X-Git-Url: http://review.tizen.org/git/?a=blobdiff_plain;f=isl_aff.c;h=d7f1e73f0859d688263dc3b7c2e763c5dc35403e;hb=79e5f558e5f9bb9a559da4eef8d181dcf053f60d;hp=4ebf7b3e0670124e7c9c66a466b52c3fb24eec1b;hpb=9043b59e4de5587c954c865b183fe5339064a19f;p=platform%2Fupstream%2Fisl.git

diff --git a/isl_aff.c b/isl_aff.c
index 4ebf7b3..d7f1e73 100644
--- a/isl_aff.c
+++ b/isl_aff.c
@@ -1,16 +1,20 @@
 /*
  * Copyright 2011      INRIA Saclay
  * Copyright 2011      Sven Verdoolaege
+ * Copyright 2012      Ecole Normale Superieure
  *
  * Use of this software is governed by the GNU LGPLv2.1 license
  *
  * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
  * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
  * 91893 Orsay, France
+ * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
  */
 
 #include <isl_ctx_private.h>
+#define ISL_DIM_H
 #include <isl_map_private.h>
+#include <isl_union_map_private.h>
 #include <isl_aff_private.h>
 #include <isl_space_private.h>
 #include <isl_local_space_private.h>
@@ -19,6 +23,7 @@
 #include <isl/constraint.h>
 #include <isl/seq.h>
 #include <isl/set.h>
+#include <isl_config.h>
 
 __isl_give isl_aff *isl_aff_alloc_vec(__isl_take isl_local_space *ls,
 	__isl_take isl_vec *v)
@@ -56,6 +61,10 @@ __isl_give isl_aff *isl_aff_alloc(__isl_take isl_local_space *ls)
 	if (!isl_local_space_divs_known(ls))
 		isl_die(ctx, isl_error_invalid, "local space has unknown divs",
 			goto error);
+	if (!isl_local_space_is_set(ls))
+		isl_die(ctx, isl_error_invalid,
+			"domain of affine expression should be a set",
+			goto error);
 
 	total = isl_local_space_dim(ls, isl_dim_all);
 	v = isl_vec_alloc(ctx, 1 + 1 + total);
@@ -65,7 +74,7 @@ error:
 	return NULL;
 }
 
-__isl_give isl_aff *isl_aff_zero(__isl_take isl_local_space *ls)
+__isl_give isl_aff *isl_aff_zero_on_domain(__isl_take isl_local_space *ls)
 {
 	isl_aff *aff;
 
@@ -129,28 +138,69 @@ isl_ctx *isl_aff_get_ctx(__isl_keep isl_aff *aff)
 	return aff ? isl_local_space_get_ctx(aff->ls) : NULL;
 }
 
+/* Externally, an isl_aff has a map space, but internally, the
+ * ls field corresponds to the domain of that space.
+ */
 int isl_aff_dim(__isl_keep isl_aff *aff, enum isl_dim_type type)
 {
-	return aff ? isl_local_space_dim(aff->ls, type) : 0;
+	if (!aff)
+		return 0;
+	if (type == isl_dim_out)
+		return 1;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_local_space_dim(aff->ls, type);
 }
 
-__isl_give isl_space *isl_aff_get_space(__isl_keep isl_aff *aff)
+__isl_give isl_space *isl_aff_get_domain_space(__isl_keep isl_aff *aff)
 {
 	return aff ? isl_local_space_get_space(aff->ls) : NULL;
 }
 
-__isl_give isl_local_space *isl_aff_get_local_space(__isl_keep isl_aff *aff)
+__isl_give isl_space *isl_aff_get_space(__isl_keep isl_aff *aff)
+{
+	isl_space *space;
+	if (!aff)
+		return NULL;
+	space = isl_local_space_get_space(aff->ls);
+	space = isl_space_from_domain(space);
+	space = isl_space_add_dims(space, isl_dim_out, 1);
+	return space;
+}
+
+__isl_give isl_local_space *isl_aff_get_domain_local_space(
+	__isl_keep isl_aff *aff)
 {
 	return aff ? isl_local_space_copy(aff->ls) : NULL;
 }
 
+__isl_give isl_local_space *isl_aff_get_local_space(__isl_keep isl_aff *aff)
+{
+	isl_local_space *ls;
+	if (!aff)
+		return NULL;
+	ls = isl_local_space_copy(aff->ls);
+	ls = isl_local_space_from_domain(ls);
+	ls = isl_local_space_add_dims(ls, isl_dim_out, 1);
+	return ls;
+}
+
+/* Externally, an isl_aff has a map space, but internally, the
+ * ls field corresponds to the domain of that space.
+ */
 const char *isl_aff_get_dim_name(__isl_keep isl_aff *aff,
 	enum isl_dim_type type, unsigned pos)
 {
-	return aff ? isl_local_space_get_dim_name(aff->ls, type, pos) : 0;
+	if (!aff)
+		return NULL;
+	if (type == isl_dim_out)
+		return NULL;
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	return isl_local_space_get_dim_name(aff->ls, type, pos);
 }
 
-__isl_give isl_aff *isl_aff_reset_space(__isl_take isl_aff *aff,
+__isl_give isl_aff *isl_aff_reset_domain_space(__isl_take isl_aff *aff,
 	__isl_take isl_space *dim)
 {
 	aff = isl_aff_cow(aff);
@@ -168,6 +218,17 @@ error:
 	return NULL;
 }
 
+/* Reset the space of "aff".  This function is called from isl_pw_templ.c
+ * and doesn't know if the space of an element object is represented
+ * directly or through its domain.  It therefore passes along both.
+ */
+__isl_give isl_aff *isl_aff_reset_space_and_domain(__isl_take isl_aff *aff,
+	__isl_take isl_space *space, __isl_take isl_space *domain)
+{
+	isl_space_free(space);
+	return isl_aff_reset_domain_space(aff, domain);
+}
+
 /* Reorder the coefficients of the affine expression based
  * on the given reodering.
  * The reordering r is assumed to have been extended with the local
@@ -198,9 +259,10 @@ error:
 	return NULL;
 }
 
-/* Reorder the dimensions of "aff" according to the given reordering.
+/* Reorder the dimensions of the domain of "aff" according
+ * to the given reordering.
  */
-__isl_give isl_aff *isl_aff_realign(__isl_take isl_aff *aff,
+__isl_give isl_aff *isl_aff_realign_domain(__isl_take isl_aff *aff,
 	__isl_take isl_reordering *r)
 {
 	aff = isl_aff_cow(aff);
@@ -222,6 +284,34 @@ error:
 	return NULL;
 }
 
+__isl_give isl_aff *isl_aff_align_params(__isl_take isl_aff *aff,
+	__isl_take isl_space *model)
+{
+	if (!aff || !model)
+		goto error;
+
+	if (!isl_space_match(aff->ls->dim, isl_dim_param,
+			     model, isl_dim_param)) {
+		isl_reordering *exp;
+
+		model = isl_space_drop_dims(model, isl_dim_in,
+					0, isl_space_dim(model, isl_dim_in));
+		model = isl_space_drop_dims(model, isl_dim_out,
+					0, isl_space_dim(model, isl_dim_out));
+		exp = isl_parameter_alignment_reordering(aff->ls->dim, model);
+		exp = isl_reordering_extend_space(exp,
+					isl_aff_get_domain_space(aff));
+		aff = isl_aff_realign_domain(aff, exp);
+	}
+
+	isl_space_free(model);
+	return aff;
+error:
+	isl_space_free(model);
+	isl_aff_free(aff);
+	return NULL;
+}
+
 int isl_aff_plain_is_zero(__isl_keep isl_aff *aff)
 {
 	if (!aff)
@@ -266,6 +356,13 @@ int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
 	if (!aff)
 		return -1;
 
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return -1);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
 	if (pos >= isl_local_space_dim(aff->ls, type))
 		isl_die(aff->v->ctx, isl_error_invalid,
 			"position out of bounds", return -1);
@@ -336,6 +433,43 @@ __isl_give isl_aff *isl_aff_add_constant_si(__isl_take isl_aff *aff, int v)
 	return aff;
 }
 
+/* Add "v" to the numerator of the constant term of "aff".
+ */
+__isl_give isl_aff *isl_aff_add_constant_num(__isl_take isl_aff *aff, isl_int v)
+{
+	if (isl_int_is_zero(v))
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_add(aff->v->el[1], aff->v->el[1], v);
+
+	return aff;
+}
+
+/* Add "v" to the numerator of the constant term of "aff".
+ */
+__isl_give isl_aff *isl_aff_add_constant_num_si(__isl_take isl_aff *aff, int v)
+{
+	isl_int t;
+
+	if (v == 0)
+		return aff;
+
+	isl_int_init(t);
+	isl_int_set_si(t, v);
+	aff = isl_aff_add_constant_num(aff, t);
+	isl_int_clear(t);
+
+	return aff;
+}
+
 __isl_give isl_aff *isl_aff_set_constant_si(__isl_take isl_aff *aff, int v)
 {
 	aff = isl_aff_cow(aff);
@@ -357,6 +491,13 @@ __isl_give isl_aff *isl_aff_set_coefficient(__isl_take isl_aff *aff,
 	if (!aff)
 		return NULL;
 
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
 	if (pos >= isl_local_space_dim(aff->ls, type))
 		isl_die(aff->v->ctx, isl_error_invalid,
 			"position out of bounds", return isl_aff_free(aff));
@@ -381,6 +522,13 @@ __isl_give isl_aff *isl_aff_set_coefficient_si(__isl_take isl_aff *aff,
 	if (!aff)
 		return NULL;
 
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
 	if (pos >= isl_local_space_dim(aff->ls, type))
 		isl_die(aff->v->ctx, isl_error_invalid,
 			"position out of bounds", return isl_aff_free(aff));
@@ -405,6 +553,13 @@ __isl_give isl_aff *isl_aff_add_coefficient(__isl_take isl_aff *aff,
 	if (!aff)
 		return NULL;
 
