[thirdparty/gcc.git] / gcc / tree-data-ref.c

/* Data references and dependences detectors.
   Copyright (C) 2003, 2004, 2005, 2006, 2007 Free Software Foundation, Inc.
   Contributed by Sebastian Pop <pop@cri.ensmp.fr>

This file is part of GCC.

GCC is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation; either version 2, or (at your option) any later
version.

GCC is distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with GCC; see the file COPYING.  If not, write to the Free
Software Foundation, 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA.  */

/* This pass walks a given loop structure searching for array
   references.  The information about the array accesses is recorded
   in DATA_REFERENCE structures. 
   
   The basic test for determining the dependences is: 
   given two access functions chrec1 and chrec2 to a same array, and 
   x and y two vectors from the iteration domain, the same element of 
   the array is accessed twice at iterations x and y if and only if:
   |             chrec1 (x) == chrec2 (y).
   
   The goals of this analysis are:
   
   - to determine the independence: the relation between two
     independent accesses is qualified with the chrec_known (this
     information allows a loop parallelization),
     
   - when two data references access the same data, to qualify the
     dependence relation with classic dependence representations:
     
       - distance vectors
       - direction vectors
       - loop carried level dependence
       - polyhedron dependence
     or with the chains of recurrences based representation,
     
   - to define a knowledge base for storing the data dependence 
     information,
     
   - to define an interface to access this data.
   
   
   Definitions:
   
   - subscript: given two array accesses a subscript is the tuple
   composed of the access functions for a given dimension.  Example:
   Given A[f1][f2][f3] and B[g1][g2][g3], there are three subscripts:
   (f1, g1), (f2, g2), (f3, g3).

   - Diophantine equation: an equation whose coefficients and
   solutions are integer constants, for example the equation 
   |   3*x + 2*y = 1
   has an integer solution x = 1 and y = -1.
     
   References:
   
   - "Advanced Compilation for High Performance Computing" by Randy
   Allen and Ken Kennedy.
   http://citeseer.ist.psu.edu/goff91practical.html 
   
   - "Loop Transformations for Restructuring Compilers - The Foundations" 
   by Utpal Banerjee.

   
*/

#include "config.h"
#include "system.h"
#include "coretypes.h"
#include "tm.h"
#include "ggc.h"
#include "tree.h"

/* These RTL headers are needed for basic-block.h.  */
#include "rtl.h"
#include "basic-block.h"
#include "diagnostic.h"
#include "tree-flow.h"
#include "tree-dump.h"
#include "timevar.h"
#include "cfgloop.h"
#include "tree-chrec.h"
#include "tree-data-ref.h"
#include "tree-scalar-evolution.h"
#include "tree-pass.h"
#include "langhooks.h"

static struct datadep_stats
{
  int num_dependence_tests;
  int num_dependence_dependent;
  int num_dependence_independent;
  int num_dependence_undetermined;

  int num_subscript_tests;
  int num_subscript_undetermined;
  int num_same_subscript_function;

  int num_ziv;
  int num_ziv_independent;
  int num_ziv_dependent;
  int num_ziv_unimplemented;

  int num_siv;
  int num_siv_independent;
  int num_siv_dependent;
  int num_siv_unimplemented;

  int num_miv;
  int num_miv_independent;
  int num_miv_dependent;
  int num_miv_unimplemented;
} dependence_stats;

static bool subscript_dependence_tester_1 (struct data_dependence_relation *,
                                           struct data_reference *,
                                           struct data_reference *);
/* Returns true iff A divides B.  */

static inline bool 
tree_fold_divides_p (tree a, tree b)
{
  gcc_assert (TREE_CODE (a) == INTEGER_CST);
  gcc_assert (TREE_CODE (b) == INTEGER_CST);
  return integer_zerop (int_const_binop (TRUNC_MOD_EXPR, b, a, 0));
}

/* Returns true iff A divides B.  */

static inline bool 
int_divides_p (int a, int b)
{
  return ((b % a) == 0);
}

\f

/* Dump into FILE all the data references from DATAREFS.  */ 

void 
dump_data_references (FILE *file, VEC (data_reference_p, heap) *datarefs)
{
  unsigned int i;
  struct data_reference *dr;

  for (i = 0; VEC_iterate (data_reference_p, datarefs, i, dr); i++)
    dump_data_reference (file, dr);
}

/* Dump into FILE all the dependence relations from DDRS.  */ 

void 
dump_data_dependence_relations (FILE *file, 
                                VEC (ddr_p, heap) *ddrs)
{
  unsigned int i;
  struct data_dependence_relation *ddr;

  for (i = 0; VEC_iterate (ddr_p, ddrs, i, ddr); i++)
    dump_data_dependence_relation (file, ddr);
}

/* Dump function for a DATA_REFERENCE structure.  */

void 
dump_data_reference (FILE *outf, 
                     struct data_reference *dr)
{
  unsigned int i;
  
  fprintf (outf, "(Data Ref: \n  stmt: ");
  print_generic_stmt (outf, DR_STMT (dr), 0);
  fprintf (outf, "  ref: ");
  print_generic_stmt (outf, DR_REF (dr), 0);
  fprintf (outf, "  base_object: ");
  print_generic_stmt (outf, DR_BASE_OBJECT (dr), 0);
  
  for (i = 0; i < DR_NUM_DIMENSIONS (dr); i++)
    {
      fprintf (outf, "  Access function %d: ", i);
      print_generic_stmt (outf, DR_ACCESS_FN (dr, i), 0);
    }
  fprintf (outf, ")\n");
}

