isl_ast_codegen.c: update_unrolling_lower_bound: use isl_val

[platform/upstream/isl.git] / isl_ast_codegen.c

/*
 * Copyright 2012      Ecole Normale Superieure
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege,
 * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 */

#include <limits.h>
#include <isl/aff.h>
#include <isl/set.h>
#include <isl/ilp.h>
#include <isl/union_map.h>
#include <isl_sort.h>
#include <isl_tarjan.h>
#include <isl_ast_private.h>
#include <isl_ast_build_expr.h>
#include <isl_ast_build_private.h>
#include <isl_ast_graft_private.h>

/* Add the constraint to the list that "user" points to, if it is not
 * a div constraint.
 */
static int collect_constraint(__isl_take isl_constraint *constraint,
        void *user)
{
        isl_constraint_list **list = user;

        if (isl_constraint_is_div_constraint(constraint))
                isl_constraint_free(constraint);
        else
                *list = isl_constraint_list_add(*list, constraint);

        return 0;
}

/* Extract the constraints of "bset" (except the div constraints)
 * and collect them in an isl_constraint_list.
 */
static __isl_give isl_constraint_list *isl_constraint_list_from_basic_set(
        __isl_take isl_basic_set *bset)
{
        int n;
        isl_ctx *ctx;
        isl_constraint_list *list;

        if (!bset)
                return NULL;

        ctx = isl_basic_set_get_ctx(bset);

        n = isl_basic_set_n_constraint(bset);
        list = isl_constraint_list_alloc(ctx, n);
        if (isl_basic_set_foreach_constraint(bset,
                                            &collect_constraint, &list) < 0)
                list = isl_constraint_list_free(list);

        isl_basic_set_free(bset);
        return list;
}

/* Data used in generate_domain.
 *
 * "build" is the input build.
 * "list" collects the results.
 */
struct isl_generate_domain_data {
        isl_ast_build *build;

        isl_ast_graft_list *list;
};

static __isl_give isl_ast_graft_list *generate_next_level(
        __isl_take isl_union_map *executed,
        __isl_take isl_ast_build *build);
static __isl_give isl_ast_graft_list *generate_code(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build,
        int internal);

/* Generate an AST for a single domain based on
 * the (non single valued) inverse schedule "executed".
 *
 * We extend the schedule with the iteration domain
 * and continue generating through a call to generate_code.
 *
 * In particular, if executed has the form
 *
 *      S -> D
 *
 * then we continue generating code on
 *
 *      [S -> D] -> D
 *
 * The extended inverse schedule is clearly single valued
 * ensuring that the nested generate_code will not reach this function,
 * but will instead create calls to all elements of D that need
 * to be executed from the current schedule domain.
 */
static int generate_non_single_valued(__isl_take isl_map *executed,
        struct isl_generate_domain_data *data)
{
        isl_map *identity;
        isl_ast_build *build;
        isl_ast_graft_list *list;

        build = isl_ast_build_copy(data->build);

        identity = isl_set_identity(isl_map_range(isl_map_copy(executed)));
        executed = isl_map_domain_product(executed, identity);
        build = isl_ast_build_set_single_valued(build, 1);

        list = generate_code(isl_union_map_from_map(executed), build, 1);

        data->list = isl_ast_graft_list_concat(data->list, list);

        return 0;
}

/* Call the at_each_domain callback, if requested by the user,
 * after recording the current inverse schedule in the build.
 */
static __isl_give isl_ast_graft *at_each_domain(__isl_take isl_ast_graft *graft,
        __isl_keep isl_map *executed, __isl_keep isl_ast_build *build)
{
        if (!graft || !build)
                return isl_ast_graft_free(graft);
        if (!build->at_each_domain)
                return graft;

        build = isl_ast_build_copy(build);
        build = isl_ast_build_set_executed(build,
                        isl_union_map_from_map(isl_map_copy(executed)));
        if (!build)
                return isl_ast_graft_free(graft);

        graft->node = build->at_each_domain(graft->node,
                                        build, build->at_each_domain_user);
        isl_ast_build_free(build);

        if (!graft->node)
                graft = isl_ast_graft_free(graft);

        return graft;
}

/* Generate an AST for a single domain based on
 * the inverse schedule "executed".
 *
 * If there is more than one domain element associated to the current
 * schedule "time", then we need to continue the generation process
 * in generate_non_single_valued.
 * Note that the inverse schedule being single-valued may depend
 * on constraints that are only available in the original context
 * domain specified by the user.  We therefore first introduce
 * the constraints from data->build->domain.
 * On the other hand, we only perform the test after having taken the gist
 * of the domain as the resulting map is the one from which the call
 * expression is constructed.  Using this map to construct the call
 * expression usually yields simpler results.
 * Because we perform the single-valuedness test on the gisted map,
 * we may in rare cases fail to recognize that the inverse schedule
 * is single-valued.  This becomes problematic if this happens
 * from the recursive call through generate_non_single_valued
 * as we would then end up in an infinite recursion.
 * We therefore check if we are inside a call to generate_non_single_valued
 * and revert to the ungisted map if the gisted map turns out not to be
 * single-valued.
 *
 * Otherwise, we generate a call expression for the single executed
 * domain element and put a guard around it based on the (simplified)
 * domain of "executed".
 *
 * If the user has set an at_each_domain callback, it is called
 * on the constructed call expression node.
 */
static int generate_domain(__isl_take isl_map *executed, void *user)
{
        struct isl_generate_domain_data *data = user;
        isl_ast_graft *graft;
        isl_ast_graft_list *list;
        isl_set *guard;
        isl_map *map;
        int sv;

        executed = isl_map_intersect_domain(executed,
                                            isl_set_copy(data->build->domain));

        executed = isl_map_coalesce(executed);
        map = isl_map_copy(executed);
        map = isl_ast_build_compute_gist_map_domain(data->build, map);
        sv = isl_map_is_single_valued(map);
        if (sv < 0)
                goto error;
        if (!sv) {
                isl_map_free(map);
                if (data->build->single_valued)
                        map = isl_map_copy(executed);
                else
                        return generate_non_single_valued(executed, data);
        }
        guard = isl_map_domain(isl_map_copy(map));
        guard = isl_set_coalesce(guard);
        guard = isl_ast_build_compute_gist(data->build, guard);
        graft = isl_ast_graft_alloc_domain(map, data->build);
        graft = at_each_domain(graft, executed, data->build);

        isl_map_free(executed);
        graft = isl_ast_graft_add_guard(graft, guard, data->build);

        list = isl_ast_graft_list_from_ast_graft(graft);
        data->list = isl_ast_graft_list_concat(data->list, list);

        return 0;
error:
        isl_map_free(map);
        isl_map_free(executed);
        return -1;
}

/* Call build->create_leaf to a create "leaf" node in the AST,
 * encapsulate the result in an isl_ast_graft and return the result
 * as a 1-element list.
 *
 * Note that the node returned by the user may be an entire tree.
 *
 * Before we pass control to the user, we first clear some information
 * from the build that is (presumbably) only meaningful
 * for the current code generation.
 * This includes the create_leaf callback itself, so we make a copy
 * of the build first.
 */
static __isl_give isl_ast_graft_list *call_create_leaf(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{
        isl_ast_node *node;
        isl_ast_graft *graft;
        isl_ast_build *user_build;

        user_build = isl_ast_build_copy(build);
        user_build = isl_ast_build_set_executed(user_build, executed);
        user_build = isl_ast_build_clear_local_info(user_build);
        if (!user_build)
                node = NULL;
        else
                node = build->create_leaf(user_build, build->create_leaf_user);
        graft = isl_ast_graft_alloc(node, build);
        isl_ast_build_free(build);
        return isl_ast_graft_list_from_ast_graft(graft);
}

/* Generate an AST after having handled the complete schedule
 * of this call to the code generator.
 *
 * If the user has specified a create_leaf callback, control
 * is passed to the user in call_create_leaf.
 *
 * Otherwise, we generate one or more calls for each individual
 * domain in generate_domain.
 */
static __isl_give isl_ast_graft_list *generate_inner_level(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{
        isl_ctx *ctx;
        struct isl_generate_domain_data data = { build };

        if (!build || !executed)
                goto error;

        if (build->create_leaf)
                return call_create_leaf(executed, build);

        ctx = isl_union_map_get_ctx(executed);
        data.list = isl_ast_graft_list_alloc(ctx, 0);
        if (isl_union_map_foreach_map(executed, &generate_domain, &data) < 0)
                data.list = isl_ast_graft_list_free(data.list);

        if (0)
error:          data.list = NULL;
        isl_ast_build_free(build);
        isl_union_map_free(executed);
        return data.list;
}

/* Call the before_each_for callback, if requested by the user.
 */
static __isl_give isl_ast_node *before_each_for(__isl_take isl_ast_node *node,
        __isl_keep isl_ast_build *build)
{
        isl_id *id;

        if (!node || !build)
                return isl_ast_node_free(node);
        if (!build->before_each_for)
                return node;
        id = build->before_each_for(build, build->before_each_for_user);
        node = isl_ast_node_set_annotation(node, id);
        return node;
}

/* Call the after_each_for callback, if requested by the user.
 */
static __isl_give isl_ast_graft *after_each_for(__isl_keep isl_ast_graft *graft,
        __isl_keep isl_ast_build *build)
{
        if (!graft || !build)
                return isl_ast_graft_free(graft);
        if (!build->after_each_for)
                return graft;
        graft->node = build->after_each_for(graft->node, build,
                                                build->after_each_for_user);
        if (!graft->node)
                return isl_ast_graft_free(graft);
        return graft;
}

/* Plug in all the know values of the current and outer dimensions
 * in the domain of "executed".  In principle, we only need to plug
 * in the known value of the current dimension since the values of
 * outer dimensions have been plugged in already.
 * However, it turns out to be easier to just plug in all known values.
 */
static __isl_give isl_union_map *plug_in_values(
        __isl_take isl_union_map *executed, __isl_keep isl_ast_build *build)
{
        return isl_ast_build_substitute_values_union_map_domain(build,
                                                                    executed);
}

/* Check if the constraint "c" is a lower bound on dimension "pos",
 * an upper bound, or independent of dimension "pos".
 */
static int constraint_type(isl_constraint *c, int pos)
{
        if (isl_constraint_is_lower_bound(c, isl_dim_set, pos))
                return 1;
        if (isl_constraint_is_upper_bound(c, isl_dim_set, pos))
                return 2;
        return 0;
}

/* Compare the types of the constraints "a" and "b",
 * resulting in constraints that are independent of "depth"
 * to be sorted before the lower bounds on "depth", which in
 * turn are sorted before the upper bounds on "depth".
 */
static int cmp_constraint(__isl_keep isl_constraint *a,
        __isl_keep isl_constraint *b, void *user)
{
        int *depth = user;
        int t1 = constraint_type(a, *depth);
        int t2 = constraint_type(b, *depth);

        return t1 - t2;
}

/* Extract a lower bound on dimension "pos" from constraint "c".
 *
 * If the constraint is of the form
 *
 *      a x + f(...) >= 0
 *
 * then we essentially return
 *
 *      l = ceil(-f(...)/a)
 *
 * However, if the current dimension is strided, then we need to make
 * sure that the lower bound we construct is of the form
 *
 *      f + s a
 *
 * with f the offset and s the stride.
 * We therefore compute
 *
 *      f + s * ceil((l - f)/s)
 */
static __isl_give isl_aff *lower_bound(__isl_keep isl_constraint *c,
        int pos, __isl_keep isl_ast_build *build)
{
        isl_aff *aff;

        aff = isl_constraint_get_bound(c, isl_dim_set, pos);
        aff = isl_aff_ceil(aff);

        if (isl_ast_build_has_stride(build, pos)) {
                isl_aff *offset;
                isl_val *stride;

                offset = isl_ast_build_get_offset(build, pos);
                stride = isl_ast_build_get_stride(build, pos);

                aff = isl_aff_sub(aff, isl_aff_copy(offset));
                aff = isl_aff_scale_down_val(aff, isl_val_copy(stride));
                aff = isl_aff_ceil(aff);
                aff = isl_aff_scale_val(aff, stride);
                aff = isl_aff_add(aff, offset);
        }

        aff = isl_ast_build_compute_gist_aff(build, aff);

        return aff;
}

/* Return the exact lower bound (or upper bound if "upper" is set)
 * of "domain" as a piecewise affine expression.
 *
 * If we are computing a lower bound (of a strided dimension), then
 * we need to make sure it is of the form
 *
 *      f + s a
 *
 * where f is the offset and s is the stride.
 * We therefore need to include the stride constraint before computing
 * the minimum.
 */
static __isl_give isl_pw_aff *exact_bound(__isl_keep isl_set *domain,
        __isl_keep isl_ast_build *build, int upper)
{
        isl_set *stride;
        isl_map *it_map;
        isl_pw_aff *pa;
        isl_pw_multi_aff *pma;

        domain = isl_set_copy(domain);
        if (!upper) {
                stride = isl_ast_build_get_stride_constraint(build);
                domain = isl_set_intersect(domain, stride);
        }
        it_map = isl_ast_build_map_to_iterator(build, domain);
        if (upper)
                pma = isl_map_lexmax_pw_multi_aff(it_map);
        else
                pma = isl_map_lexmin_pw_multi_aff(it_map);
        pa = isl_pw_multi_aff_get_pw_aff(pma, 0);
        isl_pw_multi_aff_free(pma);
        pa = isl_ast_build_compute_gist_pw_aff(build, pa);
        pa = isl_pw_aff_coalesce(pa);

        return pa;
}

/* Extract a lower bound on dimension "pos" from each constraint
 * in "constraints" and return the list of lower bounds.
 * If "constraints" has zero elements, then we extract a lower bound
 * from "domain" instead.
 */
static __isl_give isl_pw_aff_list *lower_bounds(
        __isl_keep isl_constraint_list *constraints, int pos,
        __isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
{
        isl_ctx *ctx;
        isl_pw_aff_list *list;
        int i, n;

        if (!build)
                return NULL;

        n = isl_constraint_list_n_constraint(constraints);
        if (n == 0) {
                isl_pw_aff *pa;
                pa = exact_bound(domain, build, 0);
                return isl_pw_aff_list_from_pw_aff(pa);
        }

        ctx = isl_ast_build_get_ctx(build);
        list = isl_pw_aff_list_alloc(ctx,n);

        for (i = 0; i < n; ++i) {
                isl_aff *aff;
                isl_constraint *c;

                c = isl_constraint_list_get_constraint(constraints, i);
                aff = lower_bound(c, pos, build);
                isl_constraint_free(c);
                list = isl_pw_aff_list_add(list, isl_pw_aff_from_aff(aff));
        }

        return list;
}

/* Extract an upper bound on dimension "pos" from each constraint
 * in "constraints" and return the list of upper bounds.
 * If "constraints" has zero elements, then we extract an upper bound
 * from "domain" instead.
 */
static __isl_give isl_pw_aff_list *upper_bounds(
        __isl_keep isl_constraint_list *constraints, int pos,
        __isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
{
        isl_ctx *ctx;
        isl_pw_aff_list *list;
        int i, n;

        n = isl_constraint_list_n_constraint(constraints);
        if (n == 0) {
                isl_pw_aff *pa;
                pa = exact_bound(domain, build, 1);
                return isl_pw_aff_list_from_pw_aff(pa);
        }