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
 	if (pos >= isl_local_space_dim(aff->ls, type))
 		isl_die(aff->v->ctx, isl_error_invalid,
 			"position out of bounds", return isl_aff_free(aff));
@@ -436,7 +591,7 @@ __isl_give isl_aff *isl_aff_add_coefficient_si(__isl_take isl_aff *aff,
 	return aff;
 }
 
-__isl_give isl_div *isl_aff_get_div(__isl_keep isl_aff *aff, int pos)
+__isl_give isl_aff *isl_aff_get_div(__isl_keep isl_aff *aff, int pos)
 {
 	if (!aff)
 		return NULL;
@@ -458,8 +613,55 @@ __isl_give isl_aff *isl_aff_neg(__isl_take isl_aff *aff)
 	return aff;
 }
 
+/* Remove divs from the local space that do not appear in the affine
+ * expression.
+ * We currently only remove divs at the end.
+ * Some intermediate divs may also not appear directly in the affine
+ * expression, but we would also need to check that no other divs are
+ * defined in terms of them.
+ */
+__isl_give isl_aff *isl_aff_remove_unused_divs( __isl_take isl_aff *aff)
+{
+	int pos;
+	int off;
+	int n;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+
+	pos = isl_seq_last_non_zero(aff->v->el + 1 + off, n) + 1;
+	if (pos == n)
+		return aff;
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_drop_dims(aff->ls, isl_dim_div, pos, n - pos);
+	aff->v = isl_vec_drop_els(aff->v, 1 + off + pos, n - pos);
+	if (!aff->ls || !aff->v)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+__isl_give isl_aff *isl_aff_normalize(__isl_take isl_aff *aff)
+{
+	if (!aff)
+		return NULL;
+	aff->v = isl_vec_normalize(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+	aff = isl_aff_remove_unused_divs(aff);
+	return aff;
+}
+
 /* Given f, return floor(f).
  * If f is an integer expression, then just return f.
+ * If f is a constant, then return the constant floor(f).
  * Otherwise, if f = g/m, write g = q m + r,
  * create a new div d = [r/m] and return the expression q + d.
  * The coefficients in r are taken to lie between -m/2 and m/2.
@@ -482,6 +684,15 @@ __isl_give isl_aff *isl_aff_floor(__isl_take isl_aff *aff)
 		return NULL;
 
 	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	if (isl_aff_is_cst(aff)) {
+		isl_int_fdiv_q(aff->v->el[1], aff->v->el[1], aff->v->el[0]);
+		isl_int_set_si(aff->v->el[0], 1);
+		return aff;
+	}
+
 	div = isl_vec_copy(aff->v);
 	div = isl_vec_cow(div);
 	if (!div)
@@ -728,6 +939,11 @@ __isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff, isl_int f)
 	aff = isl_aff_cow(aff);
 	if (!aff)
 		return NULL;
+
+	if (isl_int_is_zero(f))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"cannot scale down by zero", return isl_aff_free(aff));
+
 	aff->v = isl_vec_cow(aff->v);
 	if (!aff->v)
 		return isl_aff_free(aff);
@@ -764,6 +980,12 @@ __isl_give isl_aff *isl_aff_set_dim_name(__isl_take isl_aff *aff,
 	aff = isl_aff_cow(aff);
 	if (!aff)
 		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot set name of output/set dimension",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
 	aff->ls = isl_local_space_set_dim_name(aff->ls, type, pos, s);
 	if (!aff->ls)
 		return isl_aff_free(aff);
@@ -771,6 +993,29 @@ __isl_give isl_aff *isl_aff_set_dim_name(__isl_take isl_aff *aff,
 	return aff;
 }
 
+__isl_give isl_aff *isl_aff_set_dim_id(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, __isl_take isl_id *id)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return isl_id_free(id);
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot set name of output/set dimension",
+			goto error);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+	aff->ls = isl_local_space_set_dim_id(aff->ls, type, pos, id);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+error:
+	isl_id_free(id);
+	isl_aff_free(aff);
+	return NULL;
+}
+
 /* Exploit the equalities in "eq" to simplify the affine expression
  * and the expressions of the integer divisions in the local space.
  * The integer divisions in this local space are assumed to appear
@@ -811,6 +1056,7 @@ static __isl_give isl_aff *isl_aff_substitute_equalities_lifted(
 	}
 
 	isl_basic_set_free(eq);
+	aff = isl_aff_normalize(aff);
 	return aff;
 error:
 	isl_basic_set_free(eq);
@@ -854,9 +1100,10 @@ __isl_give isl_aff *isl_aff_gist(__isl_take isl_aff *aff,
 	n_div = isl_local_space_dim(aff->ls, isl_dim_div);
 	if (n_div > 0) {
 		isl_basic_set *bset;
+		isl_local_space *ls;
 		context = isl_set_add_dims(context, isl_dim_set, n_div);
-		bset = isl_basic_set_from_local_space(
-					    isl_aff_get_local_space(aff));
+		ls = isl_aff_get_domain_local_space(aff);
+		bset = isl_basic_set_from_local_space(ls);
 		bset = isl_basic_set_lift(bset);
 		bset = isl_basic_set_flatten(bset);
 		context = isl_set_intersect(context,
@@ -871,16 +1118,37 @@ error:
 	return NULL;
 }
 
+__isl_give isl_aff *isl_aff_gist_params(__isl_take isl_aff *aff,
+	__isl_take isl_set *context)
+{
+	isl_set *dom_context = isl_set_universe(isl_aff_get_domain_space(aff));
+	dom_context = isl_set_intersect_params(dom_context, context);
+	return isl_aff_gist(aff, dom_context);
+}
+
 /* Return a basic set containing those elements in the space
  * of aff where it is non-negative.
  */
 __isl_give isl_basic_set *isl_aff_nonneg_basic_set(__isl_take isl_aff *aff)
 {
 	isl_constraint *ineq;
+	isl_basic_set *bset;
 
 	ineq = isl_inequality_from_aff(aff);
 
-	return isl_basic_set_from_constraint(ineq);
+	bset = isl_basic_set_from_constraint(ineq);
+	bset = isl_basic_set_simplify(bset);
+	return bset;
+}
+
+/* Return a basic set containing those elements in the domain space
+ * of aff where it is negative.
+ */
+__isl_give isl_basic_set *isl_aff_neg_basic_set(__isl_take isl_aff *aff)
+{
+	aff = isl_aff_neg(aff);
+	aff = isl_aff_add_constant_num_si(aff, -1);
+	return isl_aff_nonneg_basic_set(aff);
 }
 
 /* Return a basic set containing those elements in the space
@@ -889,10 +1157,13 @@ __isl_give isl_basic_set *isl_aff_nonneg_basic_set(__isl_take isl_aff *aff)
 __isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff)
 {
 	isl_constraint *ineq;
+	isl_basic_set *bset;
 
 	ineq = isl_equality_from_aff(aff);
 
-	return isl_basic_set_from_constraint(ineq);
+	bset = isl_basic_set_from_constraint(ineq);
+	bset = isl_basic_set_simplify(bset);
+	return bset;
 }
 
 /* Return a basic set containing those elements in the shared space
@@ -931,34 +1202,6 @@ int isl_aff_is_empty(__isl_keep isl_aff *aff)
 	return 0;
 }
 
-/* Set active[i] to 1 if the dimension at position i is involved
- * in the affine expression.
- */
-static int set_active(__isl_keep isl_aff *aff, int *active)
-{
-	int i, j;
-	unsigned total;
-	unsigned offset;
-
-	if (!aff || !active)
-		return -1;
-
-	total = aff->v->size - 2;
-	for (i = 0; i < total; ++i)
-		active[i] = !isl_int_is_zero(aff->v->el[2 + i]);
-
-	offset = isl_local_space_offset(aff->ls, isl_dim_div) - 1;
-	for (i = aff->ls->div->n_row - 1; i >= 0; --i) {
-		if (!active[offset + i])
-			continue;
-		for (j = 0; j < total; ++j)
-			active[j] |=
-				!isl_int_is_zero(aff->ls->div->row[i][2 + j]);
-	}
-
-	return 0;
-}
-
 /* Check whether the given affine expression has non-zero coefficient
  * for any dimension in the given range or if any of these dimensions
  * appear with non-zero coefficients in any of the integer divisions
@@ -982,9 +1225,8 @@ int isl_aff_involves_dims(__isl_keep isl_aff *aff,
 		isl_die(ctx, isl_error_invalid,
 			"range out of bounds", return -1);
 
-	active = isl_calloc_array(ctx, int,
-				  isl_local_space_dim(aff->ls, isl_dim_all));
-	if (set_active(aff, active) < 0)
+	active = isl_local_space_get_active(aff->ls, aff->v->el + 2);
+	if (!active)
 		goto error;
 
 	first += isl_local_space_offset(aff->ls, type) - 1;
@@ -1009,11 +1251,17 @@ __isl_give isl_aff *isl_aff_drop_dims(__isl_take isl_aff *aff,
 
 	if (!aff)
 		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot drop output/set dimension",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
 	if (n == 0 && !isl_local_space_is_named_or_nested(aff->ls, type))
 		return aff;
 
 	ctx = isl_aff_get_ctx(aff);
-	if (first + n > isl_aff_dim(aff, type))
+	if (first + n > isl_local_space_dim(aff->ls, type))
 		isl_die(ctx, isl_error_invalid, "range out of bounds",
 			return isl_aff_free(aff));
 
@@ -1033,6 +1281,30 @@ __isl_give isl_aff *isl_aff_drop_dims(__isl_take isl_aff *aff,
 	return aff;
 }
 
