 __isl_give isl_basic_map *isl_basic_map_empty(
787 __isl_take isl_dim *dim);
788 __isl_give isl_set *isl_set_empty(
789 __isl_take isl_dim *dim);
790 __isl_give isl_map *isl_map_empty(
791 __isl_take isl_dim *dim);
792 __isl_give isl_union_set *isl_union_set_empty(
793 __isl_take isl_dim *dim);
794 __isl_give isl_union_map *isl_union_map_empty(
795 __isl_take isl_dim *dim);
797 For C<isl_union_set>s and C<isl_union_map>s, the dimensions specification
798 is only used to specify the parameters.
800 =item * Universe sets and relations
802 __isl_give isl_basic_set *isl_basic_set_universe(
803 __isl_take isl_dim *dim);
804 __isl_give isl_basic_map *isl_basic_map_universe(
805 __isl_take isl_dim *dim);
806 __isl_give isl_set *isl_set_universe(
807 __isl_take isl_dim *dim);
808 __isl_give isl_map *isl_map_universe(
809 __isl_take isl_dim *dim);
810 __isl_give isl_union_set *isl_union_set_universe(
811 __isl_take isl_union_set *uset);
812 __isl_give isl_union_map *isl_union_map_universe(
813 __isl_take isl_union_map *umap);
815 The sets and relations constructed by the functions above
816 contain all integer values, while those constructed by the
817 functions below only contain non-negative values.
819 __isl_give isl_basic_set *isl_basic_set_nat_universe(
820 __isl_take isl_dim *dim);
821 __isl_give isl_basic_map *isl_basic_map_nat_universe(
822 __isl_take isl_dim *dim);
823 __isl_give isl_set *isl_set_nat_universe(
824 __isl_take isl_dim *dim);
825 __isl_give isl_map *isl_map_nat_universe(
826 __isl_take isl_dim *dim);
828 =item * Identity relations
830 __isl_give isl_basic_map *isl_basic_map_identity(
831 __isl_take isl_dim *dim);
832 __isl_give isl_map *isl_map_identity(
833 __isl_take isl_dim *dim);
835 The number of input and output dimensions in C<dim> needs
838 =item * Lexicographic order
840 __isl_give isl_map *isl_map_lex_lt(
841 __isl_take isl_dim *set_dim);
842 __isl_give isl_map *isl_map_lex_le(
843 __isl_take isl_dim *set_dim);
844 __isl_give isl_map *isl_map_lex_gt(
845 __isl_take isl_dim *set_dim);
846 __isl_give isl_map *isl_map_lex_ge(
847 __isl_take isl_dim *set_dim);
848 __isl_give isl_map *isl_map_lex_lt_first(
849 __isl_take isl_dim *dim, unsigned n);
850 __isl_give isl_map *isl_map_lex_le_first(
851 __isl_take isl_dim *dim, unsigned n);
852 __isl_give isl_map *isl_map_lex_gt_first(
853 __isl_take isl_dim *dim, unsigned n);
854 __isl_give isl_map *isl_map_lex_ge_first(
855 __isl_take isl_dim *dim, unsigned n);
857 The first four functions take a dimension specification for a B<set>
858 and return relations that express that the elements in the domain
859 are lexicographically less
860 (C<isl_map_lex_lt>), less or equal (C<isl_map_lex_le>),
861 greater (C<isl_map_lex_gt>) or greater or equal (C<isl_map_lex_ge>)
862 than the elements in the range.
863 The last four functions take a dimension specification for a map
864 and return relations that express that the first C<n> dimensions
865 in the domain are lexicographically less
866 (C<isl_map_lex_lt_first>), less or equal (C<isl_map_lex_le_first>),
867 greater (C<isl_map_lex_gt_first>) or greater or equal (C<isl_map_lex_ge_first>)
868 than the first C<n> dimensions in the range.
872 A basic set or relation can be converted to a set or relation
873 using the following functions.
875 __isl_give isl_set *isl_set_from_basic_set(
876 __isl_take isl_basic_set *bset);
877 __isl_give isl_map *isl_map_from_basic_map(
878 __isl_take isl_basic_map *bmap);
880 Sets and relations can be converted to union sets and relations
881 using the following functions.
883 __isl_give isl_union_map *isl_union_map_from_map(
884 __isl_take isl_map *map);
885 __isl_give isl_union_set *isl_union_set_from_set(
886 __isl_take isl_set *set);
888 Sets and relations can be copied and freed again using the following
891 __isl_give isl_basic_set *isl_basic_set_copy(
892 __isl_keep isl_basic_set *bset);
893 __isl_give isl_set *isl_set_copy(__isl_keep isl_set *set);
894 __isl_give isl_union_set *isl_union_set_copy(
895 __isl_keep isl_union_set *uset);
896 __isl_give isl_basic_map *isl_basic_map_copy(
897 __isl_keep isl_basic_map *bmap);
898 __isl_give isl_map *isl_map_copy(__isl_keep isl_map *map);
899 __isl_give isl_union_map *isl_union_map_copy(
900 __isl_keep isl_union_map *umap);
901 void isl_basic_set_free(__isl_take isl_basic_set *bset);
902 void isl_set_free(__isl_take isl_set *set);
903 void isl_union_set_free(__isl_take isl_union_set *uset);
904 void isl_basic_map_free(__isl_take isl_basic_map *bmap);
905 void isl_map_free(__isl_take isl_map *map);
906 void isl_union_map_free(__isl_take isl_union_map *umap);
908 Other sets and relations can be constructed by starting
909 from a universe set or relation, adding equality and/or
910 inequality constraints and then projecting out the
911 existentially quantified variables, if any.
912 Constraints can be constructed, manipulated and
913 added to (basic) sets and relations using the following functions.
915 #include <isl/constraint.h>
916 __isl_give isl_constraint *isl_equality_alloc(
917 __isl_take isl_dim *dim);
918 __isl_give isl_constraint *isl_inequality_alloc(
919 __isl_take isl_dim *dim);
920 void isl_constraint_set_constant(
921 __isl_keep isl_constraint *constraint, isl_int v);
922 void isl_constraint_set_coefficient(
923 __isl_keep isl_constraint *constraint,
924 enum isl_dim_type type, int pos, isl_int v);
925 __isl_give isl_basic_map *isl_basic_map_add_constraint(
926 __isl_take isl_basic_map *bmap,
927 __isl_take isl_constraint *constraint);
928 __isl_give isl_basic_set *isl_basic_set_add_constraint(
929 __isl_take isl_basic_set *bset,
930 __isl_take isl_constraint *constraint);
931 __isl_give isl_map *isl_map_add_constraint(
932 __isl_take isl_map *map,
933 __isl_take isl_constraint *constraint);
934 __isl_give isl_set *isl_set_add_constraint(
935 __isl_take isl_set *set,
936 __isl_take isl_constraint *constraint);
938 For example, to create a set containing the even integers
939 between 10 and 42, you would use the following code.
943 struct isl_constraint *c;
944 struct isl_basic_set *bset;
947 dim = isl_dim_set_alloc(ctx, 0, 2);
948 bset = isl_basic_set_universe(isl_dim_copy(dim));
950 c = isl_equality_alloc(isl_dim_copy(dim));
951 isl_int_set_si(v, -1);
952 isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
953 isl_int_set_si(v, 2);
954 isl_constraint_set_coefficient(c, isl_dim_set, 1, v);
955 bset = isl_basic_set_add_constraint(bset, c);
957 c = isl_inequality_alloc(isl_dim_copy(dim));
958 isl_int_set_si(v, -10);
959 isl_constraint_set_constant(c, v);
960 isl_int_set_si(v, 1);
961 isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
962 bset = isl_basic_set_add_constraint(bset, c);
964 c = isl_inequality_alloc(dim);
965 isl_int_set_si(v, 42);
966 isl_constraint_set_constant(c, v);
967 isl_int_set_si(v, -1);
968 isl_constraint_set_coefficient(c, isl_dim_set, 0, v);
969 bset = isl_basic_set_add_constraint(bset, c);
971 bset = isl_basic_set_project_out(bset, isl_dim_set, 1, 1);
977 struct isl_basic_set *bset;
978 bset = isl_basic_set_read_from_str(ctx,
979 "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}", -1);
981 A basic set or relation can also be constructed from two matrices
982 describing the equalities and the inequalities.
984 __isl_give isl_basic_set *isl_basic_set_from_constraint_matrices(
985 __isl_take isl_dim *dim,
986 __isl_take isl_mat *eq, __isl_take isl_mat *ineq,
987 enum isl_dim_type c1,
988 enum isl_dim_type c2, enum isl_dim_type c3,
989 enum isl_dim_type c4);
990 __isl_give isl_basic_map *isl_basic_map_from_constraint_matrices(
991 __isl_take isl_dim *dim,
992 __isl_take isl_mat *eq, __isl_take isl_mat *ineq,
993 enum isl_dim_type c1,
994 enum isl_dim_type c2, enum isl_dim_type c3,
995 enum isl_dim_type c4, enum isl_dim_type c5);
997 The C<isl_dim_type> arguments indicate the order in which
998 different kinds of variables appear in the input matrices
999 and should be a permutation of C<isl_dim_cst>, C<isl_dim_param>,
1000 C<isl_dim_set> and C<isl_dim_div> for sets and
1001 of C<isl_dim_cst>, C<isl_dim_param>,
1002 C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div> for relations.
1004 =head2 Inspecting Sets and Relations
1006 Usually, the user should not have to care about the actual constraints
1007 of the sets and maps, but should instead apply the abstract operations
1008 explained in the following sections.
1009 Occasionally, however, it may be required to inspect the individual
1010 coefficients of the constraints. This section explains how to do so.
1011 In these cases, it may also be useful to have C<isl> compute
1012 an explicit representation of the existentially quantified variables.
1014 __isl_give isl_set *isl_set_compute_divs(
1015 __isl_take isl_set *set);
1016 __isl_give isl_map *isl_map_compute_divs(
1017 __isl_take isl_map *map);
1018 __isl_give isl_union_set *isl_union_set_compute_divs(
1019 __isl_take isl_union_set *uset);
1020 __isl_give isl_union_map *isl_union_map_compute_divs(
1021 __isl_take isl_union_map *umap);
1023 This explicit representation defines the existentially quantified
1024 variables as integer divisions of the other variables, possibly
1025 including earlier existentially quantified variables.
1026 An explicitly represented existentially quantified variable therefore
1027 has a unique value when the values of the other variables are known.
1028 If, furthermore, the same existentials, i.e., existentials
1029 with the same explicit representations, should appear in the
1030 same order in each of the disjuncts of a set or map, then the user should call
1031 either of the following functions.
1033 __isl_give isl_set *isl_set_align_divs(
1034 __isl_take isl_set *set);
1035 __isl_give isl_map *isl_map_align_divs(
1036 __isl_take isl_map *map);
1038 Alternatively, the existentially quantified variables can be removed
1039 using the following functions, which compute an overapproximation.
1041 __isl_give isl_basic_set *isl_basic_set_remove_divs(
1042 __isl_take isl_basic_set *bset);
1043 __isl_give isl_basic_map *isl_basic_map_remove_divs(
1044 __isl_take isl_basic_map *bmap);
1045 __isl_give isl_set *isl_set_remove_divs(
1046 __isl_take isl_set *set);
1047 __isl_give isl_map *isl_map_remove_divs(
1048 __isl_take isl_map *map);
1050 To iterate over all the sets or maps in a union set or map, use
1052 int isl_union_set_foreach_set(__isl_keep isl_union_set *uset,
1053 int (*fn)(__isl_take isl_set *set, void *user),
1055 int isl_union_map_foreach_map(__isl_keep isl_union_map *umap,
1056 int (*fn)(__isl_take isl_map *map, void *user),
1059 The number of sets or maps in a union set or map can be obtained
1062 int isl_union_set_n_set(__isl_keep isl_union_set *uset);
1063 int isl_union_map_n_map(__isl_keep isl_union_map *umap);
1065 To extract the set or map from a union with a given dimension
1068 __isl_give isl_set *isl_union_set_extract_set(
1069 __isl_keep isl_union_set *uset,
1070 __isl_take isl_dim *dim);
1071 __isl_give isl_map *isl_union_map_extract_map(
1072 __isl_keep isl_union_map *umap,
1073 __isl_take isl_dim *dim);
1075 To iterate over all the basic sets or maps in a set or map, use
1077 int isl_set_foreach_basic_set(__isl_keep isl_set *set,
1078 int (*fn)(__isl_take isl_basic_set *bset, void *user),
1080 int isl_map_foreach_basic_map(__isl_keep isl_map *map,
1081 int (*fn)(__isl_take isl_basic_map *bmap, void *user),
1084 The callback function C<fn> should return 0 if successful and
1085 -1 if an error occurs. In the latter case, or if any other error
1086 occurs, the above functions will return -1.