/* Dumps the affine function described by FN to the file OUTF.  */

static void
dump_affine_function (FILE *outf, affine_fn fn)
{
  unsigned i;
  tree coef;

  print_generic_expr (outf, VEC_index (tree, fn, 0), TDF_SLIM);
  for (i = 1; VEC_iterate (tree, fn, i, coef); i++)
    {
      fprintf (outf, " + ");
      print_generic_expr (outf, coef, TDF_SLIM);
      fprintf (outf, " * x_%u", i);
    }
}

/* Dumps the conflict function CF to the file OUTF.  */

static void
dump_conflict_function (FILE *outf, conflict_function *cf)
{
  unsigned i;

  if (cf->n == NO_DEPENDENCE)
    fprintf (outf, "no dependence\n");
  else if (cf->n == NOT_KNOWN)
    fprintf (outf, "not known\n");
  else
    {
      for (i = 0; i < cf->n; i++)
        {
          fprintf (outf, "[");
          dump_affine_function (outf, cf->fns[i]);
          fprintf (outf, "]\n");
        }
    }
}

/* Dump function for a SUBSCRIPT structure.  */

void 
dump_subscript (FILE *outf, struct subscript *subscript)
{
  conflict_function *cf = SUB_CONFLICTS_IN_A (subscript);

  fprintf (outf, "\n (subscript \n");
  fprintf (outf, "  iterations_that_access_an_element_twice_in_A: ");
  dump_conflict_function (outf, cf);
  if (CF_NONTRIVIAL_P (cf))
    {
      tree last_iteration = SUB_LAST_CONFLICT (subscript);
      fprintf (outf, "  last_conflict: ");
      print_generic_stmt (outf, last_iteration, 0);
    }
          
  cf = SUB_CONFLICTS_IN_B (subscript);
  fprintf (outf, "  iterations_that_access_an_element_twice_in_B: ");
  dump_conflict_function (outf, cf);
  if (CF_NONTRIVIAL_P (cf))
    {
      tree last_iteration = SUB_LAST_CONFLICT (subscript);
      fprintf (outf, "  last_conflict: ");
      print_generic_stmt (outf, last_iteration, 0);
    }

  fprintf (outf, "  (Subscript distance: ");
  print_generic_stmt (outf, SUB_DISTANCE (subscript), 0);
  fprintf (outf, "  )\n");
  fprintf (outf, " )\n");
}

/* Print the classic direction vector DIRV to OUTF.  */

void
print_direction_vector (FILE *outf,
                        lambda_vector dirv,
                        int length)
{
  int eq;

  for (eq = 0; eq < length; eq++)
    {
      enum data_dependence_direction dir = dirv[eq];

      switch (dir)
        {
        case dir_positive:
          fprintf (outf, "    +");
          break;
        case dir_negative:
          fprintf (outf, "    -");
          break;
        case dir_equal:
          fprintf (outf, "    =");
          break;
        case dir_positive_or_equal:
          fprintf (outf, "   +=");
          break;
        case dir_positive_or_negative:
          fprintf (outf, "   +-");
          break;
        case dir_negative_or_equal:
          fprintf (outf, "   -=");
          break;
        case dir_star:
          fprintf (outf, "    *");
          break;
        default:
          fprintf (outf, "indep");
          break;
        }
    }
  fprintf (outf, "\n");
}

/* Print a vector of direction vectors.  */

void
print_dir_vectors (FILE *outf, VEC (lambda_vector, heap) *dir_vects,
                   int length)
{
  unsigned j;
  lambda_vector v;

  for (j = 0; VEC_iterate (lambda_vector, dir_vects, j, v); j++)
    print_direction_vector (outf, v, length);
}

/* Print a vector of distance vectors.  */

void
print_dist_vectors  (FILE *outf, VEC (lambda_vector, heap) *dist_vects,
                     int length)
{
  unsigned j;
  lambda_vector v;

  for (j = 0; VEC_iterate (lambda_vector, dist_vects, j, v); j++)
    print_lambda_vector (outf, v, length);
}

/* Debug version.  */

void 
debug_data_dependence_relation (struct data_dependence_relation *ddr)
{
  dump_data_dependence_relation (stderr, ddr);
}

/* Dump function for a DATA_DEPENDENCE_RELATION structure.  */

void 
dump_data_dependence_relation (FILE *outf, 
                               struct data_dependence_relation *ddr)
{
  struct data_reference *dra, *drb;

  dra = DDR_A (ddr);
  drb = DDR_B (ddr);
  fprintf (outf, "(Data Dep: \n");
  if (DDR_ARE_DEPENDENT (ddr) == chrec_dont_know)
    fprintf (outf, "    (don't know)\n");
  
  else if (DDR_ARE_DEPENDENT (ddr) == chrec_known)
    fprintf (outf, "    (no dependence)\n");
  
  else if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE)
    {
      unsigned int i;
      struct loop *loopi;

      for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
        {
          fprintf (outf, "  access_fn_A: ");
          print_generic_stmt (outf, DR_ACCESS_FN (dra, i), 0);
          fprintf (outf, "  access_fn_B: ");
          print_generic_stmt (outf, DR_ACCESS_FN (drb, i), 0);
          dump_subscript (outf, DDR_SUBSCRIPT (ddr, i));
        }

      fprintf (outf, "  inner loop index: %d\n", DDR_INNER_LOOP (ddr));
      fprintf (outf, "  loop nest: (");
      for (i = 0; VEC_iterate (loop_p, DDR_LOOP_NEST (ddr), i, loopi); i++)
        fprintf (outf, "%d ", loopi->num);
      fprintf (outf, ")\n");

      for (i = 0; i < DDR_NUM_DIST_VECTS (ddr); i++)
        {
          fprintf (outf, "  distance_vector: ");
          print_lambda_vector (outf, DDR_DIST_VECT (ddr, i),
                               DDR_NB_LOOPS (ddr));
        }