        ctx = isl_ast_build_get_ctx(build);
        list = isl_pw_aff_list_alloc(ctx,n);

        for (i = 0; i < n; ++i) {
                isl_aff *aff;
                isl_constraint *c;

                c = isl_constraint_list_get_constraint(constraints, i);
                aff = isl_constraint_get_bound(c, isl_dim_set, pos);
                isl_constraint_free(c);
                aff = isl_aff_floor(aff);
                list = isl_pw_aff_list_add(list, isl_pw_aff_from_aff(aff));
        }

        return list;
}

/* Return an isl_ast_expr that performs the reduction of type "type"
 * on AST expressions corresponding to the elements in "list".
 *
 * The list is assumed to contain at least one element.
 * If the list contains exactly one element, then the returned isl_ast_expr
 * simply computes that affine expression.
 */
static __isl_give isl_ast_expr *reduce_list(enum isl_ast_op_type type,
        __isl_keep isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
{
        int i, n;
        isl_ctx *ctx;
        isl_ast_expr *expr;

        if (!list)
                return NULL;

        n = isl_pw_aff_list_n_pw_aff(list);

        if (n == 1)
                return isl_ast_build_expr_from_pw_aff_internal(build,
                                isl_pw_aff_list_get_pw_aff(list, 0));

        ctx = isl_pw_aff_list_get_ctx(list);
        expr = isl_ast_expr_alloc_op(ctx, type, n);
        if (!expr)
                return NULL;

        for (i = 0; i < n; ++i) {
                isl_ast_expr *expr_i;

                expr_i = isl_ast_build_expr_from_pw_aff_internal(build,
                                isl_pw_aff_list_get_pw_aff(list, i));
                if (!expr_i)
                        return isl_ast_expr_free(expr);
                expr->u.op.args[i] = expr_i;
        }

        return expr;
}

/* Add a guard to "graft" based on "bound" in the case of a degenerate
 * level (including the special case of an eliminated level).
 *
 * We eliminate the current dimension, simplify the result in the current
 * build and add the result as guards to the graft.
 *
 * Note that we cannot simply drop the constraints on the current dimension
 * even in the eliminated case, because the single affine expression may
 * not be explicitly available in "bounds".  Moreover, the single affine
 * expression may only be defined on a subset of the build domain,
 * so we do in some cases need to insert a guard even in the eliminated case.
 */
static __isl_give isl_ast_graft *add_degenerate_guard(
        __isl_take isl_ast_graft *graft, __isl_keep isl_basic_set *bounds,
        __isl_keep isl_ast_build *build)
{
        int depth;
        isl_set *dom;

        depth = isl_ast_build_get_depth(build);

        dom = isl_set_from_basic_set(isl_basic_set_copy(bounds));
        if (isl_ast_build_has_stride(build, depth)) {
                isl_set *stride;

                stride = isl_ast_build_get_stride_constraint(build);
                dom = isl_set_intersect(dom, stride);
        }
        dom = isl_set_eliminate(dom, isl_dim_set, depth, 1);
        dom = isl_ast_build_compute_gist(build, dom);

        graft = isl_ast_graft_add_guard(graft, dom, build);

        return graft;
}

/* Update "graft" based on "bounds" for the eliminated case.
 *
 * In the eliminated case, no for node is created, so we only need
 * to check if "bounds" imply any guards that need to be inserted.
 */
static __isl_give isl_ast_graft *refine_eliminated(
        __isl_take isl_ast_graft *graft, __isl_keep isl_basic_set *bounds,
        __isl_keep isl_ast_build *build)
{
        return add_degenerate_guard(graft, bounds, build);
}

/* Update "graft" based on "bounds" and "sub_build" for the degenerate case.
 *
 * "build" is the build in which graft->node was created
 * "sub_build" contains information about the current level itself,
 * including the single value attained.
 *
 * We first set the initialization part of the for loop to the single
 * value attained by the current dimension.
 * The increment and condition are not strictly needed as the are known
 * to be "1" and "iterator <= value" respectively.
 * Then we set the size of the iterator and
 * check if "bounds" imply any guards that need to be inserted.
 */
static __isl_give isl_ast_graft *refine_degenerate(
        __isl_take isl_ast_graft *graft, __isl_keep isl_basic_set *bounds,
        __isl_keep isl_ast_build *build,
        __isl_keep isl_ast_build *sub_build)
{
        isl_pw_aff *value;

        if (!graft || !sub_build)
                return isl_ast_graft_free(graft);

        value = isl_pw_aff_copy(sub_build->value);

        graft->node->u.f.init = isl_ast_build_expr_from_pw_aff_internal(build,
                                                value);
        if (!graft->node->u.f.init)
                return isl_ast_graft_free(graft);

        graft = add_degenerate_guard(graft, bounds, build);

        return graft;
}

/* Return the intersection of constraints in "list" as a set.
 */
static __isl_give isl_set *intersect_constraints(
        __isl_keep isl_constraint_list *list)
{
        int i, n;
        isl_basic_set *bset;

        n = isl_constraint_list_n_constraint(list);
        if (n < 1)
                isl_die(isl_constraint_list_get_ctx(list), isl_error_internal,
                        "expecting at least one constraint", return NULL);

        bset = isl_basic_set_from_constraint(
                                isl_constraint_list_get_constraint(list, 0));
        for (i = 1; i < n; ++i) {
                isl_basic_set *bset_i;

                bset_i = isl_basic_set_from_constraint(
                                isl_constraint_list_get_constraint(list, i));
                bset = isl_basic_set_intersect(bset, bset_i);
        }

        return isl_set_from_basic_set(bset);
}

/* Compute the constraints on the outer dimensions enforced by
 * graft->node and add those constraints to graft->enforced,
 * in case the upper bound is expressed as a set "upper".
 *
 * In particular, if l(...) is a lower bound in "lower", and
 *
 *      -a i + f(...) >= 0              or      a i <= f(...)
 *
 * is an upper bound ocnstraint on the current dimension i,
 * then the for loop enforces the constraint
 *
 *      -a l(...) + f(...) >= 0         or      a l(...) <= f(...)
 *
 * We therefore simply take each lower bound in turn, plug it into
 * the upper bounds and compute the intersection over all lower bounds.
 *
 * If a lower bound is a rational expression, then
 * isl_basic_set_preimage_multi_aff will force this rational
 * expression to have only integer values.  However, the loop
 * itself does not enforce this integrality constraint.  We therefore
 * use the ceil of the lower bounds instead of the lower bounds themselves.
 * Other constraints will make sure that the for loop is only executed
 * when each of the lower bounds attains an integral value.
 * In particular, potentially rational values only occur in
 * lower_bound if the offset is a (seemingly) rational expression,
 * but then outer conditions will make sure that this rational expression
 * only attains integer values.
 */
static __isl_give isl_ast_graft *set_enforced_from_set(
        __isl_take isl_ast_graft *graft,
        __isl_keep isl_pw_aff_list *lower, int pos, __isl_keep isl_set *upper)
{
        isl_space *space;
        isl_basic_set *enforced;
        isl_pw_multi_aff *pma;
        int i, n;

        if (!graft || !lower)
                return isl_ast_graft_free(graft);

        space = isl_set_get_space(upper);
        enforced = isl_basic_set_universe(isl_space_copy(space));

        space = isl_space_map_from_set(space);
        pma = isl_pw_multi_aff_identity(space);

        n = isl_pw_aff_list_n_pw_aff(lower);
        for (i = 0; i < n; ++i) {
                isl_pw_aff *pa;
                isl_set *enforced_i;
                isl_basic_set *hull;
                isl_pw_multi_aff *pma_i;

                pa = isl_pw_aff_list_get_pw_aff(lower, i);
                pa = isl_pw_aff_ceil(pa);
                pma_i = isl_pw_multi_aff_copy(pma);
                pma_i = isl_pw_multi_aff_set_pw_aff(pma_i, pos, pa);
                enforced_i = isl_set_copy(upper);
                enforced_i = isl_set_preimage_pw_multi_aff(enforced_i, pma_i);
                hull = isl_set_simple_hull(enforced_i);
                enforced = isl_basic_set_intersect(enforced, hull);
        }

        isl_pw_multi_aff_free(pma);

        graft = isl_ast_graft_enforce(graft, enforced);

        return graft;
}

/* Compute the constraints on the outer dimensions enforced by
 * graft->node and add those constraints to graft->enforced,
 * in case the upper bound is expressed as
 * a list of affine expressions "upper".
 *
 * The enforced condition is that each lower bound expression is less
 * than or equal to each upper bound expression.
 */
static __isl_give isl_ast_graft *set_enforced_from_list(
        __isl_take isl_ast_graft *graft,
        __isl_keep isl_pw_aff_list *lower, __isl_keep isl_pw_aff_list *upper)
{
        isl_set *cond;
        isl_basic_set *enforced;

        lower = isl_pw_aff_list_copy(lower);
        upper = isl_pw_aff_list_copy(upper);
        cond = isl_pw_aff_list_le_set(lower, upper);
        enforced = isl_set_simple_hull(cond);
        graft = isl_ast_graft_enforce(graft, enforced);

        return graft;
}

/* Does "aff" have a negative constant term?
 */
static int aff_constant_is_negative(__isl_take isl_set *set,
        __isl_take isl_aff *aff, void *user)
{
        int *neg = user;
        isl_val *v;

        v = isl_aff_get_constant_val(aff);
        *neg = isl_val_is_neg(v);
        isl_val_free(v);
        isl_set_free(set);
        isl_aff_free(aff);

        return *neg ? 0 : -1;
}

/* Does "pa" have a negative constant term over its entire domain?
 */
static int pw_aff_constant_is_negative(__isl_take isl_pw_aff *pa, void *user)
{
        int r;
        int *neg = user;

        r = isl_pw_aff_foreach_piece(pa, &aff_constant_is_negative, user);
        isl_pw_aff_free(pa);

        return *neg ? 0 : -1;
}

/* Does each element in "list" have a negative constant term?
 *
 * The callback terminates the iteration as soon an element has been
 * found that does not have a negative constant term.
 */
static int list_constant_is_negative(__isl_keep isl_pw_aff_list *list)
{
        int neg = 1;

        if (isl_pw_aff_list_foreach(list,
                                &pw_aff_constant_is_negative, &neg) < 0 && neg)
                return -1;

        return neg;
}

/* Add 1 to each of the elements in "list", where each of these elements
 * is defined over the internal schedule space of "build".
 */
static __isl_give isl_pw_aff_list *list_add_one(
        __isl_take isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
{
        int i, n;
        isl_space *space;
        isl_aff *aff;
        isl_pw_aff *one;

        space = isl_ast_build_get_space(build, 1);
        aff = isl_aff_zero_on_domain(isl_local_space_from_space(space));
        aff = isl_aff_add_constant_si(aff, 1);
        one = isl_pw_aff_from_aff(aff);

        n = isl_pw_aff_list_n_pw_aff(list);
        for (i = 0; i < n; ++i) {
                isl_pw_aff *pa;
                pa = isl_pw_aff_list_get_pw_aff(list, i);
                pa = isl_pw_aff_add(pa, isl_pw_aff_copy(one));
                list = isl_pw_aff_list_set_pw_aff(list, i, pa);
        }

        isl_pw_aff_free(one);

        return list;
}

/* Set the condition part of the for node graft->node in case
 * the upper bound is represented as a list of piecewise affine expressions.
 *
 * In particular, set the condition to
 *
 *      iterator <= min(list of upper bounds)
 *
 * If each of the upper bounds has a negative constant term, then
 * set the condition to
 *
 *      iterator < min(list of (upper bound + 1)s)
 *
 */
static __isl_give isl_ast_graft *set_for_cond_from_list(
        __isl_take isl_ast_graft *graft, __isl_keep isl_pw_aff_list *list,
        __isl_keep isl_ast_build *build)
{
        int neg;
        isl_ast_expr *bound, *iterator, *cond;
        enum isl_ast_op_type type = isl_ast_op_le;

        if (!graft || !list)
                return isl_ast_graft_free(graft);

        neg = list_constant_is_negative(list);
        if (neg < 0)
                return isl_ast_graft_free(graft);
        list = isl_pw_aff_list_copy(list);
        if (neg) {
                list = list_add_one(list, build);
                type = isl_ast_op_lt;
        }

        bound = reduce_list(isl_ast_op_min, list, build);
        iterator = isl_ast_expr_copy(graft->node->u.f.iterator);
        cond = isl_ast_expr_alloc_binary(type, iterator, bound);
        graft->node->u.f.cond = cond;

        isl_pw_aff_list_free(list);
        if (!graft->node->u.f.cond)
                return isl_ast_graft_free(graft);
        return graft;
}

/* Set the condition part of the for node graft->node in case
 * the upper bound is represented as a set.
 */
static __isl_give isl_ast_graft *set_for_cond_from_set(
        __isl_take isl_ast_graft *graft, __isl_keep isl_set *set,
        __isl_keep isl_ast_build *build)
{
        isl_ast_expr *cond;

        if (!graft)
                return NULL;

        cond = isl_ast_build_expr_from_set(build, isl_set_copy(set));
        graft->node->u.f.cond = cond;
        if (!graft->node->u.f.cond)
                return isl_ast_graft_free(graft);
        return graft;
}

/* Construct an isl_ast_expr for the increment (i.e., stride) of
 * the current dimension.
 */
static __isl_give isl_ast_expr *for_inc(__isl_keep isl_ast_build *build)
{
        int depth;
        isl_val *v;
        isl_ctx *ctx;

        if (!build)
                return NULL;
        ctx = isl_ast_build_get_ctx(build);
        depth = isl_ast_build_get_depth(build);

        if (!isl_ast_build_has_stride(build, depth))
                return isl_ast_expr_alloc_int_si(ctx, 1);

        v = isl_ast_build_get_stride(build, depth);
        return isl_ast_expr_from_val(v);
}

/* Should we express the loop condition as
 *
 *      iterator <= min(list of upper bounds)
 *
 * or as a conjunction of constraints?
 *
 * The first is constructed from a list of upper bounds.
 * The second is constructed from a set.
 *
 * If there are no upper bounds in "constraints", then this could mean
 * that "domain" simply doesn't have an upper bound or that we didn't
 * pick any upper bound.  In the first case, we want to generate the
 * loop condition as a(n empty) conjunction of constraints
 * In the second case, we will compute
 * a single upper bound from "domain" and so we use the list form.
 *
 * If there are upper bounds in "constraints",
 * then we use the list form iff the atomic_upper_bound option is set.
 */
static int use_upper_bound_list(isl_ctx *ctx, int n_upper,
        __isl_keep isl_set *domain, int depth)
{
        if (n_upper > 0)
                return isl_options_get_ast_build_atomic_upper_bound(ctx);
        else
                return isl_set_dim_has_upper_bound(domain, isl_dim_set, depth);
}