+/* Project the domain of the affine expression onto its parameter space.
+ * The affine expression may not involve any of the domain dimensions.
+ */
+__isl_give isl_aff *isl_aff_project_domain_on_params(__isl_take isl_aff *aff)
+{
+	isl_space *space;
+	unsigned n;
+	int involves;
+
+	n = isl_aff_dim(aff, isl_dim_in);
+	involves = isl_aff_involves_dims(aff, isl_dim_in, 0, n);
+	if (involves < 0)
+		return isl_aff_free(aff);
+	if (involves)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+		    "affine expression involves some of the domain dimensions",
+		    return isl_aff_free(aff));
+	aff = isl_aff_drop_dims(aff, isl_dim_in, 0, n);
+	space = isl_aff_get_domain_space(aff);
+	space = isl_space_params(space);
+	aff = isl_aff_reset_domain_space(aff, space);
+	return aff;
+}
+
 __isl_give isl_aff *isl_aff_insert_dims(__isl_take isl_aff *aff,
 	enum isl_dim_type type, unsigned first, unsigned n)
 {
@@ -1040,11 +1312,17 @@ __isl_give isl_aff *isl_aff_insert_dims(__isl_take isl_aff *aff,
 
 	if (!aff)
 		return NULL;
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"cannot insert output/set dimensions",
+			return isl_aff_free(aff));
+	if (type == isl_dim_in)
+		type = isl_dim_set;
 	if (n == 0 && !isl_local_space_is_named_or_nested(aff->ls, type))
 		return aff;
 
 	ctx = isl_aff_get_ctx(aff);
-	if (first > isl_aff_dim(aff, type))
+	if (first > isl_local_space_dim(aff->ls, type))
 		isl_die(ctx, isl_error_invalid, "position out of bounds",
 			return isl_aff_free(aff));
 
@@ -1084,20 +1362,9 @@ __isl_give isl_pw_aff *isl_pw_aff_add_dims(__isl_take isl_pw_aff *pwaff,
 	return isl_pw_aff_insert_dims(pwaff, type, pos, n);
 }
 
-__isl_give isl_pw_aff *isl_pw_aff_set_tuple_id(__isl_take isl_pw_aff *pwaff,
-	__isl_take isl_id *id)
-{
-	isl_space *dim;
-
-	dim = isl_pw_aff_get_space(pwaff);
-	dim = isl_space_set_tuple_id(dim, isl_dim_set, id);
-
-	return isl_pw_aff_reset_space(pwaff, dim);
-}
-
 __isl_give isl_pw_aff *isl_pw_aff_from_aff(__isl_take isl_aff *aff)
 {
-	isl_set *dom = isl_set_universe(isl_aff_get_space(aff));
+	isl_set *dom = isl_set_universe(isl_aff_get_domain_space(aff));
 	return isl_pw_aff_alloc(dom, aff);
 }
 
@@ -1113,6 +1380,8 @@ __isl_give isl_pw_aff *isl_pw_aff_from_aff(__isl_take isl_aff *aff)
 #define IS_ZERO is_empty
 #undef FIELD
 #define FIELD aff
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 0
 
 #define NO_EVAL
 #define NO_OPT
@@ -1243,7 +1512,8 @@ static __isl_give isl_pw_aff *pw_aff_union_max(__isl_take isl_pw_aff *pwaff1,
 __isl_give isl_pw_aff *isl_pw_aff_union_max(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	return align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_union_max);
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2,
+							&pw_aff_union_max);
 }
 
 /* Compute a piecewise quasi-affine expression with a domain that
@@ -1261,7 +1531,8 @@ static __isl_give isl_pw_aff *pw_aff_union_min(__isl_take isl_pw_aff *pwaff1,
 __isl_give isl_pw_aff *isl_pw_aff_union_min(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	return align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_union_min);
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2,
+							&pw_aff_union_min);
 }
 
 __isl_give isl_pw_aff *isl_pw_aff_union_opt(__isl_take isl_pw_aff *pwaff1,
@@ -1286,8 +1557,6 @@ static __isl_give isl_map *map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
 		return NULL;
 
 	dim = isl_pw_aff_get_space(pwaff);
-	dim = isl_space_from_domain(dim);
-	dim = isl_space_add_dims(dim, isl_dim_out, 1);
 	map = isl_map_empty(dim);
 
 	for (i = 0; i < pwaff->n; ++i) {
@@ -1311,7 +1580,9 @@ static __isl_give isl_map *map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
  */
 __isl_give isl_map *isl_map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
 {
-	if (isl_space_is_params(pwaff->dim))
+	if (!pwaff)
+		return NULL;
+	if (isl_space_is_set(pwaff->dim))
 		isl_die(isl_pw_aff_get_ctx(pwaff), isl_error_invalid,
 			"space of input is not a map",
 			return isl_pw_aff_free(pwaff));
@@ -1324,7 +1595,9 @@ __isl_give isl_map *isl_map_from_pw_aff(__isl_take isl_pw_aff *pwaff)
  */
 __isl_give isl_set *isl_set_from_pw_aff(__isl_take isl_pw_aff *pwaff)
 {
-	if (!isl_space_is_params(pwaff->dim))
+	if (!pwaff)
+		return NULL;
+	if (!isl_space_is_set(pwaff->dim))
 		isl_die(isl_pw_aff_get_ctx(pwaff), isl_error_invalid,
 			"space of input is not a set",
 			return isl_pw_aff_free(pwaff));
@@ -1342,7 +1615,7 @@ __isl_give isl_set *isl_pw_aff_nonneg_set(__isl_take isl_pw_aff *pwaff)
 	if (!pwaff)
 		return NULL;
 
-	set = isl_set_empty(isl_pw_aff_get_space(pwaff));
+	set = isl_set_empty(isl_pw_aff_get_domain_space(pwaff));
 
 	for (i = 0; i < pwaff->n; ++i) {
 		isl_basic_set *bset;
@@ -1360,9 +1633,11 @@ __isl_give isl_set *isl_pw_aff_nonneg_set(__isl_take isl_pw_aff *pwaff)
 }
 
 /* Return a set containing those elements in the domain
- * of pwaff where it is zero.
+ * of pwaff where it is zero (if complement is 0) or not zero
+ * (if complement is 1).
  */
-__isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff)
+static __isl_give isl_set *pw_aff_zero_set(__isl_take isl_pw_aff *pwaff,
+	int complement)
 {
 	int i;
 	isl_set *set;
@@ -1370,15 +1645,19 @@ __isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff)
 	if (!pwaff)
 		return NULL;
 
-	set = isl_set_empty(isl_pw_aff_get_space(pwaff));
+	set = isl_set_empty(isl_pw_aff_get_domain_space(pwaff));
 
 	for (i = 0; i < pwaff->n; ++i) {
 		isl_basic_set *bset;
-		isl_set *set_i;
+		isl_set *set_i, *zero;
 
 		bset = isl_aff_zero_basic_set(isl_aff_copy(pwaff->p[i].aff));
-		set_i = isl_set_from_basic_set(bset);
-		set_i = isl_set_intersect(set_i, isl_set_copy(pwaff->p[i].set));
+		zero = isl_set_from_basic_set(bset);
+		set_i = isl_set_copy(pwaff->p[i].set);
+		if (complement)
+			set_i = isl_set_subtract(set_i, zero);
+		else
+			set_i = isl_set_intersect(set_i, zero);
 		set = isl_set_union_disjoint(set, set_i);
 	}
 
@@ -1388,11 +1667,19 @@ __isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff)
 }
 
 /* Return a set containing those elements in the domain
+ * of pwaff where it is zero.
+ */
+__isl_give isl_set *isl_pw_aff_zero_set(__isl_take isl_pw_aff *pwaff)
+{
+	return pw_aff_zero_set(pwaff, 0);
+}
+
+/* Return a set containing those elements in the domain
  * of pwaff where it is not zero.
  */
 __isl_give isl_set *isl_pw_aff_non_zero_set(__isl_take isl_pw_aff *pwaff)
 {
-	return isl_set_complement(isl_pw_aff_zero_set(pwaff));
+	return pw_aff_zero_set(pwaff, 1);
 }
 
 /* Return a set containing those elements in the shared domain
@@ -1419,7 +1706,7 @@ static __isl_give isl_set *pw_aff_gte_set(__isl_take isl_pw_aff *pwaff1,
 	if (strict) {
 		isl_space *dim = isl_set_get_space(set1);
 		isl_aff *aff;
-		aff = isl_aff_zero(isl_local_space_from_space(dim));
+		aff = isl_aff_zero_on_domain(isl_local_space_from_space(dim));
 		aff = isl_aff_add_constant_si(aff, -1);
 		pwaff1 = isl_pw_aff_add(pwaff1, isl_pw_aff_alloc(set1, aff));
 	} else
@@ -1508,7 +1795,7 @@ static __isl_give isl_set *pw_aff_list_set(__isl_take isl_pw_aff_list *list1,
 		isl_die(ctx, isl_error_invalid,
 			"list should contain at least one element", goto error);
 
-	set = isl_set_universe(isl_pw_aff_get_space(list1->p[0]));
+	set = isl_set_universe(isl_pw_aff_get_domain_space(list1->p[0]));
 	for (i = 0; i < list1->n; ++i)
 		for (j = 0; j < list2->n; ++j) {
 			isl_set *set_ij;
@@ -1656,20 +1943,34 @@ __isl_give isl_pw_aff *isl_pw_aff_ceil(__isl_take isl_pw_aff *pwaff)
 	return pwaff;
 }
 