1088 It should be noted that C<isl> does not guarantee that
1089 the basic sets or maps passed to C<fn> are disjoint.
1090 If this is required, then the user should call one of
1091 the following functions first.
1093 __isl_give isl_set *isl_set_make_disjoint(
1094 __isl_take isl_set *set);
1095 __isl_give isl_map *isl_map_make_disjoint(
1096 __isl_take isl_map *map);
1098 The number of basic sets in a set can be obtained
1101 int isl_set_n_basic_set(__isl_keep isl_set *set);
1103 To iterate over the constraints of a basic set or map, use
1105 #include <isl/constraint.h>
1107 int isl_basic_map_foreach_constraint(
1108 __isl_keep isl_basic_map *bmap,
1109 int (*fn)(__isl_take isl_constraint *c, void *user),
1111 void isl_constraint_free(struct isl_constraint *c);
1113 Again, the callback function C<fn> should return 0 if successful and
1114 -1 if an error occurs. In the latter case, or if any other error
1115 occurs, the above functions will return -1.
1116 The constraint C<c> represents either an equality or an inequality.
1117 Use the following function to find out whether a constraint
1118 represents an equality. If not, it represents an inequality.
1120 int isl_constraint_is_equality(
1121 __isl_keep isl_constraint *constraint);
1123 The coefficients of the constraints can be inspected using
1124 the following functions.
1126 void isl_constraint_get_constant(
1127 __isl_keep isl_constraint *constraint, isl_int *v);
1128 void isl_constraint_get_coefficient(
1129 __isl_keep isl_constraint *constraint,
1130 enum isl_dim_type type, int pos, isl_int *v);
1131 int isl_constraint_involves_dims(
1132 __isl_keep isl_constraint *constraint,
1133 enum isl_dim_type type, unsigned first, unsigned n);
1135 The explicit representations of the existentially quantified
1136 variables can be inspected using the following functions.
1137 Note that the user is only allowed to use these functions
1138 if the inspected set or map is the result of a call
1139 to C<isl_set_compute_divs> or C<isl_map_compute_divs>.
1141 __isl_give isl_div *isl_constraint_div(
1142 __isl_keep isl_constraint *constraint, int pos);
1143 isl_ctx *isl_div_get_ctx(__isl_keep isl_div *div);
1144 void isl_div_get_constant(__isl_keep isl_div *div,
1146 void isl_div_get_denominator(__isl_keep isl_div *div,
1148 void isl_div_get_coefficient(__isl_keep isl_div *div,
1149 enum isl_dim_type type, int pos, isl_int *v);
1151 To obtain the constraints of a basic set or map in matrix
1152 form, use the following functions.
1154 __isl_give isl_mat *isl_basic_set_equalities_matrix(
1155 __isl_keep isl_basic_set *bset,
1156 enum isl_dim_type c1, enum isl_dim_type c2,
1157 enum isl_dim_type c3, enum isl_dim_type c4);
1158 __isl_give isl_mat *isl_basic_set_inequalities_matrix(
1159 __isl_keep isl_basic_set *bset,
1160 enum isl_dim_type c1, enum isl_dim_type c2,
1161 enum isl_dim_type c3, enum isl_dim_type c4);
1162 __isl_give isl_mat *isl_basic_map_equalities_matrix(
1163 __isl_keep isl_basic_map *bmap,
1164 enum isl_dim_type c1,
1165 enum isl_dim_type c2, enum isl_dim_type c3,
1166 enum isl_dim_type c4, enum isl_dim_type c5);
1167 __isl_give isl_mat *isl_basic_map_inequalities_matrix(
1168 __isl_keep isl_basic_map *bmap,
1169 enum isl_dim_type c1,
1170 enum isl_dim_type c2, enum isl_dim_type c3,
1171 enum isl_dim_type c4, enum isl_dim_type c5);
1173 The C<isl_dim_type> arguments dictate the order in which
1174 different kinds of variables appear in the resulting matrix
1175 and should be a permutation of C<isl_dim_cst>, C<isl_dim_param>,
1176 C<isl_dim_in>, C<isl_dim_out> and C<isl_dim_div>.
1178 The names of the domain and range spaces of a set or relation can be
1179 read off using the following functions.
1181 const char *isl_basic_set_get_tuple_name(
1182 __isl_keep isl_basic_set *bset);
1183 const char *isl_set_get_tuple_name(
1184 __isl_keep isl_set *set);
1185 const char *isl_basic_map_get_tuple_name(
1186 __isl_keep isl_basic_map *bmap,
1187 enum isl_dim_type type);
1188 const char *isl_map_get_tuple_name(
1189 __isl_keep isl_map *map,
1190 enum isl_dim_type type);
1192 As with C<isl_dim_get_tuple_name>, the value returned points to
1193 an internal data structure.
1194 The names of individual dimensions can be read off using
1195 the following functions.
1197 const char *isl_constraint_get_dim_name(
1198 __isl_keep isl_constraint *constraint,
1199 enum isl_dim_type type, unsigned pos);
1200 const char *isl_basic_set_get_dim_name(
1201 __isl_keep isl_basic_set *bset,
1202 enum isl_dim_type type, unsigned pos);
1203 const char *isl_set_get_dim_name(
1204 __isl_keep isl_set *set,
1205 enum isl_dim_type type, unsigned pos);
1206 const char *isl_basic_map_get_dim_name(
1207 __isl_keep isl_basic_map *bmap,
1208 enum isl_dim_type type, unsigned pos);
1209 const char *isl_map_get_dim_name(
1210 __isl_keep isl_map *map,
1211 enum isl_dim_type type, unsigned pos);
1213 These functions are mostly useful to obtain the names
1218 =head3 Unary Properties
1224 The following functions test whether the given set or relation
1225 contains any integer points. The ``plain'' variants do not perform
1226 any computations, but simply check if the given set or relation
1227 is already known to be empty.
1229 int isl_basic_set_plain_is_empty(__isl_keep isl_basic_set *bset);
1230 int isl_basic_set_is_empty(__isl_keep isl_basic_set *bset);
1231 int isl_set_plain_is_empty(__isl_keep isl_set *set);
1232 int isl_set_is_empty(__isl_keep isl_set *set);
1233 int isl_union_set_is_empty(__isl_keep isl_union_set *uset);
1234 int isl_basic_map_plain_is_empty(__isl_keep isl_basic_map *bmap);
1235 int isl_basic_map_is_empty(__isl_keep isl_basic_map *bmap);
1236 int isl_map_plain_is_empty(__isl_keep isl_map *map);
1237 int isl_map_is_empty(__isl_keep isl_map *map);
1238 int isl_union_map_is_empty(__isl_keep isl_union_map *umap);
1240 =item * Universality
1242 int isl_basic_set_is_universe(__isl_keep isl_basic_set *bset);
1243 int isl_basic_map_is_universe(__isl_keep isl_basic_map *bmap);
1244 int isl_set_plain_is_universe(__isl_keep isl_set *set);
1246 =item * Single-valuedness
1248 int isl_map_is_single_valued(__isl_keep isl_map *map);
1249 int isl_union_map_is_single_valued(__isl_keep isl_union_map *umap);
1253 int isl_map_plain_is_injective(__isl_keep isl_map *map);
1254 int isl_map_is_injective(__isl_keep isl_map *map);
1255 int isl_union_map_plain_is_injective(
1256 __isl_keep isl_union_map *umap);
1257 int isl_union_map_is_injective(
1258 __isl_keep isl_union_map *umap);
1262 int isl_map_is_bijective(__isl_keep isl_map *map);
1263 int isl_union_map_is_bijective(__isl_keep isl_union_map *umap);
1267 The following functions check whether the domain of the given
1268 (basic) set is a wrapped relation.
1270 int isl_basic_set_is_wrapping(
1271 __isl_keep isl_basic_set *bset);
1272 int isl_set_is_wrapping(__isl_keep isl_set *set);
1274 =item * Internal Product
1276 int isl_basic_map_can_zip(
1277 __isl_keep isl_basic_map *bmap);
1278 int isl_map_can_zip(__isl_keep isl_map *map);
1280 Check whether the product of domain and range of the given relation
1282 i.e., whether both domain and range are nested relations.
1286 =head3 Binary Properties
1292 int isl_set_plain_is_equal(__isl_keep isl_set *set1,
1293 __isl_keep isl_set *set2);
1294 int isl_set_is_equal(__isl_keep isl_set *set1,
1295 __isl_keep isl_set *set2);
1296 int isl_union_set_is_equal(
1297 __isl_keep isl_union_set *uset1,
1298 __isl_keep isl_union_set *uset2);
1299 int isl_basic_map_is_equal(
1300 __isl_keep isl_basic_map *bmap1,
1301 __isl_keep isl_basic_map *bmap2);
1302 int isl_map_is_equal(__isl_keep isl_map *map1,
1303 __isl_keep isl_map *map2);
1304 int isl_map_plain_is_equal(__isl_keep isl_map *map1,
1305 __isl_keep isl_map *map2);
1306 int isl_union_map_is_equal(
1307 __isl_keep isl_union_map *umap1,
1308 __isl_keep isl_union_map *umap2);
1310 =item * Disjointness
1312 int isl_set_plain_is_disjoint(__isl_keep isl_set *set1,
1313 __isl_keep isl_set *set2);
1317 int isl_set_is_subset(__isl_keep isl_set *set1,
1318 __isl_keep isl_set *set2);
1319 int isl_set_is_strict_subset(
1320 __isl_keep isl_set *set1,
1321 __isl_keep isl_set *set2);
1322 int isl_union_set_is_subset(
1323 __isl_keep isl_union_set *uset1,
1324 __isl_keep isl_union_set *uset2);
1325 int isl_union_set_is_strict_subset(
1326 __isl_keep isl_union_set *uset1,
1327 __isl_keep isl_union_set *uset2);
1328 int isl_basic_map_is_subset(
1329 __isl_keep isl_basic_map *bmap1,
1330 __isl_keep isl_basic_map *bmap2);
1331 int isl_basic_map_is_strict_subset(
1332 __isl_keep isl_basic_map *bmap1,
1333 __isl_keep isl_basic_map *bmap2);
1334 int isl_map_is_subset(
1335 __isl_keep isl_map *map1,
1336 __isl_keep isl_map *map2);
1337 int isl_map_is_strict_subset(
1338 __isl_keep isl_map *map1,
1339 __isl_keep isl_map *map2);
1340 int isl_union_map_is_subset(
1341 __isl_keep isl_union_map *umap1,
1342 __isl_keep isl_union_map *umap2);
1343 int isl_union_map_is_strict_subset(
1344 __isl_keep isl_union_map *umap1,
1345 __isl_keep isl_union_map *umap2);
1349 =head2 Unary Operations
1355 __isl_give isl_set *isl_set_complement(
1356 __isl_take isl_set *set);
1360 __isl_give isl_basic_map *isl_basic_map_reverse(
1361 __isl_take isl_basic_map *bmap);
1362 __isl_give isl_map *isl_map_reverse(
1363 __isl_take isl_map *map);
1364 __isl_give isl_union_map *isl_union_map_reverse(
1365 __isl_take isl_union_map *umap);
1369 __isl_give isl_basic_set *isl_basic_set_project_out(
1370 __isl_take isl_basic_set *bset,
1371 enum isl_dim_type type, unsigned first, unsigned n);
1372 __isl_give isl_basic_map *isl_basic_map_project_out(
1373 __isl_take isl_basic_map *bmap,
1374 enum isl_dim_type type, unsigned first, unsigned n);
1375 __isl_give isl_set *isl_set_project_out(__isl_take isl_set *set,
1376 enum isl_dim_type type, unsigned first, unsigned n);
1377 __isl_give isl_map *isl_map_project_out(__isl_take isl_map *map,
1378 enum isl_dim_type type, unsigned first, unsigned n);
1379 __isl_give isl_basic_set *isl_basic_map_domain(
1380 __isl_take isl_basic_map *bmap);
1381 __isl_give isl_basic_set *isl_basic_map_range(
1382 __isl_take isl_basic_map *bmap);
1383 __isl_give isl_set *isl_map_domain(
1384 __isl_take isl_map *bmap);
1385 __isl_give isl_set *isl_map_range(
1386 __isl_take isl_map *map);
1387 __isl_give isl_union_set *isl_union_map_domain(
1388 __isl_take isl_union_map *umap);
1389 __isl_give isl_union_set *isl_union_map_range(
1390 __isl_take isl_union_map *umap);
1392 __isl_give isl_basic_map *isl_basic_map_domain_map(
1393 __isl_take isl_basic_map *bmap);
1394 __isl_give isl_basic_map *isl_basic_map_range_map(
1395 __isl_take isl_basic_map *bmap);
1396 __isl_give isl_map *isl_map_domain_map(__isl_take isl_map *map);
1397 __isl_give isl_map *isl_map_range_map(__isl_take isl_map *map);
1398 __isl_give isl_union_map *isl_union_map_domain_map(
1399 __isl_take isl_union_map *umap);
1400 __isl_give isl_union_map *isl_union_map_range_map(
1401 __isl_take isl_union_map *umap);
1403 The functions above construct a (basic, regular or union) relation
1404 that maps (a wrapped version of) the input relation to its domain or range.