/* Fill in the expressions of the for node in graft->node.
 *
 * In particular,
 * - set the initialization part of the loop to the maximum of the lower bounds
 * - set the size of the iterator based on the values attained by the iterator
 * - extract the increment from the stride of the current dimension
 * - construct the for condition either based on a list of upper bounds
 *      or on a set of upper bound constraints.
 */
static __isl_give isl_ast_graft *set_for_node_expressions(
        __isl_take isl_ast_graft *graft, __isl_keep isl_pw_aff_list *lower,
        int use_list, __isl_keep isl_pw_aff_list *upper_list,
        __isl_keep isl_set *upper_set, __isl_keep isl_ast_build *build)
{
        isl_ast_node *node;

        if (!graft)
                return NULL;

        build = isl_ast_build_copy(build);
        build = isl_ast_build_set_enforced(build,
                                        isl_ast_graft_get_enforced(graft));

        node = graft->node;
        node->u.f.init = reduce_list(isl_ast_op_max, lower, build);
        node->u.f.inc = for_inc(build);

        if (use_list)
                graft = set_for_cond_from_list(graft, upper_list, build);
        else
                graft = set_for_cond_from_set(graft, upper_set, build);

        isl_ast_build_free(build);

        if (!node->u.f.iterator || !node->u.f.init ||
            !node->u.f.cond || !node->u.f.inc)
                return isl_ast_graft_free(graft);

        return graft;
}

/* Update "graft" based on "bounds" and "domain" for the generic,
 * non-degenerate, case.
 *
 * "c_lower" and "c_upper" contain the lower and upper bounds
 * that the loop node should express.
 * "domain" is the subset of the intersection of the constraints
 * for which some code is executed.
 *
 * There may be zero lower bounds or zero upper bounds in "constraints"
 * in case the list of constraints was created
 * based on the atomic option or based on separation with explicit bounds.
 * In that case, we use "domain" to derive lower and/or upper bounds.
 *
 * We first compute a list of one or more lower bounds.
 *
 * Then we decide if we want to express the condition as
 *
 *      iterator <= min(list of upper bounds)
 *
 * or as a conjunction of constraints.
 *
 * The set of enforced constraints is then computed either based on
 * a list of upper bounds or on a set of upper bound constraints.
 * We do not compute any enforced constraints if we were forced
 * to compute a lower or upper bound using exact_bound.  The domains
 * of the resulting expressions may imply some bounds on outer dimensions
 * that we do not want to appear in the enforced constraints since
 * they are not actually enforced by the corresponding code.
 *
 * Finally, we fill in the expressions of the for node.
 */
static __isl_give isl_ast_graft *refine_generic_bounds(
        __isl_take isl_ast_graft *graft,
        __isl_take isl_constraint_list *c_lower,
        __isl_take isl_constraint_list *c_upper,
        __isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
{
        int depth;
        isl_ctx *ctx;
        isl_pw_aff_list *lower;
        int use_list;
        isl_set *upper_set = NULL;
        isl_pw_aff_list *upper_list = NULL;
        int n_lower, n_upper;

        if (!graft || !c_lower || !c_upper || !build)
                goto error;

        depth = isl_ast_build_get_depth(build);
        ctx = isl_ast_graft_get_ctx(graft);

        n_lower = isl_constraint_list_n_constraint(c_lower);
        n_upper = isl_constraint_list_n_constraint(c_upper);

        use_list = use_upper_bound_list(ctx, n_upper, domain, depth);

        lower = lower_bounds(c_lower, depth, domain, build);

        if (use_list)
                upper_list = upper_bounds(c_upper, depth, domain, build);
        else if (n_upper > 0)
                upper_set = intersect_constraints(c_upper);
        else
                upper_set = isl_set_universe(isl_set_get_space(domain));

        if (n_lower == 0 || n_upper == 0)
                ;
        else if (use_list)
                graft = set_enforced_from_list(graft, lower, upper_list);
        else
                graft = set_enforced_from_set(graft, lower, depth, upper_set);

        graft = set_for_node_expressions(graft, lower, use_list, upper_list,
                                        upper_set, build);

        isl_pw_aff_list_free(lower);
        isl_pw_aff_list_free(upper_list);
        isl_set_free(upper_set);
        isl_constraint_list_free(c_lower);
        isl_constraint_list_free(c_upper);

        return graft;
error:
        isl_constraint_list_free(c_lower);
        isl_constraint_list_free(c_upper);
        return isl_ast_graft_free(graft);
}

/* Internal data structure used inside count_constraints to keep
 * track of the number of constraints that are independent of dimension "pos",
 * the lower bounds in "pos" and the upper bounds in "pos".
 */
struct isl_ast_count_constraints_data {
        int pos;

        int n_indep;
        int n_lower;
        int n_upper;
};

/* Increment data->n_indep, data->lower or data->upper depending
 * on whether "c" is independenct of dimensions data->pos,
 * a lower bound or an upper bound.
 */
static int count_constraints(__isl_take isl_constraint *c, void *user)
{
        struct isl_ast_count_constraints_data *data = user;

        if (isl_constraint_is_lower_bound(c, isl_dim_set, data->pos))
                data->n_lower++;
        else if (isl_constraint_is_upper_bound(c, isl_dim_set, data->pos))
                data->n_upper++;
        else
                data->n_indep++;

        isl_constraint_free(c);

        return 0;
}

/* Update "graft" based on "bounds" and "domain" for the generic,
 * non-degenerate, case.
 *
 * "list" respresent the list of bounds that need to be encoded by
 * the for loop (or a guard around the for loop).
 * "domain" is the subset of the intersection of the constraints
 * for which some code is executed.
 * "build" is the build in which graft->node was created.
 *
 * We separate lower bounds, upper bounds and constraints that
 * are independent of the loop iterator.
 *
 * The actual for loop bounds are generated in refine_generic_bounds.
 * If there are any constraints that are independent of the loop iterator,
 * we need to put a guard around the for loop (which may get hoisted up
 * to higher levels) and we call refine_generic_bounds in a build
 * where this guard is enforced.
 */
static __isl_give isl_ast_graft *refine_generic_split(
        __isl_take isl_ast_graft *graft, __isl_take isl_constraint_list *list,
        __isl_keep isl_set *domain, __isl_keep isl_ast_build *build)
{
        isl_ast_build *for_build;
        isl_set *guard;
        struct isl_ast_count_constraints_data data;
        isl_constraint_list *lower;
        isl_constraint_list *upper;

        if (!list)
                return isl_ast_graft_free(graft);

        data.pos = isl_ast_build_get_depth(build);

        list = isl_constraint_list_sort(list, &cmp_constraint, &data.pos);
        if (!list)
                return isl_ast_graft_free(graft);

        data.n_indep = data.n_lower = data.n_upper = 0;
        if (isl_constraint_list_foreach(list, &count_constraints, &data) < 0) {
                isl_constraint_list_free(list);
                return isl_ast_graft_free(graft);
        }

        lower = isl_constraint_list_copy(list);
        lower = isl_constraint_list_drop(lower, 0, data.n_indep);
        upper = isl_constraint_list_copy(lower);
        lower = isl_constraint_list_drop(lower, data.n_lower, data.n_upper);
        upper = isl_constraint_list_drop(upper, 0, data.n_lower);

        if (data.n_indep == 0) {
                isl_constraint_list_free(list);
                return refine_generic_bounds(graft, lower, upper,
                                                domain, build);
        }

        list = isl_constraint_list_drop(list, data.n_indep,
                                        data.n_lower + data.n_upper);
        guard = intersect_constraints(list);
        isl_constraint_list_free(list);

        for_build = isl_ast_build_copy(build);
        for_build = isl_ast_build_restrict_pending(for_build,
                                                isl_set_copy(guard));
        graft = refine_generic_bounds(graft, lower, upper, domain, for_build);
        isl_ast_build_free(for_build);

        graft = isl_ast_graft_add_guard(graft, guard, build);

        return graft;
}

/* Add the guard implied by the current stride constraint (if any),
 * but not (necessarily) enforced by the generated AST to "graft".
 */
static __isl_give isl_ast_graft *add_stride_guard(
        __isl_take isl_ast_graft *graft, __isl_keep isl_ast_build *build)
{
        int depth;
        isl_set *dom;

        depth = isl_ast_build_get_depth(build);
        if (!isl_ast_build_has_stride(build, depth))
                return graft;

        dom = isl_ast_build_get_stride_constraint(build);
        dom = isl_set_eliminate(dom, isl_dim_set, depth, 1);
        dom = isl_ast_build_compute_gist(build, dom);

        graft = isl_ast_graft_add_guard(graft, dom, build);

        return graft;
}

/* Update "graft" based on "bounds" and "domain" for the generic,
 * non-degenerate, case.
 *
 * "bounds" respresent the bounds that need to be encoded by
 * the for loop (or a guard around the for loop).
 * "domain" is the subset of "bounds" for which some code is executed.
 * "build" is the build in which graft->node was created.
 *
 * We break up "bounds" into a list of constraints and continue with
 * refine_generic_split.
 */
static __isl_give isl_ast_graft *refine_generic(
        __isl_take isl_ast_graft *graft,
        __isl_keep isl_basic_set *bounds, __isl_keep isl_set *domain,
        __isl_keep isl_ast_build *build)
{
        isl_constraint_list *list;

        if (!build || !graft)
                return isl_ast_graft_free(graft);

        bounds = isl_basic_set_copy(bounds);
        bounds = isl_ast_build_compute_gist_basic_set(build, bounds);
        list = isl_constraint_list_from_basic_set(bounds);

        graft = refine_generic_split(graft, list, domain, build);
        graft = add_stride_guard(graft, build);

        return graft;
}

/* Create a for node for the current level.
 *
 * Mark the for node degenerate if "degenerate" is set.
 */
static __isl_give isl_ast_node *create_for(__isl_keep isl_ast_build *build,
        int degenerate)
{
        int depth;
        isl_id *id;
        isl_ast_node *node;

        if (!build)
                return NULL;

        depth = isl_ast_build_get_depth(build);
        id = isl_ast_build_get_iterator_id(build, depth);
        node = isl_ast_node_alloc_for(id);
        if (degenerate)
                node = isl_ast_node_for_mark_degenerate(node);

        return node;
}

/* Create an AST node for the current dimension based on
 * the schedule domain "bounds" and return the node encapsulated
 * in an isl_ast_graft.
 *
 * "executed" is the current inverse schedule, taking into account
 * the bounds in "bounds"
 * "domain" is the domain of "executed", with inner dimensions projected out.
 * It may be a strict subset of "bounds" in case "bounds" was created
 * based on the atomic option or based on separation with explicit bounds.
 *
 * "domain" may satisfy additional equalities that result
 * from intersecting "executed" with "bounds" in add_node.
 * It may also satisfy some global constraints that were dropped out because
 * we performed separation with explicit bounds.
 * The very first step is then to copy these constraints to "bounds".
 *
 * Since we may be calling before_each_for and after_each_for
 * callbacks, we record the current inverse schedule in the build.
 *
 * We consider three builds,
 * "build" is the one in which the current level is created,
 * "body_build" is the build in which the next level is created,
 * "sub_build" is essentially the same as "body_build", except that
 * the depth has not been increased yet.
 *
 * "build" already contains information (in strides and offsets)
 * about the strides at the current level, but this information is not
 * reflected in the build->domain.
 * We first add this information and the "bounds" to the sub_build->domain.
 * isl_ast_build_set_loop_bounds checks whether the current dimension attains
 * only a single value and whether this single value can be represented using
 * a single affine expression.
 * In the first case, the current level is considered "degenerate".
 * In the second, sub-case, the current level is considered "eliminated".
 * Eliminated level don't need to be reflected in the AST since we can
 * simply plug in the affine expression.  For degenerate, but non-eliminated,
 * levels, we do introduce a for node, but mark is as degenerate so that
 * it can be printed as an assignment of the single value to the loop
 * "iterator".
 *
 * If the current level is eliminated, we explicitly plug in the value
 * for the current level found by isl_ast_build_set_loop_bounds in the
 * inverse schedule.  This ensures that if we are working on a slice
 * of the domain based on information available in the inverse schedule
 * and the build domain, that then this information is also reflected
 * in the inverse schedule.  This operation also eliminates the current
 * dimension from the inverse schedule making sure no inner dimensions depend
 * on the current dimension.  Otherwise, we create a for node, marking
 * it degenerate if appropriate.  The initial for node is still incomplete
 * and will be completed in either refine_degenerate or refine_generic.
 *
 * We then generate a sequence of grafts for the next level,
 * create a surrounding graft for the current level and insert
 * the for node we created (if the current level is not eliminated).
 *
 * Finally, we set the bounds of the for loop and insert guards
 * (either in the AST or in the graft) in one of
 * refine_eliminated, refine_degenerate or refine_generic.
 */
static __isl_give isl_ast_graft *create_node_scaled(
        __isl_take isl_union_map *executed,
        __isl_take isl_basic_set *bounds, __isl_take isl_set *domain,
        __isl_take isl_ast_build *build)
{
        int depth;
        int degenerate, eliminated;
        isl_basic_set *hull;
        isl_ast_node *node = NULL;
        isl_ast_graft *graft;
        isl_ast_graft_list *children;
        isl_ast_build *sub_build;
        isl_ast_build *body_build;

        domain = isl_ast_build_eliminate_divs(build, domain);
        domain = isl_set_detect_equalities(domain);
        hull = isl_set_unshifted_simple_hull(isl_set_copy(domain));
        bounds = isl_basic_set_intersect(bounds, hull);
        build = isl_ast_build_set_executed(build, isl_union_map_copy(executed));

        depth = isl_ast_build_get_depth(build);
        sub_build = isl_ast_build_copy(build);
        sub_build = isl_ast_build_include_stride(sub_build);
        sub_build = isl_ast_build_set_loop_bounds(sub_build,
                                                isl_basic_set_copy(bounds));
        degenerate = isl_ast_build_has_value(sub_build);
        eliminated = isl_ast_build_has_affine_value(sub_build, depth);
        if (degenerate < 0 || eliminated < 0)
                executed = isl_union_map_free(executed);
        if (eliminated)
                executed = plug_in_values(executed, sub_build);
        else
                node = create_for(build, degenerate);

        body_build = isl_ast_build_copy(sub_build);
        body_build = isl_ast_build_increase_depth(body_build);
        if (!eliminated)
                node = before_each_for(node, body_build);
        children = generate_next_level(executed,
                                    isl_ast_build_copy(body_build));

        graft = isl_ast_graft_alloc_level(children, build, sub_build);
        if (!eliminated)
                graft = isl_ast_graft_insert_for(graft, node);
        if (eliminated)
                graft = refine_eliminated(graft, bounds, build);
        else if (degenerate)
                graft = refine_degenerate(graft, bounds, build, sub_build);
        else
                graft = refine_generic(graft, bounds, domain, build);
        if (!eliminated)
                graft = after_each_for(graft, body_build);

        isl_ast_build_free(body_build);
        isl_ast_build_free(sub_build);
        isl_ast_build_free(build);
        isl_basic_set_free(bounds);
        isl_set_free(domain);

        return graft;
}

/* Internal data structure for checking if all constraints involving
 * the input dimension "depth" are such that the other coefficients
 * are multiples of "m", reducing "m" if they are not.
 * If "m" is reduced all the way down to "1", then the check has failed
 * and we break out of the iteration.
 */
struct isl_check_scaled_data {
        int depth;
        isl_val *m;
};

/* If constraint "c" involves the input dimension data->depth,
 * then make sure that all the other coefficients are multiples of data->m,
 * reducing data->m if needed.
 * Break out of the iteration if data->m has become equal to "1".
 */
static int constraint_check_scaled(__isl_take isl_constraint *c, void *user)
{
        struct isl_check_scaled_data *data = user;
        int i, j, n;
        enum isl_dim_type t[] = { isl_dim_param, isl_dim_in, isl_dim_out,
                                    isl_dim_div };

        if (!isl_constraint_involves_dims(c, isl_dim_in, data->depth, 1)) {
                isl_constraint_free(c);
                return 0;
        }

        for (i = 0; i < 4; ++i) {
                n = isl_constraint_dim(c, t[i]);
                for (j = 0; j < n; ++j) {
                        isl_val *d;