+/* Assuming that "cond1" and "cond2" are disjoint,
+ * return an affine expression that is equal to pwaff1 on cond1
+ * and to pwaff2 on cond2.
+ */
+static __isl_give isl_pw_aff *isl_pw_aff_select(
+	__isl_take isl_set *cond1, __isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_set *cond2, __isl_take isl_pw_aff *pwaff2)
+{
+	pwaff1 = isl_pw_aff_intersect_domain(pwaff1, cond1);
+	pwaff2 = isl_pw_aff_intersect_domain(pwaff2, cond2);
+
+	return isl_pw_aff_add_disjoint(pwaff1, pwaff2);
+}
+
 /* Return an affine expression that is equal to pwaff_true for elements
- * in "cond" and to pwaff_false for elements not in "cond".
+ * where "cond" is non-zero and to pwaff_false for elements where "cond"
+ * is zero.
  * That is, return cond ? pwaff_true : pwaff_false;
  */
-__isl_give isl_pw_aff *isl_pw_aff_cond(__isl_take isl_set *cond,
+__isl_give isl_pw_aff *isl_pw_aff_cond(__isl_take isl_pw_aff *cond,
 	__isl_take isl_pw_aff *pwaff_true, __isl_take isl_pw_aff *pwaff_false)
 {
-	isl_set *comp;
+	isl_set *cond_true, *cond_false;
 
-	comp = isl_set_complement(isl_set_copy(cond));
-	pwaff_true = isl_pw_aff_intersect_domain(pwaff_true, cond);
-	pwaff_false = isl_pw_aff_intersect_domain(pwaff_false, comp);
-
-	return isl_pw_aff_add_disjoint(pwaff_true, pwaff_false);
+	cond_true = isl_pw_aff_non_zero_set(isl_pw_aff_copy(cond));
+	cond_false = isl_pw_aff_zero_set(cond);
+	return isl_pw_aff_select(cond_true, pwaff_true,
+				 cond_false, pwaff_false);
 }
 
 int isl_aff_is_cst(__isl_keep isl_aff *aff)
@@ -1724,82 +2025,74 @@ error:
 	return NULL;
 }
 
-static __isl_give isl_pw_aff *pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
+static __isl_give isl_pw_aff *pw_aff_add(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	int i, j, n;
-	isl_pw_aff *res;
-
-	if (!pwaff1 || !pwaff2)
-		goto error;
-
-	n = pwaff1->n * pwaff2->n;
-	res = isl_pw_aff_alloc_size(isl_space_copy(pwaff1->dim), n);
-
-	for (i = 0; i < pwaff1->n; ++i) {
-		for (j = 0; j < pwaff2->n; ++j) {
-			isl_set *common;
-			isl_aff *prod;
-			common = isl_set_intersect(
-					isl_set_copy(pwaff1->p[i].set),
-					isl_set_copy(pwaff2->p[j].set));
-			if (isl_set_plain_is_empty(common)) {
-				isl_set_free(common);
-				continue;
-			}
+	return isl_pw_aff_on_shared_domain(pwaff1, pwaff2, &isl_aff_add);
+}
 
-			prod = isl_aff_mul(isl_aff_copy(pwaff1->p[i].aff),
-					    isl_aff_copy(pwaff2->p[j].aff));
+__isl_give isl_pw_aff *isl_pw_aff_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_add);
+}
 
-			res = isl_pw_aff_add_piece(res, common, prod);
-		}
-	}
+__isl_give isl_pw_aff *isl_pw_aff_union_add(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_union_add_(pwaff1, pwaff2);
+}
 
-	isl_pw_aff_free(pwaff1);
-	isl_pw_aff_free(pwaff2);
-	return res;
-error:
-	isl_pw_aff_free(pwaff1);
-	isl_pw_aff_free(pwaff2);
-	return NULL;
+static __isl_give isl_pw_aff *pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
+	__isl_take isl_pw_aff *pwaff2)
+{
+	return isl_pw_aff_on_shared_domain(pwaff1, pwaff2, &isl_aff_mul);
 }
 
 __isl_give isl_pw_aff *isl_pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	return align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_mul);
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_mul);
 }
 
 static __isl_give isl_pw_aff *pw_aff_min(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
 	isl_set *le;
+	isl_set *dom;
 
+	dom = isl_set_intersect(isl_pw_aff_domain(isl_pw_aff_copy(pwaff1)),
+				isl_pw_aff_domain(isl_pw_aff_copy(pwaff2)));
 	le = isl_pw_aff_le_set(isl_pw_aff_copy(pwaff1),
 				isl_pw_aff_copy(pwaff2));
-	return isl_pw_aff_cond(le, pwaff1, pwaff2);
+	dom = isl_set_subtract(dom, isl_set_copy(le));
+	return isl_pw_aff_select(le, pwaff1, dom, pwaff2);
 }
 
 __isl_give isl_pw_aff *isl_pw_aff_min(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	return align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_min);
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_min);
 }
 
 static __isl_give isl_pw_aff *pw_aff_max(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	isl_set *le;
+	isl_set *ge;
+	isl_set *dom;
 
-	le = isl_pw_aff_ge_set(isl_pw_aff_copy(pwaff1),
+	dom = isl_set_intersect(isl_pw_aff_domain(isl_pw_aff_copy(pwaff1)),
+				isl_pw_aff_domain(isl_pw_aff_copy(pwaff2)));
+	ge = isl_pw_aff_ge_set(isl_pw_aff_copy(pwaff1),
 				isl_pw_aff_copy(pwaff2));
-	return isl_pw_aff_cond(le, pwaff1, pwaff2);
+	dom = isl_set_subtract(dom, isl_set_copy(ge));
+	return isl_pw_aff_select(ge, pwaff1, dom, pwaff2);
 }
 
 __isl_give isl_pw_aff *isl_pw_aff_max(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
-	return align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_max);
+	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_max);
 }
 
