llvm/polly/lib/External/isl/isl_ast_codegen.c

/*
 * Copyright 2012-2014 Ecole Normale Superieure
 * Copyright 2014      INRIA Rocquencourt
 *
 * Use of this software is governed by the MIT license
 *
 * Written by Sven Verdoolaege,
 * Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
 * B.P. 105 - 78153 Le Chesnay, France
 */

#include <limits.h>
#include <isl/id.h>
#include <isl/val.h>
#include <isl/space.h>
#include <isl/aff.h>
#include <isl/constraint.h>
#include <isl/set.h>
#include <isl/ilp.h>
#include <isl/union_set.h>
#include <isl/union_map.h>
#include <isl/schedule_node.h>
#include <isl/options.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>

/* Try and reduce the number of disjuncts in the representation of "set",
 * without dropping explicit representations of local variables.
 */
static __isl_give isl_set *isl_set_coalesce_preserve(__isl_take isl_set *set)
{ … }

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

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 isl_stat generate_non_single_valued(__isl_take isl_map *executed,
	struct isl_generate_domain_data *data)
{ … }

/* 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)
{ … }

/* Generate a call expression for the single executed
 * domain element "map" and put a guard around it based its (simplified)
 * domain.  "executed" is the original inverse schedule from which "map"
 * has been derived.  In particular, "map" is either identical to "executed"
 * or it is the result of gisting "executed" with respect to the build domain.
 * "executed" is only used if there is an at_each_domain callback.
 *
 * At this stage, any pending constraints in the build can no longer
 * be simplified with respect to any enforced constraints since
 * the call node does not have any enforced constraints.
 * Since all pending constraints not covered by any enforced constraints
 * will be added as a guard to the graft in create_node_scaled,
 * even in the eliminated case, the pending constraints
 * can be considered to have been generated by outer constructs.
 *
 * If the user has set an at_each_domain callback, it is called
 * on the constructed call expression node.
 */
static isl_stat add_domain(__isl_take isl_map *executed,
	__isl_take isl_map *map, struct isl_generate_domain_data *data)
{ … }

/* Generate an AST for a single domain based on
 * the inverse schedule "executed" and add it to data->list.
 *
 * 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
 * some of the constraints of data->build->domain.  In particular,
 * we intersect with a single-disjunct approximation of this set.
 * We perform this approximation to avoid further splitting up
 * the executed relation, possibly introducing a disjunctive guard
 * on the statement.
 *
 * 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 in cases where the original
 * map is not obviously single-valued.
 * If the original map is obviously single-valued, then the gist
 * operation is skipped.
 *
 * 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, call add_domain to generate a call expression (with guard) and
 * to call the at_each_domain callback, if any.
 */
static isl_stat generate_domain(__isl_take isl_map *executed, void *user)
{ … }

/* 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.
 *
 * Since the node itself cannot enforce any constraints, we turn
 * all pending constraints into guards and add them to the resulting
 * graft to ensure that they will be generated.
 *
 * 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)
{ … }

static __isl_give isl_ast_graft_list *build_ast_from_child(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed);

/* Generate an AST after having handled the complete schedule
 * of this call to the code generator or the complete band
 * if we are generating an AST from a schedule tree.
 *
 * If we are inside a band node, then move on to the child of the band.
 *
 * 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)
{ … }

/* 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)
{ … }

/* Call the after_each_for callback, if requested by the user.
 */
static __isl_give isl_ast_graft *after_each_for(__isl_take isl_ast_graft *graft,
	__isl_keep isl_ast_build *build)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* Callback for sorting the isl_pw_aff_list passed to reduce_list and
 * remove_redundant_lower_bounds.
 */
static int reduce_list_cmp(__isl_keep isl_pw_aff *a, __isl_keep isl_pw_aff *b,
	void *user)
{ … }

/* Given a list of lower bounds "list", remove those that are redundant
 * with respect to the other bounds in "list" and the domain of "build".
 *
 * We first sort the bounds in the same way as they would be sorted
 * by set_for_node_expressions so that we can try and remove the last
 * bounds first.
 *
 * For a lower bound to be effective, there needs to be at least
 * one domain element for which it is larger than all other lower bounds.
 * For each lower bound we therefore intersect the domain with
 * the conditions that it is larger than all other bounds and
 * check whether the result is empty.  If so, the bound can be removed.
 */
static __isl_give isl_pw_aff_list *remove_redundant_lower_bounds(
	__isl_take isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
{ … }

/* 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.
 *
 * If the current dimension is strided, then the lower bound
 * is adjusted by lower_bound to match the stride information.
 * This modification may make one or more lower bounds redundant
 * with respect to the other lower bounds.  We therefore check
 * for this condition and remove the redundant lower bounds.
 */
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)
{ … }