                        if (t[i] == isl_dim_in && j == data->depth)
                                continue;
                        if (!isl_constraint_involves_dims(c, t[i], j, 1))
                                continue;
                        d = isl_constraint_get_coefficient_val(c, t[i], j);
                        data->m = isl_val_gcd(data->m, d);
                        if (isl_val_is_one(data->m))
                                break;
                }
                if (j < n)
                        break;
        }

        isl_constraint_free(c);

        return i < 4 ? -1 : 0;
}

/* For each constraint of "bmap" that involves the input dimension data->depth,
 * make sure that all the other coefficients are multiples of data->m,
 * reducing data->m if needed.
 * Break out of the iteration if data->m has become equal to "1".
 */
static int basic_map_check_scaled(__isl_take isl_basic_map *bmap, void *user)
{
        int r;

        r = isl_basic_map_foreach_constraint(bmap,
                                                &constraint_check_scaled, user);
        isl_basic_map_free(bmap);

        return r;
}

/* For each constraint of "map" that involves the input dimension data->depth,
 * make sure that all the other coefficients are multiples of data->m,
 * reducing data->m if needed.
 * Break out of the iteration if data->m has become equal to "1".
 */
static int map_check_scaled(__isl_take isl_map *map, void *user)
{
        int r;

        r = isl_map_foreach_basic_map(map, &basic_map_check_scaled, user);
        isl_map_free(map);

        return r;
}

/* Create an AST node for the current dimension based on
 * the schedule domain "bounds" and return the node encapsulated
 * in an isl_ast_graft.
 *
 * "executed" is the current inverse schedule, taking into account
 * the bounds in "bounds"
 * "domain" is the domain of "executed", with inner dimensions projected out.
 *
 *
 * Before moving on to the actual AST node construction in create_node_scaled,
 * we first check if the current dimension is strided and if we can scale
 * down this stride.  Note that we only do this if the ast_build_scale_strides
 * option is set.
 *
 * In particular, let the current dimension take on values
 *
 *      f + s a
 *
 * with a an integer.  We check if we can find an integer m that (obviouly)
 * divides both f and s.
 *
 * If so, we check if the current dimension only appears in constraints
 * where the coefficients of the other variables are multiples of m.
 * We perform this extra check to avoid the risk of introducing
 * divisions by scaling down the current dimension.
 *
 * If so, we scale the current dimension down by a factor of m.
 * That is, we plug in
 *
 *      i = m i'                                                        (1)
 *
 * Note that in principle we could always scale down strided loops
 * by plugging in
 *
 *      i = f + s i'
 *
 * but this may result in i' taking on larger values than the original i,
 * due to the shift by "f".
 * By constrast, the scaling in (1) can only reduce the (absolute) value "i".
 */
static __isl_give isl_ast_graft *create_node(__isl_take isl_union_map *executed,
        __isl_take isl_basic_set *bounds, __isl_take isl_set *domain,
        __isl_take isl_ast_build *build)
{
        struct isl_check_scaled_data data;
        isl_ctx *ctx;
        isl_aff *offset;
        isl_val *d;

        ctx = isl_ast_build_get_ctx(build);
        if (!isl_options_get_ast_build_scale_strides(ctx))
                return create_node_scaled(executed, bounds, domain, build);

        data.depth = isl_ast_build_get_depth(build);
        if (!isl_ast_build_has_stride(build, data.depth))
                return create_node_scaled(executed, bounds, domain, build);

        offset = isl_ast_build_get_offset(build, data.depth);
        data.m = isl_ast_build_get_stride(build, data.depth);
        if (!data.m)
                offset = isl_aff_free(offset);
        offset = isl_aff_scale_down_val(offset, isl_val_copy(data.m));
        d = isl_aff_get_denominator_val(offset);
        if (!d)
                executed = isl_union_map_free(executed);

        if (executed && isl_val_is_divisible_by(data.m, d))
                data.m = isl_val_div(data.m, d);
        else {
                data.m = isl_val_set_si(data.m, 1);
                isl_val_free(d);
        }

        if (!isl_val_is_one(data.m)) {
                if (isl_union_map_foreach_map(executed, &map_check_scaled,
                                                &data) < 0 &&
                    !isl_val_is_one(data.m))
                        executed = isl_union_map_free(executed);
        }

        if (!isl_val_is_one(data.m)) {
                isl_space *space;
                isl_multi_aff *ma;
                isl_aff *aff;
                isl_map *map;
                isl_union_map *umap;

                space = isl_ast_build_get_space(build, 1);
                space = isl_space_map_from_set(space);
                ma = isl_multi_aff_identity(space);
                aff = isl_multi_aff_get_aff(ma, data.depth);
                aff = isl_aff_scale_val(aff, isl_val_copy(data.m));
                ma = isl_multi_aff_set_aff(ma, data.depth, aff);

                bounds = isl_basic_set_preimage_multi_aff(bounds,
                                                isl_multi_aff_copy(ma));
                domain = isl_set_preimage_multi_aff(domain,
                                                isl_multi_aff_copy(ma));
                map = isl_map_reverse(isl_map_from_multi_aff(ma));
                umap = isl_union_map_from_map(map);
                executed = isl_union_map_apply_domain(executed,
                                                isl_union_map_copy(umap));
                build = isl_ast_build_scale_down(build, isl_val_copy(data.m),
                                                umap);
        }
        isl_aff_free(offset);
        isl_val_free(data.m);

        return create_node_scaled(executed, bounds, domain, build);
}

/* Add the basic set to the list that "user" points to.
 */
static int collect_basic_set(__isl_take isl_basic_set *bset, void *user)
{
        isl_basic_set_list **list = user;

        *list = isl_basic_set_list_add(*list, bset);

        return 0;
}

/* Extract the basic sets of "set" and collect them in an isl_basic_set_list.
 */
static __isl_give isl_basic_set_list *isl_basic_set_list_from_set(
        __isl_take isl_set *set)
{
        int n;
        isl_ctx *ctx;
        isl_basic_set_list *list;

        if (!set)
                return NULL;

        ctx = isl_set_get_ctx(set);

        n = isl_set_n_basic_set(set);
        list = isl_basic_set_list_alloc(ctx, n);
        if (isl_set_foreach_basic_set(set, &collect_basic_set, &list) < 0)
                list = isl_basic_set_list_free(list);

        isl_set_free(set);
        return list;
}

/* Generate code for the schedule domain "bounds"
 * and add the result to "list".
 *
 * We mainly detect strides and additional equalities here
 * and then pass over control to create_node.
 *
 * "bounds" reflects the bounds on the current dimension and possibly
 * some extra conditions on outer dimensions.
 * It does not, however, include any divs involving the current dimension,
 * so it does not capture any stride constraints.
 * We therefore need to compute that part of the schedule domain that
 * intersects with "bounds" and derive the strides from the result.
 */
static __isl_give isl_ast_graft_list *add_node(
        __isl_take isl_ast_graft_list *list, __isl_take isl_union_map *executed,
        __isl_take isl_basic_set *bounds, __isl_take isl_ast_build *build)
{
        isl_ast_graft *graft;
        isl_set *domain = NULL;
        isl_union_set *uset;
        int empty;

        uset = isl_union_set_from_basic_set(isl_basic_set_copy(bounds));
        executed = isl_union_map_intersect_domain(executed, uset);
        empty = isl_union_map_is_empty(executed);
        if (empty < 0)
                goto error;
        if (empty)
                goto done;

        uset = isl_union_map_domain(isl_union_map_copy(executed));
        domain = isl_set_from_union_set(uset);
        domain = isl_ast_build_compute_gist(build, domain);
        empty = isl_set_is_empty(domain);
        if (empty < 0)
                goto error;
        if (empty)
                goto done;

        domain = isl_ast_build_eliminate_inner(build, domain);
        build = isl_ast_build_detect_strides(build, isl_set_copy(domain));

        graft = create_node(executed, bounds, domain,
                                isl_ast_build_copy(build));
        list = isl_ast_graft_list_add(list, graft);
        isl_ast_build_free(build);
        return list;
error:
        list = isl_ast_graft_list_free(list);
done:
        isl_set_free(domain);
        isl_basic_set_free(bounds);
        isl_union_map_free(executed);
        isl_ast_build_free(build);
        return list;
}

/* Does any element of i follow or coincide with any element of j
 * at the current depth for equal values of the outer dimensions?
 */
static int domain_follows_at_depth(__isl_keep isl_basic_set *i,
        __isl_keep isl_basic_set *j, void *user)
{
        int depth = *(int *) user;
        isl_basic_map *test;
        int empty;
        int l;

        test = isl_basic_map_from_domain_and_range(isl_basic_set_copy(i),
                                                    isl_basic_set_copy(j));
        for (l = 0; l < depth; ++l)
                test = isl_basic_map_equate(test, isl_dim_in, l,
                                                isl_dim_out, l);
        test = isl_basic_map_order_ge(test, isl_dim_in, depth,
                                        isl_dim_out, depth);
        empty = isl_basic_map_is_empty(test);
        isl_basic_map_free(test);

        return empty < 0 ? -1 : !empty;
}

/* Split up each element of "list" into a part that is related to "bset"
 * according to "gt" and a part that is not.
 * Return a list that consist of "bset" and all the pieces.
 */
static __isl_give isl_basic_set_list *add_split_on(
        __isl_take isl_basic_set_list *list, __isl_take isl_basic_set *bset,
        __isl_keep isl_basic_map *gt)
{
        int i, n;
        isl_basic_set_list *res;

        gt = isl_basic_map_copy(gt);
        gt = isl_basic_map_intersect_domain(gt, isl_basic_set_copy(bset));
        n = isl_basic_set_list_n_basic_set(list);
        res = isl_basic_set_list_from_basic_set(bset);
        for (i = 0; res && i < n; ++i) {
                isl_basic_set *bset;
                isl_set *set1, *set2;
                isl_basic_map *bmap;
                int empty;

                bset = isl_basic_set_list_get_basic_set(list, i);
                bmap = isl_basic_map_copy(gt);
                bmap = isl_basic_map_intersect_range(bmap, bset);
                bset = isl_basic_map_range(bmap);
                empty = isl_basic_set_is_empty(bset);
                if (empty < 0)
                        res = isl_basic_set_list_free(res);
                if (empty)  {
                        isl_basic_set_free(bset);
                        bset = isl_basic_set_list_get_basic_set(list, i);
                        res = isl_basic_set_list_add(res, bset);
                        continue;
                }

                res = isl_basic_set_list_add(res, isl_basic_set_copy(bset));
                set1 = isl_set_from_basic_set(bset);
                bset = isl_basic_set_list_get_basic_set(list, i);
                set2 = isl_set_from_basic_set(bset);
                set1 = isl_set_subtract(set2, set1);
                set1 = isl_set_make_disjoint(set1);

                res = isl_basic_set_list_concat(res,
                                            isl_basic_set_list_from_set(set1));
        }
        isl_basic_map_free(gt);
        isl_basic_set_list_free(list);
        return res;
}

static __isl_give isl_ast_graft_list *generate_sorted_domains(
        __isl_keep isl_basic_set_list *domain_list,
        __isl_keep isl_union_map *executed,
        __isl_keep isl_ast_build *build);

/* Internal data structure for add_nodes.
 *
 * "executed" and "build" are extra arguments to be passed to add_node.
 * "list" collects the results.
 */
struct isl_add_nodes_data {
        isl_union_map *executed;
        isl_ast_build *build;

        isl_ast_graft_list *list;
};

/* Generate code for the schedule domains in "scc"
 * and add the results to "list".
 *
 * The domains in "scc" form a strongly connected component in the ordering.
 * If the number of domains in "scc" is larger than 1, then this means
 * that we cannot determine a valid ordering for the domains in the component.
 * This should be fairly rare because the individual domains
 * have been made disjoint first.
 * The problem is that the domains may be integrally disjoint but not
 * rationally disjoint.  For example, we may have domains
 *
 *      { [i,i] : 0 <= i <= 1 }         and     { [i,1-i] : 0 <= i <= 1 }
 *
 * These two domains have an empty intersection, but their rational
 * relaxations do intersect.  It is impossible to order these domains
 * in the second dimension because the first should be ordered before
 * the second for outer dimension equal to 0, while it should be ordered
 * after for outer dimension equal to 1.
 *
 * This may happen in particular in case of unrolling since the domain
 * of each slice is replaced by its simple hull.
 *
 * For each basic set i in "scc" and for each of the following basic sets j,
 * we split off that part of the basic set i that shares the outer dimensions
 * with j and lies before j in the current dimension.
 * We collect all the pieces in a new list that replaces "scc".
 */
static int add_nodes(__isl_take isl_basic_set_list *scc, void *user)
{
        struct isl_add_nodes_data *data = user;
        int i, n, depth;
        isl_basic_set *bset;
        isl_basic_set_list *list;
        isl_space *space;
        isl_basic_map *gt;

        n = isl_basic_set_list_n_basic_set(scc);
        bset = isl_basic_set_list_get_basic_set(scc, 0);
        if (n == 1) {
                isl_basic_set_list_free(scc);
                data->list = add_node(data->list,
                                isl_union_map_copy(data->executed), bset,
                                isl_ast_build_copy(data->build));
                return data->list ? 0 : -1;
        }

        depth = isl_ast_build_get_depth(data->build);
        space = isl_basic_set_get_space(bset);
        space = isl_space_map_from_set(space);
        gt = isl_basic_map_universe(space);
        for (i = 0; i < depth; ++i)
                gt = isl_basic_map_equate(gt, isl_dim_in, i, isl_dim_out, i);
        gt = isl_basic_map_order_gt(gt, isl_dim_in, depth, isl_dim_out, depth);

        list = isl_basic_set_list_from_basic_set(bset);
        for (i = 1; i < n; ++i) {
                bset = isl_basic_set_list_get_basic_set(scc, i);
                list = add_split_on(list, bset, gt);
        }
        isl_basic_map_free(gt);
        isl_basic_set_list_free(scc);
        scc = list;
        data->list = isl_ast_graft_list_concat(data->list,
                    generate_sorted_domains(scc, data->executed, data->build));
        isl_basic_set_list_free(scc);

        return data->list ? 0 : -1;
}

/* Sort the domains in "domain_list" according to the execution order
 * at the current depth (for equal values of the outer dimensions),
 * generate code for each of them, collecting the results in a list.
 * If no code is generated (because the intersection of the inverse schedule
 * with the domains turns out to be empty), then an empty list is returned.
 *
 * The caller is responsible for ensuring that the basic sets in "domain_list"
 * are pair-wise disjoint.  It can, however, in principle happen that
 * two basic sets should be ordered one way for one value of the outer
 * dimensions and the other way for some other value of the outer dimensions.
 * We therefore play safe and look for strongly connected components.
 * The function add_nodes takes care of handling non-trivial components.
 */
static __isl_give isl_ast_graft_list *generate_sorted_domains(
        __isl_keep isl_basic_set_list *domain_list,
        __isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
{
        isl_ctx *ctx;
        struct isl_add_nodes_data data;
        int depth;
        int n;

        if (!domain_list)
                return NULL;

        ctx = isl_basic_set_list_get_ctx(domain_list);
        n = isl_basic_set_list_n_basic_set(domain_list);
        data.list = isl_ast_graft_list_alloc(ctx, n);
        if (n == 0)
                return data.list;
        if (n == 1)
                return add_node(data.list, isl_union_map_copy(executed),
                        isl_basic_set_list_get_basic_set(domain_list, 0),
                        isl_ast_build_copy(build));

        depth = isl_ast_build_get_depth(build);
        data.executed = executed;
        data.build = build;
        if (isl_basic_set_list_foreach_scc(domain_list,
                                        &domain_follows_at_depth, &depth,
                                        &add_nodes, &data) < 0)
                data.list = isl_ast_graft_list_free(data.list);

        return data.list;
}

/* Do i and j share any values for the outer dimensions?
 */
static int shared_outer(__isl_keep isl_basic_set *i,
        __isl_keep isl_basic_set *j, void *user)
{
        int depth = *(int *) user;
        isl_basic_map *test;
        int empty;
        int l;

        test = isl_basic_map_from_domain_and_range(isl_basic_set_copy(i),
                                                    isl_basic_set_copy(j));
        for (l = 0; l < depth; ++l)
                test = isl_basic_map_equate(test, isl_dim_in, l,
                                                isl_dim_out, l);
        empty = isl_basic_map_is_empty(test);
        isl_basic_map_free(test);

        return empty < 0 ? -1 : !empty;
}

/* Internal data structure for generate_sorted_domains_wrap.
 *
 * "n" is the total number of basic sets
 * "executed" and "build" are extra arguments to be passed
 *      to generate_sorted_domains.
 *
 * "single" is set to 1 by generate_sorted_domains_wrap if there
 * is only a single component.
 * "list" collects the results.
 */
struct isl_ast_generate_parallel_domains_data {
        int n;
        isl_union_map *executed;
        isl_ast_build *build;

        int single;
        isl_ast_graft_list *list;
};