 static __isl_give isl_pw_aff *pw_aff_list_reduce(
@@ -1845,3 +2138,1401 @@ __isl_give isl_pw_aff *isl_pw_aff_list_max(__isl_take isl_pw_aff_list *list)
 {
 	return pw_aff_list_reduce(list, &isl_pw_aff_max);
 }
+
+#undef BASE
+#define BASE aff
+
+#include <isl_multi_templ.c>
+
+/* Construct an isl_multi_aff in the given space with value zero in
+ * each of the output dimensions.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_zero(__isl_take isl_space *space)
+{
+	int n;
+	isl_multi_aff *ma;
+
+	if (!space)
+		return NULL;
+
+	n = isl_space_dim(space , isl_dim_out);
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+
+	if (!n)
+		isl_space_free(space);
+	else {
+		int i;
+		isl_local_space *ls;
+		isl_aff *aff;
+
+		space = isl_space_domain(space);
+		ls = isl_local_space_from_space(space);
+		aff = isl_aff_zero_on_domain(ls);
+
+		for (i = 0; i < n; ++i)
+			ma = isl_multi_aff_set_aff(ma, i, isl_aff_copy(aff));
+
+		isl_aff_free(aff);
+	}
+
+	return ma;
+}
+
+/* Create an isl_multi_aff in the given space that maps each
+ * input dimension to the corresponding output dimension.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_identity(__isl_take isl_space *space)
+{
+	int n;
+	isl_multi_aff *ma;
+
+	if (!space)
+		return NULL;
+
+	if (isl_space_is_set(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"expecting map space", goto error);
+
+	n = isl_space_dim(space, isl_dim_out);
+	if (n != isl_space_dim(space, isl_dim_in))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"number of input and output dimensions needs to be "
+			"the same", goto error);
+
+	ma = isl_multi_aff_alloc(isl_space_copy(space));
+
+	if (!n)
+		isl_space_free(space);
+	else {
+		int i;
+		isl_local_space *ls;
+		isl_aff *aff;
+
+		space = isl_space_domain(space);
+		ls = isl_local_space_from_space(space);
+		aff = isl_aff_zero_on_domain(ls);
+
+		for (i = 0; i < n; ++i) {
+			isl_aff *aff_i;
+			aff_i = isl_aff_copy(aff);
+			aff_i = isl_aff_add_coefficient_si(aff_i,
+							    isl_dim_in, i, 1);
+			ma = isl_multi_aff_set_aff(ma, i, aff_i);
+		}
+
+		isl_aff_free(aff);
+	}
+
+	return ma;
+error:
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Create an isl_pw_multi_aff with the given isl_multi_aff on a universe
+ * domain.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_multi_aff(
+	__isl_take isl_multi_aff *ma)
+{
+	isl_set *dom = isl_set_universe(isl_multi_aff_get_domain_space(ma));
+	return isl_pw_multi_aff_alloc(dom, ma);
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_add(__isl_take isl_multi_aff *maff1,
+	__isl_take isl_multi_aff *maff2)
+{
+	int i;
+	isl_ctx *ctx;
+
+	maff1 = isl_multi_aff_cow(maff1);
+	if (!maff1 || !maff2)
+		goto error;
+
+	ctx = isl_multi_aff_get_ctx(maff1);
+	if (!isl_space_is_equal(maff1->space, maff2->space))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+
+	for (i = 0; i < maff1->n; ++i) {
+		maff1->p[i] = isl_aff_add(maff1->p[i],
+					    isl_aff_copy(maff2->p[i]));
+		if (!maff1->p[i])
+			goto error;
+	}
+
+	isl_multi_aff_free(maff2);
+	return maff1;
+error:
+	isl_multi_aff_free(maff1);
+	isl_multi_aff_free(maff2);
+	return NULL;
+}
+
+/* Given two multi-affine expressions A -> B and C -> D,
+ * construct a multi-affine expression [A -> C] -> [B -> D].
+ */
+__isl_give isl_multi_aff *isl_multi_aff_product(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	int i;
+	isl_aff *aff;
+	isl_space *space;
+	isl_multi_aff *res;
+	int in1, in2, out1, out2;
+
+	in1 = isl_multi_aff_dim(ma1, isl_dim_in);
+	in2 = isl_multi_aff_dim(ma2, isl_dim_in);
+	out1 = isl_multi_aff_dim(ma1, isl_dim_out);
+	out2 = isl_multi_aff_dim(ma2, isl_dim_out);
+	space = isl_space_product(isl_multi_aff_get_space(ma1),
+				  isl_multi_aff_get_space(ma2));
+	res = isl_multi_aff_alloc(isl_space_copy(space));
+	space = isl_space_domain(space);
+
+	for (i = 0; i < out1; ++i) {
+		aff = isl_multi_aff_get_aff(ma1, i);
+		aff = isl_aff_insert_dims(aff, isl_dim_in, in1, in2);
+		aff = isl_aff_reset_domain_space(aff, isl_space_copy(space));
+		res = isl_multi_aff_set_aff(res, i, aff);
+	}
+
+	for (i = 0; i < out2; ++i) {
+		aff = isl_multi_aff_get_aff(ma2, i);
+		aff = isl_aff_insert_dims(aff, isl_dim_in, 0, in1);
+		aff = isl_aff_reset_domain_space(aff, isl_space_copy(space));
+		res = isl_multi_aff_set_aff(res, out1 + i, aff);
+	}
+
+	isl_space_free(space);
+	isl_multi_aff_free(ma1);
+	isl_multi_aff_free(ma2);
+	return res;
+}
+
+/* Exploit the equalities in "eq" to simplify the affine expressions.
+ */
+static __isl_give isl_multi_aff *isl_multi_aff_substitute_equalities(
+	__isl_take isl_multi_aff *maff, __isl_take isl_basic_set *eq)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff || !eq)
+		goto error;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_substitute_equalities(maff->p[i],
+						    isl_basic_set_copy(eq));
+		if (!maff->p[i])
+			goto error;
+	}
+
+	isl_basic_set_free(eq);
+	return maff;
+error:
+	isl_basic_set_free(eq);
+	isl_multi_aff_free(maff);
+	return NULL;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_scale(__isl_take isl_multi_aff *maff,
+	isl_int f)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_scale(maff->p[i], f);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_add_on_domain(__isl_keep isl_set *dom,
+	__isl_take isl_multi_aff *maff1, __isl_take isl_multi_aff *maff2)
+{
+	maff1 = isl_multi_aff_add(maff1, maff2);
+	maff1 = isl_multi_aff_gist(maff1, isl_set_copy(dom));
+	return maff1;
+}
+
+int isl_multi_aff_is_empty(__isl_keep isl_multi_aff *maff)
+{
+	if (!maff)
+		return -1;
+
+	return 0;
+}
+
+int isl_multi_aff_plain_is_equal(__isl_keep isl_multi_aff *maff1,
+	__isl_keep isl_multi_aff *maff2)
+{
+	int i;
+	int equal;
+
+	if (!maff1 || !maff2)
+		return -1;
+	if (maff1->n != maff2->n)
+		return 0;
+	equal = isl_space_is_equal(maff1->space, maff2->space);
+	if (equal < 0 || !equal)
+		return equal;
+
+	for (i = 0; i < maff1->n; ++i) {
+		equal = isl_aff_plain_is_equal(maff1->p[i], maff2->p[i]);
+		if (equal < 0 || !equal)
+			return equal;
+	}
+
+	return 1;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_set_dim_name(
+	__isl_take isl_multi_aff *maff,
+	enum isl_dim_type type, unsigned pos, const char *s)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	maff->space = isl_space_set_dim_name(maff->space, type, pos, s);
+	if (!maff->space)
+		return isl_multi_aff_free(maff);
+
+	if (type == isl_dim_out)
+		return maff;
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_set_dim_name(maff->p[i], type, pos, s);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+__isl_give isl_multi_aff *isl_multi_aff_drop_dims(__isl_take isl_multi_aff *maff,
+	enum isl_dim_type type, unsigned first, unsigned n)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	maff->space = isl_space_drop_dims(maff->space, type, first, n);
+	if (!maff->space)
+		return isl_multi_aff_free(maff);
+
+	if (type == isl_dim_out) {
+		for (i = 0; i < n; ++i)
+			isl_aff_free(maff->p[first + i]);
+		for (i = first; i + n < maff->n; ++i)
+			maff->p[i] = maff->p[i + n];
+		maff->n -= n;
+		return maff;
+	}
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_drop_dims(maff->p[i], type, first, n);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+/* Return the set of domain elements where "ma1" is lexicographically
+ * smaller than or equal to "ma2".
+ */
+__isl_give isl_set *isl_multi_aff_lex_le_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_lex_ge_set(ma2, ma1);
+}
+
+/* Return the set of domain elements where "ma1" is lexicographically
+ * greater than or equal to "ma2".
+ */
+__isl_give isl_set *isl_multi_aff_lex_ge_set(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	isl_space *space;
+	isl_map *map1, *map2;
+	isl_map *map, *ge;
+
+	map1 = isl_map_from_multi_aff(ma1);
+	map2 = isl_map_from_multi_aff(ma2);
+	map = isl_map_range_product(map1, map2);
+	space = isl_space_range(isl_map_get_space(map));
+	space = isl_space_domain(isl_space_unwrap(space));
+	ge = isl_map_lex_ge(space);
+	map = isl_map_intersect_range(map, isl_map_wrap(ge));
+
+	return isl_map_domain(map);
+}
+
+#undef PW
+#define PW isl_pw_multi_aff
+#undef EL
+#define EL isl_multi_aff
+#undef EL_IS_ZERO
+#define EL_IS_ZERO is_empty
+#undef ZERO
+#define ZERO empty
+#undef IS_ZERO
+#define IS_ZERO is_empty
+#undef FIELD
+#define FIELD maff
+#undef DEFAULT_IS_ZERO
+#define DEFAULT_IS_ZERO 0
+
+#define NO_NEG
+#define NO_EVAL
+#define NO_OPT
+#define NO_INVOLVES_DIMS
+#define NO_MOVE_DIMS
+#define NO_INSERT_DIMS
+#define NO_LIFT
+#define NO_MORPH
+
+#include <isl_pw_templ.c>
+
+#undef UNION
+#define UNION isl_union_pw_multi_aff
+#undef PART
+#define PART isl_pw_multi_aff
+#undef PARTS
+#define PARTS pw_multi_aff
+#define ALIGN_DOMAIN
+
+#define NO_EVAL
+
+#include <isl_union_templ.c>
+
+/* Given a function "cmp" that returns the set of elements where
+ * "ma1" is "better" than "ma2", return the intersection of this
+ * set with "dom1" and "dom2".
+ */
+static __isl_give isl_set *shared_and_better(__isl_keep isl_set *dom1,
+	__isl_keep isl_set *dom2, __isl_keep isl_multi_aff *ma1,
+	__isl_keep isl_multi_aff *ma2,