1408 __isl_give isl_set *isl_set_eliminate(
1409 __isl_take isl_set *set, enum isl_dim_type type,
1410 unsigned first, unsigned n);
1412 Eliminate the coefficients for the given dimensions from the constraints,
1413 without removing the dimensions.
1417 __isl_give isl_map *isl_set_identity(
1418 __isl_take isl_set *set);
1419 __isl_give isl_union_map *isl_union_set_identity(
1420 __isl_take isl_union_set *uset);
1422 Construct an identity relation on the given (union) set.
1426 __isl_give isl_basic_set *isl_basic_map_deltas(
1427 __isl_take isl_basic_map *bmap);
1428 __isl_give isl_set *isl_map_deltas(__isl_take isl_map *map);
1429 __isl_give isl_union_set *isl_union_map_deltas(
1430 __isl_take isl_union_map *umap);
1432 These functions return a (basic) set containing the differences
1433 between image elements and corresponding domain elements in the input.
1435 __isl_give isl_basic_map *isl_basic_map_deltas_map(
1436 __isl_take isl_basic_map *bmap);
1437 __isl_give isl_map *isl_map_deltas_map(
1438 __isl_take isl_map *map);
1439 __isl_give isl_union_map *isl_union_map_deltas_map(
1440 __isl_take isl_union_map *umap);
1442 The functions above construct a (basic, regular or union) relation
1443 that maps (a wrapped version of) the input relation to its delta set.
1447 Simplify the representation of a set or relation by trying
1448 to combine pairs of basic sets or relations into a single
1449 basic set or relation.
1451 __isl_give isl_set *isl_set_coalesce(__isl_take isl_set *set);
1452 __isl_give isl_map *isl_map_coalesce(__isl_take isl_map *map);
1453 __isl_give isl_union_set *isl_union_set_coalesce(
1454 __isl_take isl_union_set *uset);
1455 __isl_give isl_union_map *isl_union_map_coalesce(
1456 __isl_take isl_union_map *umap);
1458 =item * Detecting equalities
1460 __isl_give isl_basic_set *isl_basic_set_detect_equalities(
1461 __isl_take isl_basic_set *bset);
1462 __isl_give isl_basic_map *isl_basic_map_detect_equalities(
1463 __isl_take isl_basic_map *bmap);
1464 __isl_give isl_set *isl_set_detect_equalities(
1465 __isl_take isl_set *set);
1466 __isl_give isl_map *isl_map_detect_equalities(
1467 __isl_take isl_map *map);
1468 __isl_give isl_union_set *isl_union_set_detect_equalities(
1469 __isl_take isl_union_set *uset);
1470 __isl_give isl_union_map *isl_union_map_detect_equalities(
1471 __isl_take isl_union_map *umap);
1473 Simplify the representation of a set or relation by detecting implicit
1476 =item * Removing redundant constraints
1478 __isl_give isl_basic_set *isl_basic_set_remove_redundancies(
1479 __isl_take isl_basic_set *bset);
1480 __isl_give isl_basic_map *isl_basic_map_remove_redundancies(
1481 __isl_take isl_basic_map *bmap);
1485 __isl_give isl_basic_set *isl_set_convex_hull(
1486 __isl_take isl_set *set);
1487 __isl_give isl_basic_map *isl_map_convex_hull(
1488 __isl_take isl_map *map);
1490 If the input set or relation has any existentially quantified
1491 variables, then the result of these operations is currently undefined.
1495 __isl_give isl_basic_set *isl_set_simple_hull(
1496 __isl_take isl_set *set);
1497 __isl_give isl_basic_map *isl_map_simple_hull(
1498 __isl_take isl_map *map);
1499 __isl_give isl_union_map *isl_union_map_simple_hull(
1500 __isl_take isl_union_map *umap);
1502 These functions compute a single basic set or relation
1503 that contains the whole input set or relation.
1504 In particular, the output is described by translates
1505 of the constraints describing the basic sets or relations in the input.
1509 (See \autoref{s:simple hull}.)
1515 __isl_give isl_basic_set *isl_basic_set_affine_hull(
1516 __isl_take isl_basic_set *bset);
1517 __isl_give isl_basic_set *isl_set_affine_hull(
1518 __isl_take isl_set *set);
1519 __isl_give isl_union_set *isl_union_set_affine_hull(
1520 __isl_take isl_union_set *uset);
1521 __isl_give isl_basic_map *isl_basic_map_affine_hull(
1522 __isl_take isl_basic_map *bmap);
1523 __isl_give isl_basic_map *isl_map_affine_hull(
1524 __isl_take isl_map *map);
1525 __isl_give isl_union_map *isl_union_map_affine_hull(
1526 __isl_take isl_union_map *umap);
1528 In case of union sets and relations, the affine hull is computed
1531 =item * Polyhedral hull
1533 __isl_give isl_basic_set *isl_set_polyhedral_hull(
1534 __isl_take isl_set *set);
1535 __isl_give isl_basic_map *isl_map_polyhedral_hull(
1536 __isl_take isl_map *map);
1537 __isl_give isl_union_set *isl_union_set_polyhedral_hull(
1538 __isl_take isl_union_set *uset);
1539 __isl_give isl_union_map *isl_union_map_polyhedral_hull(
1540 __isl_take isl_union_map *umap);
1542 These functions compute a single basic set or relation
1543 not involving any existentially quantified variables
1544 that contains the whole input set or relation.
1545 In case of union sets and relations, the polyhedral hull is computed
1548 =item * Optimization
1550 #include <isl/ilp.h>
1551 enum isl_lp_result isl_basic_set_max(
1552 __isl_keep isl_basic_set *bset,
1553 __isl_keep isl_aff *obj, isl_int *opt)
1554 enum isl_lp_result isl_set_max(__isl_keep isl_set *set,
1555 __isl_keep isl_aff *obj, isl_int *opt);
1557 Compute the maximum of the integer affine expression C<obj>
1558 over the points in C<set>, returning the result in C<opt>.
1559 The return value may be one of C<isl_lp_error>,
1560 C<isl_lp_ok>, C<isl_lp_unbounded> or C<isl_lp_empty>.
1564 The following functions compute either the set of (rational) coefficient
1565 values of valid constraints for the given set or the set of (rational)
1566 values satisfying the constraints with coefficients from the given set.
1567 Internally, these two sets of functions perform essentially the
1568 same operations, except that the set of coefficients is assumed to
1569 be a cone, while the set of values may be any polyhedron.
1570 The current implementation is based on the Farkas lemma and
1571 Fourier-Motzkin elimination, but this may change or be made optional
1572 in future. In particular, future implementations may use different
1573 dualization algorithms or skip the elimination step.
1575 __isl_give isl_basic_set *isl_basic_set_coefficients(
1576 __isl_take isl_basic_set *bset);
1577 __isl_give isl_basic_set *isl_set_coefficients(
1578 __isl_take isl_set *set);
1579 __isl_give isl_union_set *isl_union_set_coefficients(
1580 __isl_take isl_union_set *bset);
1581 __isl_give isl_basic_set *isl_basic_set_solutions(
1582 __isl_take isl_basic_set *bset);
1583 __isl_give isl_basic_set *isl_set_solutions(
1584 __isl_take isl_set *set);
1585 __isl_give isl_union_set *isl_union_set_solutions(
1586 __isl_take isl_union_set *bset);
1590 __isl_give isl_map *isl_map_power(__isl_take isl_map *map,
1592 __isl_give isl_union_map *isl_union_map_power(
1593 __isl_take isl_union_map *umap, int *exact);
1595 Compute a parametric representation for all positive powers I<k> of C<map>.
1596 The result maps I<k> to a nested relation corresponding to the
1597 I<k>th power of C<map>.
1598 The result may be an overapproximation. If the result is known to be exact,
1599 then C<*exact> is set to C<1>.
1601 =item * Transitive closure
1603 __isl_give isl_map *isl_map_transitive_closure(
1604 __isl_take isl_map *map, int *exact);
1605 __isl_give isl_union_map *isl_union_map_transitive_closure(
1606 __isl_take isl_union_map *umap, int *exact);
1608 Compute the transitive closure of C<map>.
1609 The result may be an overapproximation. If the result is known to be exact,
1610 then C<*exact> is set to C<1>.
1612 =item * Reaching path lengths
1614 __isl_give isl_map *isl_map_reaching_path_lengths(
1615 __isl_take isl_map *map, int *exact);
1617 Compute a relation that maps each element in the range of C<map>
1618 to the lengths of all paths composed of edges in C<map> that
1619 end up in the given element.
1620 The result may be an overapproximation. If the result is known to be exact,
1621 then C<*exact> is set to C<1>.
1622 To compute the I<maximal> path length, the resulting relation
1623 should be postprocessed by C<isl_map_lexmax>.
1624 In particular, if the input relation is a dependence relation
1625 (mapping sources to sinks), then the maximal path length corresponds
1626 to the free schedule.
1627 Note, however, that C<isl_map_lexmax> expects the maximum to be
1628 finite, so if the path lengths are unbounded (possibly due to
1629 the overapproximation), then you will get an error message.
1633 __isl_give isl_basic_set *isl_basic_map_wrap(
1634 __isl_take isl_basic_map *bmap);
1635 __isl_give isl_set *isl_map_wrap(
1636 __isl_take isl_map *map);
1637 __isl_give isl_union_set *isl_union_map_wrap(
1638 __isl_take isl_union_map *umap);
1639 __isl_give isl_basic_map *isl_basic_set_unwrap(
1640 __isl_take isl_basic_set *bset);
1641 __isl_give isl_map *isl_set_unwrap(
1642 __isl_take isl_set *set);
1643 __isl_give isl_union_map *isl_union_set_unwrap(
1644 __isl_take isl_union_set *uset);
1648 Remove any internal structure of domain (and range) of the given
1649 set or relation. If there is any such internal structure in the input,
1650 then the name of the space is also removed.