/* 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)
{ … }

/* 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.
 * If the list contains more than one element, then we sort it
 * using a fairly arbitrary but hopefully reasonably stable order.
 */
static __isl_give isl_ast_expr *reduce_list(enum isl_ast_expr_op_type type,
	__isl_keep isl_pw_aff_list *list, __isl_keep isl_ast_build *build)
{ … }

/* Add guards implied by the "generated constraints",
 * but not (necessarily) enforced by the generated AST to "guard".
 * In particular, if there is any stride constraints,
 * then add the guard implied by those constraints.
 * If we have generated a degenerate loop, then add the guard
 * implied by "bounds" on the outer dimensions, i.e., the guard
 * that ensures that the single value actually exists.
 * Since there may also be guards implied by a combination
 * of these constraints, we first combine them before
 * deriving the implied constraints.
 */
static __isl_give isl_set *add_implied_guards(__isl_take isl_set *guard,
	int degenerate, __isl_keep isl_basic_set *bounds,
	__isl_keep isl_ast_build *build)
{ … }

/* Update "graft" based on "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 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 they are known
 * to be "1" and "iterator <= value" respectively.
 */
static __isl_give isl_ast_graft *refine_degenerate(
	__isl_take isl_ast_graft *graft, __isl_keep isl_ast_build *build,
	__isl_keep isl_ast_build *sub_build)
{ … }

/* Return the intersection of constraints in "list" as a set.
 */
static __isl_give isl_set *intersect_constraints(
	__isl_keep isl_constraint_list *list)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* Does "aff" have a negative constant term?
 */
static isl_bool aff_constant_is_negative(__isl_keep isl_set *set,
	__isl_keep isl_aff *aff, void *user)
{ … }

/* Does "pa" have a negative constant term over its entire domain?
 */
static isl_bool pw_aff_constant_is_negative(__isl_keep isl_pw_aff *pa,
	void *user)
{ … }

/* Does each element in "list" have a negative constant term?
 */
static int list_constant_is_negative(__isl_keep isl_pw_aff_list *list)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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
 * - 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)
{ … }

/* 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)
{ … }

/* 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 { … };

/* Increment data->n_indep, data->lower or data->upper depending
 * on whether "c" is independent of dimensions data->pos,
 * a lower bound or an upper bound.
 */
static isl_stat count_constraints(__isl_take isl_constraint *c, void *user)
{ … }

/* 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.  Only the constraints that involve the iterator
 * are relevant here.  The other constraints are taken care of by
 * the caller and are included in the generated constraints of "build".
 * "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.
 */
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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* If the ast_build_exploit_nested_bounds option is set, then return
 * the constraints enforced by all elements in "list".
 * Otherwise, return the universe.
 */
static __isl_give isl_basic_set *extract_shared_enforced(
	__isl_keep isl_ast_graft_list *list, __isl_keep isl_ast_build *build)
{ … }

/* Return the pending constraints of "build" that are not already taken
 * care of (by a combination of "enforced" and the generated constraints
 * of "build").
 */
static __isl_give isl_set *extract_pending(__isl_keep isl_ast_build *build,
	__isl_keep isl_basic_set *enforced)
{ … }

/* 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 adds the stride information and
 * 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 levels 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).
 * Before creating a graft for the current level, we first extract
 * hoistable constraints from the child guards and combine them
 * with the pending constraints in the build.  These constraints
 * are used to simplify the child guards and then added to the guard
 * of the current graft to ensure that they will be generated.
 * If the hoisted guard is a disjunction, then we use it directly
 * to gist the guards on the children before intersect it with the
 * pending constraints.  We do so because this disjunction is typically
 * identical to the guards on the children such that these guards
 * can be effectively removed completely.  After the intersection,
 * the gist operation would have a harder time figuring this out.
 *
 * Finally, we set the bounds of the for loop in either
 * refine_degenerate or refine_generic.
 * We do so in a context where the pending constraints of the build
 * have been replaced by the guard of the current graft.
 */
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)
{ … }

/* 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 { … };

/* 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 isl_stat constraint_check_scaled(__isl_take isl_constraint *c,
	void *user)
{ … }

/* 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 isl_stat basic_map_check_scaled(__isl_take isl_basic_map *bmap,
	void *user)
{ … }

/* 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 isl_stat map_check_scaled(__isl_take isl_map *map, void *user)
{ … }

/* 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 (obviously)
 * 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)
{ … }