/* Call generate_sorted_domains on "scc", fuse the result into a list
 * with either zero or one graft and collect the these single element
 * lists into data->list.
 *
 * If there is only one component, i.e., if the number of basic sets
 * in the current component is equal to the total number of basic sets,
 * then data->single is set to 1 and the result of generate_sorted_domains
 * is not fused.
 */
static int generate_sorted_domains_wrap(__isl_take isl_basic_set_list *scc,
        void *user)
{
        struct isl_ast_generate_parallel_domains_data *data = user;
        isl_ast_graft_list *list;

        list = generate_sorted_domains(scc, data->executed, data->build);
        data->single = isl_basic_set_list_n_basic_set(scc) == data->n;
        if (!data->single)
                list = isl_ast_graft_list_fuse(list, data->build);
        if (!data->list)
                data->list = list;
        else
                data->list = isl_ast_graft_list_concat(data->list, list);

        isl_basic_set_list_free(scc);
        if (!data->list)
                return -1;

        return 0;
}

/* Look for any (weakly connected) components in the "domain_list"
 * of domains that share some values of the outer dimensions.
 * That is, domains in different components do not share any values
 * of the outer dimensions.  This means that these components
 * can be freely reordered.
 * Within each of the components, we sort the domains according
 * to the execution order at the current depth.
 *
 * If there is more than one component, then generate_sorted_domains_wrap
 * fuses the result of each call to generate_sorted_domains
 * into a list with either zero or one graft and collects these (at most)
 * single element lists into a bigger list. This means that the elements of the
 * final list can be freely reordered.  In particular, we sort them
 * according to an arbitrary but fixed ordering to ease merging of
 * graft lists from different components.
 */
static __isl_give isl_ast_graft_list *generate_parallel_domains(
        __isl_keep isl_basic_set_list *domain_list,
        __isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
{
        int depth;
        struct isl_ast_generate_parallel_domains_data data;

        if (!domain_list)
                return NULL;

        data.n = isl_basic_set_list_n_basic_set(domain_list);
        if (data.n <= 1)
                return generate_sorted_domains(domain_list, executed, build);

        depth = isl_ast_build_get_depth(build);
        data.list = NULL;
        data.executed = executed;
        data.build = build;
        data.single = 0;
        if (isl_basic_set_list_foreach_scc(domain_list, &shared_outer, &depth,
                                            &generate_sorted_domains_wrap,
                                            &data) < 0)
                data.list = isl_ast_graft_list_free(data.list);

        if (!data.single)
                data.list = isl_ast_graft_list_sort_guard(data.list);

        return data.list;
}

/* Internal data for separate_domain.
 *
 * "explicit" is set if we only want to use explicit bounds.
 *
 * "domain" collects the separated domains.
 */
struct isl_separate_domain_data {
        isl_ast_build *build;
        int explicit;
        isl_set *domain;
};

/* Extract implicit bounds on the current dimension for the executed "map".
 *
 * The domain of "map" may involve inner dimensions, so we
 * need to eliminate them.
 */
static __isl_give isl_set *implicit_bounds(__isl_take isl_map *map,
        __isl_keep isl_ast_build *build)
{
        isl_set *domain;

        domain = isl_map_domain(map);
        domain = isl_ast_build_eliminate(build, domain);

        return domain;
}

/* Extract explicit bounds on the current dimension for the executed "map".
 *
 * Rather than eliminating the inner dimensions as in implicit_bounds,
 * we simply drop any constraints involving those inner dimensions.
 * The idea is that most bounds that are implied by constraints on the
 * inner dimensions will be enforced by for loops and not by explicit guards.
 * There is then no need to separate along those bounds.
 */
static __isl_give isl_set *explicit_bounds(__isl_take isl_map *map,
        __isl_keep isl_ast_build *build)
{
        isl_set *domain;
        int depth, dim;

        dim = isl_map_dim(map, isl_dim_out);
        map = isl_map_drop_constraints_involving_dims(map, isl_dim_out, 0, dim);

        domain = isl_map_domain(map);
        depth = isl_ast_build_get_depth(build);
        dim = isl_set_dim(domain, isl_dim_set);
        domain = isl_set_detect_equalities(domain);
        domain = isl_set_drop_constraints_involving_dims(domain,
                                isl_dim_set, depth + 1, dim - (depth + 1));
        domain = isl_set_remove_divs_involving_dims(domain,
                                isl_dim_set, depth, 1);
        domain = isl_set_remove_unknown_divs(domain);

        return domain;
}

/* Split data->domain into pieces that intersect with the range of "map"
 * and pieces that do not intersect with the range of "map"
 * and then add that part of the range of "map" that does not intersect
 * with data->domain.
 */
static int separate_domain(__isl_take isl_map *map, void *user)
{
        struct isl_separate_domain_data *data = user;
        isl_set *domain;
        isl_set *d1, *d2;

        if (data->explicit)
                domain = explicit_bounds(map, data->build);
        else
                domain = implicit_bounds(map, data->build);

        domain = isl_set_coalesce(domain);
        domain = isl_set_make_disjoint(domain);
        d1 = isl_set_subtract(isl_set_copy(domain), isl_set_copy(data->domain));
        d2 = isl_set_subtract(isl_set_copy(data->domain), isl_set_copy(domain));
        data->domain = isl_set_intersect(data->domain, domain);
        data->domain = isl_set_union(data->domain, d1);
        data->domain = isl_set_union(data->domain, d2);

        return 0;
}

/* Separate the schedule domains of "executed".
 *
 * That is, break up the domain of "executed" into basic sets,
 * such that for each basic set S, every element in S is associated with
 * the same domain spaces.
 *
 * "space" is the (single) domain space of "executed".
 */
static __isl_give isl_set *separate_schedule_domains(
        __isl_take isl_space *space, __isl_take isl_union_map *executed,
        __isl_keep isl_ast_build *build)
{
        struct isl_separate_domain_data data = { build };
        isl_ctx *ctx;

        ctx = isl_ast_build_get_ctx(build);
        data.explicit = isl_options_get_ast_build_separation_bounds(ctx) ==
                                    ISL_AST_BUILD_SEPARATION_BOUNDS_EXPLICIT;
        data.domain = isl_set_empty(space);
        if (isl_union_map_foreach_map(executed, &separate_domain, &data) < 0)
                data.domain = isl_set_free(data.domain);

        isl_union_map_free(executed);
        return data.domain;
}

/* Temporary data used during the search for a lower bound for unrolling.
 *
 * "domain" is the original set for which to find a lower bound
 * "depth" is the dimension for which to find a lower boudn
 *
 * "lower" is the best lower bound found so far.  It is NULL if we have not
 * found any yet.
 * "n" is the corresponding size.  If lower is NULL, then the value of n
 * is undefined.
 */
struct isl_find_unroll_data {
        isl_set *domain;
        int depth;

        isl_aff *lower;
        int *n;
};

/* Check if we can use "c" as a lower bound and if it is better than
 * any previously found lower bound.
 *
 * If "c" does not involve the dimension at the current depth,
 * then we cannot use it.
 * Otherwise, let "c" be of the form
 *
 *      i >= f(j)/a
 *
 * We compute the maximal value of
 *
 *      -ceil(f(j)/a)) + i + 1
 *
 * over the domain.  If there is such a value "n", then we know
 *
 *      -ceil(f(j)/a)) + i + 1 <= n
 *
 * or
 *
 *      i < ceil(f(j)/a)) + n
 *
 * meaning that we can use ceil(f(j)/a)) as a lower bound for unrolling.
 * We just need to check if we have found any lower bound before and
 * if the new lower bound is better (smaller n) than the previously found
 * lower bounds.
 */
static int update_unrolling_lower_bound(struct isl_find_unroll_data *data,
        __isl_keep isl_constraint *c)
{
        isl_aff *aff, *lower;
        isl_val *max;

        if (!isl_constraint_is_lower_bound(c, isl_dim_set, data->depth))
                return 0;

        lower = isl_constraint_get_bound(c, isl_dim_set, data->depth);
        lower = isl_aff_ceil(lower);
        aff = isl_aff_copy(lower);
        aff = isl_aff_neg(aff);
        aff = isl_aff_add_coefficient_si(aff, isl_dim_in, data->depth, 1);
        aff = isl_aff_add_constant_si(aff, 1);
        max = isl_set_max_val(data->domain, aff);
        isl_aff_free(aff);

        if (!max)
                goto error;
        if (isl_val_is_infty(max)) {
                isl_val_free(max);
                isl_aff_free(lower);
                return 0;
        }

        if (isl_val_cmp_si(max, INT_MAX) <= 0 &&
            (!data->lower || isl_val_cmp_si(max, *data->n) < 0)) {
                isl_aff_free(data->lower);
                data->lower = lower;
                *data->n = isl_val_get_num_si(max);
        } else
                isl_aff_free(lower);
        isl_val_free(max);

        return 1;
error:
        isl_aff_free(lower);
        return -1;
}

/* Check if we can use "c" as a lower bound and if it is better than
 * any previously found lower bound.
 */
static int constraint_find_unroll(__isl_take isl_constraint *c, void *user)
{
        struct isl_find_unroll_data *data;
        int r;

        data = (struct isl_find_unroll_data *) user;
        r = update_unrolling_lower_bound(data, c);
        isl_constraint_free(c);

        return r;
}

/* Look for a lower bound l(i) on the dimension at "depth"
 * and a size n such that "domain" is a subset of
 *
 *      { [i] : l(i) <= i_d < l(i) + n }
 *
 * where d is "depth" and l(i) depends only on earlier dimensions.
 * Furthermore, try and find a lower bound such that n is as small as possible.
 * In particular, "n" needs to be finite.
 *
 * Inner dimensions have been eliminated from "domain" by the caller.
 *
 * We first construct a collection of lower bounds on the input set
 * by computing its simple hull.  We then iterate through them,
 * discarding those that we cannot use (either because they do not
 * involve the dimension at "depth" or because they have no corresponding
 * upper bound, meaning that "n" would be unbounded) and pick out the
 * best from the remaining ones.
 *
 * If we cannot find a suitable lower bound, then we consider that
 * to be an error.
 */
static __isl_give isl_aff *find_unroll_lower_bound(__isl_keep isl_set *domain,
        int depth, int *n)
{
        struct isl_find_unroll_data data = { domain, depth, NULL, n };
        isl_basic_set *hull;

        hull = isl_set_simple_hull(isl_set_copy(domain));

        if (isl_basic_set_foreach_constraint(hull,
                                            &constraint_find_unroll, &data) < 0)
                goto error;

        isl_basic_set_free(hull);

        if (!data.lower)
                isl_die(isl_set_get_ctx(domain), isl_error_invalid,
                        "cannot find lower bound for unrolling", return NULL);

        return data.lower;
error:
        isl_basic_set_free(hull);
        return isl_aff_free(data.lower);
}

/* Return the constraint
 *
 *      i_"depth" = aff + offset
 */
static __isl_give isl_constraint *at_offset(int depth, __isl_keep isl_aff *aff,
        int offset)
{
        aff = isl_aff_copy(aff);
        aff = isl_aff_add_coefficient_si(aff, isl_dim_in, depth, -1);
        aff = isl_aff_add_constant_si(aff, offset);
        return isl_equality_from_aff(aff);
}

/* Return a list of basic sets, one for each value of the current dimension
 * in "domain".
 * The divs that involve the current dimension have not been projected out
 * from this domain.
 *
 * Since we are going to be iterating over the individual values,
 * we first check if there are any strides on the current dimension.
 * If there is, we rewrite the current dimension i as
 *
 *              i = stride i' + offset
 *
 * and then iterate over individual values of i' instead.
 *
 * We then look for a lower bound on i' and a size such that the domain
 * is a subset of
 *
 *      { [j,i'] : l(j) <= i' < l(j) + n }
 *
 * and then take slices of the domain at values of i'
 * between l(j) and l(j) + n - 1.
 *
 * We compute the unshifted simple hull of each slice to ensure that
 * we have a single basic set per offset.  The slicing constraint
 * may get simplified away before the unshifted simple hull is taken
 * and may therefore in some rare cases disappear from the result.
 * We therefore explicitly add the constraint back after computing
 * the unshifted simple hull to ensure that the basic sets
 * remain disjoint.  The constraints that are dropped by taking the hull
 * will be taken into account at the next level, as in the case of the
 * atomic option.
 *
 * Finally, we map i' back to i and add each basic set to the list.
 */
static __isl_give isl_basic_set_list *do_unroll(__isl_take isl_set *domain,
        __isl_keep isl_ast_build *build)
{
        int i, n;
        int depth;
        isl_ctx *ctx;
        isl_aff *lower;
        isl_basic_set_list *list;
        isl_multi_aff *expansion;
        isl_basic_map *bmap;

        if (!domain)
                return NULL;

        ctx = isl_set_get_ctx(domain);
        depth = isl_ast_build_get_depth(build);
        build = isl_ast_build_copy(build);
        domain = isl_ast_build_eliminate_inner(build, domain);
        build = isl_ast_build_detect_strides(build, isl_set_copy(domain));
        expansion = isl_ast_build_get_stride_expansion(build);

        domain = isl_set_preimage_multi_aff(domain,
                                            isl_multi_aff_copy(expansion));
        domain = isl_ast_build_eliminate_divs(build, domain);

        isl_ast_build_free(build);

        list = isl_basic_set_list_alloc(ctx, 0);

        lower = find_unroll_lower_bound(domain, depth, &n);
        if (!lower)
                list = isl_basic_set_list_free(list);

        bmap = isl_basic_map_from_multi_aff(expansion);

        for (i = 0; list && i < n; ++i) {
                isl_set *set;
                isl_basic_set *bset;
                isl_constraint *slice;

                slice = at_offset(depth, lower, i);
                set = isl_set_copy(domain);
                set = isl_set_add_constraint(set, isl_constraint_copy(slice));
                bset = isl_set_unshifted_simple_hull(set);
                bset = isl_basic_set_add_constraint(bset, slice);
                bset = isl_basic_set_apply(bset, isl_basic_map_copy(bmap));
                list = isl_basic_set_list_add(list, bset);
        }

        isl_aff_free(lower);
        isl_set_free(domain);
        isl_basic_map_free(bmap);

        return list;
}

/* Data structure for storing the results and the intermediate objects
 * of compute_domains.
 *
 * "list" is the main result of the function and contains a list
 * of disjoint basic sets for which code should be generated.
 *
 * "executed" and "build" are inputs to compute_domains.
 * "schedule_domain" is the domain of "executed".
 *
 * "option" constains the domains at the current depth that should by
 * atomic, separated or unrolled.  These domains are as specified by
 * the user, except that inner dimensions have been eliminated and
 * that they have been made pair-wise disjoint.
 *
 * "sep_class" contains the user-specified split into separation classes
 * specialized to the current depth.
 * "done" contains the union of the separation domains that have already
 * been handled.
 * "atomic" contains the domain that has effectively been made atomic.
 * This domain may be larger than the intersection of option[atomic]
 * and the schedule domain.
 */
struct isl_codegen_domains {
        isl_basic_set_list *list;

        isl_union_map *executed;
        isl_ast_build *build;
        isl_set *schedule_domain;

        isl_set *option[3];

        isl_map *sep_class;
        isl_set *done;
        isl_set *atomic;
};