1652 __isl_give isl_basic_set *isl_basic_set_flatten(
1653 __isl_take isl_basic_set *bset);
1654 __isl_give isl_set *isl_set_flatten(
1655 __isl_take isl_set *set);
1656 __isl_give isl_basic_map *isl_basic_map_flatten(
1657 __isl_take isl_basic_map *bmap);
1658 __isl_give isl_map *isl_map_flatten(
1659 __isl_take isl_map *map);
1661 __isl_give isl_map *isl_set_flatten_map(
1662 __isl_take isl_set *set);
1664 The function above constructs a relation
1665 that maps the input set to a flattened version of the set.
1669 Lift the input set to a space with extra dimensions corresponding
1670 to the existentially quantified variables in the input.
1671 In particular, the result lives in a wrapped map where the domain
1672 is the original space and the range corresponds to the original
1673 existentially quantified variables.
1675 __isl_give isl_basic_set *isl_basic_set_lift(
1676 __isl_take isl_basic_set *bset);
1677 __isl_give isl_set *isl_set_lift(
1678 __isl_take isl_set *set);
1679 __isl_give isl_union_set *isl_union_set_lift(
1680 __isl_take isl_union_set *uset);
1682 =item * Internal Product
1684 __isl_give isl_basic_map *isl_basic_map_zip(
1685 __isl_take isl_basic_map *bmap);
1686 __isl_give isl_map *isl_map_zip(
1687 __isl_take isl_map *map);
1688 __isl_give isl_union_map *isl_union_map_zip(
1689 __isl_take isl_union_map *umap);
1691 Given a relation with nested relations for domain and range,
1692 interchange the range of the domain with the domain of the range.
1694 =item * Aligning parameters
1696 __isl_give isl_set *isl_set_align_params(
1697 __isl_take isl_set *set,
1698 __isl_take isl_dim *model);
1699 __isl_give isl_map *isl_map_align_params(
1700 __isl_take isl_map *map,
1701 __isl_take isl_dim *model);
1703 Change the order of the parameters of the given set or relation
1704 such that the first parameters match those of C<model>.
1705 This may involve the introduction of extra parameters.
1706 All parameters need to be named.
1708 =item * Dimension manipulation
1710 __isl_give isl_set *isl_set_add_dims(
1711 __isl_take isl_set *set,
1712 enum isl_dim_type type, unsigned n);
1713 __isl_give isl_map *isl_map_add_dims(
1714 __isl_take isl_map *map,
1715 enum isl_dim_type type, unsigned n);
1717 It is usually not advisable to directly change the (input or output)
1718 space of a set or a relation as this removes the name and the internal
1719 structure of the space. However, the above functions can be useful
1720 to add new parameters, assuming
1721 C<isl_set_align_params> and C<isl_map_align_params>
1726 =head2 Binary Operations
1728 The two arguments of a binary operation not only need to live
1729 in the same C<isl_ctx>, they currently also need to have
1730 the same (number of) parameters.
1732 =head3 Basic Operations
1736 =item * Intersection
1738 __isl_give isl_basic_set *isl_basic_set_intersect(
1739 __isl_take isl_basic_set *bset1,
1740 __isl_take isl_basic_set *bset2);
1741 __isl_give isl_set *isl_set_intersect(
1742 __isl_take isl_set *set1,
1743 __isl_take isl_set *set2);
1744 __isl_give isl_union_set *isl_union_set_intersect(
1745 __isl_take isl_union_set *uset1,
1746 __isl_take isl_union_set *uset2);
1747 __isl_give isl_basic_map *isl_basic_map_intersect_domain(
1748 __isl_take isl_basic_map *bmap,
1749 __isl_take isl_basic_set *bset);
1750 __isl_give isl_basic_map *isl_basic_map_intersect_range(
1751 __isl_take isl_basic_map *bmap,
1752 __isl_take isl_basic_set *bset);
1753 __isl_give isl_basic_map *isl_basic_map_intersect(
1754 __isl_take isl_basic_map *bmap1,
1755 __isl_take isl_basic_map *bmap2);
1756 __isl_give isl_map *isl_map_intersect_domain(
1757 __isl_take isl_map *map,
1758 __isl_take isl_set *set);
1759 __isl_give isl_map *isl_map_intersect_range(
1760 __isl_take isl_map *map,
1761 __isl_take isl_set *set);
1762 __isl_give isl_map *isl_map_intersect(
1763 __isl_take isl_map *map1,
1764 __isl_take isl_map *map2);
1765 __isl_give isl_union_map *isl_union_map_intersect_domain(
1766 __isl_take isl_union_map *umap,
1767 __isl_take isl_union_set *uset);
1768 __isl_give isl_union_map *isl_union_map_intersect_range(
1769 __isl_take isl_union_map *umap,
1770 __isl_take isl_union_set *uset);
1771 __isl_give isl_union_map *isl_union_map_intersect(
1772 __isl_take isl_union_map *umap1,
1773 __isl_take isl_union_map *umap2);
1777 __isl_give isl_set *isl_basic_set_union(
1778 __isl_take isl_basic_set *bset1,
1779 __isl_take isl_basic_set *bset2);
1780 __isl_give isl_map *isl_basic_map_union(
1781 __isl_take isl_basic_map *bmap1,
1782 __isl_take isl_basic_map *bmap2);
1783 __isl_give isl_set *isl_set_union(
1784 __isl_take isl_set *set1,
1785 __isl_take isl_set *set2);
1786 __isl_give isl_map *isl_map_union(
1787 __isl_take isl_map *map1,
1788 __isl_take isl_map *map2);
1789 __isl_give isl_union_set *isl_union_set_union(
1790 __isl_take isl_union_set *uset1,
1791 __isl_take isl_union_set *uset2);
1792 __isl_give isl_union_map *isl_union_map_union(
1793 __isl_take isl_union_map *umap1,
1794 __isl_take isl_union_map *umap2);
1796 =item * Set difference
1798 __isl_give isl_set *isl_set_subtract(
1799 __isl_take isl_set *set1,
1800 __isl_take isl_set *set2);
1801 __isl_give isl_map *isl_map_subtract(
1802 __isl_take isl_map *map1,
1803 __isl_take isl_map *map2);
1804 __isl_give isl_union_set *isl_union_set_subtract(
1805 __isl_take isl_union_set *uset1,
1806 __isl_take isl_union_set *uset2);
1807 __isl_give isl_union_map *isl_union_map_subtract(
1808 __isl_take isl_union_map *umap1,
1809 __isl_take isl_union_map *umap2);
1813 __isl_give isl_basic_set *isl_basic_set_apply(
1814 __isl_take isl_basic_set *bset,
1815 __isl_take isl_basic_map *bmap);
1816 __isl_give isl_set *isl_set_apply(
1817 __isl_take isl_set *set,
1818 __isl_take isl_map *map);
1819 __isl_give isl_union_set *isl_union_set_apply(
1820 __isl_take isl_union_set *uset,
1821 __isl_take isl_union_map *umap);
1822 __isl_give isl_basic_map *isl_basic_map_apply_domain(
1823 __isl_take isl_basic_map *bmap1,
1824 __isl_take isl_basic_map *bmap2);
1825 __isl_give isl_basic_map *isl_basic_map_apply_range(
1826 __isl_take isl_basic_map *bmap1,
1827 __isl_take isl_basic_map *bmap2);
1828 __isl_give isl_map *isl_map_apply_domain(
1829 __isl_take isl_map *map1,
1830 __isl_take isl_map *map2);
1831 __isl_give isl_union_map *isl_union_map_apply_domain(
1832 __isl_take isl_union_map *umap1,
1833 __isl_take isl_union_map *umap2);
1834 __isl_give isl_map *isl_map_apply_range(
1835 __isl_take isl_map *map1,
1836 __isl_take isl_map *map2);
1837 __isl_give isl_union_map *isl_union_map_apply_range(
1838 __isl_take isl_union_map *umap1,
1839 __isl_take isl_union_map *umap2);
1841 =item * Cartesian Product
1843 __isl_give isl_set *isl_set_product(
1844 __isl_take isl_set *set1,
1845 __isl_take isl_set *set2);
1846 __isl_give isl_union_set *isl_union_set_product(
1847 __isl_take isl_union_set *uset1,
1848 __isl_take isl_union_set *uset2);
1849 __isl_give isl_basic_map *isl_basic_map_range_product(
1850 __isl_take isl_basic_map *bmap1,
1851 __isl_take isl_basic_map *bmap2);
1852 __isl_give isl_map *isl_map_range_product(
1853 __isl_take isl_map *map1,
1854 __isl_take isl_map *map2);
1855 __isl_give isl_union_map *isl_union_map_range_product(
1856 __isl_take isl_union_map *umap1,
1857 __isl_take isl_union_map *umap2);
1858 __isl_give isl_map *isl_map_product(
1859 __isl_take isl_map *map1,
1860 __isl_take isl_map *map2);
1861 __isl_give isl_union_map *isl_union_map_product(
1862 __isl_take isl_union_map *umap1,
1863 __isl_take isl_union_map *umap2);
1865 The above functions compute the cross product of the given
1866 sets or relations. The domains and ranges of the results
1867 are wrapped maps between domains and ranges of the inputs.
1868 To obtain a ``flat'' product, use the following functions
1871 __isl_give isl_basic_set *isl_basic_set_flat_product(
1872 __isl_take isl_basic_set *bset1,
1873 __isl_take isl_basic_set *bset2);
1874 __isl_give isl_set *isl_set_flat_product(
1875 __isl_take isl_set *set1,
1876 __isl_take isl_set *set2);
1877 __isl_give isl_basic_map *isl_basic_map_flat_product(
1878 __isl_take isl_basic_map *bmap1,
1879 __isl_take isl_basic_map *bmap2);
1880 __isl_give isl_map *isl_map_flat_product(
1881 __isl_take isl_map *map1,
1882 __isl_take isl_map *map2);
1884 =item * Simplification
1886 __isl_give isl_basic_set *isl_basic_set_gist(
1887 __isl_take isl_basic_set *bset,
1888 __isl_take isl_basic_set *context);
1889 __isl_give isl_set *isl_set_gist(__isl_take isl_set *set,
1890 __isl_take isl_set *context);
1891 __isl_give isl_union_set *isl_union_set_gist(
1892 __isl_take isl_union_set *uset,
1893 __isl_take isl_union_set *context);
1894 __isl_give isl_basic_map *isl_basic_map_gist(
1895 __isl_take isl_basic_map *bmap,
1896 __isl_take isl_basic_map *context);
1897 __isl_give isl_map *isl_map_gist(__isl_take isl_map *map,
1898 __isl_take isl_map *context);
1899 __isl_give isl_union_map *isl_union_map_gist(
1900 __isl_take isl_union_map *umap,
1901 __isl_take isl_union_map *context);
1903 The gist operation returns a set or relation that has the
1904 same intersection with the context as the input set or relation.
1905 Any implicit equality in the intersection is made explicit in the result,
1906 while all inequalities that are redundant with respect to the intersection
1908 In case of union sets and relations, the gist operation is performed
1913 =head3 Lexicographic Optimization
1915 Given a (basic) set C<set> (or C<bset>) and a zero-dimensional domain C<dom>,
1916 the following functions
1917 compute a set that contains the lexicographic minimum or maximum
1918 of the elements in C<set> (or C<bset>) for those values of the parameters
1919 that satisfy C<dom>.
1920 If C<empty> is not C<NULL>, then C<*empty> is assigned a set
1921 that contains the parameter values in C<dom> for which C<set> (or C<bset>)
1923 In other words, the union of the parameter values
1924 for which the result is non-empty and of C<*empty>
1927 __isl_give isl_set *isl_basic_set_partial_lexmin(
1928 __isl_take isl_basic_set *bset,
1929 __isl_take isl_basic_set *dom,
1930 __isl_give isl_set **empty);
1931 __isl_give isl_set *isl_basic_set_partial_lexmax(
1932 __isl_take isl_basic_set *bset,
1933 __isl_take isl_basic_set *dom,
1934 __isl_give isl_set **empty);
1935 __isl_give isl_set *isl_set_partial_lexmin(
1936 __isl_take isl_set *set, __isl_take isl_set *dom,
1937 __isl_give isl_set **empty);
1938 __isl_give isl_set *isl_set_partial_lexmax(
1939 __isl_take isl_set *set, __isl_take isl_set *dom,
1940 __isl_give isl_set **empty);
1942 Given a (basic) set C<set> (or C<bset>), the following functions simply
1943 return a set containing the lexicographic minimum or maximum
1944 of the elements in C<set> (or C<bset>).