      for (i = 0; i < DDR_NUM_DIR_VECTS (ddr); i++)
        {
          fprintf (outf, "  direction_vector: ");
          print_direction_vector (outf, DDR_DIR_VECT (ddr, i),
                                  DDR_NB_LOOPS (ddr));
        }
    }

  fprintf (outf, ")\n");
}

/* Dump function for a DATA_DEPENDENCE_DIRECTION structure.  */

void
dump_data_dependence_direction (FILE *file, 
                                enum data_dependence_direction dir)
{
  switch (dir)
    {
    case dir_positive: 
      fprintf (file, "+");
      break;
      
    case dir_negative:
      fprintf (file, "-");
      break;
      
    case dir_equal:
      fprintf (file, "=");
      break;
      
    case dir_positive_or_negative:
      fprintf (file, "+-");
      break;
      
    case dir_positive_or_equal: 
      fprintf (file, "+=");
      break;
      
    case dir_negative_or_equal: 
      fprintf (file, "-=");
      break;
      
    case dir_star: 
      fprintf (file, "*"); 
      break;
      
    default: 
      break;
    }
}

/* Dumps the distance and direction vectors in FILE.  DDRS contains
   the dependence relations, and VECT_SIZE is the size of the
   dependence vectors, or in other words the number of loops in the
   considered nest.  */

void 
dump_dist_dir_vectors (FILE *file, VEC (ddr_p, heap) *ddrs)
{
  unsigned int i, j;
  struct data_dependence_relation *ddr;
  lambda_vector v;

  for (i = 0; VEC_iterate (ddr_p, ddrs, i, ddr); i++)
    if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE && DDR_AFFINE_P (ddr))
      {
        for (j = 0; VEC_iterate (lambda_vector, DDR_DIST_VECTS (ddr), j, v); j++)
          {
            fprintf (file, "DISTANCE_V (");
            print_lambda_vector (file, v, DDR_NB_LOOPS (ddr));
            fprintf (file, ")\n");
          }

        for (j = 0; VEC_iterate (lambda_vector, DDR_DIR_VECTS (ddr), j, v); j++)
          {
            fprintf (file, "DIRECTION_V (");
            print_direction_vector (file, v, DDR_NB_LOOPS (ddr));
            fprintf (file, ")\n");
          }
      }

  fprintf (file, "\n\n");
}

/* Dumps the data dependence relations DDRS in FILE.  */

void 
dump_ddrs (FILE *file, VEC (ddr_p, heap) *ddrs)
{
  unsigned int i;
  struct data_dependence_relation *ddr;

  for (i = 0; VEC_iterate (ddr_p, ddrs, i, ddr); i++)
    dump_data_dependence_relation (file, ddr);

  fprintf (file, "\n\n");
}

/* Expresses EXP as VAR + OFF, where off is a constant.  The type of OFF
   will be ssizetype.  */

static void
split_constant_offset (tree exp, tree *var, tree *off)
{
  tree type = TREE_TYPE (exp), otype;
  tree var0, var1;
  tree off0, off1;

  *var = exp;
  STRIP_NOPS (exp);
  otype = TREE_TYPE (exp);

  switch (TREE_CODE (exp))
    {
    case INTEGER_CST:
      *var = build_int_cst (type, 0);
      *off = fold_convert (ssizetype, exp);
      return;

    case PLUS_EXPR:
    case MINUS_EXPR:
      split_constant_offset (TREE_OPERAND (exp, 0), &var0, &off0);
      split_constant_offset (TREE_OPERAND (exp, 1), &var1, &off1);
      *var = fold_convert (type, fold_build2 (TREE_CODE (exp), otype, 
                                              var0, var1));
      *off = size_binop (TREE_CODE (exp), off0, off1);
      return;

    case MULT_EXPR:
      off1 = TREE_OPERAND (exp, 1);
      if (TREE_CODE (off1) != INTEGER_CST)
        break;

      split_constant_offset (TREE_OPERAND (exp, 0), &var0, &off0);
      *var = fold_convert (type, fold_build2 (MULT_EXPR, otype,
                                              var0, off1));
      *off = size_binop (MULT_EXPR, off0, fold_convert (ssizetype, off1));
      return;

    case ADDR_EXPR:
      {
        tree op, base, poffset;
        HOST_WIDE_INT pbitsize, pbitpos;
        enum machine_mode pmode;
        int punsignedp, pvolatilep;

        op = TREE_OPERAND (exp, 0);
        if (!handled_component_p (op))
          break;

        base = get_inner_reference (op, &pbitsize, &pbitpos, &poffset,
                                    &pmode, &punsignedp, &pvolatilep, false);

        if (pbitpos % BITS_PER_UNIT != 0)
          break;
        base = build_fold_addr_expr (base);
        off0 = ssize_int (pbitpos / BITS_PER_UNIT);

        if (poffset)
          {
            split_constant_offset (poffset, &poffset, &off1);
            off0 = size_binop (PLUS_EXPR, off0, off1);
            base = fold_build2 (PLUS_EXPR, TREE_TYPE (base),
                                base,
                                fold_convert (TREE_TYPE (base), poffset));
          }

        *var = fold_convert (type, base);
        *off = off0;
        return;
      }

    default:
      break;
    }

  *off = ssize_int (0);
}

/* Returns the address ADDR of an object in a canonical shape (without nop
   casts, and with type of pointer to the object).  */

static tree
canonicalize_base_object_address (tree addr)
{
  tree orig = addr;

  STRIP_NOPS (addr);

