X-Git-Url: http://review.tizen.org/git/?a=blobdiff_plain;f=isl_aff.c;h=13060f0f82471253453c9bcc9d278ea7c37f935c;hb=c53ceda9cc1f9556507c9956647975a1d661a380;hp=becf73bb1a22c174fa27d0ddd17764112878764c;hpb=6817c5f804b1f77ab174af03b960e7c2e10fe342;p=platform%2Fupstream%2Fisl.git

diff --git a/isl_aff.c b/isl_aff.c
index becf73b..13060f0 100644
--- a/isl_aff.c
+++ b/isl_aff.c
@@ -1,9 +1,9 @@
 /*
  * Copyright 2011      INRIA Saclay
  * Copyright 2011      Sven Verdoolaege
- * Copyright 2012      Ecole Normale Superieure
+ * Copyright 2012-2013 Ecole Normale Superieure
  *
- * Use of this software is governed by the GNU LGPLv2.1 license
+ * Use of this software is governed by the MIT license
  *
  * Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
  * Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
@@ -12,17 +12,29 @@
  */
 
 #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>
 #include <isl_mat_private.h>
-#include <isl_list_private.h>
 #include <isl/constraint.h>
 #include <isl/seq.h>
 #include <isl/set.h>
+#include <isl_val_private.h>
 #include <isl_config.h>
 
+#undef BASE
+#define BASE aff
+
+#include <isl_list_templ.c>
+
+#undef BASE
+#define BASE pw_aff
+
+#include <isl_list_templ.c>
+
 __isl_give isl_aff *isl_aff_alloc_vec(__isl_take isl_local_space *ls,
 	__isl_take isl_vec *v)
 {
@@ -86,6 +98,63 @@ __isl_give isl_aff *isl_aff_zero_on_domain(__isl_take isl_local_space *ls)
 	return aff;
 }
 
+/* Return a piecewise affine expression defined on the specified domain
+ * that is equal to zero.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_zero_on_domain(__isl_take isl_local_space *ls)
+{
+	return isl_pw_aff_from_aff(isl_aff_zero_on_domain(ls));
+}
+
+/* Return an affine expression that is equal to the specified dimension
+ * in "ls".
+ */
+__isl_give isl_aff *isl_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	isl_space *space;
+	isl_aff *aff;
+
+	if (!ls)
+		return NULL;
+
+	space = isl_local_space_get_space(ls);
+	if (!space)
+		goto error;
+	if (isl_space_is_map(space))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"expecting (parameter) set space", goto error);
+	if (pos >= isl_local_space_dim(ls, type))
+		isl_die(isl_space_get_ctx(space), isl_error_invalid,
+			"position out of bounds", goto error);
+
+	isl_space_free(space);
+	aff = isl_aff_alloc(ls);
+	if (!aff)
+		return NULL;
+
+	pos += isl_local_space_offset(aff->ls, type);
+
+	isl_int_set_si(aff->v->el[0], 1);
+	isl_seq_clr(aff->v->el + 1, aff->v->size - 1);
+	isl_int_set_si(aff->v->el[1 + pos], 1);
+
+	return aff;
+error:
+	isl_local_space_free(ls);
+	isl_space_free(space);
+	return NULL;
+}
+
+/* Return a piecewise affine expression that is equal to
+ * the specified dimension in "ls".
+ */
+__isl_give isl_pw_aff *isl_pw_aff_var_on_domain(__isl_take isl_local_space *ls,
+	enum isl_dim_type type, unsigned pos)
+{
+	return isl_pw_aff_from_aff(isl_aff_var_on_domain(ls, type, pos));
+}
+
 __isl_give isl_aff *isl_aff_copy(__isl_keep isl_aff *aff)
 {
 	if (!aff)
@@ -340,6 +409,19 @@ int isl_aff_get_denominator(__isl_keep isl_aff *aff, isl_int *v)
 	return 0;
 }
 
+/* Return the common denominator of "aff".
+ */
+__isl_give isl_val *isl_aff_get_denominator_val(__isl_keep isl_aff *aff)
+{
+	isl_ctx *ctx;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	return isl_val_int_from_isl_int(ctx, aff->v->el[0]);
+}
+
 int isl_aff_get_constant(__isl_keep isl_aff *aff, isl_int *v)
 {
 	if (!aff)
@@ -348,6 +430,21 @@ int isl_aff_get_constant(__isl_keep isl_aff *aff, isl_int *v)
 	return 0;
 }
 
+/* Return the constant term of "aff".
+ */
+__isl_give isl_val *isl_aff_get_constant_val(__isl_keep isl_aff *aff)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	v = isl_val_rat_from_isl_int(ctx, aff->v->el[1], aff->v->el[0]);
+	return isl_val_normalize(v);
+}
+
 int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
 	enum isl_dim_type type, int pos, isl_int *v)
 {
@@ -371,6 +468,35 @@ int isl_aff_get_coefficient(__isl_keep isl_aff *aff,
 	return 0;
 }
 
+/* Return the coefficient of the variable of type "type" at position "pos"
+ * of "aff".
+ */
+__isl_give isl_val *isl_aff_get_coefficient_val(__isl_keep isl_aff *aff,
+	enum isl_dim_type type, int pos)
+{
+	isl_ctx *ctx;
+	isl_val *v;
+
+	if (!aff)
+		return NULL;
+
+	ctx = isl_aff_get_ctx(aff);
+	if (type == isl_dim_out)
+		isl_die(ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			return NULL);
+	if (type == isl_dim_in)
+		type = isl_dim_set;
+
+	if (pos >= isl_local_space_dim(aff->ls, type))
+		isl_die(ctx, isl_error_invalid,
+			"position out of bounds", return NULL);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	v = isl_val_rat_from_isl_int(ctx, aff->v->el[1 + pos], aff->v->el[0]);
+	return isl_val_normalize(v);
+}
+
 __isl_give isl_aff *isl_aff_set_denominator(__isl_take isl_aff *aff, isl_int v)
 {
 	aff = isl_aff_cow(aff);
@@ -401,6 +527,53 @@ __isl_give isl_aff *isl_aff_set_constant(__isl_take isl_aff *aff, isl_int v)
 	return aff;
 }
 
+/* Replace the constant term of "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_set_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	if (isl_int_eq(aff->v->el[1], v->n) &&
+	    isl_int_eq(aff->v->el[0], v->d)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_set(aff->v->el[1], v->n);
+	} else if (isl_int_is_one(v->d)) {
+		isl_int_mul(aff->v->el[1], aff->v->el[0], v->n);
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_mul(aff->v->el[1], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_add_constant(__isl_take isl_aff *aff, isl_int v)
 {
 	if (isl_int_is_zero(v))
@@ -419,6 +592,56 @@ __isl_give isl_aff *isl_aff_add_constant(__isl_take isl_aff *aff, isl_int v)
 	return aff;
 }
 
+/* Add "v" to the constant term of "aff".
+ */
+__isl_give isl_aff *isl_aff_add_constant_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_is_one(v->d)) {
+		isl_int_addmul(aff->v->el[1], aff->v->el[0], v->n);
+	} else if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_add(aff->v->el[1], aff->v->el[1], v->n);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_addmul(aff->v->el[1], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_add_constant_si(__isl_take isl_aff *aff, int v)
 {
 	isl_int t;
@@ -431,6 +654,43 @@ __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);
@@ -508,6 +768,66 @@ __isl_give isl_aff *isl_aff_set_coefficient_si(__isl_take isl_aff *aff,
 	return aff;
 }
 
+/* Replace the coefficient of the variable of type "type" at position "pos"
+ * of "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_set_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			goto error);
+	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", goto error);
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	pos += isl_local_space_offset(aff->ls, type);
+	if (isl_int_eq(aff->v->el[1 + pos], v->n) &&
+	    isl_int_eq(aff->v->el[0], v->d)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_set(aff->v->el[1 + pos], v->n);
+	} else if (isl_int_is_one(v->d)) {
+		isl_int_mul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_mul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_add_coefficient(__isl_take isl_aff *aff,
 	enum isl_dim_type type, int pos, isl_int v)
 {
@@ -539,6 +859,69 @@ __isl_give isl_aff *isl_aff_add_coefficient(__isl_take isl_aff *aff,
 	return aff;
 }
 