/* Add the basic set to the list that "user" points to.
 */
static isl_stat collect_basic_set(__isl_take isl_basic_set *bset, void *user)
{ … }

/* 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)
{ … }

/* Generate code for the schedule domain "bounds"
 * and add the result to "list".
 *
 * We mainly detect strides here and check if the bounds do not
 * conflict with the current build domain
 * 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)
{ … }

/* 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 isl_bool domain_follows_at_depth(__isl_keep isl_basic_set *i,
	__isl_keep isl_basic_set *j, void *user)
{ … }

/* 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)
{ … }

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 { … };

/* 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".
 *
 * While the elements in "scc" should be disjoint, we double-check
 * this property to avoid running into an infinite recursion in case
 * they intersect due to some internal error.
 */
static isl_stat add_nodes(__isl_take isl_basic_set_list *scc, void *user)
{ … }

/* 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)
{ … }

/* Do i and j share any values for the outer dimensions?
 */
static isl_bool shared_outer(__isl_keep isl_basic_set *i,
	__isl_keep isl_basic_set *j, void *user)
{ … }

/* 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 { … };

/* 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 isl_stat generate_sorted_domains_wrap(__isl_take isl_basic_set_list *scc,
	void *user)
{ … }

/* 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)
{ … }

/* 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 { … };

/* 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)
{ … }

/* 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)
{ … }

/* 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 isl_stat separate_domain(__isl_take isl_map *map, void *user)
{ … }

/* 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)
{ … }

/* Temporary data used during the search for a lower bound for unrolling.
 *
 * "build" is the build in which the unrolling will be performed
 * "domain" is the original set for which to find a lower bound
 * "depth" is the dimension for which to find a lower boudn
 * "expansion" is the expansion that needs to be applied to "domain"
 * in the unrolling that will be performed
 *
 * "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.
 * "n_div" is the maximal number of integer divisions in the first
 * unrolled iteration (after expansion).  It is set to -1 if it hasn't
 * been computed yet.
 */
struct isl_find_unroll_data { … };

/* Return the constraint
 *
 *	i_"depth" = aff + offset
 */
static __isl_give isl_constraint *at_offset(int depth, __isl_keep isl_aff *aff,
	int offset)
{ … }

/* Update *user to the number of integer divisions in the first element
 * of "ma", if it is larger than the current value.
 */
static isl_stat update_n_div(__isl_take isl_set *set,
	__isl_take isl_multi_aff *ma, void *user)
{ … }

/* Get the number of integer divisions in the expression for the iterator
 * value at the first slice in the unrolling based on lower bound "lower",
 * taking into account the expansion that needs to be performed on this slice.
 */
static int get_expanded_n_div(struct isl_find_unroll_data *data,
	__isl_keep isl_aff *lower)
{ … }

/* Is the lower bound "lower" with corresponding iteration count "n"
 * better than the one stored in "data"?
 * If there is no upper bound on the iteration count ("n" is infinity) or
 * if the count is too large, then we cannot use this lower bound.
 * Otherwise, if there was no previous lower bound or
 * if the iteration count of the new lower bound is smaller than
 * the iteration count of the previous lower bound, then we consider
 * the new lower bound to be better.
 * If the iteration count is the same, then compare the number
 * of integer divisions that would be needed to express
 * the iterator value at the first slice in the unrolling
 * according to the lower bound.  If we end up computing this
 * number, then store the lowest value in data->n_div.
 */
static int is_better_lower_bound(struct isl_find_unroll_data *data,
	__isl_keep isl_aff *lower, __isl_keep isl_val *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 or fewer integer divisions)
 * than the previously found lower bounds.
 */
static isl_stat update_unrolling_lower_bound(struct isl_find_unroll_data *data,
	__isl_keep isl_constraint *c)
{ … }

/* Check if we can use "c" as a lower bound and if it is better than
 * any previously found lower bound.
 */
static isl_stat constraint_find_unroll(__isl_take isl_constraint *c, void *user)
{ … }

/* 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.
 * "build" is the build in which the unrolling will be performed.
 * "expansion" is the expansion that needs to be applied to "domain"
 * in the unrolling that will be performed.
 *
 * 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_ast_build *build, __isl_keep isl_set *domain,
	int depth, __isl_keep isl_basic_map *expansion, int *n)
{ … }

/* Call "fn" on each iteration of the current dimension of "domain".
 * If "init" is not NULL, then it is called with the number of
 * iterations before any call to "fn".
 * Return -1 on failure.
 *
 * 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 call "fn".
 */
static int foreach_iteration(__isl_take isl_set *domain,
	__isl_keep isl_ast_build *build, int (*init)(int n, void *user),
	int (*fn)(__isl_take isl_basic_set *bset, void *user), void *user)
{ … }