+	__isl_give isl_set *(*cmp)(__isl_take isl_multi_aff *ma1,
+				    __isl_take isl_multi_aff *ma2))
+{
+	isl_set *common;
+	isl_set *better;
+	int is_empty;
+
+	common = isl_set_intersect(isl_set_copy(dom1), isl_set_copy(dom2));
+	is_empty = isl_set_plain_is_empty(common);
+	if (is_empty >= 0 && is_empty)
+		return common;
+	if (is_empty < 0)
+		return isl_set_free(common);
+	better = cmp(isl_multi_aff_copy(ma1), isl_multi_aff_copy(ma2));
+	better = isl_set_intersect(common, better);
+
+	return better;
+}
+
+/* Given a function "cmp" that returns the set of elements where
+ * "ma1" is "better" than "ma2", return a piecewise multi affine
+ * expression defined on the union of the definition domains
+ * of "pma1" and "pma2" that maps to the "best" of "pma1" and
+ * "pma2" on each cell.  If only one of the two input functions
+ * is defined on a given cell, then it is considered the best.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_opt(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2,
+	__isl_give isl_set *(*cmp)(__isl_take isl_multi_aff *ma1,
+				    __isl_take isl_multi_aff *ma2))
+{
+	int i, j, n;
+	isl_pw_multi_aff *res = NULL;
+	isl_ctx *ctx;
+	isl_set *set = NULL;
+
+	if (!pma1 || !pma2)
+		goto error;
+
+	ctx = isl_space_get_ctx(pma1->dim);
+	if (!isl_space_is_equal(pma1->dim, pma2->dim))
+		isl_die(ctx, isl_error_invalid,
+			"arguments should live in the same space", goto error);
+
+	if (isl_pw_multi_aff_is_empty(pma1)) {
+		isl_pw_multi_aff_free(pma1);
+		return pma2;
+	}
+
+	if (isl_pw_multi_aff_is_empty(pma2)) {
+		isl_pw_multi_aff_free(pma2);
+		return pma1;
+	}
+
+	n = 2 * (pma1->n + 1) * (pma2->n + 1);
+	res = isl_pw_multi_aff_alloc_size(isl_space_copy(pma1->dim), n);
+
+	for (i = 0; i < pma1->n; ++i) {
+		set = isl_set_copy(pma1->p[i].set);
+		for (j = 0; j < pma2->n; ++j) {
+			isl_set *better;
+			int is_empty;
+
+			better = shared_and_better(pma2->p[j].set,
+					pma1->p[i].set, pma2->p[j].maff,
+					pma1->p[i].maff, cmp);
+			is_empty = isl_set_plain_is_empty(better);
+			if (is_empty < 0 || is_empty) {
+				isl_set_free(better);
+				if (is_empty < 0)
+					goto error;
+				continue;
+			}
+			set = isl_set_subtract(set, isl_set_copy(better));
+
+			res = isl_pw_multi_aff_add_piece(res, better,
+					isl_multi_aff_copy(pma2->p[j].maff));
+		}
+		res = isl_pw_multi_aff_add_piece(res, set,
+					isl_multi_aff_copy(pma1->p[i].maff));
+	}
+
+	for (j = 0; j < pma2->n; ++j) {
+		set = isl_set_copy(pma2->p[j].set);
+		for (i = 0; i < pma1->n; ++i)
+			set = isl_set_subtract(set,
+					isl_set_copy(pma1->p[i].set));
+		res = isl_pw_multi_aff_add_piece(res, set,
+					isl_multi_aff_copy(pma2->p[j].maff));
+	}
+
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+
+	return res;
+error:
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	isl_set_free(set);
+	return isl_pw_multi_aff_free(res);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_lexmax(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return pw_multi_aff_union_opt(pma1, pma2, &isl_multi_aff_lex_ge_set);
+}
+
+/* Given two piecewise multi affine expressions, return a piecewise
+ * multi-affine expression defined on the union of the definition domains
+ * of the inputs that is equal to the lexicographic maximum of the two
+ * inputs on each cell.  If only one of the two inputs is defined on
+ * a given cell, then it is considered to be the maximum.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmax(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						    &pw_multi_aff_union_lexmax);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_union_lexmin(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return pw_multi_aff_union_opt(pma1, pma2, &isl_multi_aff_lex_le_set);
+}
+
+/* Given two piecewise multi affine expressions, return a piecewise
+ * multi-affine expression defined on the union of the definition domains
+ * of the inputs that is equal to the lexicographic minimum of the two
+ * inputs on each cell.  If only one of the two inputs is defined on
+ * a given cell, then it is considered to be the minimum.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_lexmin(
+	__isl_take isl_pw_multi_aff *pma1,
+	__isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						    &pw_multi_aff_union_lexmin);
+}
+
+static __isl_give isl_pw_multi_aff *pw_multi_aff_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_on_shared_domain(pma1, pma2,
+						&isl_multi_aff_add);
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						&pw_multi_aff_add);
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_union_add(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_union_add_(pma1, pma2);
+}
+
+/* Given two piecewise multi-affine expressions A -> B and C -> D,
+ * construct a piecewise multi-affine expression [A -> C] -> [B -> D].
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	int i, j, n;
+	isl_space *space;
+	isl_pw_multi_aff *res;
+
+	if (!pma1 || !pma2)
+		goto error;
+
+	n = pma1->n * pma2->n;
+	space = isl_space_product(isl_space_copy(pma1->dim),
+				  isl_space_copy(pma2->dim));
+	res = isl_pw_multi_aff_alloc_size(space, n);
+
+	for (i = 0; i < pma1->n; ++i) {
+		for (j = 0; j < pma2->n; ++j) {
+			isl_set *domain;
+			isl_multi_aff *ma;
+
+			domain = isl_set_product(isl_set_copy(pma1->p[i].set),
+						 isl_set_copy(pma2->p[j].set));
+			ma = isl_multi_aff_product(
+					isl_multi_aff_copy(pma1->p[i].maff),
+					isl_multi_aff_copy(pma2->p[i].maff));
+			res = isl_pw_multi_aff_add_piece(res, domain, ma);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma1);
+	isl_pw_multi_aff_free(pma2);
+	return NULL;
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+						&pw_multi_aff_product);
+}
+
+/* Construct a map mapping the domain of the piecewise multi-affine expression
+ * to its range, with each dimension in the range equated to the
+ * corresponding affine expression on its cell.
+ */
+__isl_give isl_map *isl_map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
+{
+	int i;
+	isl_map *map;
+
+	if (!pma)
+		return NULL;
+
+	map = isl_map_empty(isl_pw_multi_aff_get_space(pma));
+
+	for (i = 0; i < pma->n; ++i) {
+		isl_multi_aff *maff;
+		isl_basic_map *bmap;
+		isl_map *map_i;
+
+		maff = isl_multi_aff_copy(pma->p[i].maff);
+		bmap = isl_basic_map_from_multi_aff(maff);
+		map_i = isl_map_from_basic_map(bmap);
+		map_i = isl_map_intersect_domain(map_i,
+						isl_set_copy(pma->p[i].set));
+		map = isl_map_union_disjoint(map, map_i);
+	}
+
+	isl_pw_multi_aff_free(pma);
+	return map;
+}
+
+__isl_give isl_set *isl_set_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
+{
+	if (!isl_space_is_set(pma->dim))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"isl_pw_multi_aff cannot be converted into an isl_set",
+			return isl_pw_multi_aff_free(pma));
+
+	return isl_map_from_pw_multi_aff(pma);
+}
+
+/* Given a basic map with a single output dimension that is defined
+ * in terms of the parameters and input dimensions using an equality,
+ * extract an isl_aff that expresses the output dimension in terms
+ * of the parameters and input dimensions.
+ *
+ * Since some applications expect the result of isl_pw_multi_aff_from_map
+ * to only contain integer affine expressions, we compute the floor
+ * of the expression before returning.
+ *
+ * This function shares some similarities with
+ * isl_basic_map_has_defining_equality and isl_constraint_get_bound.
+ */
+static __isl_give isl_aff *extract_isl_aff_from_basic_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+	unsigned offset;
+	unsigned total;
+	isl_local_space *ls;
+	isl_aff *aff;
+
+	if (!bmap)
+		return NULL;
+	if (isl_basic_map_dim(bmap, isl_dim_out) != 1)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"basic map should have a single output dimension",
+			goto error);
+	offset = isl_basic_map_offset(bmap, isl_dim_out);
+	total = isl_basic_map_total_dim(bmap);
+	for (i = 0; i < bmap->n_eq; ++i) {
+		if (isl_int_is_zero(bmap->eq[i][offset]))
+			continue;
+		if (isl_seq_first_non_zero(bmap->eq[i] + offset + 1,
+					   1 + total - (offset + 1)) != -1)
+			continue;
+		break;
+	}
+	if (i >= bmap->n_eq)
+		isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid,
+			"unable to find suitable equality", goto error);
+	ls = isl_basic_map_get_local_space(bmap);
+	aff = isl_aff_alloc(isl_local_space_domain(ls));
+	if (!aff)
+		goto error;
+	if (isl_int_is_neg(bmap->eq[i][offset]))
+		isl_seq_cpy(aff->v->el + 1, bmap->eq[i], offset);
+	else
+		isl_seq_neg(aff->v->el + 1, bmap->eq[i], offset);
+	isl_seq_clr(aff->v->el + 1 + offset, aff->v->size - (1 + offset));
+	isl_int_abs(aff->v->el[0], bmap->eq[i][offset]);
+	isl_basic_map_free(bmap);
+
+	aff = isl_aff_remove_unused_divs(aff);
+	aff = isl_aff_floor(aff);
+	return aff;
+error:
+	isl_basic_map_free(bmap);
+	return NULL;
+}
+
+/* Given a basic map where each output dimension is defined
+ * in terms of the parameters and input dimensions using an equality,
+ * extract an isl_multi_aff that expresses the output dimensions in terms
+ * of the parameters and input dimensions.
+ */
+static __isl_give isl_multi_aff *extract_isl_multi_aff_from_basic_map(