+/* Add "v" to the coefficient of the variable of type "type"
+ * at position "pos" of "aff".
+ */
+__isl_give isl_aff *isl_aff_add_coefficient_val(__isl_take isl_aff *aff,
+	enum isl_dim_type type, int pos, __isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_zero(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (type == isl_dim_out)
+		isl_die(aff->v->ctx, isl_error_invalid,
+			"output/set dimension does not have a coefficient",
+			goto error);
+	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", goto error);
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational value", goto error);
+
+	aff = isl_aff_cow(aff);
+	if (!aff)
+		goto error;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		goto error;
+
+	pos += isl_local_space_offset(aff->ls, type);
+	if (isl_int_is_one(v->d)) {
+		isl_int_addmul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+	} else if (isl_int_eq(aff->v->el[0], v->d)) {
+		isl_int_add(aff->v->el[1 + pos], aff->v->el[1 + pos], v->n);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	} else {
+		isl_seq_scale(aff->v->el + 1,
+				aff->v->el + 1, v->d, aff->v->size - 1);
+		isl_int_addmul(aff->v->el[1 + pos], aff->v->el[0], v->n);
+		isl_int_mul(aff->v->el[0], aff->v->el[0], v->d);
+		aff->v = isl_vec_normalize(aff->v);
+		if (!aff->v)
+			goto error;
+	}
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_add_coefficient_si(__isl_take isl_aff *aff,
 	enum isl_dim_type type, int pos, int v)
 {
@@ -609,19 +992,243 @@ __isl_give isl_aff *isl_aff_remove_unused_divs( __isl_take isl_aff *aff)
 	return aff;
 }
 
-__isl_give isl_aff *isl_aff_normalize(__isl_take isl_aff *aff)
+/* Given two affine expressions "p" of length p_len (including the
+ * denominator and the constant term) and "subs" of length subs_len,
+ * plug in "subs" for the variable at position "pos".
+ * The variables of "subs" and "p" are assumed to match up to subs_len,
+ * but "p" may have additional variables.
+ * "v" is an initialized isl_int that can be used internally.
+ *
+ * In particular, if "p" represents the expression
+ *
+ *	(a i + g)/m
+ *
+ * with i the variable at position "pos" and "subs" represents the expression
+ *
+ *	f/d
+ *
+ * then the result represents the expression
+ *
+ *	(a f + d g)/(m d)
+ *
+ */
+void isl_seq_substitute(isl_int *p, int pos, isl_int *subs,
+	int p_len, int subs_len, isl_int v)
 {
-	if (!aff)
-		return NULL;
-	aff->v = isl_vec_normalize(aff->v);
-	if (!aff->v)
+	isl_int_set(v, p[1 + pos]);
+	isl_int_set_si(p[1 + pos], 0);
+	isl_seq_combine(p + 1, subs[0], p + 1, v, subs + 1, subs_len - 1);
+	isl_seq_scale(p + subs_len, p + subs_len, subs[0], p_len - subs_len);
+	isl_int_mul(p[0], p[0], subs[0]);
+}
+
+/* Look for any divs in the aff->ls with a denominator equal to one
+ * and plug them into the affine expression and any subsequent divs
+ * that may reference the div.
+ */
+static __isl_give isl_aff *plug_in_integral_divs(__isl_take isl_aff *aff)
+{
+	int i, n;
+	int len;
+	isl_int v;
+	isl_vec *vec;
+	isl_local_space *ls;
+	unsigned pos;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	len = aff->v->size;
+	for (i = 0; i < n; ++i) {
+		if (!isl_int_is_one(aff->ls->div->row[i][0]))
+			continue;
+		ls = isl_local_space_copy(aff->ls);
+		ls = isl_local_space_substitute_seq(ls, isl_dim_div, i,
+				aff->ls->div->row[i], len, i + 1, n - (i + 1));
+		vec = isl_vec_copy(aff->v);
+		vec = isl_vec_cow(vec);
+		if (!ls || !vec)
+			goto error;
+
+		isl_int_init(v);
+
+		pos = isl_local_space_offset(aff->ls, isl_dim_div) + i;
+		isl_seq_substitute(vec->el, pos, aff->ls->div->row[i],
+					len, len, v);
+
+		isl_int_clear(v);
+
+		isl_vec_free(aff->v);
+		aff->v = vec;
+		isl_local_space_free(aff->ls);
+		aff->ls = ls;
+	}
+
+	return aff;
+error:
+	isl_vec_free(vec);
+	isl_local_space_free(ls);
+	return isl_aff_free(aff);
+}
+
+/* Look for any divs j that appear with a unit coefficient inside
+ * the definitions of other divs i and plug them into the definitions
+ * of the divs i.
+ *
+ * In particular, an expression of the form
+ *
+ *	floor((f(..) + floor(g(..)/n))/m)
+ *
+ * is simplified to
+ *
+ *	floor((n * f(..) + g(..))/(n * m))
+ *
+ * This simplification is correct because we can move the expression
+ * f(..) into the inner floor in the original expression to obtain
+ *
+ *	floor(floor((n * f(..) + g(..))/n)/m)
+ *
+ * from which we can derive the simplified expression.
+ */
+static __isl_give isl_aff *plug_in_unit_divs(__isl_take isl_aff *aff)
+{
+	int i, j, n;
+	int off;
+
+	if (!aff)
+		return NULL;
+
+	n = isl_local_space_dim(aff->ls, isl_dim_div);
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+	for (i = 1; i < n; ++i) {
+		for (j = 0; j < i; ++j) {
+			if (!isl_int_is_one(aff->ls->div->row[i][1 + off + j]))
+				continue;
+			aff->ls = isl_local_space_substitute_seq(aff->ls,
+				isl_dim_div, j, aff->ls->div->row[j],
+				aff->v->size, i, 1);
+			if (!aff->ls)
+				return isl_aff_free(aff);
+		}
+	}
+
+	return aff;
+}
+
+/* Swap divs "a" and "b" in "aff", which is assumed to be non-NULL.
+ *
+ * Even though this function is only called on isl_affs with a single
+ * reference, we are careful to only change aff->v and aff->ls together.
+ */
+static __isl_give isl_aff *swap_div(__isl_take isl_aff *aff, int a, int b)
+{
+	unsigned off = isl_local_space_offset(aff->ls, isl_dim_div);
+	isl_local_space *ls;
+	isl_vec *v;
+
+	ls = isl_local_space_copy(aff->ls);
+	ls = isl_local_space_swap_div(ls, a, b);
+	v = isl_vec_copy(aff->v);
+	v = isl_vec_cow(v);
+	if (!ls || !v)
+		goto error;
+
+	isl_int_swap(v->el[1 + off + a], v->el[1 + off + b]);
+	isl_vec_free(aff->v);
+	aff->v = v;
+	isl_local_space_free(aff->ls);
+	aff->ls = ls;
+
+	return aff;
+error:
+	isl_vec_free(v);
+	isl_local_space_free(ls);
+	return isl_aff_free(aff);
+}
+
+/* Merge divs "a" and "b" in "aff", which is assumed to be non-NULL.
+ *
+ * We currently do not actually remove div "b", but simply add its
+ * coefficient to that of "a" and then zero it out.
+ */
+static __isl_give isl_aff *merge_divs(__isl_take isl_aff *aff, int a, int b)
+{
+	unsigned off = isl_local_space_offset(aff->ls, isl_dim_div);
+
+	if (isl_int_is_zero(aff->v->el[1 + off + b]))
+		return aff;
+
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->v)
+		return isl_aff_free(aff);
+
+	isl_int_add(aff->v->el[1 + off + a],
+		    aff->v->el[1 + off + a], aff->v->el[1 + off + b]);
+	isl_int_set_si(aff->v->el[1 + off + b], 0);
+
+	return aff;
+}
+
+/* Sort the divs in the local space of "aff" according to
+ * the comparison function "cmp_row" in isl_local_space.c,
+ * combining the coefficients of identical divs.
+ *
+ * Reordering divs does not change the semantics of "aff",
+ * so there is no need to call isl_aff_cow.
+ * Moreover, this function is currently only called on isl_affs
+ * with a single reference.
+ */
+static __isl_give isl_aff *sort_divs(__isl_take isl_aff *aff)
+{
+	int i, j, n;
+	unsigned off;
+
+	if (!aff)
+		return NULL;
+
+	off = isl_local_space_offset(aff->ls, isl_dim_div);
+	n = isl_aff_dim(aff, isl_dim_div);
+	for (i = 1; i < n; ++i) {
+		for (j = i - 1; j >= 0; --j) {
+			int cmp = isl_mat_cmp_div(aff->ls->div, j, j + 1);
+			if (cmp < 0)
+				break;
+			if (cmp == 0)
+				aff = merge_divs(aff, j, j + 1);
+			else
+				aff = swap_div(aff, j, j + 1);
+			if (!aff)
+				return NULL;
+		}
+	}
+
+	return aff;
+}
+
+/* Normalize the representation of "aff".
+ *
+ * This function should only be called of "new" isl_affs, i.e.,
+ * with only a single reference.  We therefore do not need to
+ * worry about affecting other instances.
+ */
+__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 = plug_in_integral_divs(aff);
+	aff = plug_in_unit_divs(aff);
+	aff = sort_divs(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.
@@ -644,6 +1251,15 @@ __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)
@@ -671,6 +1287,8 @@ __isl_give isl_aff *isl_aff_floor(__isl_take isl_aff *aff)
 	isl_int_set_si(aff->v->el[0], 1);
 	isl_int_set_si(aff->v->el[size], 1);
 