/* Add domains to domains->list for each individual value of the current
 * dimension, for that part of the schedule domain that lies in the
 * intersection of the option domain and the class domain.
 *
 * "domain" is the intersection of the class domain and the schedule domain.
 * The divs that involve the current dimension have not been projected out
 * from this domain.
 *
 * We first break up the unroll option domain into individual pieces
 * and then handle each of them separately.  The unroll option domain
 * has been made disjoint in compute_domains_init_options,
 *
 * Note that we actively want to combine different pieces of the
 * schedule domain that have the same value at the current dimension.
 * We therefore need to break up the unroll option domain before
 * intersecting with class and schedule domain, hoping that the
 * unroll option domain specified by the user is relatively simple.
 */
static int compute_unroll_domains(struct isl_codegen_domains *domains,
        __isl_keep isl_set *domain)
{
        isl_set *unroll_domain;
        isl_basic_set_list *unroll_list;
        int i, n;
        int empty;

        empty = isl_set_is_empty(domains->option[unroll]);
        if (empty < 0)
                return -1;
        if (empty)
                return 0;

        unroll_domain = isl_set_copy(domains->option[unroll]);
        unroll_list = isl_basic_set_list_from_set(unroll_domain);

        n = isl_basic_set_list_n_basic_set(unroll_list);
        for (i = 0; i < n; ++i) {
                isl_basic_set *bset;
                isl_basic_set_list *list;

                bset = isl_basic_set_list_get_basic_set(unroll_list, i);
                unroll_domain = isl_set_from_basic_set(bset);
                unroll_domain = isl_set_intersect(unroll_domain,
                                                    isl_set_copy(domain));

                empty = isl_set_is_empty(unroll_domain);
                if (empty >= 0 && empty) {
                        isl_set_free(unroll_domain);
                        continue;
                }

                list = do_unroll(unroll_domain, domains->build);
                domains->list = isl_basic_set_list_concat(domains->list, list);
        }

        isl_basic_set_list_free(unroll_list);

        return 0;
}

/* Construct a single basic set that includes the intersection of
 * the schedule domain, the atomic option domain and the class domain.
 * Add the resulting basic set to domains->list and save a copy
 * in domains->atomic for use in compute_partial_domains.
 *
 * We construct a single domain rather than trying to combine
 * the schedule domains of individual domains because we are working
 * within a single component so that non-overlapping schedule domains
 * should already have been separated.
 * Note, though, that this does not take into account the class domain.
 * So, it is possible for a class domain to carve out a piece of the
 * schedule domain with independent pieces and then we would only
 * generate a single domain for them.  If this proves to be problematic
 * for some users, then this function will have to be adjusted.
 *
 * "domain" is the intersection of the schedule domain and the class domain,
 * with inner dimensions projected out.
 */
static int compute_atomic_domain(struct isl_codegen_domains *domains,
        __isl_keep isl_set *domain)
{
        isl_basic_set *bset;
        isl_set *atomic_domain;
        int empty;

        atomic_domain = isl_set_copy(domains->option[atomic]);
        atomic_domain = isl_set_intersect(atomic_domain, isl_set_copy(domain));
        empty = isl_set_is_empty(atomic_domain);
        if (empty < 0 || empty) {
                domains->atomic = atomic_domain;
                return empty < 0 ? -1 : 0;
        }

        atomic_domain = isl_set_coalesce(atomic_domain);
        bset = isl_set_unshifted_simple_hull(atomic_domain);
        domains->atomic = isl_set_from_basic_set(isl_basic_set_copy(bset));
        domains->list = isl_basic_set_list_add(domains->list, bset);

        return 0;
}

/* Split up the schedule domain into uniform basic sets,
 * in the sense that each element in a basic set is associated to
 * elements of the same domains, and add the result to domains->list.
 * Do this for that part of the schedule domain that lies in the
 * intersection of "class_domain" and the separate option domain.
 *
 * "class_domain" may or may not include the constraints
 * of the schedule domain, but this does not make a difference
 * since we are going to intersect it with the domain of the inverse schedule.
 * If it includes schedule domain constraints, then they may involve
 * inner dimensions, but we will eliminate them in separation_domain.
 */
static int compute_separate_domain(struct isl_codegen_domains *domains,
        __isl_keep isl_set *class_domain)
{
        isl_space *space;
        isl_set *domain;
        isl_union_map *executed;
        isl_basic_set_list *list;
        int empty;

        domain = isl_set_copy(domains->option[separate]);
        domain = isl_set_intersect(domain, isl_set_copy(class_domain));
        executed = isl_union_map_copy(domains->executed);
        executed = isl_union_map_intersect_domain(executed,
                                    isl_union_set_from_set(domain));
        empty = isl_union_map_is_empty(executed);
        if (empty < 0 || empty) {
                isl_union_map_free(executed);
                return empty < 0 ? -1 : 0;
        }

        space = isl_set_get_space(class_domain);
        domain = separate_schedule_domains(space, executed, domains->build);

        list = isl_basic_set_list_from_set(domain);
        domains->list = isl_basic_set_list_concat(domains->list, list);

        return 0;
}

/* Split up the domain at the current depth into disjoint
 * basic sets for which code should be generated separately
 * for the given separation class domain.
 *
 * If any separation classes have been defined, then "class_domain"
 * is the domain of the current class and does not refer to inner dimensions.
 * Otherwise, "class_domain" is the universe domain.
 *
 * We first make sure that the class domain is disjoint from
 * previously considered class domains.
 *
 * The separate domains can be computed directly from the "class_domain".
 *
 * The unroll, atomic and remainder domains need the constraints
 * from the schedule domain.
 *
 * For unrolling, the actual schedule domain is needed (with divs that
 * may refer to the current dimension) so that stride detection can be
 * performed.
 *
 * For atomic and remainder domains, inner dimensions and divs involving
 * the current dimensions should be eliminated.
 * In case we are working within a separation class, we need to intersect
 * the result with the current "class_domain" to ensure that the domains
 * are disjoint from those generated from other class domains.
 *
 * The domain that has been made atomic may be larger than specified
 * by the user since it needs to be representable as a single basic set.
 * This possibly larger domain is stored in domains->atomic by
 * compute_atomic_domain.
 *
 * If anything is left after handling separate, unroll and atomic,
 * we split it up into basic sets and append the basic sets to domains->list.
 */
static int compute_partial_domains(struct isl_codegen_domains *domains,
        __isl_take isl_set *class_domain)
{
        isl_basic_set_list *list;
        isl_set *domain;

        class_domain = isl_set_subtract(class_domain,
                                        isl_set_copy(domains->done));
        domains->done = isl_set_union(domains->done,
                                        isl_set_copy(class_domain));

        domain = isl_set_copy(class_domain);

        if (compute_separate_domain(domains, domain) < 0)
                goto error;
        domain = isl_set_subtract(domain,
                                    isl_set_copy(domains->option[separate]));

        domain = isl_set_intersect(domain,
                                isl_set_copy(domains->schedule_domain));

        if (compute_unroll_domains(domains, domain) < 0)
                goto error;
        domain = isl_set_subtract(domain,
                                    isl_set_copy(domains->option[unroll]));

        domain = isl_ast_build_eliminate(domains->build, domain);
        domain = isl_set_intersect(domain, isl_set_copy(class_domain));

        if (compute_atomic_domain(domains, domain) < 0)
                domain = isl_set_free(domain);
        domain = isl_set_subtract(domain, domains->atomic);

        domain = isl_set_coalesce(domain);
        domain = isl_set_make_disjoint(domain);

        list = isl_basic_set_list_from_set(domain);
        domains->list = isl_basic_set_list_concat(domains->list, list);

        isl_set_free(class_domain);

        return 0;
error:
        isl_set_free(domain);
        isl_set_free(class_domain);
        return -1;
}

/* Split up the domain at the current depth into disjoint
 * basic sets for which code should be generated separately
 * for the separation class identified by "pnt".
 *
 * We extract the corresponding class domain from domains->sep_class,
 * eliminate inner dimensions and pass control to compute_partial_domains.
 */
static int compute_class_domains(__isl_take isl_point *pnt, void *user)
{
        struct isl_codegen_domains *domains = user;
        isl_set *class_set;
        isl_set *domain;
        int disjoint;

        class_set = isl_set_from_point(pnt);
        domain = isl_map_domain(isl_map_intersect_range(
                                isl_map_copy(domains->sep_class), class_set));
        domain = isl_ast_build_compute_gist(domains->build, domain);
        domain = isl_ast_build_eliminate(domains->build, domain);

        disjoint = isl_set_plain_is_disjoint(domain, domains->schedule_domain);
        if (disjoint < 0)
                return -1;
        if (disjoint) {
                isl_set_free(domain);
                return 0;
        }

        return compute_partial_domains(domains, domain);
}

/* Extract the domains at the current depth that should be atomic,
 * separated or unrolled and store them in option.
 *
 * The domains specified by the user might overlap, so we make
 * them disjoint by subtracting earlier domains from later domains.
 */
static void compute_domains_init_options(isl_set *option[3],
        __isl_keep isl_ast_build *build)
{
        enum isl_ast_build_domain_type type, type2;

        for (type = atomic; type <= separate; ++type) {
                option[type] = isl_ast_build_get_option_domain(build, type);
                for (type2 = atomic; type2 < type; ++type2)
                        option[type] = isl_set_subtract(option[type],
                                                isl_set_copy(option[type2]));
        }

        option[unroll] = isl_set_coalesce(option[unroll]);
        option[unroll] = isl_set_make_disjoint(option[unroll]);
}

/* Split up the domain at the current depth into disjoint
 * basic sets for which code should be generated separately,
 * based on the user-specified options.
 * Return the list of disjoint basic sets.
 *
 * There are three kinds of domains that we need to keep track of.
 * - the "schedule domain" is the domain of "executed"
 * - the "class domain" is the domain corresponding to the currrent
 *      separation class
 * - the "option domain" is the domain corresponding to one of the options
 *      atomic, unroll or separate
 *
 * We first consider the individial values of the separation classes
 * and split up the domain for each of them separately.
 * Finally, we consider the remainder.  If no separation classes were
 * specified, then we call compute_partial_domains with the universe
 * "class_domain".  Otherwise, we take the "schedule_domain" as "class_domain",
 * with inner dimensions removed.  We do this because we want to
 * avoid computing the complement of the class domains (i.e., the difference
 * between the universe and domains->done).
 */
static __isl_give isl_basic_set_list *compute_domains(
        __isl_keep isl_union_map *executed, __isl_keep isl_ast_build *build)
{
        struct isl_codegen_domains domains;
        isl_ctx *ctx;
        isl_set *domain;
        isl_union_set *schedule_domain;
        isl_set *classes;
        isl_space *space;
        int n_param;
        enum isl_ast_build_domain_type type;
        int empty;

        if (!executed)
                return NULL;

        ctx = isl_union_map_get_ctx(executed);
        domains.list = isl_basic_set_list_alloc(ctx, 0);

        schedule_domain = isl_union_map_domain(isl_union_map_copy(executed));
        domain = isl_set_from_union_set(schedule_domain);

        compute_domains_init_options(domains.option, build);

        domains.sep_class = isl_ast_build_get_separation_class(build);
        classes = isl_map_range(isl_map_copy(domains.sep_class));
        n_param = isl_set_dim(classes, isl_dim_param);
        classes = isl_set_project_out(classes, isl_dim_param, 0, n_param);

        space = isl_set_get_space(domain);
        domains.build = build;
        domains.schedule_domain = isl_set_copy(domain);
        domains.executed = executed;
        domains.done = isl_set_empty(space);

        if (isl_set_foreach_point(classes, &compute_class_domains, &domains) < 0)
                domains.list = isl_basic_set_list_free(domains.list);
        isl_set_free(classes);

        empty = isl_set_is_empty(domains.done);
        if (empty < 0) {
                domains.list = isl_basic_set_list_free(domains.list);
                domain = isl_set_free(domain);
        } else if (empty) {
                isl_set_free(domain);
                domain = isl_set_universe(isl_set_get_space(domains.done));
        } else {
                domain = isl_ast_build_eliminate(build, domain);
        }
        if (compute_partial_domains(&domains, domain) < 0)
                domains.list = isl_basic_set_list_free(domains.list);

        isl_set_free(domains.schedule_domain);
        isl_set_free(domains.done);
        isl_map_free(domains.sep_class);
        for (type = atomic; type <= separate; ++type)
                isl_set_free(domains.option[type]);

        return domains.list;
}

/* Generate code for a single component, after shifting (if any)
 * has been applied.
 *
 * We first split up the domain at the current depth into disjoint
 * basic sets based on the user-specified options.
 * Then we generated code for each of them and concatenate the results.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{
        isl_basic_set_list *domain_list;
        isl_ast_graft_list *list = NULL;

        domain_list = compute_domains(executed, build);
        list = generate_parallel_domains(domain_list, executed, build);

        isl_basic_set_list_free(domain_list);
        isl_union_map_free(executed);
        isl_ast_build_free(build);

        return list;
}

struct isl_set_map_pair {
        isl_set *set;
        isl_map *map;
};

/* Given an array "domain" of isl_set_map_pairs and an array "order"
 * of indices into the "domain" array,
 * return the union of the "map" fields of the elements
 * indexed by the first "n" elements of "order".
 */
static __isl_give isl_union_map *construct_component_executed(
        struct isl_set_map_pair *domain, int *order, int n)
{
        int i;
        isl_map *map;
        isl_union_map *executed;

        map = isl_map_copy(domain[order[0]].map);
        executed = isl_union_map_from_map(map);
        for (i = 1; i < n; ++i) {
                map = isl_map_copy(domain[order[i]].map);
                executed = isl_union_map_add_map(executed, map);
        }

        return executed;
}

/* Generate code for a single component, after shifting (if any)
 * has been applied.
 *
 * The component inverse schedule is specified as the "map" fields
 * of the elements of "domain" indexed by the first "n" elements of "order".
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_from_list(
        struct isl_set_map_pair *domain, int *order, int n,
        __isl_take isl_ast_build *build)
{
        isl_union_map *executed;

        executed = construct_component_executed(domain, order, n);
        return generate_shifted_component(executed, build);
}