1945 In case of union sets, the optimum is computed per space.
1947 __isl_give isl_set *isl_basic_set_lexmin(
1948 __isl_take isl_basic_set *bset);
1949 __isl_give isl_set *isl_basic_set_lexmax(
1950 __isl_take isl_basic_set *bset);
1951 __isl_give isl_set *isl_set_lexmin(
1952 __isl_take isl_set *set);
1953 __isl_give isl_set *isl_set_lexmax(
1954 __isl_take isl_set *set);
1955 __isl_give isl_union_set *isl_union_set_lexmin(
1956 __isl_take isl_union_set *uset);
1957 __isl_give isl_union_set *isl_union_set_lexmax(
1958 __isl_take isl_union_set *uset);
1960 Given a (basic) relation C<map> (or C<bmap>) and a domain C<dom>,
1961 the following functions
1962 compute a relation that maps each element of C<dom>
1963 to the single lexicographic minimum or maximum
1964 of the elements that are associated to that same
1965 element in C<map> (or C<bmap>).
1966 If C<empty> is not C<NULL>, then C<*empty> is assigned a set
1967 that contains the elements in C<dom> that do not map
1968 to any elements in C<map> (or C<bmap>).
1969 In other words, the union of the domain of the result and of C<*empty>
1972 __isl_give isl_map *isl_basic_map_partial_lexmax(
1973 __isl_take isl_basic_map *bmap,
1974 __isl_take isl_basic_set *dom,
1975 __isl_give isl_set **empty);
1976 __isl_give isl_map *isl_basic_map_partial_lexmin(
1977 __isl_take isl_basic_map *bmap,
1978 __isl_take isl_basic_set *dom,
1979 __isl_give isl_set **empty);
1980 __isl_give isl_map *isl_map_partial_lexmax(
1981 __isl_take isl_map *map, __isl_take isl_set *dom,
1982 __isl_give isl_set **empty);
1983 __isl_give isl_map *isl_map_partial_lexmin(
1984 __isl_take isl_map *map, __isl_take isl_set *dom,
1985 __isl_give isl_set **empty);
1987 Given a (basic) map C<map> (or C<bmap>), the following functions simply
1988 return a map mapping each element in the domain of
1989 C<map> (or C<bmap>) to the lexicographic minimum or maximum
1990 of all elements associated to that element.
1991 In case of union relations, the optimum is computed per space.
1993 __isl_give isl_map *isl_basic_map_lexmin(
1994 __isl_take isl_basic_map *bmap);
1995 __isl_give isl_map *isl_basic_map_lexmax(
1996 __isl_take isl_basic_map *bmap);
1997 __isl_give isl_map *isl_map_lexmin(
1998 __isl_take isl_map *map);
1999 __isl_give isl_map *isl_map_lexmax(
2000 __isl_take isl_map *map);
2001 __isl_give isl_union_map *isl_union_map_lexmin(
2002 __isl_take isl_union_map *umap);
2003 __isl_give isl_union_map *isl_union_map_lexmax(
2004 __isl_take isl_union_map *umap);
2008 Lists are defined over several element types, including
2009 C<isl_aff>, C<isl_basic_set> and C<isl_set>.
2010 Here we take lists of C<isl_set>s as an example.
2011 Lists can be created, copied and freed using the following functions.
2013 #include <isl/list.h>
2014 __isl_give isl_set_list *isl_set_list_alloc(
2015 isl_ctx *ctx, int n);
2016 __isl_give isl_set_list *isl_set_list_copy(
2017 __isl_keep isl_set_list *list);
2018 __isl_give isl_set_list *isl_set_list_add(
2019 __isl_take isl_set_list *list,
2020 __isl_take isl_set *el);
2021 void isl_set_list_free(__isl_take isl_set_list *list);
2023 C<isl_set_list_alloc> creates an empty list with a capacity for
2026 Lists can be inspected using the following functions.
2028 #include <isl/list.h>
2029 isl_ctx *isl_set_list_get_ctx(__isl_keep isl_set_list *list);
2030 int isl_set_list_n_set(__isl_keep isl_set_list *list);
2031 __isl_give struct isl_set *isl_set_list_get_set(
2032 __isl_keep isl_set_list *list, int index);
2033 int isl_set_list_foreach(__isl_keep isl_set_list *list,
2034 int (*fn)(__isl_take struct isl_set *el, void *user),
2037 Lists can be printed using
2039 #include <isl/list.h>
2040 __isl_give isl_printer *isl_printer_print_set_list(
2041 __isl_take isl_printer *p,
2042 __isl_keep isl_set_list *list);
2046 Matrices can be created, copied and freed using the following functions.
2048 #include <isl/mat.h>
2049 __isl_give isl_mat *isl_mat_alloc(struct isl_ctx *ctx,
2050 unsigned n_row, unsigned n_col);
2051 __isl_give isl_mat *isl_mat_copy(__isl_keep isl_mat *mat);
2052 void isl_mat_free(__isl_take isl_mat *mat);
2054 Note that the elements of a newly created matrix may have arbitrary values.
2055 The elements can be changed and inspected using the following functions.
2057 isl_ctx *isl_mat_get_ctx(__isl_keep isl_mat *mat);
2058 int isl_mat_rows(__isl_keep isl_mat *mat);
2059 int isl_mat_cols(__isl_keep isl_mat *mat);
2060 int isl_mat_get_element(__isl_keep isl_mat *mat,
2061 int row, int col, isl_int *v);
2062 __isl_give isl_mat *isl_mat_set_element(__isl_take isl_mat *mat,
2063 int row, int col, isl_int v);
2064 __isl_give isl_mat *isl_mat_set_element_si(__isl_take isl_mat *mat,
2065 int row, int col, int v);
2067 C<isl_mat_get_element> will return a negative value if anything went wrong.
2068 In that case, the value of C<*v> is undefined.
2070 The following function can be used to compute the (right) inverse
2071 of a matrix, i.e., a matrix such that the product of the original
2072 and the inverse (in that order) is a multiple of the identity matrix.
2073 The input matrix is assumed to be of full row-rank.
2075 __isl_give isl_mat *isl_mat_right_inverse(__isl_take isl_mat *mat);
2077 The following function can be used to compute the (right) kernel
2078 (or null space) of a matrix, i.e., a matrix such that the product of
2079 the original and the kernel (in that order) is the zero matrix.
2081 __isl_give isl_mat *isl_mat_right_kernel(__isl_take isl_mat *mat);
2083 =head2 Quasi Affine Expressions
2085 The zero quasi affine expression can be created using
2087 __isl_give isl_aff *isl_aff_zero(
2088 __isl_take isl_local_space *ls);
2090 Quasi affine expressions can be copied and free using
2092 #include <isl/aff.h>
2093 __isl_give isl_aff *isl_aff_copy(__isl_keep isl_aff *aff);
2094 void *isl_aff_free(__isl_take isl_aff *aff);
2096 A (rational) bound on a dimension can be extracted from an C<isl_constraint>
2097 using the following function. The constraint is required to have
2098 a non-zero coefficient for the specified dimension.
2100 #include <isl/constraint.h>
2101 __isl_give isl_aff *isl_constraint_get_bound(
2102 __isl_keep isl_constraint *constraint,
2103 enum isl_dim_type type, int pos);
2105 Conversely, an equality constraint equating
2106 the affine expression to zero or an inequality constraint enforcing
2107 the affine expression to be non-negative, can be constructed using
2109 __isl_give isl_constraint *isl_equality_from_aff(
2110 __isl_take isl_aff *aff);
2111 __isl_give isl_constraint *isl_inequality_from_aff(
2112 __isl_take isl_aff *aff);
2114 The expression can be inspected using
2116 #include <isl/aff.h>
2117 isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff);
2118 int isl_aff_dim(__isl_keep isl_aff *aff,
2119 enum isl_dim_type type);
2120 __isl_give isl_local_space *isl_aff_get_local_space(
2121 __isl_keep isl_aff *aff);
2122 const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
2123 enum isl_dim_type type, unsigned pos);
2124 int isl_aff_get_constant(__isl_keep isl_aff *aff,
2126 int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
2127 enum isl_dim_type type, int pos, isl_int *v);
2128 int isl_aff_get_denominator(__isl_keep isl_aff *aff,
2130 __isl_give isl_div *isl_aff_get_div(
2131 __isl_keep isl_aff *aff, int pos);
2133 It can be modified using
2135 #include <isl/aff.h>
2136 __isl_give isl_aff *isl_aff_set_constant(
2137 __isl_take isl_aff *aff, isl_int v);
2138 __isl_give isl_aff *isl_aff_set_constant_si(
2139 __isl_take isl_aff *aff, int v);
2140 __isl_give isl_aff *isl_aff_set_coefficient(
2141 __isl_take isl_aff *aff,
2142 enum isl_dim_type type, int pos, isl_int v);
2143 __isl_give isl_aff *isl_aff_set_coefficient_si(
2144 __isl_take isl_aff *aff,
2145 enum isl_dim_type type, int pos, int v);
2146 __isl_give isl_aff *isl_aff_set_denominator(
2147 __isl_take isl_aff *aff, isl_int v);
2149 __isl_give isl_aff *isl_aff_add_constant(
2150 __isl_take isl_aff *aff, isl_int v);
2151 __isl_give isl_aff *isl_aff_add_coefficient_si(
2152 __isl_take isl_aff *aff,
2153 enum isl_dim_type type, int pos, int v);
2155 Note that the C<set_constant> and C<set_coefficient> functions
2156 set the I<numerator> of the constant or coefficient, while
2157 C<add_constant> and C<add_coefficient> add an integer value to
2158 the possibly rational constant or coefficient.
2162 #include <isl/aff.h>
2163 __isl_give isl_aff *isl_aff_add(__isl_take isl_aff *aff1,
2164 __isl_take isl_aff *aff2);
2165 __isl_give isl_aff *isl_aff_sub(__isl_take isl_aff *aff1,
2166 __isl_take isl_aff *aff2);
2167 __isl_give isl_aff *isl_aff_neg(__isl_take isl_aff *aff);
2168 __isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff);
2169 __isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff,
2171 __isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff,
2174 An expression can be printed using
2176 #include <isl/aff.h>
2177 __isl_give isl_printer *isl_printer_print_aff(
2178 __isl_take isl_printer *p, __isl_keep isl_aff *aff);
2182 Points are elements of a set. They can be used to construct
2183 simple sets (boxes) or they can be used to represent the
2184 individual elements of a set.
2185 The zero point (the origin) can be created using
2187 __isl_give isl_point *isl_point_zero(__isl_take isl_dim *dim);
2189 The coordinates of a point can be inspected, set and changed
2192 void isl_point_get_coordinate(__isl_keep isl_point *pnt,
2193 enum isl_dim_type type, int pos, isl_int *v);
2194 __isl_give isl_point *isl_point_set_coordinate(
2195 __isl_take isl_point *pnt,
2196 enum isl_dim_type type, int pos, isl_int v);
2198 __isl_give isl_point *isl_point_add_ui(
2199 __isl_take isl_point *pnt,
2200 enum isl_dim_type type, int pos, unsigned val);
2201 __isl_give isl_point *isl_point_sub_ui(
2202 __isl_take isl_point *pnt,
2203 enum isl_dim_type type, int pos, unsigned val);
2205 Points can be copied or freed using
2207 __isl_give isl_point *isl_point_copy(
2208 __isl_keep isl_point *pnt);
2209 void isl_point_free(__isl_take isl_point *pnt);
2211 A singleton set can be created from a point using
2213 __isl_give isl_basic_set *isl_basic_set_from_point(
2214 __isl_take isl_point *pnt);
2215 __isl_give isl_set *isl_set_from_point(
2216 __isl_take isl_point *pnt);
2218 and a box can be created from two opposite extremal points using
2220 __isl_give isl_basic_set *isl_basic_set_box_from_points(
2221 __isl_take isl_point *pnt1,
2222 __isl_take isl_point *pnt2);
2223 __isl_give isl_set *isl_set_box_from_points(
2224 __isl_take isl_point *pnt1,
2225 __isl_take isl_point *pnt2);
2227 All elements of a B<bounded> (union) set can be enumerated using
2228 the following functions.