+	aff = isl_aff_normalize(aff);
+
 	return aff;
 }
 
@@ -710,7 +1328,13 @@ __isl_give isl_pw_aff *isl_pw_aff_mod(__isl_take isl_pw_aff *pwaff, isl_int m)
 
 /* Given f, return ceil(f).
  * If f is an integer expression, then just return f.
- * Otherwise, create a new div d = [-f] and return the expression -d.
+ * Otherwise, let f be the expression
+ *
+ *	e/m
+ *
+ * then return
+ *
+ *	floor((e + m - 1)/m)
  */
 __isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff)
 {
@@ -720,9 +1344,16 @@ __isl_give isl_aff *isl_aff_ceil(__isl_take isl_aff *aff)
 	if (isl_int_is_one(aff->v->el[0]))
 		return aff;
 
-	aff = isl_aff_neg(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], aff->v->el[0]);
+	isl_int_sub_ui(aff->v->el[1], aff->v->el[1], 1);
 	aff = isl_aff_floor(aff);
-	aff = isl_aff_neg(aff);
 
 	return aff;
 }
@@ -870,6 +1501,11 @@ __isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff, isl_int f)
 	if (!aff->v)
 		return isl_aff_free(aff);
 
+	if (isl_int_is_pos(f) && isl_int_is_divisible_by(aff->v->el[0], f)) {
+		isl_int_divexact(aff->v->el[0], aff->v->el[0], f);
+		return aff;
+	}
+
 	isl_int_init(gcd);
 	isl_int_gcd(gcd, aff->v->el[0], f);
 	isl_int_divexact(aff->v->el[0], aff->v->el[0], gcd);
@@ -880,6 +1516,34 @@ __isl_give isl_aff *isl_aff_scale(__isl_take isl_aff *aff, isl_int f)
 	return aff;
 }
 
+/* Multiple "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_scale_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational factor", goto error);
+
+	aff = isl_aff_scale(aff, v->n);
+	aff = isl_aff_scale_down(aff, v->d);
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff, isl_int f)
 {
 	isl_int gcd;
@@ -890,6 +1554,11 @@ __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);
@@ -905,6 +1574,37 @@ __isl_give isl_aff *isl_aff_scale_down(__isl_take isl_aff *aff, isl_int f)
 	return aff;
 }
 
+/* Divide "aff" by "v".
+ */
+__isl_give isl_aff *isl_aff_scale_down_val(__isl_take isl_aff *aff,
+	__isl_take isl_val *v)
+{
+	if (!aff || !v)
+		goto error;
+
+	if (isl_val_is_one(v)) {
+		isl_val_free(v);
+		return aff;
+	}
+
+	if (!isl_val_is_rat(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"expecting rational factor", goto error);
+	if (!isl_val_is_pos(v))
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"factor needs to be positive", goto error);
+
+	aff = isl_aff_scale(aff, v->d);
+	aff = isl_aff_scale_down(aff, v->n);
+
+	isl_val_free(v);
+	return aff;
+error:
+	isl_aff_free(aff);
+	isl_val_free(v);
+	return NULL;
+}
+
 __isl_give isl_aff *isl_aff_scale_down_ui(__isl_take isl_aff *aff, unsigned f)
 {
 	isl_int v;
@@ -987,7 +1687,8 @@ static __isl_give isl_aff *isl_aff_substitute_equalities_lifted(
 
 	aff->ls = isl_local_space_substitute_equalities(aff->ls,
 							isl_basic_set_copy(eq));
-	if (!aff->ls)
+	aff->v = isl_vec_cow(aff->v);
+	if (!aff->ls || !aff->v)
 		goto error;
 
 	total = 1 + isl_space_dim(eq->dim, isl_dim_all);
@@ -1022,7 +1723,7 @@ static __isl_give isl_aff *isl_aff_substitute_equalities(
 		goto error;
 	n_div = isl_local_space_dim(aff->ls, isl_dim_div);
 	if (n_div > 0)
-		eq = isl_basic_set_add(eq, isl_dim_set, n_div);
+		eq = isl_basic_set_add_dims(eq, isl_dim_set, n_div);
 	return isl_aff_substitute_equalities_lifted(aff, eq);
 error:
 	isl_basic_set_free(eq);
@@ -1074,8 +1775,10 @@ __isl_give isl_aff *isl_aff_gist_params(__isl_take isl_aff *aff,
 
 /* Return a basic set containing those elements in the space
  * of aff where it is non-negative.
+ * If "rational" is set, then return a rational basic set.
  */
-__isl_give isl_basic_set *isl_aff_nonneg_basic_set(__isl_take isl_aff *aff)
+static __isl_give isl_basic_set *aff_nonneg_basic_set(
+	__isl_take isl_aff *aff, int rational)
 {
 	isl_constraint *ineq;
 	isl_basic_set *bset;
@@ -1083,14 +1786,36 @@ __isl_give isl_basic_set *isl_aff_nonneg_basic_set(__isl_take isl_aff *aff)
 	ineq = isl_inequality_from_aff(aff);
 
 	bset = isl_basic_set_from_constraint(ineq);
+	if (rational)
+		bset = isl_basic_set_set_rational(bset);
 	bset = isl_basic_set_simplify(bset);
 	return bset;
 }
 
 /* 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)
+{
+	return aff_nonneg_basic_set(aff, 0);
+}
+
+/* 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
  * of aff where it is zero.
+ * If "rational" is set, then return a rational basic set.
  */
-__isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff)
+static __isl_give isl_basic_set *aff_zero_basic_set(__isl_take isl_aff *aff,
+	int rational)
 {
 	isl_constraint *ineq;
 	isl_basic_set *bset;
@@ -1098,10 +1823,20 @@ __isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff)
 	ineq = isl_equality_from_aff(aff);
 
 	bset = isl_basic_set_from_constraint(ineq);
+	if (rational)
+		bset = isl_basic_set_set_rational(bset);
 	bset = isl_basic_set_simplify(bset);
 	return bset;
 }
 
+/* Return a basic set containing those elements in the space
+ * of aff where it is zero.
+ */
+__isl_give isl_basic_set *isl_aff_zero_basic_set(__isl_take isl_aff *aff)
+{
+	return aff_zero_basic_set(aff, 0);
+}
+
 /* Return a basic set containing those elements in the shared space
  * of aff1 and aff2 where aff1 is greater than or equal to aff2.
  */
@@ -1556,8 +2291,11 @@ __isl_give isl_set *isl_pw_aff_nonneg_set(__isl_take isl_pw_aff *pwaff)
 	for (i = 0; i < pwaff->n; ++i) {
 		isl_basic_set *bset;
 		isl_set *set_i;
+		int rational;
 
-		bset = isl_aff_nonneg_basic_set(isl_aff_copy(pwaff->p[i].aff));
+		rational = isl_set_has_rational(pwaff->p[i].set);
+		bset = aff_nonneg_basic_set(isl_aff_copy(pwaff->p[i].aff),
+						rational);
 		set_i = isl_set_from_basic_set(bset);
 		set_i = isl_set_intersect(set_i, isl_set_copy(pwaff->p[i].set));
 		set = isl_set_union_disjoint(set, set_i);
@@ -1586,8 +2324,11 @@ static __isl_give isl_set *pw_aff_zero_set(__isl_take isl_pw_aff *pwaff,
 	for (i = 0; i < pwaff->n; ++i) {
 		isl_basic_set *bset;
 		isl_set *set_i, *zero;
+		int rational;
 
-		bset = isl_aff_zero_basic_set(isl_aff_copy(pwaff->p[i].aff));
+		rational = isl_set_has_rational(pwaff->p[i].set);
+		bset = aff_zero_basic_set(isl_aff_copy(pwaff->p[i].aff),
+						rational);
 		zero = isl_set_from_basic_set(bset);
 		set_i = isl_set_copy(pwaff->p[i].set);
 		if (complement)
@@ -1961,6 +2702,46 @@ error:
 	return NULL;
 }
 