/* 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" contains 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.
 */
struct isl_codegen_domains { … };

/* Internal data structure for do_unroll.
 *
 * "domains" stores the results of compute_domains.
 * "class_domain" is the original class domain passed to do_unroll.
 * "unroll_domain" collects the unrolled iterations.
 */
struct isl_ast_unroll_data { … };

/* Given an iteration of an unrolled domain represented by "bset",
 * add it to data->domains->list.
 * Since we may have dropped some constraints, we intersect with
 * the class domain again to ensure that each element in the list
 * is disjoint from the other class domains.
 */
static int do_unroll_iteration(__isl_take isl_basic_set *bset, void *user)
{ … }

/* Extend domains->list with a list of basic sets, one for each value
 * of the current dimension in "domain" and remove the corresponding
 * sets from the class domain.  Return the updated class domain.
 * The divs that involve the current dimension have not been projected out
 * from this domain.
 *
 * We call foreach_iteration to iterate over the individual values and
 * in do_unroll_iteration we collect the individual basic sets in
 * domains->list and their union in data->unroll_domain, which is then
 * used to update the class domain.
 */
static __isl_give isl_set *do_unroll(struct isl_codegen_domains *domains,
	__isl_take isl_set *domain, __isl_take isl_set *class_domain)
{ … }

/* 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.
 * Remove the corresponding sets from the class domain and
 * return the updated class 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 __isl_give isl_set *compute_unroll_domains(
	struct isl_codegen_domains *domains, __isl_take isl_set *class_domain)
{ … }

/* Try and 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(s) to domains->list and remove them
 * from class_domain.  Return the updated class domain.
 *
 * 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.
 * We do however need to make sure that this single domains is a subset
 * of the class domain so that it would not intersect with any other
 * class domains.  This means that we may end up splitting up the atomic
 * domain in case separation classes are being used.
 *
 * "domain" is the intersection of the schedule domain and the class domain,
 * with inner dimensions projected out.
 */
static __isl_give isl_set *compute_atomic_domain(
	struct isl_codegen_domains *domains, __isl_take isl_set *class_domain)
{ … }

/* 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)
{ … }

/* 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 removed from class_domain by
 * compute_atomic_domain.  It is computed first so that the extended domain
 * would not overlap with any domains computed before.
 * Similary, the unrolled domains may have some constraints removed and
 * may therefore also be larger than specified by the user.
 *
 * 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 isl_stat compute_partial_domains(struct isl_codegen_domains *domains,
	__isl_take isl_set *class_domain)
{ … }

/* 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 isl_stat compute_class_domains(__isl_take isl_point *pnt, void *user)
{ … }

/* 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[4],
	__isl_keep isl_ast_build *build)
{ … }

/* 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)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a union map.
 *
 * 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_flat(
	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree
 * and the separate option was specified.
 *
 * We perform separation on the domain of "executed" and then generate
 * an AST for each of the resulting disjoint basic sets.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_tree_separate(
	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{ … }

/* Internal data structure for generate_shifted_component_tree_unroll.
 *
 * "executed" and "build" are inputs to generate_shifted_component_tree_unroll.
 * "list" collects the constructs grafts.
 */
struct isl_ast_unroll_tree_data { … };

/* Initialize data->list to a list of "n" elements.
 */
static int init_unroll_tree(int n, void *user)
{ … }

/* Given an iteration of an unrolled domain represented by "bset",
 * generate the corresponding AST and add the result to data->list.
 */
static int do_unroll_tree_iteration(__isl_take isl_basic_set *bset, void *user)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree
 * and the unroll option was specified.
 *
 * We call foreach_iteration to iterate over the individual values and
 * construct and collect the corresponding grafts in do_unroll_tree_iteration.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_tree_unroll(
	__isl_take isl_union_map *executed, __isl_take isl_set *domain,
	__isl_take isl_ast_build *build)
{ … }