2230 int isl_set_foreach_point(__isl_keep isl_set *set,
2231 int (*fn)(__isl_take isl_point *pnt, void *user),
2233 int isl_union_set_foreach_point(__isl_keep isl_union_set *uset,
2234 int (*fn)(__isl_take isl_point *pnt, void *user),
2237 The function C<fn> is called for each integer point in
2238 C<set> with as second argument the last argument of
2239 the C<isl_set_foreach_point> call. The function C<fn>
2240 should return C<0> on success and C<-1> on failure.
2241 In the latter case, C<isl_set_foreach_point> will stop
2242 enumerating and return C<-1> as well.
2243 If the enumeration is performed successfully and to completion,
2244 then C<isl_set_foreach_point> returns C<0>.
2246 To obtain a single point of a (basic) set, use
2248 __isl_give isl_point *isl_basic_set_sample_point(
2249 __isl_take isl_basic_set *bset);
2250 __isl_give isl_point *isl_set_sample_point(
2251 __isl_take isl_set *set);
2253 If C<set> does not contain any (integer) points, then the
2254 resulting point will be ``void'', a property that can be
2257 int isl_point_is_void(__isl_keep isl_point *pnt);
2259 =head2 Piecewise Quasipolynomials
2261 A piecewise quasipolynomial is a particular kind of function that maps
2262 a parametric point to a rational value.
2263 More specifically, a quasipolynomial is a polynomial expression in greatest
2264 integer parts of affine expressions of parameters and variables.
2265 A piecewise quasipolynomial is a subdivision of a given parametric
2266 domain into disjoint cells with a quasipolynomial associated to
2267 each cell. The value of the piecewise quasipolynomial at a given
2268 point is the value of the quasipolynomial associated to the cell
2269 that contains the point. Outside of the union of cells,
2270 the value is assumed to be zero.
2271 For example, the piecewise quasipolynomial
2273 [n] -> { [x] -> ((1 + n) - x) : x <= n and x >= 0 }
2275 maps C<x> to C<1 + n - x> for values of C<x> between C<0> and C<n>.
2276 A given piecewise quasipolynomial has a fixed domain dimension.
2277 Union piecewise quasipolynomials are used to contain piecewise quasipolynomials
2278 defined over different domains.
2279 Piecewise quasipolynomials are mainly used by the C<barvinok>
2280 library for representing the number of elements in a parametric set or map.
2281 For example, the piecewise quasipolynomial above represents
2282 the number of points in the map
2284 [n] -> { [x] -> [y] : x,y >= 0 and 0 <= x + y <= n }
2286 =head3 Printing (Piecewise) Quasipolynomials
2288 Quasipolynomials and piecewise quasipolynomials can be printed
2289 using the following functions.
2291 __isl_give isl_printer *isl_printer_print_qpolynomial(
2292 __isl_take isl_printer *p,
2293 __isl_keep isl_qpolynomial *qp);
2295 __isl_give isl_printer *isl_printer_print_pw_qpolynomial(
2296 __isl_take isl_printer *p,
2297 __isl_keep isl_pw_qpolynomial *pwqp);
2299 __isl_give isl_printer *isl_printer_print_union_pw_qpolynomial(
2300 __isl_take isl_printer *p,
2301 __isl_keep isl_union_pw_qpolynomial *upwqp);
2303 The output format of the printer
2304 needs to be set to either C<ISL_FORMAT_ISL> or C<ISL_FORMAT_C>.
2305 For C<isl_printer_print_union_pw_qpolynomial>, only C<ISL_FORMAT_ISL>
2307 In case of printing in C<ISL_FORMAT_C>, the user may want
2308 to set the names of all dimensions
2310 __isl_give isl_qpolynomial *isl_qpolynomial_set_dim_name(
2311 __isl_take isl_qpolynomial *qp,
2312 enum isl_dim_type type, unsigned pos,
2314 __isl_give isl_pw_qpolynomial *
2315 isl_pw_qpolynomial_set_dim_name(
2316 __isl_take isl_pw_qpolynomial *pwqp,
2317 enum isl_dim_type type, unsigned pos,
2320 =head3 Creating New (Piecewise) Quasipolynomials
2322 Some simple quasipolynomials can be created using the following functions.
2323 More complicated quasipolynomials can be created by applying
2324 operations such as addition and multiplication
2325 on the resulting quasipolynomials
2327 __isl_give isl_qpolynomial *isl_qpolynomial_zero(
2328 __isl_take isl_dim *dim);
2329 __isl_give isl_qpolynomial *isl_qpolynomial_one(
2330 __isl_take isl_dim *dim);
2331 __isl_give isl_qpolynomial *isl_qpolynomial_infty(
2332 __isl_take isl_dim *dim);
2333 __isl_give isl_qpolynomial *isl_qpolynomial_neginfty(
2334 __isl_take isl_dim *dim);
2335 __isl_give isl_qpolynomial *isl_qpolynomial_nan(
2336 __isl_take isl_dim *dim);
2337 __isl_give isl_qpolynomial *isl_qpolynomial_rat_cst(
2338 __isl_take isl_dim *dim,
2339 const isl_int n, const isl_int d);
2340 __isl_give isl_qpolynomial *isl_qpolynomial_div(
2341 __isl_take isl_div *div);
2342 __isl_give isl_qpolynomial *isl_qpolynomial_var(
2343 __isl_take isl_dim *dim,
2344 enum isl_dim_type type, unsigned pos);
2345 __isl_give isl_qpolynomial *isl_qpolynomial_from_aff(
2346 __isl_take isl_aff *aff);
2348 The zero piecewise quasipolynomial or a piecewise quasipolynomial
2349 with a single cell can be created using the following functions.
2350 Multiple of these single cell piecewise quasipolynomials can
2351 be combined to create more complicated piecewise quasipolynomials.
2353 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_zero(
2354 __isl_take isl_dim *dim);
2355 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_alloc(
2356 __isl_take isl_set *set,
2357 __isl_take isl_qpolynomial *qp);
2359 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_zero(
2360 __isl_take isl_dim *dim);
2361 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_from_pw_qpolynomial(
2362 __isl_take isl_pw_qpolynomial *pwqp);
2363 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add_pw_qpolynomial(
2364 __isl_take isl_union_pw_qpolynomial *upwqp,
2365 __isl_take isl_pw_qpolynomial *pwqp);
2367 Quasipolynomials can be copied and freed again using the following
2370 __isl_give isl_qpolynomial *isl_qpolynomial_copy(
2371 __isl_keep isl_qpolynomial *qp);
2372 void isl_qpolynomial_free(__isl_take isl_qpolynomial *qp);
2374 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_copy(
2375 __isl_keep isl_pw_qpolynomial *pwqp);
2376 void isl_pw_qpolynomial_free(
2377 __isl_take isl_pw_qpolynomial *pwqp);
2379 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_copy(
2380 __isl_keep isl_union_pw_qpolynomial *upwqp);
2381 void isl_union_pw_qpolynomial_free(
2382 __isl_take isl_union_pw_qpolynomial *upwqp);
2384 =head3 Inspecting (Piecewise) Quasipolynomials
2386 To iterate over all piecewise quasipolynomials in a union
2387 piecewise quasipolynomial, use the following function
2389 int isl_union_pw_qpolynomial_foreach_pw_qpolynomial(
2390 __isl_keep isl_union_pw_qpolynomial *upwqp,
2391 int (*fn)(__isl_take isl_pw_qpolynomial *pwqp, void *user),
2394 To extract the piecewise quasipolynomial from a union with a given dimension
2397 __isl_give isl_pw_qpolynomial *
2398 isl_union_pw_qpolynomial_extract_pw_qpolynomial(
2399 __isl_keep isl_union_pw_qpolynomial *upwqp,
2400 __isl_take isl_dim *dim);
2402 To iterate over the cells in a piecewise quasipolynomial,
2403 use either of the following two functions
2405 int isl_pw_qpolynomial_foreach_piece(
2406 __isl_keep isl_pw_qpolynomial *pwqp,
2407 int (*fn)(__isl_take isl_set *set,
2408 __isl_take isl_qpolynomial *qp,
2409 void *user), void *user);
2410 int isl_pw_qpolynomial_foreach_lifted_piece(
2411 __isl_keep isl_pw_qpolynomial *pwqp,
2412 int (*fn)(__isl_take isl_set *set,
2413 __isl_take isl_qpolynomial *qp,
2414 void *user), void *user);
2416 As usual, the function C<fn> should return C<0> on success
2417 and C<-1> on failure. The difference between
2418 C<isl_pw_qpolynomial_foreach_piece> and
2419 C<isl_pw_qpolynomial_foreach_lifted_piece> is that
2420 C<isl_pw_qpolynomial_foreach_lifted_piece> will first
2421 compute unique representations for all existentially quantified
2422 variables and then turn these existentially quantified variables
2423 into extra set variables, adapting the associated quasipolynomial
2424 accordingly. This means that the C<set> passed to C<fn>
2425 will not have any existentially quantified variables, but that
2426 the dimensions of the sets may be different for different
2427 invocations of C<fn>.
2429 To iterate over all terms in a quasipolynomial,
2432 int isl_qpolynomial_foreach_term(
2433 __isl_keep isl_qpolynomial *qp,
2434 int (*fn)(__isl_take isl_term *term,
2435 void *user), void *user);
2437 The terms themselves can be inspected and freed using
2440 unsigned isl_term_dim(__isl_keep isl_term *term,
2441 enum isl_dim_type type);
2442 void isl_term_get_num(__isl_keep isl_term *term,
2444 void isl_term_get_den(__isl_keep isl_term *term,
2446 int isl_term_get_exp(__isl_keep isl_term *term,
2447 enum isl_dim_type type, unsigned pos);
2448 __isl_give isl_div *isl_term_get_div(
2449 __isl_keep isl_term *term, unsigned pos);
2450 void isl_term_free(__isl_take isl_term *term);
2452 Each term is a product of parameters, set variables and
2453 integer divisions. The function C<isl_term_get_exp>
2454 returns the exponent of a given dimensions in the given term.
2455 The C<isl_int>s in the arguments of C<isl_term_get_num>
2456 and C<isl_term_get_den> need to have been initialized
2457 using C<isl_int_init> before calling these functions.
2459 =head3 Properties of (Piecewise) Quasipolynomials
2461 To check whether a quasipolynomial is actually a constant,
2462 use the following function.
2464 int isl_qpolynomial_is_cst(__isl_keep isl_qpolynomial *qp,
2465 isl_int *n, isl_int *d);
2467 If C<qp> is a constant and if C<n> and C<d> are not C<NULL>
2468 then the numerator and denominator of the constant
2469 are returned in C<*n> and C<*d>, respectively.