+/* Divide "aff1" by "aff2", assuming "aff2" is a piecewise constant.
+ */
+__isl_give isl_aff *isl_aff_div(__isl_take isl_aff *aff1,
+	__isl_take isl_aff *aff2)
+{
+	int is_cst;
+	int neg;
+
+	is_cst = isl_aff_is_cst(aff2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_aff_get_ctx(aff2), isl_error_invalid,
+			"second argument should be a constant", goto error);
+
+	if (!aff2)
+		goto error;
+
+	neg = isl_int_is_neg(aff2->v->el[1]);
+	if (neg) {
+		isl_int_neg(aff2->v->el[0], aff2->v->el[0]);
+		isl_int_neg(aff2->v->el[1], aff2->v->el[1]);
+	}
+
+	aff1 = isl_aff_scale(aff1, aff2->v->el[0]);
+	aff1 = isl_aff_scale_down(aff1, aff2->v->el[1]);
+
+	if (neg) {
+		isl_int_neg(aff2->v->el[0], aff2->v->el[0]);
+		isl_int_neg(aff2->v->el[1], aff2->v->el[1]);
+	}
+
+	isl_aff_free(aff2);
+	return aff1;
+error:
+	isl_aff_free(aff1);
+	isl_aff_free(aff2);
+	return NULL;
+}
+
 static __isl_give isl_pw_aff *pw_aff_add(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
@@ -1991,6 +2772,101 @@ __isl_give isl_pw_aff *isl_pw_aff_mul(__isl_take isl_pw_aff *pwaff1,
 	return isl_pw_aff_align_params_pw_pw_and(pwaff1, pwaff2, &pw_aff_mul);
 }
 
+static __isl_give isl_pw_aff *pw_aff_div(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	return isl_pw_aff_on_shared_domain(pa1, pa2, &isl_aff_div);
+}
+
+/* Divide "pa1" by "pa2", assuming "pa2" is a piecewise constant.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_div(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+	return isl_pw_aff_align_params_pw_pw_and(pa1, pa2, &pw_aff_div);
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
+/* Compute the quotient of the integer division of "pa1" by "pa2"
+ * with rounding towards zero.
+ * "pa2" is assumed to be a piecewise constant.
+ *
+ * In particular, return
+ *
+ *	pa1 >= 0 ? floor(pa1/pa2) : ceil(pa1/pa2)
+ *
+ */
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_q(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+	isl_set *cond;
+	isl_pw_aff *f, *c;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+
+	pa1 = isl_pw_aff_div(pa1, pa2);
+
+	cond = isl_pw_aff_nonneg_set(isl_pw_aff_copy(pa1));
+	f = isl_pw_aff_floor(isl_pw_aff_copy(pa1));
+	c = isl_pw_aff_ceil(pa1);
+	return isl_pw_aff_cond(isl_set_indicator_function(cond), f, c);
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
+/* Compute the remainder of the integer division of "pa1" by "pa2"
+ * with rounding towards zero.
+ * "pa2" is assumed to be a piecewise constant.
+ *
+ * In particular, return
+ *
+ *	pa1 - pa2 * (pa1 >= 0 ? floor(pa1/pa2) : ceil(pa1/pa2))
+ *
+ */
+__isl_give isl_pw_aff *isl_pw_aff_tdiv_r(__isl_take isl_pw_aff *pa1,
+	__isl_take isl_pw_aff *pa2)
+{
+	int is_cst;
+	isl_pw_aff *res;
+
+	is_cst = isl_pw_aff_is_cst(pa2);
+	if (is_cst < 0)
+		goto error;
+	if (!is_cst)
+		isl_die(isl_pw_aff_get_ctx(pa2), isl_error_invalid,
+			"second argument should be a piecewise constant",
+			goto error);
+	res = isl_pw_aff_tdiv_q(isl_pw_aff_copy(pa1), isl_pw_aff_copy(pa2));
+	res = isl_pw_aff_mul(pa2, res);
+	res = isl_pw_aff_sub(pa1, res);
+	return res;
+error:
+	isl_pw_aff_free(pa1);
+	isl_pw_aff_free(pa2);
+	return NULL;
+}
+
 static __isl_give isl_pw_aff *pw_aff_min(__isl_take isl_pw_aff *pwaff1,
 	__isl_take isl_pw_aff *pwaff2)
 {
@@ -2075,39 +2951,163 @@ __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>
+/* Mark the domains of "pwaff" as rational.
+ */
+__isl_give isl_pw_aff *isl_pw_aff_set_rational(__isl_take isl_pw_aff *pwaff)
+{
+	int i;
+
+	pwaff = isl_pw_aff_cow(pwaff);
+	if (!pwaff)
+		return NULL;
+	if (pwaff->n == 0)
+		return pwaff;
+
+	for (i = 0; i < pwaff->n; ++i) {
+		pwaff->p[i].set = isl_set_set_rational(pwaff->p[i].set);
+		if (!pwaff->p[i].set)
+			return isl_pw_aff_free(pwaff);
+	}
+
+	return pwaff;
+}
+
+/* Mark the domains of the elements of "list" as rational.
+ */
+__isl_give isl_pw_aff_list *isl_pw_aff_list_set_rational(
+	__isl_take isl_pw_aff_list *list)
+{
+	int i, n;
+
+	if (!list)
+		return NULL;
+	if (list->n == 0)
+		return list;
+
+	n = list->n;
+	for (i = 0; i < n; ++i) {
+		isl_pw_aff *pa;
+
+		pa = isl_pw_aff_list_get_pw_aff(list, i);
+		pa = isl_pw_aff_set_rational(pa);
+		list = isl_pw_aff_list_set_pw_aff(list, i, pa);
+	}
+
+	return list;
+}
+
+/* Check that the domain space of "aff" matches "space".
+ *
+ * Return 0 on success and -1 on error.
+ */
+int isl_aff_check_match_domain_space(__isl_keep isl_aff *aff,
+	__isl_keep isl_space *space)
+{
+	isl_space *aff_space;
+	int match;
+
+	if (!aff || !space)
+		return -1;
+
+	aff_space = isl_aff_get_domain_space(aff);
+
+	match = isl_space_match(space, isl_dim_param, aff_space, isl_dim_param);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"parameters don't match", goto error);
+	match = isl_space_tuple_match(space, isl_dim_in,
+					aff_space, isl_dim_set);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_aff_get_ctx(aff), isl_error_invalid,
+			"domains don't match", goto error);
+	isl_space_free(aff_space);
+	return 0;
+error:
+	isl_space_free(aff_space);
+	return -1;
+}
+
+#undef BASE
+#define BASE aff
+
+#include <isl_multi_templ.c>
+
+/* 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);
+}
+
+/* Create a piecewise multi-affine expression in the given space that maps each
+ * input dimension to the corresponding output dimension.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_identity(
+	__isl_take isl_space *space)
+{
+	return isl_pw_multi_aff_from_multi_aff(isl_multi_aff_identity(space));
+}
 
 __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;
+	return isl_multi_aff_bin_op(maff1, maff2, &isl_aff_add);
+}
 
-	maff1 = isl_multi_aff_cow(maff1);
-	if (!maff1 || !maff2)
-		goto error;
+/* Subtract "ma2" from "ma1" and return the result.
+ */
+__isl_give isl_multi_aff *isl_multi_aff_sub(__isl_take isl_multi_aff *ma1,
+	__isl_take isl_multi_aff *ma2)
+{
+	return isl_multi_aff_bin_op(ma1, ma2, &isl_aff_sub);
+}
 
-	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);
+/* 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 < maff1->n; ++i) {
-		maff1->p[i] = isl_aff_add(maff1->p[i],
-					    isl_aff_copy(maff2->p[i]));
-		if (!maff1->p[i])
-			goto error;
+	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_multi_aff_free(maff2);
-	return maff1;
-error:
-	isl_multi_aff_free(maff1);
-	isl_multi_aff_free(maff2);
-	return NULL;
+	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.
@@ -2193,57 +3193,34 @@ int isl_multi_aff_plain_is_equal(__isl_keep isl_multi_aff *maff1,
 	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)
+/* 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)
 {
-	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);
-	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;
+	return isl_multi_aff_lex_ge_set(ma2, ma1);
 }
 
-__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)
+/* 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)
 {
-	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;
-	}
+	isl_space *space;
+	isl_map *map1, *map2;
+	isl_map *map, *ge;
 
-	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);
-	}
+	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 maff;
+	return isl_map_domain(map);
 }
 
 #undef PW
@@ -2272,6 +3249,170 @@ __isl_give isl_multi_aff *isl_multi_aff_drop_dims(__isl_take isl_multi_aff *maff
 