/* Given an array "domain" of isl_set_map_pairs and an array "order"
 * of indices into the "domain" array,
 * do all (except for at most one) of the "set" field of the elements
 * indexed by the first "n" elements of "order" have a fixed value
 * at position "depth"?
 */
static int at_most_one_non_fixed(struct isl_set_map_pair *domain,
        int *order, int n, int depth)
{
        int i;
        int non_fixed = -1;

        for (i = 0; i < n; ++i) {
                int f;

                f = isl_set_plain_is_fixed(domain[order[i]].set,
                                                isl_dim_set, depth, NULL);
                if (f < 0)
                        return -1;
                if (f)
                        continue;
                if (non_fixed >= 0)
                        return 0;
                non_fixed = i;
        }

        return 1;
}

/* Given an array "domain" of isl_set_map_pairs and an array "order"
 * of indices into the "domain" array,
 * eliminate the inner dimensions from the "set" field of the elements
 * indexed by the first "n" elements of "order", provided the current
 * dimension does not have a fixed value.
 *
 * Return the index of the first element in "order" with a corresponding
 * "set" field that does not have an (obviously) fixed value.
 */
static int eliminate_non_fixed(struct isl_set_map_pair *domain,
        int *order, int n, int depth, __isl_keep isl_ast_build *build)
{
        int i;
        int base = -1;

        for (i = n - 1; i >= 0; --i) {
                int f;
                f = isl_set_plain_is_fixed(domain[order[i]].set,
                                                isl_dim_set, depth, NULL);
                if (f < 0)
                        return -1;
                if (f)
                        continue;
                domain[order[i]].set = isl_ast_build_eliminate_inner(build,
                                                        domain[order[i]].set);
                base = i;
        }

        return base;
}

/* Given an array "domain" of isl_set_map_pairs and an array "order"
 * of indices into the "domain" array,
 * find the element of "domain" (amongst those indexed by the first "n"
 * elements of "order") with the "set" field that has the smallest
 * value for the current iterator.
 *
 * Note that the domain with the smallest value may depend on the parameters
 * and/or outer loop dimension.  Since the result of this function is only
 * used as heuristic, we only make a reasonable attempt at finding the best
 * domain, one that should work in case a single domain provides the smallest
 * value for the current dimension over all values of the parameters
 * and outer dimensions.
 *
 * In particular, we compute the smallest value of the first domain
 * and replace it by that of any later domain if that later domain
 * has a smallest value that is smaller for at least some value
 * of the parameters and outer dimensions.
 */
static int first_offset(struct isl_set_map_pair *domain, int *order, int n,
        __isl_keep isl_ast_build *build)
{
        int i;
        isl_map *min_first;
        int first = 0;

        min_first = isl_ast_build_map_to_iterator(build,
                                        isl_set_copy(domain[order[0]].set));
        min_first = isl_map_lexmin(min_first);

        for (i = 1; i < n; ++i) {
                isl_map *min, *test;
                int empty;

                min = isl_ast_build_map_to_iterator(build,
                                        isl_set_copy(domain[order[i]].set));
                min = isl_map_lexmin(min);
                test = isl_map_copy(min);
                test = isl_map_apply_domain(isl_map_copy(min_first), test);
                test = isl_map_order_lt(test, isl_dim_in, 0, isl_dim_out, 0);
                empty = isl_map_is_empty(test);
                isl_map_free(test);
                if (empty >= 0 && !empty) {
                        isl_map_free(min_first);
                        first = i;
                        min_first = min;
                } else
                        isl_map_free(min);

                if (empty < 0)
                        break;
        }

        isl_map_free(min_first);

        return i < n ? -1 : first;
}

/* Construct a shifted inverse schedule based on the original inverse schedule,
 * the stride and the offset.
 *
 * The original inverse schedule is specified as the "map" fields
 * of the elements of "domain" indexed by the first "n" elements of "order".
 *
 * "stride" and "offset" are such that the difference
 * between the values of the current dimension of domain "i"
 * and the values of the current dimension for some reference domain are
 * equal to
 *
 *      stride * integer + offset[i]
 *
 * Moreover, 0 <= offset[i] < stride.
 *
 * For each domain, we create a map
 *
 *      { [..., j, ...] -> [..., j - offset[i], offset[i], ....] }
 *
 * where j refers to the current dimension and the other dimensions are
 * unchanged, and apply this map to the original schedule domain.
 *
 * For example, for the original schedule
 *
 *      { A[i] -> [2i]: 0 <= i < 10; B[i] -> [2i+1] : 0 <= i < 10 }
 *
 * and assuming the offset is 0 for the A domain and 1 for the B domain,
 * we apply the mapping
 *
 *      { [j] -> [j, 0] }
 *
 * to the schedule of the "A" domain and the mapping
 *
 *      { [j - 1] -> [j, 1] }
 *
 * to the schedule of the "B" domain.
 *
 *
 * Note that after the transformation, the differences between pairs
 * of values of the current dimension over all domains are multiples
 * of stride and that we have therefore exposed the stride.
 *
 *
 * To see that the mapping preserves the lexicographic order,
 * first note that each of the individual maps above preserves the order.
 * If the value of the current iterator is j1 in one domain and j2 in another,
 * then if j1 = j2, we know that the same map is applied to both domains
 * and the order is preserved.
 * Otherwise, let us assume, without loss of generality, that j1 < j2.
 * If c1 >= c2 (with c1 and c2 the corresponding offsets), then
 *
 *      j1 - c1 < j2 - c2
 *
 * and the order is preserved.
 * If c1 < c2, then we know
 *
 *      0 <= c2 - c1 < s
 *
 * We also have
 *
 *      j2 - j1 = n * s + r
 *
 * with n >= 0 and 0 <= r < s.
 * In other words, r = c2 - c1.
 * If n > 0, then
 *
 *      j1 - c1 < j2 - c2
 *
 * If n = 0, then
 *
 *      j1 - c1 = j2 - c2
 *
 * and so
 *
 *      (j1 - c1, c1) << (j2 - c2, c2)
 *
 * with "<<" the lexicographic order, proving that the order is preserved
 * in all cases.
 */
static __isl_give isl_union_map *contruct_shifted_executed(
        struct isl_set_map_pair *domain, int *order, int n, isl_int stride,
        __isl_keep isl_vec *offset, __isl_keep isl_ast_build *build)
{
        int i;
        isl_int v;
        isl_union_map *executed;
        isl_space *space;
        isl_map *map;
        int depth;
        isl_constraint *c;

        depth = isl_ast_build_get_depth(build);
        space = isl_ast_build_get_space(build, 1);
        executed = isl_union_map_empty(isl_space_copy(space));
        space = isl_space_map_from_set(space);
        map = isl_map_identity(isl_space_copy(space));
        map = isl_map_eliminate(map, isl_dim_out, depth, 1);
        map = isl_map_insert_dims(map, isl_dim_out, depth + 1, 1);
        space = isl_space_insert_dims(space, isl_dim_out, depth + 1, 1);

        c = isl_equality_alloc(isl_local_space_from_space(space));
        c = isl_constraint_set_coefficient_si(c, isl_dim_in, depth, 1);
        c = isl_constraint_set_coefficient_si(c, isl_dim_out, depth, -1);

        isl_int_init(v);

        for (i = 0; i < n; ++i) {
                isl_map *map_i;

                if (isl_vec_get_element(offset, i, &v) < 0)
                        break;
                map_i = isl_map_copy(map);
                map_i = isl_map_fix(map_i, isl_dim_out, depth + 1, v);
                isl_int_neg(v, v);
                c = isl_constraint_set_constant(c, v);
                map_i = isl_map_add_constraint(map_i, isl_constraint_copy(c));

                map_i = isl_map_apply_domain(isl_map_copy(domain[order[i]].map),
                                                map_i);
                executed = isl_union_map_add_map(executed, map_i);
        }

        isl_constraint_free(c);
        isl_map_free(map);

        isl_int_clear(v);

        if (i < n)
                executed = isl_union_map_free(executed);

        return executed;
}

/* Generate code for a single component, after exposing the stride,
 * given that the schedule domain is "shifted strided".
 *
 * The component inverse schedule is specified as the "map" fields
 * of the elements of "domain" indexed by the first "n" elements of "order".
 *
 * The schedule domain being "shifted strided" means that the differences
 * between the values of the current dimension of domain "i"
 * and the values of the current dimension for some reference domain are
 * equal to
 *
 *      stride * integer + offset[i]
 *
 * We first look for the domain with the "smallest" value for the current
 * dimension and adjust the offsets such that the offset of the "smallest"
 * domain is equal to zero.  The other offsets are reduced modulo stride.
 *
 * Based on this information, we construct a new inverse schedule in
 * contruct_shifted_executed that exposes the stride.
 * Since this involves the introduction of a new schedule dimension,
 * the build needs to be changed accodingly.
 * After computing the AST, the newly introduced dimension needs
 * to be removed again from the list of grafts.  We do this by plugging
 * in a mapping that represents the new schedule domain in terms of the
 * old schedule domain.
 */
static __isl_give isl_ast_graft_list *generate_shift_component(
        struct isl_set_map_pair *domain, int *order, int n, isl_int stride,
        __isl_keep isl_vec *offset, __isl_take isl_ast_build *build)
{
        isl_ast_graft_list *list;
        int first;
        int depth;
        isl_ctx *ctx;
        isl_int val;
        isl_vec *v;
        isl_space *space;
        isl_multi_aff *ma, *zero;
        isl_union_map *executed;

        ctx = isl_ast_build_get_ctx(build);
        depth = isl_ast_build_get_depth(build);

        first = first_offset(domain, order, n, build);
        if (first < 0)
                return isl_ast_build_free(build);

        isl_int_init(val);
        v = isl_vec_alloc(ctx, n);
        if (isl_vec_get_element(offset, first, &val) < 0)
                v = isl_vec_free(v);
        isl_int_neg(val, val);
        v = isl_vec_set(v, val);
        v = isl_vec_add(v, isl_vec_copy(offset));
        v = isl_vec_fdiv_r(v, stride);

        executed = contruct_shifted_executed(domain, order, n, stride, v,
                                                build);
        space = isl_ast_build_get_space(build, 1);
        space = isl_space_map_from_set(space);
        ma = isl_multi_aff_identity(isl_space_copy(space));
        space = isl_space_from_domain(isl_space_domain(space));
        space = isl_space_add_dims(space, isl_dim_out, 1);
        zero = isl_multi_aff_zero(space);
        ma = isl_multi_aff_range_splice(ma, depth + 1, zero);
        build = isl_ast_build_insert_dim(build, depth + 1);
        list = generate_shifted_component(executed, build);

        list = isl_ast_graft_list_preimage_multi_aff(list, ma);

        isl_vec_free(v);
        isl_int_clear(val);

        return list;
}

/* Generate code for a single component.
 *
 * The component inverse schedule is specified as the "map" fields
 * of the elements of "domain" indexed by the first "n" elements of "order".
 *
 * This function may modify the "set" fields of "domain".
 *
 * Before proceeding with the actual code generation for the component,
 * we first check if there are any "shifted" strides, meaning that
 * the schedule domains of the individual domains are all strided,
 * but that they have different offsets, resulting in the union
 * of schedule domains not being strided anymore.
 *
 * The simplest example is the schedule
 *
 *      { A[i] -> [2i]: 0 <= i < 10; B[i] -> [2i+1] : 0 <= i < 10 }
 *
 * Both schedule domains are strided, but their union is not.
 * This function detects such cases and then rewrites the schedule to
 *
 *      { A[i] -> [2i, 0]: 0 <= i < 10; B[i] -> [2i, 1] : 0 <= i < 10 }
 *
 * In the new schedule, the schedule domains have the same offset (modulo
 * the stride), ensuring that the union of schedule domains is also strided.
 *
 *
 * If there is only a single domain in the component, then there is
 * nothing to do.   Similarly, if the current schedule dimension has
 * a fixed value for almost all domains then there is nothing to be done.
 * In particular, we need at least two domains where the current schedule
 * dimension does not have a fixed value.
 * Finally, if any of the options refer to the current schedule dimension,
 * then we bail out as well.  It would be possible to reformulate the options
 * in terms of the new schedule domain, but that would introduce constraints
 * that separate the domains in the options and that is something we would
 * like to avoid.
 *
 *
 * To see if there is any shifted stride, we look at the differences
 * between the values of the current dimension in pairs of domains
 * for equal values of outer dimensions.  These differences should be
 * of the form
 *
 *      m x + r
 *
 * with "m" the stride and "r" a constant.  Note that we cannot perform
 * this analysis on individual domains as the lower bound in each domain
 * may depend on parameters or outer dimensions and so the current dimension
 * itself may not have a fixed remainder on division by the stride.
 *
 * In particular, we compare the first domain that does not have an
 * obviously fixed value for the current dimension to itself and all
 * other domains and collect the offsets and the gcd of the strides.
 * If the gcd becomes one, then we failed to find shifted strides.
 * If all the offsets are the same (for those domains that do not have
 * an obviously fixed value for the current dimension), then we do not
 * apply the transformation.
 * If none of the domains were skipped, then there is nothing to do.
 * If some of them were skipped, then if we apply separation, the schedule
 * domain should get split in pieces with a (non-shifted) stride.
 *
 * Otherwise, we apply a shift to expose the stride in
 * generate_shift_component.
 */
static __isl_give isl_ast_graft_list *generate_component(
        struct isl_set_map_pair *domain, int *order, int n,
        __isl_take isl_ast_build *build)
{
        int i, d;
        int depth;
        isl_ctx *ctx;
        isl_map *map;
        isl_set *deltas;
        isl_int m, r, gcd;
        isl_vec *v;
        int fixed, skip;
        int base;
        isl_ast_graft_list *list;
        int res = 0;

        depth = isl_ast_build_get_depth(build);

        skip = n == 1;
        if (skip >= 0 && !skip)
                skip = at_most_one_non_fixed(domain, order, n, depth);
        if (skip >= 0 && !skip)
                skip = isl_ast_build_options_involve_depth(build);
        if (skip < 0)
                return isl_ast_build_free(build);
        if (skip)
                return generate_shifted_component_from_list(domain,
                                                            order, n, build);

        base = eliminate_non_fixed(domain, order, n, depth, build);
        if (base < 0)
                return isl_ast_build_free(build);

        ctx = isl_ast_build_get_ctx(build);

        isl_int_init(m);
        isl_int_init(r);
        isl_int_init(gcd);
        v = isl_vec_alloc(ctx, n);

        fixed = 1;
        for (i = 0; i < n; ++i) {
                map = isl_map_from_domain_and_range(
                                        isl_set_copy(domain[order[base]].set),
                                        isl_set_copy(domain[order[i]].set));
                for (d = 0; d < depth; ++d)
                        map = isl_map_equate(map, isl_dim_in, d,
                                                    isl_dim_out, d);
                deltas = isl_map_deltas(map);
                res = isl_set_dim_residue_class(deltas, depth, &m, &r);
                isl_set_free(deltas);
                if (res < 0)
                        break;

                if (i == 0)
                        isl_int_set(gcd, m);
                else
                        isl_int_gcd(gcd, gcd, m);
                if (isl_int_is_one(gcd))
                        break;
                v = isl_vec_set_element(v, i, r);

                res = isl_set_plain_is_fixed(domain[order[i]].set,
                                                isl_dim_set, depth, NULL);
                if (res < 0)
                        break;
                if (res)
                        continue;

                if (fixed && i > base) {
                        isl_vec_get_element(v, base, &m);
                        if (isl_int_ne(m, r))
                                fixed = 0;
                }
        }

        if (res < 0) {
                isl_ast_build_free(build);
                list = NULL;
        } else if (i < n || fixed) {
                list = generate_shifted_component_from_list(domain,
                                                            order, n, build);
        } else {
                list = generate_shift_component(domain, order, n, gcd, v,
                                                build);
        }

        isl_vec_free(v);
        isl_int_clear(gcd);
        isl_int_clear(r);
        isl_int_clear(m);

        return list;
}

/* Store both "map" itself and its domain in the
 * structure pointed to by *next and advance to the next array element.
 */
static int extract_domain(__isl_take isl_map *map, void *user)
{
        struct isl_set_map_pair **next = user;