2471 =head3 Operations on (Piecewise) Quasipolynomials
2473 __isl_give isl_qpolynomial *isl_qpolynomial_neg(
2474 __isl_take isl_qpolynomial *qp);
2475 __isl_give isl_qpolynomial *isl_qpolynomial_add(
2476 __isl_take isl_qpolynomial *qp1,
2477 __isl_take isl_qpolynomial *qp2);
2478 __isl_give isl_qpolynomial *isl_qpolynomial_sub(
2479 __isl_take isl_qpolynomial *qp1,
2480 __isl_take isl_qpolynomial *qp2);
2481 __isl_give isl_qpolynomial *isl_qpolynomial_mul(
2482 __isl_take isl_qpolynomial *qp1,
2483 __isl_take isl_qpolynomial *qp2);
2484 __isl_give isl_qpolynomial *isl_qpolynomial_pow(
2485 __isl_take isl_qpolynomial *qp, unsigned exponent);
2487 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add(
2488 __isl_take isl_pw_qpolynomial *pwqp1,
2489 __isl_take isl_pw_qpolynomial *pwqp2);
2490 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_sub(
2491 __isl_take isl_pw_qpolynomial *pwqp1,
2492 __isl_take isl_pw_qpolynomial *pwqp2);
2493 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_add_disjoint(
2494 __isl_take isl_pw_qpolynomial *pwqp1,
2495 __isl_take isl_pw_qpolynomial *pwqp2);
2496 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_neg(
2497 __isl_take isl_pw_qpolynomial *pwqp);
2498 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_mul(
2499 __isl_take isl_pw_qpolynomial *pwqp1,
2500 __isl_take isl_pw_qpolynomial *pwqp2);
2502 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_add(
2503 __isl_take isl_union_pw_qpolynomial *upwqp1,
2504 __isl_take isl_union_pw_qpolynomial *upwqp2);
2505 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_sub(
2506 __isl_take isl_union_pw_qpolynomial *upwqp1,
2507 __isl_take isl_union_pw_qpolynomial *upwqp2);
2508 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_mul(
2509 __isl_take isl_union_pw_qpolynomial *upwqp1,
2510 __isl_take isl_union_pw_qpolynomial *upwqp2);
2512 __isl_give isl_qpolynomial *isl_pw_qpolynomial_eval(
2513 __isl_take isl_pw_qpolynomial *pwqp,
2514 __isl_take isl_point *pnt);
2516 __isl_give isl_qpolynomial *isl_union_pw_qpolynomial_eval(
2517 __isl_take isl_union_pw_qpolynomial *upwqp,
2518 __isl_take isl_point *pnt);
2520 __isl_give isl_set *isl_pw_qpolynomial_domain(
2521 __isl_take isl_pw_qpolynomial *pwqp);
2522 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_intersect_domain(
2523 __isl_take isl_pw_qpolynomial *pwpq,
2524 __isl_take isl_set *set);
2526 __isl_give isl_union_set *isl_union_pw_qpolynomial_domain(
2527 __isl_take isl_union_pw_qpolynomial *upwqp);
2528 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_intersect_domain(
2529 __isl_take isl_union_pw_qpolynomial *upwpq,
2530 __isl_take isl_union_set *uset);
2532 __isl_give isl_qpolynomial *isl_qpolynomial_align_params(
2533 __isl_take isl_qpolynomial *qp,
2534 __isl_take isl_dim *model);
2536 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_coalesce(
2537 __isl_take isl_union_pw_qpolynomial *upwqp);
2539 __isl_give isl_qpolynomial *isl_qpolynomial_gist(
2540 __isl_take isl_qpolynomial *qp,
2541 __isl_take isl_set *context);
2543 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_gist(
2544 __isl_take isl_pw_qpolynomial *pwqp,
2545 __isl_take isl_set *context);
2547 __isl_give isl_union_pw_qpolynomial *isl_union_pw_qpolynomial_gist(
2548 __isl_take isl_union_pw_qpolynomial *upwqp,
2549 __isl_take isl_union_set *context);
2551 The gist operation applies the gist operation to each of
2552 the cells in the domain of the input piecewise quasipolynomial.
2553 The context is also exploited
2554 to simplify the quasipolynomials associated to each cell.
2556 __isl_give isl_pw_qpolynomial *isl_pw_qpolynomial_to_polynomial(
2557 __isl_take isl_pw_qpolynomial *pwqp, int sign);
2558 __isl_give isl_union_pw_qpolynomial *
2559 isl_union_pw_qpolynomial_to_polynomial(
2560 __isl_take isl_union_pw_qpolynomial *upwqp, int sign);
2562 Approximate each quasipolynomial by a polynomial. If C<sign> is positive,
2563 the polynomial will be an overapproximation. If C<sign> is negative,
2564 it will be an underapproximation. If C<sign> is zero, the approximation
2565 will lie somewhere in between.
2567 =head2 Bounds on Piecewise Quasipolynomials and Piecewise Quasipolynomial Reductions
2569 A piecewise quasipolynomial reduction is a piecewise
2570 reduction (or fold) of quasipolynomials.
2571 In particular, the reduction can be maximum or a minimum.
2572 The objects are mainly used to represent the result of
2573 an upper or lower bound on a quasipolynomial over its domain,
2574 i.e., as the result of the following function.
2576 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_bound(
2577 __isl_take isl_pw_qpolynomial *pwqp,
2578 enum isl_fold type, int *tight);
2580 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_bound(
2581 __isl_take isl_union_pw_qpolynomial *upwqp,
2582 enum isl_fold type, int *tight);
2584 The C<type> argument may be either C<isl_fold_min> or C<isl_fold_max>.
2585 If C<tight> is not C<NULL>, then C<*tight> is set to C<1>
2586 is the returned bound is known be tight, i.e., for each value
2587 of the parameters there is at least
2588 one element in the domain that reaches the bound.
2589 If the domain of C<pwqp> is not wrapping, then the bound is computed
2590 over all elements in that domain and the result has a purely parametric
2591 domain. If the domain of C<pwqp> is wrapping, then the bound is
2592 computed over the range of the wrapped relation. The domain of the
2593 wrapped relation becomes the domain of the result.
2595 A (piecewise) quasipolynomial reduction can be copied or freed using the
2596 following functions.
2598 __isl_give isl_qpolynomial_fold *isl_qpolynomial_fold_copy(
2599 __isl_keep isl_qpolynomial_fold *fold);
2600 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_copy(
2601 __isl_keep isl_pw_qpolynomial_fold *pwf);
2602 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_copy(
2603 __isl_keep isl_union_pw_qpolynomial_fold *upwf);
2604 void isl_qpolynomial_fold_free(
2605 __isl_take isl_qpolynomial_fold *fold);
2606 void isl_pw_qpolynomial_fold_free(
2607 __isl_take isl_pw_qpolynomial_fold *pwf);
2608 void isl_union_pw_qpolynomial_fold_free(
2609 __isl_take isl_union_pw_qpolynomial_fold *upwf);
2611 =head3 Printing Piecewise Quasipolynomial Reductions
2613 Piecewise quasipolynomial reductions can be printed
2614 using the following function.
2616 __isl_give isl_printer *isl_printer_print_pw_qpolynomial_fold(
2617 __isl_take isl_printer *p,
2618 __isl_keep isl_pw_qpolynomial_fold *pwf);
2619 __isl_give isl_printer *isl_printer_print_union_pw_qpolynomial_fold(
2620 __isl_take isl_printer *p,
2621 __isl_keep isl_union_pw_qpolynomial_fold *upwf);
2623 For C<isl_printer_print_pw_qpolynomial_fold>,
2624 output format of the printer
2625 needs to be set to either C<ISL_FORMAT_ISL> or C<ISL_FORMAT_C>.
2626 For C<isl_printer_print_union_pw_qpolynomial_fold>,
2627 output format of the printer
2628 needs to be set to C<ISL_FORMAT_ISL>.
2629 In case of printing in C<ISL_FORMAT_C>, the user may want
2630 to set the names of all dimensions
2632 __isl_give isl_pw_qpolynomial_fold *
2633 isl_pw_qpolynomial_fold_set_dim_name(
2634 __isl_take isl_pw_qpolynomial_fold *pwf,
2635 enum isl_dim_type type, unsigned pos,
2638 =head3 Inspecting (Piecewise) Quasipolynomial Reductions
2640 To iterate over all piecewise quasipolynomial reductions in a union
2641 piecewise quasipolynomial reduction, use the following function
2643 int isl_union_pw_qpolynomial_fold_foreach_pw_qpolynomial_fold(
2644 __isl_keep isl_union_pw_qpolynomial_fold *upwf,
2645 int (*fn)(__isl_take isl_pw_qpolynomial_fold *pwf,
2646 void *user), void *user);
2648 To iterate over the cells in a piecewise quasipolynomial reduction,
2649 use either of the following two functions
2651 int isl_pw_qpolynomial_fold_foreach_piece(
2652 __isl_keep isl_pw_qpolynomial_fold *pwf,
2653 int (*fn)(__isl_take isl_set *set,
2654 __isl_take isl_qpolynomial_fold *fold,
2655 void *user), void *user);
2656 int isl_pw_qpolynomial_fold_foreach_lifted_piece(
2657 __isl_keep isl_pw_qpolynomial_fold *pwf,
2658 int (*fn)(__isl_take isl_set *set,
2659 __isl_take isl_qpolynomial_fold *fold,
2660 void *user), void *user);
2662 See L<Inspecting (Piecewise) Quasipolynomials> for an explanation
2663 of the difference between these two functions.
2665 To iterate over all quasipolynomials in a reduction, use
2667 int isl_qpolynomial_fold_foreach_qpolynomial(
2668 __isl_keep isl_qpolynomial_fold *fold,
2669 int (*fn)(__isl_take isl_qpolynomial *qp,
2670 void *user), void *user);
2672 =head3 Operations on Piecewise Quasipolynomial Reductions
2674 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_add(
2675 __isl_take isl_pw_qpolynomial_fold *pwf1,
2676 __isl_take isl_pw_qpolynomial_fold *pwf2);
2678 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_fold(
2679 __isl_take isl_pw_qpolynomial_fold *pwf1,
2680 __isl_take isl_pw_qpolynomial_fold *pwf2);
2682 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_fold(
2683 __isl_take isl_union_pw_qpolynomial_fold *upwf1,
2684 __isl_take isl_union_pw_qpolynomial_fold *upwf2);
2686 __isl_give isl_qpolynomial *isl_pw_qpolynomial_fold_eval(
2687 __isl_take isl_pw_qpolynomial_fold *pwf,
2688 __isl_take isl_point *pnt);
2690 __isl_give isl_qpolynomial *isl_union_pw_qpolynomial_fold_eval(
2691 __isl_take isl_union_pw_qpolynomial_fold *upwf,
2692 __isl_take isl_point *pnt);
2694 __isl_give isl_union_set *isl_union_pw_qpolynomial_fold_domain(
2695 __isl_take isl_union_pw_qpolynomial_fold *upwf);
2696 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_intersect_domain(
2697 __isl_take isl_union_pw_qpolynomial_fold *upwf,
2698 __isl_take isl_union_set *uset);
2700 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_coalesce(
2701 __isl_take isl_pw_qpolynomial_fold *pwf);
2703 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_coalesce(
2704 __isl_take isl_union_pw_qpolynomial_fold *upwf);
2706 __isl_give isl_pw_qpolynomial_fold *isl_pw_qpolynomial_fold_gist(
2707 __isl_take isl_pw_qpolynomial_fold *pwf,
2708 __isl_take isl_set *context);
2710 __isl_give isl_union_pw_qpolynomial_fold *isl_union_pw_qpolynomial_fold_gist(
2711 __isl_take isl_union_pw_qpolynomial_fold *upwf,
2712 __isl_take isl_union_set *context);
2714 The gist operation applies the gist operation to each of
2715 the cells in the domain of the input piecewise quasipolynomial reduction.
2716 In future, the operation will also exploit the context
2717 to simplify the quasipolynomial reductions associated to each cell.
2719 __isl_give isl_pw_qpolynomial_fold *
2720 isl_set_apply_pw_qpolynomial_fold(
2721 __isl_take isl_set *set,
2722 __isl_take isl_pw_qpolynomial_fold *pwf,
2724 __isl_give isl_pw_qpolynomial_fold *
2725 isl_map_apply_pw_qpolynomial_fold(
2726 __isl_take isl_map *map,
2727 __isl_take isl_pw_qpolynomial_fold *pwf,
2729 __isl_give isl_union_pw_qpolynomial_fold *
2730 isl_union_set_apply_union_pw_qpolynomial_fold(
2731 __isl_take isl_union_set *uset,
2732 __isl_take isl_union_pw_qpolynomial_fold *upwf,
2734 __isl_give isl_union_pw_qpolynomial_fold *
2735 isl_union_map_apply_union_pw_qpolynomial_fold(
2736 __isl_take isl_union_map *umap,
2737 __isl_take isl_union_pw_qpolynomial_fold *upwf,
2740 The functions taking a map
2741 compose the given map with the given piecewise quasipolynomial reduction.