 #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)
 {
@@ -2286,13 +3427,77 @@ __isl_give isl_pw_multi_aff *isl_pw_multi_aff_add(
 						&pw_multi_aff_add);
 }
 
+static __isl_give isl_pw_multi_aff *pw_multi_aff_sub(
+	__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_sub);
+}
+
+/* Subtract "pma2" from "pma1" and return the result.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_sub(
+	__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_sub);
+}
+
 __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);
 }
 
-/* Construct a map mapping the domain the piecewise multi-affine expression
+/* 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.
  */
@@ -2325,6 +3530,9 @@ __isl_give isl_map *isl_map_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
 
 __isl_give isl_set *isl_set_from_pw_multi_aff(__isl_take isl_pw_multi_aff *pma)
 {
+	if (!pma)
+		return NULL;
+
 	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",
@@ -2443,34 +3651,18 @@ static __isl_give isl_pw_multi_aff *plain_pw_multi_aff_from_map(
 }
 
 /* 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.
+ * This obviously 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)
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_base(
+	__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)
@@ -2499,48 +3691,544 @@ error:
 	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".
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map,
+ * taking into account that the output dimension at position "d"
+ * can be represented as
  *
- * Let i be the dimension to replace and let "subs" be of the form
+ *	x = floor((e(...) + c1) / m)
  *
- *	f/d
+ * given that constraint "i" is of the form
  *
- * and "aff" of the form
+ *	e(...) + c1 - m x >= 0
  *
- *	(a i + g)/m
  *
- * The result is
+ * Let "map" be of the form
  *
- *	floor((a f + d g')/(m d))
+ *	A -> B
  *
- * where g' is the result of plugging in "subs" in each of the integer
- * divisions in g.
+ * We construct a mapping
+ *
+ *	A -> [A -> x = floor(...)]
+ *
+ * apply that to the map, obtaining
+ *
+ *	[A -> x = floor(...)] -> B
+ *
+ * and equate dimension "d" to x.
+ * We then compute a isl_pw_multi_aff representation of the resulting map
+ * and plug in the mapping above.
  */
-__isl_give isl_aff *isl_aff_substitute(__isl_take isl_aff *aff,
-	enum isl_dim_type type, unsigned pos, __isl_keep isl_aff *subs)
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_div(
+	__isl_take isl_map *map, __isl_take isl_basic_map *hull, int d, int i)
+{
+	isl_ctx *ctx;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_multi_aff *ma;
+	isl_aff *aff;
+	isl_vec *v;
+	isl_map *insert;
+	int offset;
+	int n;
+	int n_in;
+	isl_pw_multi_aff *pma;
+	int is_set;
+
+	is_set = isl_map_is_set(map);
+
+	offset = isl_basic_map_offset(hull, isl_dim_out);
+	ctx = isl_map_get_ctx(map);
+	space = isl_space_domain(isl_map_get_space(map));
+	n_in = isl_space_dim(space, isl_dim_set);
+	n = isl_space_dim(space, isl_dim_all);
+
+	v = isl_vec_alloc(ctx, 1 + 1 + n);
+	if (v) {
+		isl_int_neg(v->el[0], hull->ineq[i][offset + d]);
+		isl_seq_cpy(v->el + 1, hull->ineq[i], 1 + n);
+	}
+	isl_basic_map_free(hull);
+
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	aff = isl_aff_alloc_vec(ls, v);
+	aff = isl_aff_floor(aff);
+	if (is_set) {
+		isl_space_free(space);
+		ma = isl_multi_aff_from_aff(aff);
+	} else {
+		ma = isl_multi_aff_identity(isl_space_map_from_set(space));
+		ma = isl_multi_aff_range_product(ma,
+						isl_multi_aff_from_aff(aff));
+	}
+
+	insert = isl_map_from_multi_aff(isl_multi_aff_copy(ma));
+	map = isl_map_apply_domain(map, insert);
+	map = isl_map_equate(map, isl_dim_in, n_in, isl_dim_out, d);
+	pma = isl_pw_multi_aff_from_map(map);
+	pma = isl_pw_multi_aff_pullback_multi_aff(pma, ma);
+
+	return pma;
+}
+
+/* Is constraint "c" of the form
+ *
+ *	e(...) + c1 - m x >= 0
+ *
+ * or
+ *
+ *	-e(...) + c2 + m x >= 0
+ *
+ * where m > 1 and e only depends on parameters and input dimemnsions?
+ *
+ * "offset" is the offset of the output dimensions
+ * "pos" is the position of output dimension x.
+ */
+static int is_potential_div_constraint(isl_int *c, int offset, int d, int total)
+{
+	if (isl_int_is_zero(c[offset + d]))
+		return 0;
+	if (isl_int_is_one(c[offset + d]))
+		return 0;
+	if (isl_int_is_negone(c[offset + d]))
+		return 0;
+	if (isl_seq_first_non_zero(c + offset, d) != -1)
+		return 0;
+	if (isl_seq_first_non_zero(c + offset + d + 1,
+				    total - (offset + d + 1)) != -1)
+		return 0;
+	return 1;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ *
+ * As a special case, we first check if there is any pair of constraints,
+ * shared by all the basic maps in "map" that force a given dimension
+ * to be equal to the floor of some affine combination of the input dimensions.
+ *
+ * In particular, if we can find two constraints
+ *
+ *	e(...) + c1 - m x >= 0		i.e.,		m x <= e(...) + c1
+ *
+ * and
+ *
+ *	-e(...) + c2 + m x >= 0		i.e.,		m x >= e(...) - c2
+ *
+ * where m > 1 and e only depends on parameters and input dimemnsions,
+ * and such that
+ *
+ *	c1 + c2 < m			i.e.,		-c2 >= c1 - (m - 1)
+ *
+ * then we know that we can take
+ *
+ *	x = floor((e(...) + c1) / m)
+ *
+ * without having to perform any computation.
+ *
+ * Note that we know that
+ *
+ *	c1 + c2 >= 1
+ *
+ * If c1 + c2 were 0, then we would have detected an equality during
+ * simplification.  If c1 + c2 were negative, then we would have detected
+ * a contradiction.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_check_div(
+	__isl_take isl_map *map)
+{
+	int d, dim;
+	int i, j, n;
+	int offset, total;
+	isl_int sum;
+	isl_basic_map *hull;
+
+	hull = isl_map_unshifted_simple_hull(isl_map_copy(map));
+	if (!hull)
+		goto error;
+
+	isl_int_init(sum);
+	dim = isl_map_dim(map, isl_dim_out);
+	offset = isl_basic_map_offset(hull, isl_dim_out);
+	total = 1 + isl_basic_map_total_dim(hull);
+	n = hull->n_ineq;
+	for (d = 0; d < dim; ++d) {
+		for (i = 0; i < n; ++i) {
+			if (!is_potential_div_constraint(hull->ineq[i],
+							offset, d, total))
+				continue;
+			for (j = i + 1; j < n; ++j) {
+				if (!isl_seq_is_neg(hull->ineq[i] + 1,
+						hull->ineq[j] + 1, total - 1))
+					continue;
+				isl_int_add(sum, hull->ineq[i][0],
+						hull->ineq[j][0]);
+				if (isl_int_abs_lt(sum,
+						    hull->ineq[i][offset + d]))
+					break;
+
+			}
+			if (j >= n)
+				continue;
+			isl_int_clear(sum);
+			if (isl_int_is_pos(hull->ineq[j][offset + d]))
+				j = i;
+			return pw_multi_aff_from_map_div(map, hull, d, j);
+		}
+	}
+	isl_int_clear(sum);
+	isl_basic_map_free(hull);
+	return pw_multi_aff_from_map_base(map);
+error:
+	isl_map_free(map);
+	isl_basic_map_free(hull);
+	return NULL;
+}
+
+/* Given an affine expression
+ *
+ *	[A -> B] -> f(A,B)
+ *
+ * construct an isl_multi_aff
+ *
+ *	[A -> B] -> B'
+ *
+ * such that dimension "d" in B' is set to "aff" and the remaining
+ * dimensions are set equal to the corresponding dimensions in B.
+ * "n_in" is the dimension of the space A.
+ * "n_out" is the dimension of the space B.
+ *
+ * If "is_set" is set, then the affine expression is of the form
+ *
+ *	[B] -> f(B)
+ *
+ * and we construct an isl_multi_aff
+ *
+ *	B -> B'
+ */
+static __isl_give isl_multi_aff *range_map(__isl_take isl_aff *aff, int d,
+	unsigned n_in, unsigned n_out, int is_set)
+{
+	int i;
+	isl_multi_aff *ma;
+	isl_space *space, *space2;
+	isl_local_space *ls;
+
+	space = isl_aff_get_domain_space(aff);
+	ls = isl_local_space_from_space(isl_space_copy(space));
+	space2 = isl_space_copy(space);