  /* The base address may be obtained by casting from integer, in that case
     keep the cast.  */
  if (!POINTER_TYPE_P (TREE_TYPE (addr)))
    return orig;

  if (TREE_CODE (addr) != ADDR_EXPR)
    return addr;

  return build_fold_addr_expr (TREE_OPERAND (addr, 0));
}

/* Analyzes the behavior of the memory reference DR in the innermost loop that
   contains it.  */

void
dr_analyze_innermost (struct data_reference *dr)
{
  tree stmt = DR_STMT (dr);
  struct loop *loop = loop_containing_stmt (stmt);
  tree ref = DR_REF (dr);
  HOST_WIDE_INT pbitsize, pbitpos;
  tree base, poffset;
  enum machine_mode pmode;
  int punsignedp, pvolatilep;
  affine_iv base_iv, offset_iv;
  tree init, dinit, step;

  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "analyze_innermost: ");

  base = get_inner_reference (ref, &pbitsize, &pbitpos, &poffset,
                              &pmode, &punsignedp, &pvolatilep, false);
  gcc_assert (base != NULL_TREE);

  if (pbitpos % BITS_PER_UNIT != 0)
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "failed: bit offset alignment.\n");
      return;
    }

  base = build_fold_addr_expr (base);
  if (!simple_iv (loop, stmt, base, &base_iv, false))
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "failed: evolution of base is not affine.\n");
      return;
    }
  if (!poffset)
    {
      offset_iv.base = ssize_int (0);
      offset_iv.step = ssize_int (0);
    }
  else if (!simple_iv (loop, stmt, poffset, &offset_iv, false))
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "failed: evolution of offset is not affine.\n");
      return;
    }

  init = ssize_int (pbitpos / BITS_PER_UNIT);
  split_constant_offset (base_iv.base, &base_iv.base, &dinit);
  init =  size_binop (PLUS_EXPR, init, dinit);
  split_constant_offset (offset_iv.base, &offset_iv.base, &dinit);
  init =  size_binop (PLUS_EXPR, init, dinit);

  step = size_binop (PLUS_EXPR,
                     fold_convert (ssizetype, base_iv.step),
                     fold_convert (ssizetype, offset_iv.step));

  DR_BASE_ADDRESS (dr) = canonicalize_base_object_address (base_iv.base);

  DR_OFFSET (dr) = fold_convert (ssizetype, offset_iv.base);
  DR_INIT (dr) = init;
  DR_STEP (dr) = step;

  DR_ALIGNED_TO (dr) = size_int (highest_pow2_factor (offset_iv.base));

  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "success.\n");
}

/* Determines the base object and the list of indices of memory reference
   DR, analyzed in loop nest NEST.  */

static void
dr_analyze_indices (struct data_reference *dr, struct loop *nest)
{
  tree stmt = DR_STMT (dr);
  struct loop *loop = loop_containing_stmt (stmt);
  VEC (tree, heap) *access_fns = NULL;
  tree ref = unshare_expr (DR_REF (dr)), aref = ref, op;
  tree base, off, access_fn;

  while (handled_component_p (aref))
    {
      if (TREE_CODE (aref) == ARRAY_REF)
        {
          op = TREE_OPERAND (aref, 1);
          access_fn = analyze_scalar_evolution (loop, op);
          access_fn = resolve_mixers (nest, access_fn);
          VEC_safe_push (tree, heap, access_fns, access_fn);

          TREE_OPERAND (aref, 1) = build_int_cst (TREE_TYPE (op), 0);
        }
      
      aref = TREE_OPERAND (aref, 0);
    }

  if (INDIRECT_REF_P (aref))
    {
      op = TREE_OPERAND (aref, 0);
      access_fn = analyze_scalar_evolution (loop, op);
      access_fn = resolve_mixers (nest, access_fn);
      base = initial_condition (access_fn);
      split_constant_offset (base, &base, &off);
      access_fn = chrec_replace_initial_condition (access_fn,
                        fold_convert (TREE_TYPE (base), off));

      TREE_OPERAND (aref, 0) = base;
      VEC_safe_push (tree, heap, access_fns, access_fn);
    }

  DR_BASE_OBJECT (dr) = ref;
  DR_ACCESS_FNS (dr) = access_fns;
}

/* Extracts the alias analysis information from the memory reference DR.  */

static void
dr_analyze_alias (struct data_reference *dr)
{
  tree stmt = DR_STMT (dr);
  tree ref = DR_REF (dr);
  tree base = get_base_address (ref), addr, smt = NULL_TREE;
  ssa_op_iter it;
  tree op;
  bitmap vops;

  if (DECL_P (base))
    smt = base;
  else if (INDIRECT_REF_P (base))
    {
      addr = TREE_OPERAND (base, 0);
      if (TREE_CODE (addr) == SSA_NAME)
        {
          smt = symbol_mem_tag (SSA_NAME_VAR (addr));
          DR_PTR_INFO (dr) = SSA_NAME_PTR_INFO (addr);
        }
    }

  DR_SYMBOL_TAG (dr) = smt;
  if (smt && var_can_have_subvars (smt))
    DR_SUBVARS (dr) = get_subvars_for_var (smt);

  vops = BITMAP_ALLOC (NULL);
  FOR_EACH_SSA_TREE_OPERAND (op, stmt, it, SSA_OP_VIRTUAL_USES)
    {
      bitmap_set_bit (vops, DECL_UID (SSA_NAME_VAR (op)));
    }

  DR_VOPS (dr) = vops;
}

/* Returns true if the address of DR is invariant.  */

static bool
dr_address_invariant_p (struct data_reference *dr)
{
  unsigned i;
  tree idx;

  for (i = 0; VEC_iterate (tree, DR_ACCESS_FNS (dr), i, idx); i++)
    if (tree_contains_chrecs (idx, NULL))
      return false;

  return true;
}

/* Frees data reference DR.  */

static void
free_data_ref (data_reference_p dr)
{
  BITMAP_FREE (DR_VOPS (dr));
  VEC_free (tree, heap, DR_ACCESS_FNS (dr));
  free (dr);
}