        (*next)->map = isl_map_copy(map);
        (*next)->set = isl_map_domain(map);
        (*next)++;

        return 0;
}

/* Internal data for any_scheduled_after.
 *
 * "depth" is the number of loops that have already been generated
 * "group_coscheduled" is a local copy of options->ast_build_group_coscheduled
 * "domain" is an array of set-map pairs corresponding to the different
 * iteration domains.  The set is the schedule domain, i.e., the domain
 * of the inverse schedule, while the map is the inverse schedule itself.
 */
struct isl_any_scheduled_after_data {
        int depth;
        int group_coscheduled;
        struct isl_set_map_pair *domain;
};

/* Is any element of domain "i" scheduled after any element of domain "j"
 * (for a common iteration of the first data->depth loops)?
 *
 * data->domain[i].set contains the domain of the inverse schedule
 * for domain "i", i.e., elements in the schedule domain.
 *
 * If data->group_coscheduled is set, then we also return 1 if there
 * is any pair of elements in the two domains that are scheduled together.
 */
static int any_scheduled_after(int i, int j, void *user)
{
        struct isl_any_scheduled_after_data *data = user;
        int dim = isl_set_dim(data->domain[i].set, isl_dim_set);
        int pos;

        for (pos = data->depth; pos < dim; ++pos) {
                int follows;

                follows = isl_set_follows_at(data->domain[i].set,
                                                data->domain[j].set, pos);

                if (follows < -1)
                        return -1;
                if (follows > 0)
                        return 1;
                if (follows < 0)
                        return 0;
        }

        return data->group_coscheduled;
}

/* Look for independent components at the current depth and generate code
 * for each component separately.  The resulting lists of grafts are
 * merged in an attempt to combine grafts with identical guards.
 *
 * Code for two domains can be generated separately if all the elements
 * of one domain are scheduled before (or together with) all the elements
 * of the other domain.  We therefore consider the graph with as nodes
 * the domains and an edge between two nodes if any element of the first
 * node is scheduled after any element of the second node.
 * If the ast_build_group_coscheduled is set, then we also add an edge if
 * there is any pair of elements in the two domains that are scheduled
 * together.
 * Code is then generated (by generate_component)
 * for each of the strongly connected components in this graph
 * in their topological order.
 *
 * Since the test is performed on the domain of the inverse schedules of
 * the different domains, we precompute these domains and store
 * them in data.domain.
 */
static __isl_give isl_ast_graft_list *generate_components(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{
        int i;
        isl_ctx *ctx = isl_ast_build_get_ctx(build);
        int n = isl_union_map_n_map(executed);
        struct isl_any_scheduled_after_data data;
        struct isl_set_map_pair *next;
        struct isl_tarjan_graph *g = NULL;
        isl_ast_graft_list *list = NULL;
        int n_domain = 0;

        data.domain = isl_calloc_array(ctx, struct isl_set_map_pair, n);
        if (!data.domain)
                goto error;
        n_domain = n;

        next = data.domain;
        if (isl_union_map_foreach_map(executed, &extract_domain, &next) < 0)
                goto error;

        if (!build)
                goto error;
        data.depth = isl_ast_build_get_depth(build);
        data.group_coscheduled = isl_options_get_ast_build_group_coscheduled(ctx);
        g = isl_tarjan_graph_init(ctx, n, &any_scheduled_after, &data);

        list = isl_ast_graft_list_alloc(ctx, 0);

        i = 0;
        while (list && n) {
                isl_ast_graft_list *list_c;
                int first = i;

                if (g->order[i] == -1)
                        isl_die(ctx, isl_error_internal, "cannot happen",
                                goto error);
                ++i; --n;
                while (g->order[i] != -1) {
                        ++i; --n;
                }

                list_c = generate_component(data.domain,
                                            g->order + first, i - first,
                                            isl_ast_build_copy(build));
                list = isl_ast_graft_list_merge(list, list_c, build);

                ++i;
        }

        if (0)
error:          list = isl_ast_graft_list_free(list);
        isl_tarjan_graph_free(g);
        for (i = 0; i < n_domain; ++i) {
                isl_map_free(data.domain[i].map);
                isl_set_free(data.domain[i].set);
        }
        free(data.domain);
        isl_union_map_free(executed);
        isl_ast_build_free(build);

        return list;
}

/* Generate code for the next level (and all inner levels).
 *
 * If "executed" is empty, i.e., no code needs to be generated,
 * then we return an empty list.
 *
 * If we have already generated code for all loop levels, then we pass
 * control to generate_inner_level.
 *
 * If "executed" lives in a single space, i.e., if code needs to be
 * generated for a single domain, then there can only be a single
 * component and we go directly to generate_shifted_component.
 * Otherwise, we call generate_components to detect the components
 * and to call generate_component on each of them separately.
 */
static __isl_give isl_ast_graft_list *generate_next_level(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{
        int depth;

        if (!build || !executed)
                goto error;

        if (isl_union_map_is_empty(executed)) {
                isl_ctx *ctx = isl_ast_build_get_ctx(build);
                isl_union_map_free(executed);
                isl_ast_build_free(build);
                return isl_ast_graft_list_alloc(ctx, 0);
        }

        depth = isl_ast_build_get_depth(build);
        if (depth >= isl_set_dim(build->domain, isl_dim_set))
                return generate_inner_level(executed, build);

        if (isl_union_map_n_map(executed) == 1)
                return generate_shifted_component(executed, build);

        return generate_components(executed, build);
error:
        isl_union_map_free(executed);
        isl_ast_build_free(build);
        return NULL;
}

/* Internal data structure used by isl_ast_build_ast_from_schedule.
 * internal, executed and build are the inputs to generate_code.
 * list collects the output.
 */
struct isl_generate_code_data {
        int internal;
        isl_union_map *executed;
        isl_ast_build *build;

        isl_ast_graft_list *list;
};

/* Given an inverse schedule in terms of the external build schedule, i.e.,
 *
 *      [E -> S] -> D
 *
 * with E the external build schedule and S the additional schedule "space",
 * reformulate the inverse schedule in terms of the internal schedule domain,
 * i.e., return
 *
 *      [I -> S] -> D
 *
 * We first obtain a mapping
 *
 *      I -> E
 *
 * take the inverse and the product with S -> S, resulting in
 *
 *      [I -> S] -> [E -> S]
 *
 * Applying the map to the input produces the desired result.
 */
static __isl_give isl_union_map *internal_executed(
        __isl_take isl_union_map *executed, __isl_keep isl_space *space,
        __isl_keep isl_ast_build *build)
{
        isl_map *id, *proj;

        proj = isl_ast_build_get_schedule_map(build);
        proj = isl_map_reverse(proj);
        space = isl_space_map_from_set(isl_space_copy(space));
        id = isl_map_identity(space);
        proj = isl_map_product(proj, id);
        executed = isl_union_map_apply_domain(executed,
                                                isl_union_map_from_map(proj));
        return executed;
}

/* Generate an AST that visits the elements in the range of data->executed
 * in the relative order specified by the corresponding image element(s)
 * for those image elements that belong to "set".
 * Add the result to data->list.
 *
 * The caller ensures that "set" is a universe domain.
 * "space" is the space of the additional part of the schedule.
 * It is equal to the space of "set" if build->domain is parametric.
 * Otherwise, it is equal to the range of the wrapped space of "set".
 *
 * If the build space is not parametric and if isl_ast_build_ast_from_schedule
 * was called from an outside user (data->internal not set), then
 * the (inverse) schedule refers to the external build domain and needs to
 * be transformed to refer to the internal build domain.
 *
 * The build is extended to include the additional part of the schedule.
 * If the original build space was not parametric, then the options
 * in data->build refer only to the additional part of the schedule
 * and they need to be adjusted to refer to the complete AST build
 * domain.
 *
 * After having adjusted inverse schedule and build, we start generating
 * code with the outer loop of the current code generation
 * in generate_next_level.
 *
 * If the original build space was not parametric, we undo the embedding
 * on the resulting isl_ast_node_list so that it can be used within
 * the outer AST build.
 */
static int generate_code_in_space(struct isl_generate_code_data *data,
        __isl_take isl_set *set, __isl_take isl_space *space)
{
        isl_union_map *executed;
        isl_ast_build *build;
        isl_ast_graft_list *list;
        int embed;

        executed = isl_union_map_copy(data->executed);
        executed = isl_union_map_intersect_domain(executed,
                                                 isl_union_set_from_set(set));

        embed = !isl_set_is_params(data->build->domain);
        if (embed && !data->internal)
                executed = internal_executed(executed, space, data->build);

        build = isl_ast_build_copy(data->build);
        build = isl_ast_build_product(build, space);

        list = generate_next_level(executed, build);

        list = isl_ast_graft_list_unembed(list, embed);

        data->list = isl_ast_graft_list_concat(data->list, list);

        return 0;
}

/* Generate an AST that visits the elements in the range of data->executed
 * in the relative order specified by the corresponding domain element(s)
 * for those domain elements that belong to "set".
 * Add the result to data->list.
 *
 * The caller ensures that "set" is a universe domain.
 *
 * If the build space S is not parametric, then the space of "set"
 * need to be a wrapped relation with S as domain.  That is, it needs
 * to be of the form
 *
 *      [S -> T]
 *
 * Check this property and pass control to generate_code_in_space
 * passing along T.
 * If the build space is not parametric, then T is the space of "set".
 */
static int generate_code_set(__isl_take isl_set *set, void *user)
{
        struct isl_generate_code_data *data = user;
        isl_space *space, *build_space;
        int is_domain;

        space = isl_set_get_space(set);

        if (isl_set_is_params(data->build->domain))
                return generate_code_in_space(data, set, space);

        build_space = isl_ast_build_get_space(data->build, data->internal);
        space = isl_space_unwrap(space);
        is_domain = isl_space_is_domain(build_space, space);
        isl_space_free(build_space);
        space = isl_space_range(space);

        if (is_domain < 0)
                goto error;
        if (!is_domain)
                isl_die(isl_set_get_ctx(set), isl_error_invalid,
                        "invalid nested schedule space", goto error);

        return generate_code_in_space(data, set, space);
error:
        isl_set_free(set);
        isl_space_free(space);
        return -1;
}

/* Generate an AST that visits the elements in the range of "executed"
 * in the relative order specified by the corresponding domain element(s).
 *
 * "build" is an isl_ast_build that has either been constructed by
 * isl_ast_build_from_context or passed to a callback set by
 * isl_ast_build_set_create_leaf.
 * In the first case, the space of the isl_ast_build is typically
 * a parametric space, although this is currently not enforced.
 * In the second case, the space is never a parametric space.
 * If the space S is not parametric, then the domain space(s) of "executed"
 * need to be wrapped relations with S as domain.
 *
 * If the domain of "executed" consists of several spaces, then an AST
 * is generated for each of them (in arbitrary order) and the results
 * are concatenated.
 *
 * If "internal" is set, then the domain "S" above refers to the internal
 * schedule domain representation.  Otherwise, it refers to the external
 * representation, as returned by isl_ast_build_get_schedule_space.
 *
 * We essentially run over all the spaces in the domain of "executed"
 * and call generate_code_set on each of them.
 */
static __isl_give isl_ast_graft_list *generate_code(
        __isl_take isl_union_map *executed, __isl_take isl_ast_build *build,
        int internal)
{
        isl_ctx *ctx;
        struct isl_generate_code_data data = { 0 };
        isl_space *space;
        isl_union_set *schedule_domain;
        isl_union_map *universe;

        if (!build)
                goto error;
        space = isl_ast_build_get_space(build, 1);
        space = isl_space_align_params(space,
                                    isl_union_map_get_space(executed));
        space = isl_space_align_params(space,
                                    isl_union_map_get_space(build->options));
        build = isl_ast_build_align_params(build, isl_space_copy(space));
        executed = isl_union_map_align_params(executed, space);
        if (!executed || !build)
                goto error;

        ctx = isl_ast_build_get_ctx(build);

        data.internal = internal;
        data.executed = executed;
        data.build = build;
        data.list = isl_ast_graft_list_alloc(ctx, 0);

        universe = isl_union_map_universe(isl_union_map_copy(executed));
        schedule_domain = isl_union_map_domain(universe);
        if (isl_union_set_foreach_set(schedule_domain, &generate_code_set,
                                        &data) < 0)
                data.list = isl_ast_graft_list_free(data.list);

        isl_union_set_free(schedule_domain);
        isl_union_map_free(executed);

        isl_ast_build_free(build);
        return data.list;
error:
        isl_union_map_free(executed);
        isl_ast_build_free(build);
        return NULL;
}

/* Generate an AST that visits the elements in the domain of "schedule"
 * in the relative order specified by the corresponding image element(s).
 *
 * "build" is an isl_ast_build that has either been constructed by
 * isl_ast_build_from_context or passed to a callback set by
 * isl_ast_build_set_create_leaf.
 * In the first case, the space of the isl_ast_build is typically
 * a parametric space, although this is currently not enforced.
 * In the second case, the space is never a parametric space.
 * If the space S is not parametric, then the range space(s) of "schedule"
 * need to be wrapped relations with S as domain.
 *
 * If the range of "schedule" consists of several spaces, then an AST
 * is generated for each of them (in arbitrary order) and the results
 * are concatenated.
 *
 * We first initialize the local copies of the relevant options.
 * We do this here rather than when the isl_ast_build is created
 * because the options may have changed between the construction
 * of the isl_ast_build and the call to isl_generate_code.
 *
 * The main computation is performed on an inverse schedule (with
 * the schedule domain in the domain and the elements to be executed
 * in the range) called "executed".
 */
__isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
        __isl_keep isl_ast_build *build, __isl_take isl_union_map *schedule)
{
        isl_ast_graft_list *list;
        isl_ast_node *node;
        isl_union_map *executed;

        build = isl_ast_build_copy(build);
        build = isl_ast_build_set_single_valued(build, 0);
        executed = isl_union_map_reverse(schedule);
        list = generate_code(executed, isl_ast_build_copy(build), 0);
        node = isl_ast_node_from_graft_list(list, build);
        isl_ast_build_free(build);

        return node;
}