+	if (!is_set)
+		space2 = isl_space_range(isl_space_unwrap(space2));
+	space = isl_space_map_from_domain_and_range(space, space2);
+	ma = isl_multi_aff_alloc(space);
+	ma = isl_multi_aff_set_aff(ma, d, aff);
+
+	for (i = 0; i < n_out; ++i) {
+		if (i == d)
+			continue;
+		aff = isl_aff_var_on_domain(isl_local_space_copy(ls),
+						isl_dim_set, n_in + i);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+
+	isl_local_space_free(ls);
+
+	return ma;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map,
+ * taking into account that the dimension at position "d" can be written as
+ *
+ *	x = m a + f(..)						(1)
+ *
+ * where m is equal to "gcd".
+ * "i" is the index of the equality in "hull" that defines f(..).
+ * In particular, the equality is of the form
+ *
+ *	f(..) - x + m g(existentials) = 0
+ *
+ * or
+ *
+ *	-f(..) + x + m g(existentials) = 0
+ *
+ * We basically plug (1) into "map", resulting in a map with "a"
+ * in the range instead of "x".  The corresponding isl_pw_multi_aff
+ * defining "a" is then plugged back into (1) to obtain a definition fro "x".
+ *
+ * Specifically, given the input map
+ *
+ *	A -> B
+ *
+ * We first wrap it into a set
+ *
+ *	[A -> B]
+ *
+ * and define (1) on top of the corresponding space, resulting in "aff".
+ * We use this to create an isl_multi_aff that maps the output position "d"
+ * from "a" to "x", leaving all other (intput and output) dimensions unchanged.
+ * We plug this into the wrapped map, unwrap the result and compute the
+ * corresponding isl_pw_multi_aff.
+ * The result is an expression
+ *
+ *	A -> T(A)
+ *
+ * We adjust that to
+ *
+ *	A -> [A -> T(A)]
+ *
+ * so that we can plug that into "aff", after extending the latter to
+ * a mapping
+ *
+ *	[A -> B] -> B'
+ *
+ *
+ * If "map" is actually a set, then there is no "A" space, meaning
+ * that we do not need to perform any wrapping, and that the result
+ * of the recursive call is of the form
+ *
+ *	[T]
+ *
+ * which is plugged into a mapping of the form
+ *
+ *	B -> B'
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_from_map_stride(
+	__isl_take isl_map *map, __isl_take isl_basic_map *hull, int d, int i,
+	isl_int gcd)
+{
+	isl_set *set;
+	isl_space *space;
+	isl_local_space *ls;
+	isl_aff *aff;
+	isl_multi_aff *ma;
+	isl_pw_multi_aff *pma, *id;
+	unsigned n_in;
+	unsigned o_out;
+	unsigned n_out;
+	int is_set;
+
+	is_set = isl_map_is_set(map);
+
+	n_in = isl_basic_map_dim(hull, isl_dim_in);
+	n_out = isl_basic_map_dim(hull, isl_dim_out);
+	o_out = isl_basic_map_offset(hull, isl_dim_out);
+
+	if (is_set)
+		set = map;
+	else
+		set = isl_map_wrap(map);
+	space = isl_space_map_from_set(isl_set_get_space(set));
+	ma = isl_multi_aff_identity(space);
+	ls = isl_local_space_from_space(isl_set_get_space(set));
+	aff = isl_aff_alloc(ls);
+	if (aff) {
+		isl_int_set_si(aff->v->el[0], 1);
+		if (isl_int_is_one(hull->eq[i][o_out + d]))
+			isl_seq_neg(aff->v->el + 1, hull->eq[i],
+				    aff->v->size - 1);
+		else
+			isl_seq_cpy(aff->v->el + 1, hull->eq[i],
+				    aff->v->size - 1);
+		isl_int_set(aff->v->el[1 + o_out + d], gcd);
+	}
+	ma = isl_multi_aff_set_aff(ma, n_in + d, isl_aff_copy(aff));
+	set = isl_set_preimage_multi_aff(set, ma);
+
+	ma = range_map(aff, d, n_in, n_out, is_set);
+
+	if (is_set)
+		map = set;
+	else
+		map = isl_set_unwrap(set);
+	pma = isl_pw_multi_aff_from_map(set);
+
+	if (!is_set) {
+		space = isl_pw_multi_aff_get_domain_space(pma);
+		space = isl_space_map_from_set(space);
+		id = isl_pw_multi_aff_identity(space);
+		pma = isl_pw_multi_aff_range_product(id, pma);
+	}
+	id = isl_pw_multi_aff_from_multi_aff(ma);
+	pma = isl_pw_multi_aff_pullback_pw_multi_aff(id, pma);
+
+	isl_basic_map_free(hull);
+	return pma;
+}
+
+/* Try and create an isl_pw_multi_aff that is equivalent to the given isl_map.
+ *
+ * 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.
+ *
+ * Otherwise, we check if any of the output dimensions is "strided".
+ * That is, we check if can be written as
+ *
+ *	x = m a + f(..)
+ *
+ * with m greater than 1, a some combination of existentiall quantified
+ * variables and f and expression in the parameters and input dimensions.
+ * If so, we remove the stride in pw_multi_aff_from_map_stride.
+ *
+ * Otherwise, we continue with pw_multi_aff_from_map_check_div for a further
+ * special case.
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_from_map(__isl_take isl_map *map)
+{
+	int i, j;
+	int sv;
+	isl_basic_map *hull;
+	unsigned n_out;
+	unsigned o_out;
+	unsigned n_div;
+	unsigned o_div;
+	isl_int gcd;
+
+	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);
+	if (sv < 0)
+		hull = isl_basic_map_free(hull);
+	if (!hull)
+		goto error;
+
+	n_div = isl_basic_map_dim(hull, isl_dim_div);
+	o_div = isl_basic_map_offset(hull, isl_dim_div);
+
+	if (n_div == 0) {
+		isl_basic_map_free(hull);
+		return pw_multi_aff_from_map_check_div(map);
+	}
+
+	isl_int_init(gcd);
+
+	n_out = isl_basic_map_dim(hull, isl_dim_out);
+	o_out = isl_basic_map_offset(hull, isl_dim_out);
+
+	for (i = 0; i < n_out; ++i) {
+		for (j = 0; j < hull->n_eq; ++j) {
+			isl_int *eq = hull->eq[j];
+			isl_pw_multi_aff *res;
+
+			if (!isl_int_is_one(eq[o_out + i]) &&
+			    !isl_int_is_negone(eq[o_out + i]))
+				continue;
+			if (isl_seq_first_non_zero(eq + o_out, i) != -1)
+				continue;
+			if (isl_seq_first_non_zero(eq + o_out + i + 1,
+						    n_out - (i + 1)) != -1)
+				continue;
+			isl_seq_gcd(eq + o_div, n_div, &gcd);
+			if (isl_int_is_zero(gcd))
+				continue;
+			if (isl_int_is_one(gcd))
+				continue;
+
+			res = pw_multi_aff_from_map_stride(map, hull,
+								i, j, gcd);
+			isl_int_clear(gcd);
+			return res;
+		}
+	}
+
+	isl_int_clear(gcd);
+	isl_basic_map_free(hull);
+	return pw_multi_aff_from_map_check_div(map);
+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);
+}
+
+/* Convert "map" into an isl_pw_multi_aff (if possible) and
+ * add it to *user.
+ */
+static int pw_multi_aff_from_map(__isl_take isl_map *map, void *user)
+{
+	isl_union_pw_multi_aff **upma = user;
+	isl_pw_multi_aff *pma;
+
+	pma = isl_pw_multi_aff_from_map(map);
+	*upma = isl_union_pw_multi_aff_add_pw_multi_aff(*upma, pma);
+
+	return *upma ? 0 : -1;
+}
+
+/* Try and create an isl_union_pw_multi_aff that is equivalent
+ * to the given isl_union_map.
+ * The isl_union_map is required to be single-valued in each space.
+ * Otherwise, an error is produced.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_map(
+	__isl_take isl_union_map *umap)
+{
+	isl_space *space;
+	isl_union_pw_multi_aff *upma;
+
+	space = isl_union_map_get_space(umap);
+	upma = isl_union_pw_multi_aff_empty(space);
+	if (isl_union_map_foreach_map(umap, &pw_multi_aff_from_map, &upma) < 0)
+		upma = isl_union_pw_multi_aff_free(upma);
+	isl_union_map_free(umap);
+
+	return upma;
+}
+
+/* Try and create an isl_union_pw_multi_aff that is equivalent
+ * to the given isl_union_set.
+ * The isl_union_set is required to be a singleton in each space.
+ * Otherwise, an error is produced.
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_from_union_set(
+	__isl_take isl_union_set *uset)
+{
+	return isl_union_pw_multi_aff_from_union_map(uset);
+}
+
+/* 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
+ *
+ *	(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;
@@ -2568,11 +4256,8 @@ __isl_give isl_aff *isl_aff_substitute(__isl_take isl_aff *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_seq_substitute(aff->v->el, pos, subs->v->el,
+			    aff->v->size, subs->v->size, v);
 	isl_int_clear(v);
 