/* Does "domain" involve a disjunction that is purely based on
 * constraints involving only outer dimension?
 *
 * In particular, is there a disjunction such that the constraints
 * involving the current and later dimensions are the same over
 * all the disjuncts?
 */
static isl_bool has_pure_outer_disjunction(__isl_keep isl_set *domain,
	__isl_keep isl_ast_build *build)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree.
 * In particular, handle the base case where there is either no isolated
 * set or we are within the isolated set (in which case "isolated" is set)
 * or the iterations that precede or follow the isolated set.
 *
 * The schedule domain is broken up or combined into basic sets
 * according to the AST generation option specified in the current
 * schedule node, which may be either atomic, separate, unroll or
 * unspecified.  If the option is unspecified, then we currently simply
 * split the schedule domain into disjoint basic sets.
 *
 * In case the separate option is specified, the AST generation is
 * handled by generate_shifted_component_tree_separate.
 * In the other cases, we need the global schedule domain.
 * In the unroll case, the AST generation is then handled by
 * generate_shifted_component_tree_unroll which needs the actual
 * schedule domain (with divs that may refer to the current dimension)
 * so that stride detection can be performed.
 * In the atomic or unspecified case, inner dimensions and divs involving
 * the current dimensions should be eliminated.
 * The result is then either combined into a single basic set or
 * split up into disjoint basic sets.
 * Finally an AST is generated for each basic set and the results are
 * concatenated.
 *
 * If the schedule domain involves a disjunction that is purely based on
 * constraints involving only outer dimension, then it is treated as
 * if atomic was specified.  This ensures that only a single loop
 * is generated instead of a sequence of identical loops with
 * different guards.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_tree_base(
	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build,
	int isolated)
{ … }

/* Extract out the disjunction imposed by "domain" on the outer
 * schedule dimensions.
 *
 * In particular, remove all inner dimensions from "domain" (including
 * the current dimension) and then remove the constraints that are shared
 * by all disjuncts in the result.
 */
static __isl_give isl_set *extract_disjunction(__isl_take isl_set *domain,
	__isl_keep isl_ast_build *build)
{ … }

/* Add "guard" to the grafts in "list".
 * "build" is the outer AST build, while "sub_build" includes "guard"
 * in its generated domain.
 *
 * First combine the grafts into a single graft and then add the guard.
 * If the list is empty, or if some error occurred, then simply return
 * the list.
 */
static __isl_give isl_ast_graft_list *list_add_guard(
	__isl_take isl_ast_graft_list *list, __isl_keep isl_set *guard,
	__isl_keep isl_ast_build *build, __isl_keep isl_ast_build *sub_build)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree.
 * In particular, do so for the specified subset of the schedule domain.
 *
 * If we are outside of the isolated part, then "domain" may include
 * a disjunction.  Explicitly generate this disjunction at this point
 * instead of relying on the disjunction getting hoisted back up
 * to this level.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_tree_part(
	__isl_keep isl_union_map *executed, __isl_take isl_set *domain,
	__isl_keep isl_ast_build *build, int isolated)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree.
 * In particular, do so for the specified sequence of subsets
 * of the schedule domain, "before", "isolated", "after" and "other",
 * where only the "isolated" part is considered to be isolated.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_parts(
	__isl_take isl_union_map *executed, __isl_take isl_set *before,
	__isl_take isl_set *isolated, __isl_take isl_set *after,
	__isl_take isl_set *other, __isl_take isl_ast_build *build)
{ … }

/* Does "set" intersect "first", but not "second"?
 */
static isl_bool only_intersects_first(__isl_keep isl_set *set,
	__isl_keep isl_set *first, __isl_keep isl_set *second)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree.
 * In particular, do so in case of isolation where there is
 * only an "isolated" part and an "after" part.
 * "dead1" and "dead2" are freed by this function in order to simplify
 * the caller.
 *
 * The "before" and "other" parts are set to empty sets.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_only_after(
	__isl_take isl_union_map *executed, __isl_take isl_set *isolated,
	__isl_take isl_set *after, __isl_take isl_ast_build *build,
	__isl_take isl_set *dead1, __isl_take isl_set *dead2)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied, in case the schedule was specified as a schedule tree.
 *
 * We first check if the user has specified an isolated schedule domain
 * and that we are not already outside of this isolated schedule domain.
 * If so, we break up the schedule domain into iterations that
 * precede the isolated domain, the isolated domain itself,
 * the iterations that follow the isolated domain and
 * the remaining iterations (those that are incomparable
 * to the isolated domain).
 * We generate an AST for each piece and concatenate the results.
 *
 * If the isolated domain is not convex, then it is replaced
 * by a convex superset to ensure that the sets of preceding and
 * following iterations are properly defined and, in particular,
 * that there are no intermediate iterations that do not belong
 * to the isolated domain.
 *
 * In the special case where at least one element of the schedule
 * domain that does not belong to the isolated domain needs
 * to be scheduled after this isolated domain, but none of those
 * elements need to be scheduled before, break up the schedule domain
 * in only two parts, the isolated domain, and a part that will be
 * scheduled after the isolated domain.
 *
 * If no isolated set has been specified, then we generate an
 * AST for the entire inverse schedule.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component_tree(
	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{ … }