/* Analyzes memory reference MEMREF accessed in STMT.  The reference
   is read if IS_READ is true, write otherwise.  Returns the
   data_reference description of MEMREF.  NEST is the outermost loop of the
   loop nest in that the reference should be analyzed.  */

struct data_reference *
create_data_ref (struct loop *nest, tree memref, tree stmt, bool is_read)
{
  struct data_reference *dr;

  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "Creating dr for ");
      print_generic_expr (dump_file, memref, TDF_SLIM);
      fprintf (dump_file, "\n");
    }

  dr = XCNEW (struct data_reference);
  DR_STMT (dr) = stmt;
  DR_REF (dr) = memref;
  DR_IS_READ (dr) = is_read;

  dr_analyze_innermost (dr);
  dr_analyze_indices (dr, nest);
  dr_analyze_alias (dr);

  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "\tbase_address: ");
      print_generic_expr (dump_file, DR_BASE_ADDRESS (dr), TDF_SLIM);
      fprintf (dump_file, "\n\toffset from base address: ");
      print_generic_expr (dump_file, DR_OFFSET (dr), TDF_SLIM);
      fprintf (dump_file, "\n\tconstant offset from base address: ");
      print_generic_expr (dump_file, DR_INIT (dr), TDF_SLIM);
      fprintf (dump_file, "\n\tstep: ");
      print_generic_expr (dump_file, DR_STEP (dr), TDF_SLIM);
      fprintf (dump_file, "\n\taligned to: ");
      print_generic_expr (dump_file, DR_ALIGNED_TO (dr), TDF_SLIM);
      fprintf (dump_file, "\n\tbase_object: ");
      print_generic_expr (dump_file, DR_BASE_OBJECT (dr), TDF_SLIM);
      fprintf (dump_file, "\n\tsymbol tag: ");
      print_generic_expr (dump_file, DR_SYMBOL_TAG (dr), TDF_SLIM);
      fprintf (dump_file, "\n");
    }

  return dr;  
}

/* Returns true if FNA == FNB.  */

static bool
affine_function_equal_p (affine_fn fna, affine_fn fnb)
{
  unsigned i, n = VEC_length (tree, fna);

  if (n != VEC_length (tree, fnb))
    return false;

  for (i = 0; i < n; i++)
    if (!operand_equal_p (VEC_index (tree, fna, i),
                          VEC_index (tree, fnb, i), 0))
      return false;

  return true;
}

/* If all the functions in CF are the same, returns one of them,
   otherwise returns NULL.  */

static affine_fn
common_affine_function (conflict_function *cf)
{
  unsigned i;
  affine_fn comm;

  if (!CF_NONTRIVIAL_P (cf))
    return NULL;

  comm = cf->fns[0];

  for (i = 1; i < cf->n; i++)
    if (!affine_function_equal_p (comm, cf->fns[i]))
      return NULL;

  return comm;
}

/* Returns the base of the affine function FN.  */

static tree
affine_function_base (affine_fn fn)
{
  return VEC_index (tree, fn, 0);
}

/* Returns true if FN is a constant.  */

static bool
affine_function_constant_p (affine_fn fn)
{
  unsigned i;
  tree coef;

  for (i = 1; VEC_iterate (tree, fn, i, coef); i++)
    if (!integer_zerop (coef))
      return false;

  return true;
}

/* Returns true if FN is the zero constant function.  */

static bool
affine_function_zero_p (affine_fn fn)
{
  return (integer_zerop (affine_function_base (fn))
          && affine_function_constant_p (fn));
}

/* Applies operation OP on affine functions FNA and FNB, and returns the
   result.  */

static affine_fn
affine_fn_op (enum tree_code op, affine_fn fna, affine_fn fnb)
{
  unsigned i, n, m;
  affine_fn ret;
  tree coef;

  if (VEC_length (tree, fnb) > VEC_length (tree, fna))
    {
      n = VEC_length (tree, fna);
      m = VEC_length (tree, fnb);
    }
  else
    {
      n = VEC_length (tree, fnb);
      m = VEC_length (tree, fna);
    }

  ret = VEC_alloc (tree, heap, m);
  for (i = 0; i < n; i++)
    VEC_quick_push (tree, ret,
                    fold_build2 (op, integer_type_node,
                                 VEC_index (tree, fna, i), 
                                 VEC_index (tree, fnb, i)));

  for (; VEC_iterate (tree, fna, i, coef); i++)
    VEC_quick_push (tree, ret,
                    fold_build2 (op, integer_type_node,
                                 coef, integer_zero_node));
  for (; VEC_iterate (tree, fnb, i, coef); i++)
    VEC_quick_push (tree, ret,
                    fold_build2 (op, integer_type_node,
                                 integer_zero_node, coef));

  return ret;
}

/* Returns the sum of affine functions FNA and FNB.  */

static affine_fn
affine_fn_plus (affine_fn fna, affine_fn fnb)
{
  return affine_fn_op (PLUS_EXPR, fna, fnb);
}

/* Returns the difference of affine functions FNA and FNB.  */

static affine_fn
affine_fn_minus (affine_fn fna, affine_fn fnb)
{
  return affine_fn_op (MINUS_EXPR, fna, fnb);
}