 	return aff;
@@ -2634,30 +4319,236 @@ __isl_give isl_pw_multi_aff *isl_pw_multi_aff_substitute(
 	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;
-			}
+	for (i = 0; i < pma->n; ++i) {
+		for (j = 0; j < subs->n; ++j) {
+			isl_set *common;
+			isl_multi_aff *res_ij;
+			int empty;
+
+			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);
+			empty = isl_set_plain_is_empty(common);
+			if (empty < 0 || empty) {
+				isl_set_free(common);
+				if (empty < 0)
+					goto error;
+				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;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_multi_aff_free(res);
+	return NULL;
+}
+
+/* Compute the preimage of a range of dimensions in the affine expression "src"
+ * under "ma" and put the result in "dst".  The number of dimensions in "src"
+ * that precede the range is given by "n_before".  The number of dimensions
+ * in the range is given by the number of output dimensions of "ma".
+ * The number of dimensions that follow the range is given by "n_after".
+ * If "has_denom" is set (to one),
+ * then "src" and "dst" have an extra initial denominator.
+ * "n_div_ma" is the number of existentials in "ma"
+ * "n_div_bset" is the number of existentials in "src"
+ * The resulting "dst" (which is assumed to have been allocated by
+ * the caller) contains coefficients for both sets of existentials,
+ * first those in "ma" and then those in "src".
+ * f, c1, c2 and g are temporary objects that have been initialized
+ * by the caller.
+ *
+ * Let src represent the expression
+ *
+ *	(a(p) + f_u u + b v + f_w w + c(divs))/d
+ *
+ * and let ma represent the expressions
+ *
+ *	v_i = (r_i(p) + s_i(y) + t_i(divs'))/m_i
+ *
+ * We start out with the following expression for dst:
+ *
+ *	(a(p) + f_u u + 0 y + f_w w + 0 divs' + c(divs) + f \sum_i b_i v_i)/d
+ *
+ * with the multiplication factor f initially equal to 1
+ * and f \sum_i b_i v_i kept separately.
+ * For each x_i that we substitute, we multiply the numerator
+ * (and denominator) of dst by c_1 = m_i and add the numerator
+ * of the x_i expression multiplied by c_2 = f b_i,
+ * after removing the common factors of c_1 and c_2.
+ * The multiplication factor f also needs to be multiplied by c_1
+ * for the next x_j, j > i.
+ */
+void isl_seq_preimage(isl_int *dst, isl_int *src,
+	__isl_keep isl_multi_aff *ma, int n_before, int n_after,
+	int n_div_ma, int n_div_bmap,
+	isl_int f, isl_int c1, isl_int c2, isl_int g, int has_denom)
+{
+	int i;
+	int n_param, n_in, n_out;
+	int o_dst, o_src;
+
+	n_param = isl_multi_aff_dim(ma, isl_dim_param);
+	n_in = isl_multi_aff_dim(ma, isl_dim_in);
+	n_out = isl_multi_aff_dim(ma, isl_dim_out);
+
+	isl_seq_cpy(dst, src, has_denom + 1 + n_param + n_before);
+	o_dst = o_src = has_denom + 1 + n_param + n_before;
+	isl_seq_clr(dst + o_dst, n_in);
+	o_dst += n_in;
+	o_src += n_out;
+	isl_seq_cpy(dst + o_dst, src + o_src, n_after);
+	o_dst += n_after;
+	o_src += n_after;
+	isl_seq_clr(dst + o_dst, n_div_ma);
+	o_dst += n_div_ma;
+	isl_seq_cpy(dst + o_dst, src + o_src, n_div_bmap);
+
+	isl_int_set_si(f, 1);
+
+	for (i = 0; i < n_out; ++i) {
+		int offset = has_denom + 1 + n_param + n_before + i;
+
+		if (isl_int_is_zero(src[offset]))
+			continue;
+		isl_int_set(c1, ma->p[i]->v->el[0]);
+		isl_int_mul(c2, f, src[offset]);
+		isl_int_gcd(g, c1, c2);
+		isl_int_divexact(c1, c1, g);
+		isl_int_divexact(c2, c2, g);
+
+		isl_int_mul(f, f, c1);
+		o_dst = has_denom;
+		o_src = 1;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, 1 + n_param);
+		o_dst += 1 + n_param;
+		o_src += 1 + n_param;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_before);
+		o_dst += n_before;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, n_in);
+		o_dst += n_in;
+		o_src += n_in;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_after);
+		o_dst += n_after;
+		isl_seq_combine(dst + o_dst, c1, dst + o_dst,
+				c2, ma->p[i]->v->el + o_src, n_div_ma);
+		o_dst += n_div_ma;
+		o_src += n_div_ma;
+		isl_seq_scale(dst + o_dst, dst + o_dst, c1, n_div_bmap);
+		if (has_denom)
+			isl_int_mul(dst[0], dst[0], c1);
+	}
+}
+
+/* Compute the pullback of "aff" by the function represented by "ma".
+ * In other words, plug in "ma" in "aff".  The result is an affine expression
+ * defined over the domain space of "ma".
+ *
+ * If "aff" is represented by
+ *
+ *	(a(p) + b x + c(divs))/d
+ *
+ * and ma is represented by
+ *
+ *	x = D(p) + F(y) + G(divs')
+ *
+ * then the result is
+ *
+ *	(a(p) + b D(p) + b F(y) + b G(divs') + c(divs))/d
+ *
+ * The divs in the local space of the input are similarly adjusted
+ * through a call to isl_local_space_preimage_multi_aff.
+ */
+__isl_give isl_aff *isl_aff_pullback_multi_aff(__isl_take isl_aff *aff,
+	__isl_take isl_multi_aff *ma)
+{
+	isl_aff *res = NULL;
+	isl_local_space *ls;
+	int n_div_aff, n_div_ma;
+	isl_int f, c1, c2, g;
+
+	ma = isl_multi_aff_align_divs(ma);
+	if (!aff || !ma)
+		goto error;
+
+	n_div_aff = isl_aff_dim(aff, isl_dim_div);
+	n_div_ma = ma->n ? isl_aff_dim(ma->p[0], isl_dim_div) : 0;
+
+	ls = isl_aff_get_domain_local_space(aff);
+	ls = isl_local_space_preimage_multi_aff(ls, isl_multi_aff_copy(ma));
+	res = isl_aff_alloc(ls);
+	if (!res)
+		goto error;
+
+	isl_int_init(f);
+	isl_int_init(c1);
+	isl_int_init(c2);
+	isl_int_init(g);
+
+	isl_seq_preimage(res->v->el, aff->v->el, ma, 0, 0, n_div_ma, n_div_aff,
+			f, c1, c2, g, 1);
+
+	isl_int_clear(f);
+	isl_int_clear(c1);
+	isl_int_clear(c2);
+	isl_int_clear(g);
+
+	isl_aff_free(aff);
+	isl_multi_aff_free(ma);
+	res = isl_aff_normalize(res);
+	return res;
+error:
+	isl_aff_free(aff);
+	isl_multi_aff_free(ma);
+	isl_aff_free(res);
+	return NULL;
+}
+
+/* Compute the pullback of "ma1" by the function represented by "ma2".
+ * In other words, plug in "ma2" in "ma1".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_pullback_multi_aff(
+	__isl_take isl_multi_aff *ma1, __isl_take isl_multi_aff *ma2)
+{
+	int i;
+	isl_space *space = NULL;
+
+	ma2 = isl_multi_aff_align_divs(ma2);
+	ma1 = isl_multi_aff_cow(ma1);
+	if (!ma1 || !ma2)
+		goto error;
 
-			res_ij = isl_multi_aff_substitute(
-					isl_multi_aff_copy(pma->p[i].maff),
-					type, pos, subs->p[j].aff);
+	space = isl_space_join(isl_multi_aff_get_space(ma2),
+				isl_multi_aff_get_space(ma1));
 