/* Generate code for a single component, after shifting (if any)
 * has been applied.
 *
 * Call generate_shifted_component_tree or generate_shifted_component_flat
 * depending on whether the schedule was specified as a schedule tree.
 */
static __isl_give isl_ast_graft_list *generate_shifted_component(
	__isl_take isl_union_map *executed, __isl_take isl_ast_build *build)
{ … }

struct isl_set_map_pair { … };

/* 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)
{ … }

/* 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)
{ … }

/* Does set dimension "pos" of "set" have an obviously fixed value?
 */
static int dim_is_fixed(__isl_keep isl_set *set, int pos)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* 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 *construct_shifted_executed(
	struct isl_set_map_pair *domain, int *order, int n,
	__isl_keep isl_val *stride, __isl_keep isl_multi_val *offset,
	__isl_keep isl_ast_build *build)
{ … }

/* 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
 * construct_shifted_executed that exposes the stride.
 * Since this involves the introduction of a new schedule dimension,
 * the build needs to be changed accordingly.
 * 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_keep isl_val *stride, __isl_keep isl_multi_val *offset,
	__isl_take isl_ast_build *build)
{ … }

/* Does any node in the schedule tree rooted at the current schedule node
 * of "build" depend on outer schedule nodes?
 */
static int has_anchored_subtree(__isl_keep isl_ast_build *build)
{ … }

/* 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, in case of a schedule map input,
 * 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.
 * In the case of a schedule tree input, we bail out if any of
 * the descendants of the current schedule node refer to outer
 * schedule nodes in any way.
 *
 *
 * 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 the gcd is zero, then the differences were all fixed, meaning
 * that some domains had non-obviously fixed values for the current dimension.
 * 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)
{ … }

/* Store both "map" itself and its domain in the
 * structure pointed to by *next and advance to the next array element.
 */
static isl_stat extract_domain(__isl_take isl_map *map, void *user)
{ … }

static isl_bool after_in_tree(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node);

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the child of "node"?
 */
static isl_bool after_in_child(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the band node "node"?
 *
 * We first check if any domain element is scheduled after any
 * of the corresponding image elements by the band node itself.
 * If not, we restrict "map" to those pairs of element that
 * are scheduled together by the band node and continue with
 * the child of the band node.
 * If there are no such pairs then the map passed to after_in_child
 * will be empty causing it to return 0.
 */
static isl_bool after_in_band(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the context node "node"?
 *
 * The context constraints apply to the schedule domain,
 * so we cannot apply them directly to "umap", which contains
 * pairs of statement instances.  Instead, we add them
 * to the range of the prefix schedule for both domain and
 * range of "umap".
 */
static isl_bool after_in_context(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the expansion node "node"?
 *
 * We apply the expansion to domain and range of "umap" and
 * continue with its child.
 */
static isl_bool after_in_expansion(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the extension node "node"?
 *
 * Since the extension node may add statement instances before or
 * after the pairs of statement instances in "umap", we return isl_bool_true
 * to ensure that these pairs are not broken up.
 */
static isl_bool after_in_extension(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the filter node "node"?
 *
 * We intersect domain and range of "umap" with the filter and
 * continue with its child.
 */
static isl_bool after_in_filter(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the set node "node"?
 *
 * This is only the case if this condition holds in any
 * of the (filter) children of the set node.
 * In particular, if the domain and the range of "umap"
 * are contained in different children, then the condition
 * does not hold.
 */
static isl_bool after_in_set(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Return the filter of child "i" of "node".
 */
static __isl_give isl_union_set *child_filter(
	__isl_keep isl_schedule_node *node, int i)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at
 * the sequence node "node"?
 *
 * This happens in particular if any domain element is
 * contained in a later child than one containing a range element or
 * if the condition holds within a given child in the sequence.
 * The later part of the condition is checked by after_in_set.
 */
static isl_bool after_in_sequence(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "umap" scheduled after any of
 * the corresponding image elements by the tree rooted at "node"?
 *
 * If "umap" is empty, then clearly there is no such element.
 * Otherwise, consider the different types of nodes separately.
 */
static isl_bool after_in_tree(__isl_keep isl_union_map *umap,
	__isl_keep isl_schedule_node *node)
{ … }

/* Is any domain element of "map1" scheduled after any domain
 * element of "map2" by the subtree underneath the current band node,
 * while at the same time being scheduled together by the current
 * band node, i.e., by "map1" and "map2?
 *
 * If the child of the current band node is a leaf, then
 * no element can be scheduled after any other element.
 *
 * Otherwise, we construct a relation between domain elements
 * of "map1" and domain elements of "map2" that are scheduled
 * together and then check if the subtree underneath the current
 * band node determines their relative order.
 */
static isl_bool after_in_subtree(__isl_keep isl_ast_build *build,
	__isl_keep isl_map *map1, __isl_keep isl_map *map2)
{ … }