/* Frees affine function FN.  */

static void
affine_fn_free (affine_fn fn)
{
  VEC_free (tree, heap, fn);
}

/* Determine for each subscript in the data dependence relation DDR
   the distance.  */

static void
compute_subscript_distance (struct data_dependence_relation *ddr)
{
  conflict_function *cf_a, *cf_b;
  affine_fn fn_a, fn_b, diff;

  if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE)
    {
      unsigned int i;
      
      for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
        {
          struct subscript *subscript;
          
          subscript = DDR_SUBSCRIPT (ddr, i);
          cf_a = SUB_CONFLICTS_IN_A (subscript);
          cf_b = SUB_CONFLICTS_IN_B (subscript);

          fn_a = common_affine_function (cf_a);
          fn_b = common_affine_function (cf_b);
          if (!fn_a || !fn_b)
            {
              SUB_DISTANCE (subscript) = chrec_dont_know;
              return;
            }
          diff = affine_fn_minus (fn_a, fn_b);
          
          if (affine_function_constant_p (diff))
            SUB_DISTANCE (subscript) = affine_function_base (diff);
          else
            SUB_DISTANCE (subscript) = chrec_dont_know;

          affine_fn_free (diff);
        }
    }
}

/* Returns the conflict function for "unknown".  */

static conflict_function *
conflict_fn_not_known (void)
{
  conflict_function *fn = XCNEW (conflict_function);
  fn->n = NOT_KNOWN;

  return fn;
}

/* Returns the conflict function for "independent".  */

static conflict_function *
conflict_fn_no_dependence (void)
{
  conflict_function *fn = XCNEW (conflict_function);
  fn->n = NO_DEPENDENCE;

  return fn;
}

/* Returns true if the address of OBJ is invariant in LOOP.  */

static bool
object_address_invariant_in_loop_p (struct loop *loop, tree obj)
{
  while (handled_component_p (obj))
    {
      if (TREE_CODE (obj) == ARRAY_REF)
        {
          /* Index of the ARRAY_REF was zeroed in analyze_indices, thus we only
             need to check the stride and the lower bound of the reference.  */
          if (chrec_contains_symbols_defined_in_loop (TREE_OPERAND (obj, 2),
                                                      loop->num)
              || chrec_contains_symbols_defined_in_loop (TREE_OPERAND (obj, 3),
                                                         loop->num))
            return false;
        }
      else if (TREE_CODE (obj) == COMPONENT_REF)
        {
          if (chrec_contains_symbols_defined_in_loop (TREE_OPERAND (obj, 2),
                                                      loop->num))
            return false;
        }
      obj = TREE_OPERAND (obj, 0);
    }

  if (!INDIRECT_REF_P (obj))
    return true;

  return !chrec_contains_symbols_defined_in_loop (TREE_OPERAND (obj, 0),
                                                  loop->num);
}

/* Returns true if A and B are accesses to different objects, or to different
   fields of the same object.  */

static bool
disjoint_objects_p (tree a, tree b)
{
  tree base_a, base_b;
  VEC (tree, heap) *comp_a = NULL, *comp_b = NULL;
  bool ret;

  base_a = get_base_address (a);
  base_b = get_base_address (b);

  if (DECL_P (base_a)
      && DECL_P (base_b)
      && base_a != base_b)
    return true;

  if (!operand_equal_p (base_a, base_b, 0))
    return false;

  /* Compare the component references of A and B.  We must start from the inner
     ones, so record them to the vector first.  */
  while (handled_component_p (a))
    {
      VEC_safe_push (tree, heap, comp_a, a);
      a = TREE_OPERAND (a, 0);
    }
  while (handled_component_p (b))
    {
      VEC_safe_push (tree, heap, comp_b, b);
      b = TREE_OPERAND (b, 0);
    }

  ret = false;
  while (1)
    {
      if (VEC_length (tree, comp_a) == 0
          || VEC_length (tree, comp_b) == 0)
        break;

      a = VEC_pop (tree, comp_a);
      b = VEC_pop (tree, comp_b);

      /* Real and imaginary part of a variable do not alias.  */
      if ((TREE_CODE (a) == REALPART_EXPR
           && TREE_CODE (b) == IMAGPART_EXPR)
          || (TREE_CODE (a) == IMAGPART_EXPR
              && TREE_CODE (b) == REALPART_EXPR))
        {
          ret = true;
          break;
        }

      if (TREE_CODE (a) != TREE_CODE (b))
        break;

      /* Nothing to do for ARRAY_REFs, as the indices of array_refs in
         DR_BASE_OBJECT are always zero.  */
      if (TREE_CODE (a) == ARRAY_REF)
        continue;
      else if (TREE_CODE (a) == COMPONENT_REF)
        {
          if (operand_equal_p (TREE_OPERAND (a, 1), TREE_OPERAND (b, 1), 0))
            continue;

          /* Different fields of unions may overlap.  */
          base_a = TREE_OPERAND (a, 0);
          if (TREE_CODE (TREE_TYPE (base_a)) == UNION_TYPE)
            break;

          /* Different fields of structures cannot.  */
          ret = true;
          break;
        }
      else
        break;
    }

  VEC_free (tree, heap, comp_a);
  VEC_free (tree, heap, comp_b);

  return ret;
}

/* Returns false if we can prove that data references A and B do not alias,
   true otherwise.  */

static bool
dr_may_alias_p (struct data_reference *a, struct data_reference *b)
{
  tree addr_a = DR_BASE_ADDRESS (a);
  tree addr_b = DR_BASE_ADDRESS (b);
  tree type_a, type_b;
  tree decl_a = NULL_TREE, decl_b = NULL_TREE;

  /* If the sets of virtual operands are disjoint, the memory references do not
     alias.  */
  if (!bitmap_intersect_p (DR_VOPS (a), DR_VOPS (b)))
    return false;

  /* If the accessed objects are disjoint, the memory references do not
     alias.  */
  if (disjoint_objects_p (DR_BASE_OBJECT (a), DR_BASE_OBJECT (b)))
    return false;

  if (!addr_a || !addr_b)
    return true;