-			res = isl_pw_multi_aff_add_piece(res, common, res_ij);
-		}
+	for (i = 0; i < ma1->n; ++i) {
+		ma1->p[i] = isl_aff_pullback_multi_aff(ma1->p[i],
+						    isl_multi_aff_copy(ma2));
+		if (!ma1->p[i])
+			goto error;
 	}
 
-	isl_pw_multi_aff_free(pma);
-	return res;
+	ma1 = isl_multi_aff_reset_space(ma1, space);
+	isl_multi_aff_free(ma2);
+	return ma1;
+error:
+	isl_space_free(space);
+	isl_multi_aff_free(ma2);
+	isl_multi_aff_free(ma1);
+	return NULL;
 }
 
 /* Extend the local space of "dst" to include the divs
@@ -2834,3 +4725,464 @@ __isl_give isl_pw_aff *isl_pw_multi_aff_get_pw_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 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_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));
+	return isl_pw_multi_aff_on_shared_domain_in(pma1, pma2, space,
+					    &isl_multi_aff_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_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_range_product);
+}
+
+/* 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);
+}
+
+/* Replace the affine expressions at position "pos" in "pma" by "pa".
+ * The parameters are assumed to have been aligned.
+ *
+ * The implementation essentially performs an isl_pw_*_on_shared_domain,
+ * except that it works on two different isl_pw_* types.
+ */
+static __isl_give isl_pw_multi_aff *pw_multi_aff_set_pw_aff(
+	__isl_take isl_pw_multi_aff *pma, unsigned pos,
+	__isl_take isl_pw_aff *pa)
+{
+	int i, j, n;
+	isl_pw_multi_aff *res = NULL;
+
+	if (!pma || !pa)
+		goto error;
+
+	if (!isl_space_tuple_match(pma->dim, isl_dim_in, pa->dim, isl_dim_in))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"domains don't match", goto error);
+	if (pos >= isl_pw_multi_aff_dim(pma, isl_dim_out))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"index out of bounds", goto error);
+
+	n = pma->n * pa->n;
+	res = isl_pw_multi_aff_alloc_size(isl_pw_multi_aff_get_space(pma), n);
+
+	for (i = 0; i < pma->n; ++i) {
+		for (j = 0; j < pa->n; ++j) {
+			isl_set *common;
+			isl_multi_aff *res_ij;
+			int empty;
+
+			common = isl_set_intersect(isl_set_copy(pma->p[i].set),
+						   isl_set_copy(pa->p[j].set));
+			empty = isl_set_plain_is_empty(common);
+			if (empty < 0 || empty) {
+				isl_set_free(common);
+				if (empty < 0)
+					goto error;
+				continue;
+			}
+
+			res_ij = isl_multi_aff_set_aff(
+					isl_multi_aff_copy(pma->p[i].maff), pos,
+					isl_aff_copy(pa->p[j].aff));
+			res_ij = isl_multi_aff_gist(res_ij,
+					isl_set_copy(common));
+
+			res = isl_pw_multi_aff_add_piece(res, common, res_ij);
+		}
+	}
+
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return res;
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return isl_pw_multi_aff_free(res);
+}
+
+/* Replace the affine expressions at position "pos" in "pma" by "pa".
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_set_pw_aff(
+	__isl_take isl_pw_multi_aff *pma, unsigned pos,
+	__isl_take isl_pw_aff *pa)
+{
+	if (!pma || !pa)
+		goto error;
+	if (isl_space_match(pma->dim, isl_dim_param, pa->dim, isl_dim_param))
+		return pw_multi_aff_set_pw_aff(pma, pos, pa);
+	if (!isl_space_has_named_params(pma->dim) ||
+	    !isl_space_has_named_params(pa->dim))
+		isl_die(isl_pw_multi_aff_get_ctx(pma), isl_error_invalid,
+			"unaligned unnamed parameters", goto error);
+	pma = isl_pw_multi_aff_align_params(pma, isl_pw_aff_get_space(pa));
+	pa = isl_pw_aff_align_params(pa, isl_pw_multi_aff_get_space(pma));
+	return pw_multi_aff_set_pw_aff(pma, pos, pa);
+error:
+	isl_pw_multi_aff_free(pma);
+	isl_pw_aff_free(pa);
+	return NULL;
+}
+
+/* Check that the domain space of "pa" matches "space".
+ *
+ * Return 0 on success and -1 on error.
+ */
+int isl_pw_aff_check_match_domain_space(__isl_keep isl_pw_aff *pa,
+	__isl_keep isl_space *space)
+{
+	isl_space *pa_space;
+	int match;
+
+	if (!pa || !space)
+		return -1;
+
+	pa_space = isl_pw_aff_get_space(pa);
+
+	match = isl_space_match(space, isl_dim_param, pa_space, isl_dim_param);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"parameters don't match", goto error);
+	match = isl_space_tuple_match(space, isl_dim_in, pa_space, isl_dim_in);
+	if (match < 0)
+		goto error;
+	if (!match)
+		isl_die(isl_pw_aff_get_ctx(pa), isl_error_invalid,
+			"domains don't match", goto error);
+	isl_space_free(pa_space);
+	return 0;
+error:
+	isl_space_free(pa_space);
+	return -1;
+}
+
+#undef BASE
+#define BASE pw_aff
+
+#include <isl_multi_templ.c>
+
+/* Scale the first elements of "ma" by the corresponding elements of "vec".
+ */
+__isl_give isl_multi_aff *isl_multi_aff_scale_vec(__isl_take isl_multi_aff *ma,
+	__isl_take isl_vec *vec)
+{
+	int i, n;
+	isl_int v;
+
+	if (!ma || !vec)
+		goto error;
+
+	n = isl_multi_aff_dim(ma, isl_dim_out);
+	if (isl_vec_size(vec) < n)
+		n = isl_vec_size(vec);
+
+	isl_int_init(v);
+	for (i = 0; i < n; ++i) {
+		isl_aff *aff;
+
+		isl_vec_get_element(vec, i, &v);
+
+		aff = isl_multi_aff_get_aff(ma, i);
+		aff = isl_aff_scale(aff, v);
+		ma = isl_multi_aff_set_aff(ma, i, aff);
+	}
+	isl_int_clear(v);
+
+	isl_vec_free(vec);
+	return ma;
+error:
+	isl_vec_free(vec);
+	isl_multi_aff_free(ma);
+	return NULL;
+}
+
+/* Scale the first elements of "pma" by the corresponding elements of "vec".
+ */
+__isl_give isl_pw_multi_aff *isl_pw_multi_aff_scale_vec(
+	__isl_take isl_pw_multi_aff *pma, __isl_take isl_vec *v)
+{
+	int i;
+
+	pma = isl_pw_multi_aff_cow(pma);
+	if (!pma || !v)
+		goto error;
+
+	for (i = 0; i < pma->n; ++i) {
+		pma->p[i].maff = isl_multi_aff_scale_vec(pma->p[i].maff,
+							isl_vec_copy(v));
+		if (!pma->p[i].maff)
+			goto error;
+	}
+
+	isl_vec_free(v);
+	return pma;
+error:
+	isl_vec_free(v);
+	isl_pw_multi_aff_free(pma);
+	return NULL;
+}
+
+/* This function is called for each entry of an isl_union_pw_multi_aff.
+ * Replace the entry by the result of applying isl_pw_multi_aff_scale_vec
+ * to the original entry with the isl_vec in "user" as extra argument.
+ */
+static int union_pw_multi_aff_scale_vec_entry(void **entry, void *user)
+{
+	isl_pw_multi_aff **pma = (isl_pw_multi_aff **) entry;
+	isl_vec *v = user;
+
+	*pma = isl_pw_multi_aff_scale_vec(*pma, isl_vec_copy(v));
+	if (!*pma)
+		return -1;
+
+	return 0;
+}
+
+/* Scale the first elements of "upma" by the corresponding elements of "vec".
+ */
+__isl_give isl_union_pw_multi_aff *isl_union_pw_multi_aff_scale_vec(
+	__isl_take isl_union_pw_multi_aff *upma, __isl_take isl_vec *v)
+{
+	upma = isl_union_pw_multi_aff_cow(upma);
+	if (!upma || !v)
+		goto error;
+
+	if (isl_hash_table_foreach(upma->dim->ctx, &upma->table,
+				   &union_pw_multi_aff_scale_vec_entry, v) < 0)
+		goto error;
+
+	isl_vec_free(v);
+	return upma;
+error:
+	isl_vec_free(v);
+	isl_union_pw_multi_aff_free(upma);
+	return NULL;
+}