/* Internal data for any_scheduled_after.
 *
 * "build" is the build in which the AST is constructed.
 * "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 { … };

/* 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 we are inside a band of a schedule tree and there is a pair
 * of elements in the two domains that is schedule together by
 * the current band, then we check if any element of "i" may be schedule
 * after element of "j" by the descendants of the band node.
 *
 * 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 isl_bool any_scheduled_after(int i, int j, void *user)
{ … }

/* 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)
{ … }

/* 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)
{ … }

/* Internal data structure used by isl_ast_build_node_from_schedule_map.
 * internal, executed and build are the inputs to generate_code.
 * list collects the output.
 */
struct isl_generate_code_data { … };

/* 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)
{ … }

/* 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.
 * "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_node_from_schedule_map
 * 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.
 *
 * If the build space is parametric, then we add some of the parameter
 * constraints to the executed relation.  Adding these constraints
 * allows for an earlier detection of conflicts in some cases.
 * However, we do not want to divide the executed relation into
 * more disjuncts than necessary.  We therefore approximate
 * the constraints on the parameters by a single disjunct set.
 *
 * 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 isl_stat generate_code_in_space(struct isl_generate_code_data *data,
	__isl_take isl_set *set, __isl_take isl_space *space)
{ … }

/* 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 isl_stat generate_code_set(__isl_take isl_set *set, void *user)
{ … }

/* 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)
{ … }

/* 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_node_from_schedule_map(
	__isl_keep isl_ast_build *build, __isl_take isl_union_map *schedule)
{ … }

/* The old name for isl_ast_build_node_from_schedule_map.
 * It is being kept for backward compatibility, but
 * it will be removed in the future.
 */
__isl_give isl_ast_node *isl_ast_build_ast_from_schedule(
	__isl_keep isl_ast_build *build, __isl_take isl_union_map *schedule)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the leaf node "node".
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * Simply pass control to generate_inner_level.
 * Note that the current build does not refer to any band node, so
 * that generate_inner_level will not try to visit the child of
 * the leaf node.
 *
 * If multiple statement instances reach a leaf,
 * then they can be executed in any order.
 * Group the list of grafts based on shared guards
 * such that identical guards are only generated once
 * when the list is eventually passed on to isl_ast_graft_list_fuse.
 */
static __isl_give isl_ast_graft_list *build_ast_from_leaf(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Check that the band partial schedule "partial" does not filter out
 * any statement instances, as specified by the range of "executed".
 */
static isl_stat check_band_schedule_total_on_instances(
	__isl_keep isl_multi_union_pw_aff *partial,
	__isl_keep isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the band node "node" and its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * If the band is empty, we continue with its descendants.
 * Otherwise, we extend the build and the inverse schedule with
 * the additional space/partial schedule and continue generating
 * an AST in generate_next_level.
 * As soon as we have extended the inverse schedule with the additional
 * partial schedule, we look for equalities that may exists between
 * the old and the new part.
 */
static __isl_give isl_ast_graft_list *build_ast_from_band(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Hoist a list of grafts (in practice containing a single graft)
 * from "sub_build" (which includes extra context information)
 * to "build".
 *
 * In particular, project out all additional parameters introduced
 * by the context node from the enforced constraints and the guard
 * of the single graft.
 */
static __isl_give isl_ast_graft_list *hoist_out_of_context(
	__isl_take isl_ast_graft_list *list, __isl_keep isl_ast_build *build,
	__isl_keep isl_ast_build *sub_build)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the context node "node"
 * and its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * The context node may introduce additional parameters as well as
 * constraints on the outer schedule dimensions or original parameters.
 *
 * We add the extra parameters to a new build and the context
 * constraints to both the build and (as a single disjunct)
 * to the domain of "executed".  Since the context constraints
 * are specified in terms of the input schedule, we first need
 * to map them to the internal schedule domain.
 *
 * After constructing the AST from the descendants of "node",
 * we combine the list of grafts into a single graft within
 * the new build, in order to be able to exploit the additional
 * context constraints during this combination.
 *
 * Additionally, if the current node is the outermost node in
 * the schedule tree (apart from the root domain node), we generate
 * all pending guards, again to be able to exploit the additional
 * context constraints.  We currently do not do this for internal
 * context nodes since we may still want to hoist conditions
 * to outer AST nodes.
 *
 * If the context node introduced any new parameters, then they
 * are removed from the set of enforced constraints and guard
 * in hoist_out_of_context.
 */
static __isl_give isl_ast_graft_list *build_ast_from_context(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the expansion node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * We expand the domain elements by the expansion and
 * continue with the descendants of the node.
 */
static __isl_give isl_ast_graft_list *build_ast_from_expansion(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the extension node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * Extend the inverse schedule with the extension applied to current
 * set of generated constraints.  Since the extension if formulated
 * in terms of the input schedule, it first needs to be transformed
 * to refer to the internal schedule.
 */
static __isl_give isl_ast_graft_list *build_ast_from_extension(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the filter node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * We simply intersect the iteration domain (i.e., the range of "executed")
 * with the filter and continue with the descendants of the node,
 * unless the resulting inverse schedule is empty, in which
 * case we return an empty list.
 *
 * If the result of the intersection is equal to the original "executed"
 * relation, then keep the original representation since the intersection
 * may have unnecessarily broken up the relation into a greater number
 * of disjuncts.
 */
static __isl_give isl_ast_graft_list *build_ast_from_filter(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the guard node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * Ensure that the associated guard is enforced by the outer AST
 * constructs by adding it to the guard of the graft.
 * Since we know that we will enforce the guard, we can also include it
 * in the generated constraints used to construct an AST for
 * the descendant nodes.
 */
static __isl_give isl_ast_graft_list *build_ast_from_guard(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Call the before_each_mark callback, if requested by the user.
 *
 * Return 0 on success and -1 on error.
 *
 * The caller is responsible for recording the current inverse schedule
 * in "build".
 */
static isl_stat before_each_mark(__isl_keep isl_id *mark,
	__isl_keep isl_ast_build *build)
{ … }