  /* If the references are based on different static objects, they cannot alias
     (PTA should be able to disambiguate such accesses, but often it fails to,
     since currently we cannot distinguish between pointer and offset in pointer
     arithmetics).  */
  if (TREE_CODE (addr_a) == ADDR_EXPR
      && TREE_CODE (addr_b) == ADDR_EXPR)
    return TREE_OPERAND (addr_a, 0) == TREE_OPERAND (addr_b, 0);

  /* An instruction writing through a restricted pointer is "independent" of any 
     instruction reading or writing through a different restricted pointer, 
     in the same block/scope.  */

  type_a = TREE_TYPE (addr_a);
  type_b = TREE_TYPE (addr_b);
  gcc_assert (POINTER_TYPE_P (type_a) && POINTER_TYPE_P (type_b));

  if (TREE_CODE (addr_a) == SSA_NAME)
    decl_a = SSA_NAME_VAR (addr_a);
  if (TREE_CODE (addr_b) == SSA_NAME)
    decl_b = SSA_NAME_VAR (addr_b);

  if (TYPE_RESTRICT (type_a) && TYPE_RESTRICT (type_b) 
      && (!DR_IS_READ (a) || !DR_IS_READ (b))
      && decl_a && DECL_P (decl_a)
      && decl_b && DECL_P (decl_b)
      && decl_a != decl_b
      && TREE_CODE (DECL_CONTEXT (decl_a)) == FUNCTION_DECL
      && DECL_CONTEXT (decl_a) == DECL_CONTEXT (decl_b))
    return false;

  return true;
}

/* Initialize a data dependence relation between data accesses A and
   B.  NB_LOOPS is the number of loops surrounding the references: the
   size of the classic distance/direction vectors.  */

static struct data_dependence_relation *
initialize_data_dependence_relation (struct data_reference *a, 
                                     struct data_reference *b,
                                     VEC (loop_p, heap) *loop_nest)
{
  struct data_dependence_relation *res;
  unsigned int i;
  
  res = XNEW (struct data_dependence_relation);
  DDR_A (res) = a;
  DDR_B (res) = b;
  DDR_LOOP_NEST (res) = NULL;

  if (a == NULL || b == NULL)
    {
      DDR_ARE_DEPENDENT (res) = chrec_dont_know;    
      return res;
    }   

  /* If the data references do not alias, then they are independent.  */
  if (!dr_may_alias_p (a, b))
    {
      DDR_ARE_DEPENDENT (res) = chrec_known;    
      return res;
    }

  /* If the references do not access the same object, we do not know
     whether they alias or not.  */
  if (!operand_equal_p (DR_BASE_OBJECT (a), DR_BASE_OBJECT (b), 0))
    {
      DDR_ARE_DEPENDENT (res) = chrec_dont_know;    
      return res;
    }

  /* If the base of the object is not invariant in the loop nest, we cannot
     analyze it.  TODO -- in fact, it would suffice to record that there may
     be arbitrary dependences in the loops where the base object varies.  */
  if (!object_address_invariant_in_loop_p (VEC_index (loop_p, loop_nest, 0),
                                           DR_BASE_OBJECT (a)))
    {
      DDR_ARE_DEPENDENT (res) = chrec_dont_know;    
      return res;
    }

  gcc_assert (DR_NUM_DIMENSIONS (a) == DR_NUM_DIMENSIONS (b));

  DDR_AFFINE_P (res) = true;
  DDR_ARE_DEPENDENT (res) = NULL_TREE;
  DDR_SUBSCRIPTS (res) = VEC_alloc (subscript_p, heap, DR_NUM_DIMENSIONS (a));
  DDR_LOOP_NEST (res) = loop_nest;
  DDR_INNER_LOOP (res) = 0;
  DDR_DIR_VECTS (res) = NULL;
  DDR_DIST_VECTS (res) = NULL;

  for (i = 0; i < DR_NUM_DIMENSIONS (a); i++)
    {
      struct subscript *subscript;
          
      subscript = XNEW (struct subscript);
      SUB_CONFLICTS_IN_A (subscript) = conflict_fn_not_known ();
      SUB_CONFLICTS_IN_B (subscript) = conflict_fn_not_known ();
      SUB_LAST_CONFLICT (subscript) = chrec_dont_know;
      SUB_DISTANCE (subscript) = chrec_dont_know;
      VEC_safe_push (subscript_p, heap, DDR_SUBSCRIPTS (res), subscript);
    }

  return res;
}

/* Frees memory used by the conflict function F.  */

static void
free_conflict_function (conflict_function *f)
{
  unsigned i;

  if (CF_NONTRIVIAL_P (f))
    {
      for (i = 0; i < f->n; i++)
        affine_fn_free (f->fns[i]);
    }
  free (f);
}

/* Frees memory used by SUBSCRIPTS.  */

static void
free_subscripts (VEC (subscript_p, heap) *subscripts)
{
  unsigned i;
  subscript_p s;

  for (i = 0; VEC_iterate (subscript_p, subscripts, i, s); i++)
    {
      free_conflict_function (s->conflicting_iterations_in_a);
      free_conflict_function (s->conflicting_iterations_in_b);
    }
  VEC_free (subscript_p, heap, subscripts);
}

/* Set DDR_ARE_DEPENDENT to CHREC and finalize the subscript overlap
   description.  */

static inline void
finalize_ddr_dependent (struct data_dependence_relation *ddr, 
                        tree chrec)
{
  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "(dependence classified: ");
      print_generic_expr (dump_file, chrec, 0);
      fprintf (dump_file, ")\n");
    }

  DDR_ARE_DEPENDENT (ddr) = chrec;  
  free_subscripts (DDR_SUBSCRIPTS (ddr));
}

/* The dependence relation DDR cannot be represented by a distance
   vector.  */

static inline void
non_affine_dependence_relation (struct data_dependence_relation *ddr)
{
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(Dependence relation cannot be represented by distance vector.) \n");