2742 That is, compute a bound (of the same type as C<pwf> or C<upwf> itself)
2743 over all elements in the intersection of the range of the map
2744 and the domain of the piecewise quasipolynomial reduction
2745 as a function of an element in the domain of the map.
2746 The functions taking a set compute a bound over all elements in the
2747 intersection of the set and the domain of the
2748 piecewise quasipolynomial reduction.
2750 =head2 Dependence Analysis
2752 C<isl> contains specialized functionality for performing
2753 array dataflow analysis. That is, given a I<sink> access relation
2754 and a collection of possible I<source> access relations,
2755 C<isl> can compute relations that describe
2756 for each iteration of the sink access, which iteration
2757 of which of the source access relations was the last
2758 to access the same data element before the given iteration
2760 To compute standard flow dependences, the sink should be
2761 a read, while the sources should be writes.
2762 If any of the source accesses are marked as being I<may>
2763 accesses, then there will be a dependence to the last
2764 I<must> access B<and> to any I<may> access that follows
2765 this last I<must> access.
2766 In particular, if I<all> sources are I<may> accesses,
2767 then memory based dependence analysis is performed.
2768 If, on the other hand, all sources are I<must> accesses,
2769 then value based dependence analysis is performed.
2771 #include <isl/flow.h>
2773 typedef int (*isl_access_level_before)(void *first, void *second);
2775 __isl_give isl_access_info *isl_access_info_alloc(
2776 __isl_take isl_map *sink,
2777 void *sink_user, isl_access_level_before fn,
2779 __isl_give isl_access_info *isl_access_info_add_source(
2780 __isl_take isl_access_info *acc,
2781 __isl_take isl_map *source, int must,
2783 void isl_access_info_free(__isl_take isl_access_info *acc);
2785 __isl_give isl_flow *isl_access_info_compute_flow(
2786 __isl_take isl_access_info *acc);
2788 int isl_flow_foreach(__isl_keep isl_flow *deps,
2789 int (*fn)(__isl_take isl_map *dep, int must,
2790 void *dep_user, void *user),
2792 __isl_give isl_map *isl_flow_get_no_source(
2793 __isl_keep isl_flow *deps, int must);
2794 void isl_flow_free(__isl_take isl_flow *deps);
2796 The function C<isl_access_info_compute_flow> performs the actual
2797 dependence analysis. The other functions are used to construct
2798 the input for this function or to read off the output.
2800 The input is collected in an C<isl_access_info>, which can
2801 be created through a call to C<isl_access_info_alloc>.
2802 The arguments to this functions are the sink access relation
2803 C<sink>, a token C<sink_user> used to identify the sink
2804 access to the user, a callback function for specifying the
2805 relative order of source and sink accesses, and the number
2806 of source access relations that will be added.
2807 The callback function has type C<int (*)(void *first, void *second)>.
2808 The function is called with two user supplied tokens identifying
2809 either a source or the sink and it should return the shared nesting
2810 level and the relative order of the two accesses.
2811 In particular, let I<n> be the number of loops shared by
2812 the two accesses. If C<first> precedes C<second> textually,
2813 then the function should return I<2 * n + 1>; otherwise,
2814 it should return I<2 * n>.
2815 The sources can be added to the C<isl_access_info> by performing
2816 (at most) C<max_source> calls to C<isl_access_info_add_source>.
2817 C<must> indicates whether the source is a I<must> access
2818 or a I<may> access. Note that a multi-valued access relation
2819 should only be marked I<must> if every iteration in the domain
2820 of the relation accesses I<all> elements in its image.
2821 The C<source_user> token is again used to identify
2822 the source access. The range of the source access relation
2823 C<source> should have the same dimension as the range
2824 of the sink access relation.
2825 The C<isl_access_info_free> function should usually not be
2826 called explicitly, because it is called implicitly by
2827 C<isl_access_info_compute_flow>.
2829 The result of the dependence analysis is collected in an
2830 C<isl_flow>. There may be elements of
2831 the sink access for which no preceding source access could be
2832 found or for which all preceding sources are I<may> accesses.
2833 The relations containing these elements can be obtained through
2834 calls to C<isl_flow_get_no_source>, the first with C<must> set
2835 and the second with C<must> unset.
2836 In the case of standard flow dependence analysis,
2837 with the sink a read and the sources I<must> writes,
2838 the first relation corresponds to the reads from uninitialized
2839 array elements and the second relation is empty.
2840 The actual flow dependences can be extracted using
2841 C<isl_flow_foreach>. This function will call the user-specified
2842 callback function C<fn> for each B<non-empty> dependence between
2843 a source and the sink. The callback function is called
2844 with four arguments, the actual flow dependence relation
2845 mapping source iterations to sink iterations, a boolean that
2846 indicates whether it is a I<must> or I<may> dependence, a token
2847 identifying the source and an additional C<void *> with value
2848 equal to the third argument of the C<isl_flow_foreach> call.
2849 A dependence is marked I<must> if it originates from a I<must>
2850 source and if it is not followed by any I<may> sources.
2852 After finishing with an C<isl_flow>, the user should call
2853 C<isl_flow_free> to free all associated memory.
2855 A higher-level interface to dependence analysis is provided
2856 by the following function.
2858 #include <isl/flow.h>
2860 int isl_union_map_compute_flow(__isl_take isl_union_map *sink,
2861 __isl_take isl_union_map *must_source,
2862 __isl_take isl_union_map *may_source,
2863 __isl_take isl_union_map *schedule,
2864 __isl_give isl_union_map **must_dep,
2865 __isl_give isl_union_map **may_dep,
2866 __isl_give isl_union_map **must_no_source,
2867 __isl_give isl_union_map **may_no_source);
2869 The arrays are identified by the tuple names of the ranges
2870 of the accesses. The iteration domains by the tuple names
2871 of the domains of the accesses and of the schedule.
2872 The relative order of the iteration domains is given by the
2873 schedule. The relations returned through C<must_no_source>
2874 and C<may_no_source> are subsets of C<sink>.
2875 Any of C<must_dep>, C<may_dep>, C<must_no_source>
2876 or C<may_no_source> may be C<NULL>, but a C<NULL> value for
2877 any of the other arguments is treated as an error.
2881 B<The functionality described in this section is fairly new
2882 and may be subject to change.>
2884 The following function can be used to compute a schedule
2885 for a union of domains. The generated schedule respects
2886 all C<validity> dependences. That is, all dependence distances
2887 over these dependences in the scheduled space are lexicographically
2888 positive. The generated schedule schedule also tries to minimize
2889 the dependence distances over C<proximity> dependences.
2890 Moreover, it tries to obtain sequences (bands) of schedule dimensions
2891 for groups of domains where the dependence distances have only
2892 non-negative values.
2893 The algorithm used to construct the schedule is similar to that
2896 #include <isl/schedule.h>
2897 __isl_give isl_schedule *isl_union_set_compute_schedule(
2898 __isl_take isl_union_set *domain,
2899 __isl_take isl_union_map *validity,
2900 __isl_take isl_union_map *proximity);
2901 void *isl_schedule_free(__isl_take isl_schedule *sched);
2903 A mapping from the domains to the scheduled space can be obtained
2904 from an C<isl_schedule> using the following function.
2906 __isl_give isl_union_map *isl_schedule_get_map(
2907 __isl_keep isl_schedule *sched);
2909 This mapping can also be obtained in pieces using the following functions.
2911 int isl_schedule_n_band(__isl_keep isl_schedule *sched);
2912 __isl_give isl_union_map *isl_schedule_get_band(
2913 __isl_keep isl_schedule *sched, unsigned band);
2915 C<isl_schedule_n_band> returns the maximal number of bands.
2916 C<isl_schedule_get_band> returns a union of mappings from a domain to
2917 the band of consecutive schedule dimensions with the given sequence
2918 number for that domain. Bands with the same sequence number but for
2919 different domains may be completely unrelated.
2920 Within a band, the corresponding coordinates of the distance vectors
2921 are all non-negative, assuming that the coordinates for all previous
2924 =head2 Parametric Vertex Enumeration
2926 The parametric vertex enumeration described in this section
2927 is mainly intended to be used internally and by the C<barvinok>
2930 #include <isl/vertices.h>
2931 __isl_give isl_vertices *isl_basic_set_compute_vertices(
2932 __isl_keep isl_basic_set *bset);
2934 The function C<isl_basic_set_compute_vertices> performs the
2935 actual computation of the parametric vertices and the chamber
2936 decomposition and store the result in an C<isl_vertices> object.
2937 This information can be queried by either iterating over all
2938 the vertices or iterating over all the chambers or cells
2939 and then iterating over all vertices that are active on the chamber.
2941 int isl_vertices_foreach_vertex(
2942 __isl_keep isl_vertices *vertices,
2943 int (*fn)(__isl_take isl_vertex *vertex, void *user),
2946 int isl_vertices_foreach_cell(
2947 __isl_keep isl_vertices *vertices,
2948 int (*fn)(__isl_take isl_cell *cell, void *user),
2950 int isl_cell_foreach_vertex(__isl_keep isl_cell *cell,
2951 int (*fn)(__isl_take isl_vertex *vertex, void *user),
2954 Other operations that can be performed on an C<isl_vertices> object are
2957 isl_ctx *isl_vertices_get_ctx(
2958 __isl_keep isl_vertices *vertices);
2959 int isl_vertices_get_n_vertices(
2960 __isl_keep isl_vertices *vertices);
2961 void isl_vertices_free(__isl_take isl_vertices *vertices);
2963 Vertices can be inspected and destroyed using the following functions.
2965 isl_ctx *isl_vertex_get_ctx(__isl_keep isl_vertex *vertex);
2966 int isl_vertex_get_id(__isl_keep isl_vertex *vertex);
2967 __isl_give isl_basic_set *isl_vertex_get_domain(
2968 __isl_keep isl_vertex *vertex);
2969 __isl_give isl_basic_set *isl_vertex_get_expr(
2970 __isl_keep isl_vertex *vertex);
2971 void isl_vertex_free(__isl_take isl_vertex *vertex);
2973 C<isl_vertex_get_expr> returns a singleton parametric set describing
2974 the vertex, while C<isl_vertex_get_domain> returns the activity domain
2976 Note that C<isl_vertex_get_domain> and C<isl_vertex_get_expr> return
2977 B<rational> basic sets, so they should mainly be used for inspection
2978 and should not be mixed with integer sets.
2980 Chambers can be inspected and destroyed using the following functions.
2982 isl_ctx *isl_cell_get_ctx(__isl_keep isl_cell *cell);
2983 __isl_give isl_basic_set *isl_cell_get_domain(
2984 __isl_keep isl_cell *cell);
2985 void isl_cell_free(__isl_take isl_cell *cell);
2989 Although C<isl> is mainly meant to be used as a library,
2990 it also contains some basic applications that use some
2991 of the functionality of C<isl>.
2992 The input may be specified in either the L<isl format>
2993 or the L<PolyLib format>.
2995 =head2 C<isl_polyhedron_sample>
2997 C<isl_polyhedron_sample> takes a polyhedron as input and prints
2998 an integer element of the polyhedron, if there is any.
2999 The first column in the output is the denominator and is always
3000 equal to 1. If the polyhedron contains no integer points,
3001 then a vector of length zero is printed.
3005 C<isl_pip> takes the same input as the C<example> program
3006 from the C<piplib> distribution, i.e., a set of constraints
3007 on the parameters, a line containing only -1 and finally a set
3008 of constraints on a parametric polyhedron.
3009 The coefficients of the parameters appear in the last columns
3010 (but before the final constant column).
3011 The output is the lexicographic minimum of the parametric polyhedron.
3012 As C<isl> currently does not have its own output format, the output
3013 is just a dump of the internal state.
3015 =head2 C<isl_polyhedron_minimize>
3017 C<isl_polyhedron_minimize> computes the minimum of some linear
3018 or affine objective function over the integer points in a polyhedron.
3019 If an affine objective function
3020 is given, then the constant should appear in the last column.
3022 =head2 C<isl_polytope_scan>
3024 Given a polytope, C<isl_polytope_scan> prints
3025 all integer points in the polytope.