+	__isl_take isl_basic_map *bmap)
+{
+	int i;
+	unsigned n_out;
+	isl_multi_aff *ma;
+
+	if (!bmap)
+		return NULL;
+
+	ma = isl_multi_aff_alloc(isl_basic_map_get_space(bmap));
+	n_out = isl_basic_map_dim(bmap, isl_dim_out);
+
+	for (i = 0; i < n_out; ++i) {
+		isl_basic_map *bmap_i;
+		isl_aff *aff;
+
+		bmap_i = isl_basic_map_copy(bmap);
+		bmap_i = isl_basic_map_project_out(bmap_i, isl_dim_out,
+							i + 1, n_out - (1 + i));
+		bmap_i = isl_basic_map_project_out(bmap_i, isl_dim_out, 0, i);
+		aff = extract_isl_aff_from_basic_map(bmap_i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_basic_map_free(bmap);
+
+	return ma;
+}
+
+/* Create an isl_pw_multi_aff that is equivalent to
+ * isl_map_intersect_domain(isl_map_from_basic_map(bmap), domain).
+ * The given basic map is such that each output dimension is defined
+ * in terms of the parameters and input dimensions using an equality.
+ */
+static __isl_give isl_pw_multi_aff *plain_pw_multi_aff_from_map(
+	__isl_take isl_set *domain, __isl_take isl_basic_map *bmap)
+{
+	isl_multi_aff *ma;
+
+	ma = extract_isl_multi_aff_from_basic_map(bmap);
+	return isl_pw_multi_aff_alloc(domain, ma);
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ * This obivously only works if the input "map" is single-valued.
+ * If so, we compute the lexicographic minimum of the image in the form
+ * of an isl_pw_multi_aff.  Since the image is unique, it is equal
+ * to its lexicographic minimum.
+ * If the input is not single-valued, we produce an error.
+ *
+ * As a special case, we first check if all output dimensions are uniquely
+ * defined in terms of the parameters and input dimensions over the entire
+ * domain.  If so, we extract the desired isl_pw_multi_aff directly
+ * from the affine hull of "map" and its domain.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(__isl_take isl_map *map)
+{
+	int i;
+	int sv;
+	isl_pw_multi_aff *pma;
+	isl_basic_map *hull;
+
+	if (!map)
+		return NULL;
+
+	hull = isl_map_affine_hull(isl_map_copy(map));
+	sv = isl_basic_map_plain_is_single_valued(hull);
+	if (sv >= 0 && sv)
+		return plain_pw_multi_aff_from_map(isl_map_domain(map), hull);
+	isl_basic_map_free(hull);
+	if (sv < 0)
+		goto error;
+
+	sv = isl_map_is_single_valued(map);
+	if (sv < 0)
+		goto error;
+	if (!sv)
+		isl_die(isl_map_get_ctx(map), isl_error_invalid,
+			"map is not single-valued", goto error);
+	map = isl_map_make_disjoint(map);
+	if (!map)
+		return NULL;
+
+	pma = isl_pw_multi_aff_empty(isl_map_get_space(map));
+
+	for (i = 0; i < map->n; ++i) {
+		isl_pw_multi_aff *pma_i;
+		isl_basic_map *bmap;
+		bmap = isl_basic_map_copy(map->p[i]);
+		pma_i = isl_basic_map_lexmin_pw_multi_aff(bmap);
+		pma = isl_pw_multi_aff_add_disjoint(pma, pma_i);
+	}
+
+	isl_map_free(map);
+	return pma;
+error:
+	isl_map_free(map);
+	return NULL;
+}
+
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_set(__isl_take isl_set *set)
+{
+	return isl_pw_multi_aff_from_map(set);
+}
+
+/* Return the piecewise affine expression "set ? 1 : 0".
+ */
+__isl_give isl_pw_aff *isl_set_indicator_function(__isl_take isl_set *set)
+{
+	isl_pw_aff *pa;
+	isl_space *space = isl_set_get_space(set);
+	isl_local_space *ls = isl_local_space_from_space(space);
+	isl_aff *zero = isl_aff_zero_on_domain(isl_local_space_copy(ls));
+	isl_aff *one = isl_aff_zero_on_domain(ls);
+
+	one = isl_aff_add_constant_si(one, 1);
+	pa = isl_pw_aff_alloc(isl_set_copy(set), one);
+	set = isl_set_complement(set);
+	pa = isl_pw_aff_add_disjoint(pa, isl_pw_aff_alloc(set, zero));
+
+	return pa;
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "aff".
+ *
+ * Let i be the dimension to replace and let "subs" be of the form
+ *
+ *	f/d
+ *
+ * and "aff" of the form
+ *
+ *	(a i + g)/m
+ *
+ * The result is
+ *
+ *	floor((a f + d g')/(m d))
+ *
+ * where g' is the result of plugging in "subs" in each of the integer
+ * divisions in g.
+ */
+__isl_give isl_aff *isl_aff_substitute(__isl_take isl_aff *aff,
+	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+{
+	isl_ctx *ctx;
+	isl_int v;
+
+	aff = isl_aff_cow(aff);
+	if (!aff || !subs)
+		return isl_aff_free(aff);
+
+	ctx = isl_aff_get_ctx(aff);
+	if (!isl_space_is_equal(aff->ls->dim, subs->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", return isl_aff_free(aff));
+	if (isl_local_space_dim(subs->ls, isl_dim_div) != 0)
+		isl_die(ctx, isl_error_unsupported,
+			"cannot handle divs yet", return isl_aff_free(aff));
+
+	aff->ls = isl_local_space_substitute(aff->ls, type, pos, subs);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	pos += isl_local_space_offset(aff->ls, type);
+
+	isl_int_init(v);
+	isl_int_set(v, aff->v->el[1 + pos]);
+	isl_int_set_si(aff->v->el[1 + pos], 0);
+	isl_seq_combine(aff->v->el + 1, subs->v->el[0], aff->v->el + 1,
+			v, subs->v->el + 1, subs->v->size - 1);
+	isl_int_mul(aff->v->el[0], aff->v->el[0], subs->v->el[0]);
+	isl_int_clear(v);
+
+	return aff;
+}
+
+/* Plug in "subs" for dimension "type", "pos" in each of the affine
+ * expressions in "maff".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_substitute(
+	__isl_take isl_multi_aff *maff, enum isl_dim_type type, unsigned pos,
+	__isl_keep isl_aff *subs)
+{
+	int i;
+
+	maff = isl_multi_aff_cow(maff);
+	if (!maff || !subs)
+		return isl_multi_aff_free(maff);
+
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_substitute(maff->p[i], type, pos, subs);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+/* Plug in "subs" for dimension "type", "pos" of "pma".
+ *
+ * pma is of the form
+ *
+ *	A_i(v) -> M_i(v)
+ *
+ * while subs is of the form
+ *
+ *	v' = B_j(v) -> S_j
+ *
+ * Each pair i,j such that C_ij = A_i \cap B_i is non-empty
+ * has a contribution in the result, in particular
+ *
+ *	C_ij(S_j) -> M_i(S_j)
+ *
+ * Note that plugging in S_j in C_ij may also result in an empty set
+ * and this contribution should simply be discarded.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_substitute(
+	__isl_take isl_pw_multi_aff *pma, enum isl_dim_type type, unsigned pos,
+	__isl_keep isl_pw_aff *subs)
+{
+	int i, j, n;
+	isl_pw_multi_aff *res;
+
+	if (!pma || !subs)
+		return isl_pw_multi_aff_free(pma);
+
+	n = pma->n * subs->n;
+	res = isl_pw_multi_aff_alloc_size(isl_space_copy(pma->dim), n);
+
+	for (i = 0; i < pma->n; ++i) {
+		for (j = 0; j < subs->n; ++j) {
+			isl_set *common;
+			isl_multi_aff *res_ij;
+			common = isl_set_intersect(
+					isl_set_copy(pma->p[i].set),
+					isl_set_copy(subs->p[j].set));
+			common = isl_set_substitute(common,
+					type, pos, subs->p[j].aff);
+			if (isl_set_plain_is_empty(common)) {
+				isl_set_free(common);
+				continue;
+			}
+
+			res_ij = isl_multi_aff_substitute(
+					isl_multi_aff_copy(pma->p[i].maff),
+					type, pos, subs->p[j].aff);
+
+			res = isl_pw_multi_aff_add_piece(res, common, res_ij);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma);
+	return res;
+}
+
+/* Extend the local space of "dst" to include the divs
+ * in the local space of "src".
+ */
+__isl_give isl_aff *isl_aff_align_divs(__isl_take isl_aff *dst,
+	__isl_keep isl_aff *src)
+{
+	isl_ctx *ctx;
+	int *exp1 = NULL;
+	int *exp2 = NULL;
+	isl_mat *div;
+
+	if (!src || !dst)
+		return isl_aff_free(dst);
+
+	ctx = isl_aff_get_ctx(src);
+	if (!isl_space_is_equal(src->ls->dim, dst->ls->dim))
+		isl_die(ctx, isl_error_invalid,
+			"spaces don't match", goto error);
+
+	if (src->ls->div->n_row == 0)
+		return dst;
+
+	exp1 = isl_alloc_array(ctx, int, src->ls->div->n_row);
+	exp2 = isl_alloc_array(ctx, int, dst->ls->div->n_row);
+	if (!exp1 || !exp2)
+		goto error;
+
+	div = isl_merge_divs(src->ls->div, dst->ls->div, exp1, exp2);
+	dst = isl_aff_expand_divs(dst, div, exp2);
+	free(exp1);
+	free(exp2);
+
+	return dst;
+error:
+	free(exp1);
+	free(exp2);
+	return isl_aff_free(dst);
+}
+
+/* Adjust the local spaces of the affine expressions in "maff"
+ * such that they all have the save divs.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_align_divs(
+	__isl_take isl_multi_aff *maff)
+{
+	int i;
+
+	if (!maff)
+		return NULL;
+	if (maff->n == 0)
+		return maff;
+	maff = isl_multi_aff_cow(maff);
+	if (!maff)
+		return NULL;
+
+	for (i = 1; i < maff->n; ++i)
+		maff->p[0] = isl_aff_align_divs(maff->p[0], maff->p[i]);
+	for (i = 1; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_align_divs(maff->p[i], maff->p[0]);
+		if (!maff->p[i])
+			return isl_multi_aff_free(maff);
+	}
+
+	return maff;
+}
+
+__isl_give isl_aff *isl_aff_lift(__isl_take isl_aff *aff)
+{
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		return NULL;
+
+	aff->ls = isl_local_space_lift(aff->ls);
+	if (!aff->ls)
+		return isl_aff_free(aff);
+
+	return aff;
+}
+
+/* Lift "maff" to a space with extra dimensions such that the result
+ * has no more existentially quantified variables.
+ * If "ls" is not NULL, then *ls is assigned the local space that lies
+ * at the basis of the lifting applied to "maff".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_lift(__isl_take isl_multi_aff *maff,
+	__isl_give isl_local_space **ls)
+{
+	int i;
+	isl_space *space;
+	unsigned n_div;
+
+	if (ls)
+		*ls = NULL;
+