/* Call the after_each_mark callback, if requested by the user.
 *
 * The caller is responsible for recording the current inverse schedule
 * in "build".
 */
static __isl_give isl_ast_graft *after_each_mark(
	__isl_take isl_ast_graft *graft, __isl_keep isl_ast_build *build)
{ … }


/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the mark node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.

 * Since we may be calling before_each_mark and after_each_mark
 * callbacks, we record the current inverse schedule in the build.
 *
 * We generate an AST for the child of the mark node, combine
 * the graft list into a single graft and then insert the mark
 * in the AST of that single graft.
 */
static __isl_give isl_ast_graft_list *build_ast_from_mark(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

static __isl_give isl_ast_graft_list *build_ast_from_schedule_node(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed);

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the sequence (or set) node "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * We simply generate an AST for each of the children and concatenate
 * the results.
 */
static __isl_give isl_ast_graft_list *build_ast_from_sequence(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the node "node" and its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * The node types are handled in separate functions.
 * Set nodes are currently treated in the same way as sequence nodes.
 * The children of a set node may be executed in any order,
 * including the order of the children.
 */
static __isl_give isl_ast_graft_list *build_ast_from_schedule_node(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of "executed"
 * in the relative order specified by the (single) child of "node" and
 * its descendants.
 *
 * The relation "executed" maps the outer generated loop iterators
 * to the domain elements executed by those iterations.
 *
 * This function is never called on a leaf, set or sequence node,
 * so the node always has exactly one child.
 */
static __isl_give isl_ast_graft_list *build_ast_from_child(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node,
	__isl_take isl_union_map *executed)
{ … }

/* Generate an AST that visits the elements in the domain of the domain
 * node "node" in the relative order specified by its descendants.
 *
 * An initial inverse schedule is created that maps a zero-dimensional
 * schedule space to the node domain.
 * The input "build" is assumed to have a parametric domain and
 * is replaced by the same zero-dimensional schedule space.
 *
 * We also add some of the parameter constraints in the build domain
 * to the executed relation.  Adding these constraints
 * allows for an earlier detection of conflicts in some cases.
 * However, we do not want to divide the executed relation into
 * more disjuncts than necessary.  We therefore approximate
 * the constraints on the parameters by a single disjunct set.
 */
static __isl_give isl_ast_node *build_ast_from_domain(
	__isl_take isl_ast_build *build, __isl_take isl_schedule_node *node)
{ … }

/* Generate an AST that visits the elements in the domain of "schedule"
 * in the relative order specified by the schedule tree.
 *
 * "build" is an isl_ast_build that has been created using
 * isl_ast_build_alloc or isl_ast_build_from_context based
 * on a parametric set.
 *
 * The construction starts at the root node of the schedule,
 * which is assumed to be a domain node.
 */
__isl_give isl_ast_node *isl_ast_build_node_from_schedule(
	__isl_keep isl_ast_build *build, __isl_take isl_schedule *schedule)
{ … }