+	if (!maff)
+		return NULL;
+
+	if (maff->n == 0) {
+		if (ls) {
+			isl_space *space = isl_multi_aff_get_domain_space(maff);
+			*ls = isl_local_space_from_space(space);
+			if (!*ls)
+				return isl_multi_aff_free(maff);
+		}
+		return maff;
+	}
+
+	maff = isl_multi_aff_cow(maff);
+	maff = isl_multi_aff_align_divs(maff);
+	if (!maff)
+		return NULL;
+
+	n_div = isl_aff_dim(maff->p[0], isl_dim_div);
+	space = isl_multi_aff_get_space(maff);
+	space = isl_space_lift(isl_space_domain(space), n_div);
+	space = isl_space_extend_domain_with_range(space,
+						isl_multi_aff_get_space(maff));
+	if (!space)
+		return isl_multi_aff_free(maff);
+	isl_space_free(maff->space);
+	maff->space = space;
+
+	if (ls) {
+		*ls = isl_aff_get_domain_local_space(maff->p[0]);
+		if (!*ls)
+			return isl_multi_aff_free(maff);
+	}
+
+	for (i = 0; i < maff->n; ++i) {
+		maff->p[i] = isl_aff_lift(maff->p[i]);
+		if (!maff->p[i])
+			goto error;
+	}
+
+	return maff;
+error:
+	if (ls)
+		isl_local_space_free(*ls);
+	return isl_multi_aff_free(maff);
+}
+
+
+/* Extract an isl_pw_aff corresponding to output dimension "pos" of "pma".
+ */
+__isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_aff(
+	__isl_keep isl_pw_multi_aff *pma, int pos)
+{
+	int i;
+	int n_out;
+	isl_space *space;
+	isl_pw_aff *pa;
+
+	if (!pma)
+		return NULL;
+
+	n_out = isl_pw_multi_aff_dim(pma, isl_dim_out);
+	if (pos < 0 || pos >= n_out)
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"index out of bounds", return NULL);
+
+	space = isl_pw_multi_aff_get_space(pma);
+	space = isl_space_drop_dims(space, isl_dim_out,
+				    pos + 1, n_out - pos - 1);
+	space = isl_space_drop_dims(space, isl_dim_out, 0, pos);
+
+	pa = isl_pw_aff_alloc_size(space, pma->n);
+	for (i = 0; i < pma->n; ++i) {
+		isl_aff *aff;
+		aff = isl_multi_aff_get_aff(pma->p[i].maff, pos);
+		pa = isl_pw_aff_add_piece(pa, isl_set_copy(pma->p[i].set), aff);
+	}
+
+	return pa;
+}
+
+/* Return an isl_pw_multi_aff with the given "set" as domain and
+ * an unnamed zero-dimensional range.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_domain(
+	__isl_take isl_set *set)
+{
+	isl_multi_aff *ma;
+	isl_space *space;
+
+	space = isl_set_get_space(set);
+	space = isl_space_from_domain(space);
+	ma = isl_multi_aff_zero(space);
+	return isl_pw_multi_aff_alloc(set, ma);
+}
+
+/* Add an isl_pw_multi_aff with the given "set" as domain and
+ * an unnamed zero-dimensional range to *user.
+ */
+static int add_pw_multi_aff_from_domain(__isl_take isl_set *set, void *user)
+{
+	isl_union_pw_multi_aff **upma = user;
+	isl_pw_multi_aff *pma;
+
+	pma = isl_pw_multi_aff_from_domain(set);
+	*upma = isl_union_pw_multi_aff_add_pw_multi_aff(*upma, pma);
+
+	return 0;
+}
+
+/* Return an isl_union_pw_multi_aff with the given "uset" as domain and
+ * an unnamed zero-dimensional range.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_domain(
+	__isl_take isl_union_set *uset)
+{
+	isl_space *space;
+	isl_union_pw_multi_aff *upma;
+
+	if (!uset)
+		return NULL;
+
+	space = isl_union_set_get_space(uset);
+	upma = isl_union_pw_multi_aff_empty(space);
+
+	if (isl_union_set_foreach_set(uset,
+				    &add_pw_multi_aff_from_domain, &upma) < 0)
+		goto error;
+
+	isl_union_set_free(uset);
+	return upma;
+error:
+	isl_union_set_free(uset);
+	isl_union_pw_multi_aff_free(upma);
+	return NULL;
+}
+
+/* Convert "pma" to an isl_map and add it to *umap.
+ */
+static int map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma, void *user)
+{
+	isl_union_map **umap = user;
+	isl_map *map;
+
+	map = isl_map_from_pw_multi_aff(pma);
+	*umap = isl_union_map_add_map(*umap, map);
+
+	return 0;
+}
+
+/* Construct a union map mapping the domain of the union
+ * piecewise multi-affine expression to its range, with each dimension
+ * in the range equated to the corresponding affine expression on its cell.
+ */
+__isl_give isl_union_map *isl_union_map_from_union_pw_multi_aff(
+	__isl_take isl_union_pw_multi_aff *upma)
+{
+	isl_space *space;
+	isl_union_map *umap;
+
+	if (!upma)
+		return NULL;
+
+	space = isl_union_pw_multi_aff_get_space(upma);
+	umap = isl_union_map_empty(space);
+
+	if (isl_union_pw_multi_aff_foreach_pw_multi_aff(upma,
+					&map_from_pw_multi_aff, &umap) < 0)
+		goto error;
+
+	isl_union_pw_multi_aff_free(upma);
+	return umap;
+error:
+	isl_union_pw_multi_aff_free(upma);
+	isl_union_map_free(umap);
+	return NULL;
+}
+
+/* Local data for bin_entry and the callback "fn".
+ */
+struct isl_union_pw_multi_aff_bin_data {
+	isl_union_pw_multi_aff *upma2;
+	isl_union_pw_multi_aff *res;
+	isl_pw_multi_aff *pma;
+	int (*fn)(void **entry, void *user);
+};
+
+/* Given an isl_pw_multi_aff from upma1, store it in data->pma
+ * and call data->fn for each isl_pw_multi_aff in data->upma2.
+ */
+static int bin_entry(void **entry, void *user)
+{
+	struct isl_union_pw_multi_aff_bin_data *data = user;
+	isl_pw_multi_aff *pma = *entry;
+
+	data->pma = pma;
+	if (isl_hash_table_foreach(data->upma2->dim->ctx, &data->upma2->table,
+				   data->fn, data) < 0)
+		return -1;
+
+	return 0;
+}
+
+/* Call "fn" on each pair of isl_pw_multi_affs in "upma1" and "upma2".
+ * The isl_pw_multi_aff from upma1 is stored in data->pma (where data is
+ * passed as user field) and the isl_pw_multi_aff from upma2 is available
+ * as *entry.  The callback should adjust data->res if desired.
+ */
+static __isl_give isl_union_pw_multi_aff *bin_op(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2,
+	int (*fn)(void **entry, void *user))
+{
+	isl_space *space;
+	struct isl_union_pw_multi_aff_bin_data data = { NULL, NULL, NULL, fn };
+
+	space = isl_union_pw_multi_aff_get_space(upma2);
+	upma1 = isl_union_pw_multi_aff_align_params(upma1, space);
+	space = isl_union_pw_multi_aff_get_space(upma1);
+	upma2 = isl_union_pw_multi_aff_align_params(upma2, space);
+
+	if (!upma1 || !upma2)
+		goto error;
+
+	data.upma2 = upma2;
+	data.res = isl_union_pw_multi_aff_alloc(isl_space_copy(upma1->dim),
+				       upma1->table.n);
+	if (isl_hash_table_foreach(upma1->dim->ctx, &upma1->table,
+				   &bin_entry, &data) < 0)
+		goto error;
+
+	isl_union_pw_multi_aff_free(upma1);
+	isl_union_pw_multi_aff_free(upma2);
+	return data.res;
+error:
+	isl_union_pw_multi_aff_free(upma1);
+	isl_union_pw_multi_aff_free(upma2);
+	isl_union_pw_multi_aff_free(data.res);
+	return NULL;
+}
+
+/* Given two isl_multi_affs A -> B and C -> D,
+ * construct an isl_multi_aff (A * C) -> (B, D).
+ */
+__isl_give isl_multi_aff *isl_multi_aff_flat_range_product(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	int i, n1, n2;
+	isl_aff *aff;
+	isl_space *space;
+	isl_multi_aff *res;
+
+	if (!ma1 || !ma2)
+		goto error;
+
+	space = isl_space_range_product(isl_multi_aff_get_space(ma1),
+					isl_multi_aff_get_space(ma2));
+	space = isl_space_flatten_range(space);
+	res = isl_multi_aff_alloc(space);
+
+	n1 = isl_multi_aff_dim(ma1, isl_dim_out);
+	n2 = isl_multi_aff_dim(ma2, isl_dim_out);
+
+	for (i = 0; i < n1; ++i) {
+		aff = isl_multi_aff_get_aff(ma1, i);
+		res = isl_multi_aff_set_aff(res, i, aff);
+	}
+
+	for (i = 0; i < n2; ++i) {
+		aff = isl_multi_aff_get_aff(ma2, i);
+		res = isl_multi_aff_set_aff(res, n1 + i, aff);
+	}
+
+	isl_multi_aff_free(ma1);
+	isl_multi_aff_free(ma2);
+	return res;
+error:
+	isl_multi_aff_free(ma1);
+	isl_multi_aff_free(ma2);
+	return NULL;
+}
+
+/* Given two aligned isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> (B, D).
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_flat_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	isl_space *space;
+
+	space = isl_space_range_product(isl_pw_multi_aff_get_space(pma1),
+					isl_pw_multi_aff_get_space(pma2));
+	space = isl_space_flatten_range(space);
+	return isl_pw_multi_aff_on_shared_domain_in(pma1, pma2, space,
+					    &isl_multi_aff_flat_range_product);
+}
+
+/* Given two isl_pw_multi_affs A -> B and C -> D,
+ * construct an isl_pw_multi_aff (A * C) -> (B, D).
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_flat_range_product(
+	__isl_take isl_pw_multi_aff *pma1, __isl_take isl_pw_multi_aff *pma2)
+{
+	return isl_pw_multi_aff_align_params_pw_pw_and(pma1, pma2,
+					    &pw_multi_aff_flat_range_product);
+}
+
+/* If data->pma and *entry have the same domain space, then compute
+ * their flat range product and the result to data->res.
+ */
+static int flat_range_product_entry(void **entry, void *user)
+{
+	struct isl_union_pw_multi_aff_bin_data *data = user;
+	isl_pw_multi_aff *pma2 = *entry;
+
+	if (!isl_space_tuple_match(data->pma->dim, isl_dim_in,
+				 pma2->dim, isl_dim_in))
+		return 0;
+
+	pma2 = isl_pw_multi_aff_flat_range_product(
+					isl_pw_multi_aff_copy(data->pma),
+					isl_pw_multi_aff_copy(pma2));
+
+	data->res = isl_union_pw_multi_aff_add_pw_multi_aff(data->res, pma2);
+
+	return 0;
+}
+
+/* Given two isl_union_pw_multi_affs A -> B and C -> D,
+ * construct an isl_union_pw_multi_aff (A * C) -> (B, D).
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_flat_range_product(
+	__isl_take isl_union_pw_multi_aff *upma1,
+	__isl_take isl_union_pw_multi_aff *upma2)
+{
+	return bin_op(upma1, upma2, &flat_range_product_entry);
+}