  DDR_AFFINE_P (ddr) = false;
}

\f

/* This section contains the classic Banerjee tests.  */

/* Returns true iff CHREC_A and CHREC_B are not dependent on any index
   variables, i.e., if the ZIV (Zero Index Variable) test is true.  */

static inline bool
ziv_subscript_p (tree chrec_a, 
                 tree chrec_b)
{
  return (evolution_function_is_constant_p (chrec_a)
          && evolution_function_is_constant_p (chrec_b));
}

/* Returns true iff CHREC_A and CHREC_B are dependent on an index
   variable, i.e., if the SIV (Single Index Variable) test is true.  */

static bool
siv_subscript_p (tree chrec_a,
                 tree chrec_b)
{
  if ((evolution_function_is_constant_p (chrec_a)
       && evolution_function_is_univariate_p (chrec_b))
      || (evolution_function_is_constant_p (chrec_b)
          && evolution_function_is_univariate_p (chrec_a)))
    return true;
  
  if (evolution_function_is_univariate_p (chrec_a)
      && evolution_function_is_univariate_p (chrec_b))
    {
      switch (TREE_CODE (chrec_a))
        {
        case POLYNOMIAL_CHREC:
          switch (TREE_CODE (chrec_b))
            {
            case POLYNOMIAL_CHREC:
              if (CHREC_VARIABLE (chrec_a) != CHREC_VARIABLE (chrec_b))
                return false;
              
            default:
              return true;
            }
          
        default:
          return true;
        }
    }
  
  return false;
}

/* Creates a conflict function with N dimensions.  The affine functions
   in each dimension follow.  */

static conflict_function *
conflict_fn (unsigned n, ...)
{
  unsigned i;
  conflict_function *ret = XCNEW (conflict_function);
  va_list ap;

  gcc_assert (0 < n && n <= MAX_DIM);
  va_start(ap, n);
                       
  ret->n = n;
  for (i = 0; i < n; i++)
    ret->fns[i] = va_arg (ap, affine_fn);
  va_end(ap);

  return ret;
}

/* Returns constant affine function with value CST.  */

static affine_fn
affine_fn_cst (tree cst)
{
  affine_fn fn = VEC_alloc (tree, heap, 1);
  VEC_quick_push (tree, fn, cst);
  return fn;
}

/* Returns affine function with single variable, CST + COEF * x_DIM.  */

static affine_fn
affine_fn_univar (tree cst, unsigned dim, tree coef)
{
  affine_fn fn = VEC_alloc (tree, heap, dim + 1);
  unsigned i;

  gcc_assert (dim > 0);
  VEC_quick_push (tree, fn, cst);
  for (i = 1; i < dim; i++)
    VEC_quick_push (tree, fn, integer_zero_node);
  VEC_quick_push (tree, fn, coef);
  return fn;
}

/* Analyze a ZIV (Zero Index Variable) subscript.  *OVERLAPS_A and
   *OVERLAPS_B are initialized to the functions that describe the
   relation between the elements accessed twice by CHREC_A and
   CHREC_B.  For k >= 0, the following property is verified:

   CHREC_A (*OVERLAPS_A (k)) = CHREC_B (*OVERLAPS_B (k)).  */

static void 
analyze_ziv_subscript (tree chrec_a, 
                       tree chrec_b, 
                       conflict_function **overlaps_a,
                       conflict_function **overlaps_b, 
                       tree *last_conflicts)
{
  tree difference;
  dependence_stats.num_ziv++;
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(analyze_ziv_subscript \n");
  
  chrec_a = chrec_convert (integer_type_node, chrec_a, NULL_TREE);
  chrec_b = chrec_convert (integer_type_node, chrec_b, NULL_TREE);
  difference = chrec_fold_minus (integer_type_node, chrec_a, chrec_b);
  
  switch (TREE_CODE (difference))
    {
    case INTEGER_CST:
      if (integer_zerop (difference))
        {
          /* The difference is equal to zero: the accessed index
             overlaps for each iteration in the loop.  */
          *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
          *overlaps_b = conflict_fn (1, affine_fn_cst (integer_zero_node));
          *last_conflicts = chrec_dont_know;
          dependence_stats.num_ziv_dependent++;
        }
      else
        {
          /* The accesses do not overlap.  */
          *overlaps_a = conflict_fn_no_dependence ();
          *overlaps_b = conflict_fn_no_dependence ();
          *last_conflicts = integer_zero_node;
          dependence_stats.num_ziv_independent++;
        }
      break;
      
    default:
      /* We're not sure whether the indexes overlap.  For the moment, 
         conservatively answer "don't know".  */
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "ziv test failed: difference is non-integer.\n");

      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      dependence_stats.num_ziv_unimplemented++;
      break;
    }
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, ")\n");
}

/* Sets NIT to the estimated number of executions of the statements in
   LOOP.  If CONSERVATIVE is true, we must be sure that NIT is at least as
   large as the number of iterations.  If we have no reliable estimate,
   the function returns false, otherwise returns true.  */

bool
estimated_loop_iterations (struct loop *loop, bool conservative,
                           double_int *nit)
{
  estimate_numbers_of_iterations_loop (loop);
  if (conservative)
    {
      if (!loop->any_upper_bound)
        return false;

      *nit = loop->nb_iterations_upper_bound;
    }
  else
    {
      if (!loop->any_estimate)
        return false;

      *nit = loop->nb_iterations_estimate;
    }

  return true;
}

/* Similar to estimated_loop_iterations, but returns the estimate only
   if it fits to HOST_WIDE_INT.  If this is not the case, or the estimate
   on the number of iterations of LOOP could not be derived, returns -1.  */

HOST_WIDE_INT
estimated_loop_iterations_int (struct loop *loop, bool conservative)
{
  double_int nit;
  HOST_WIDE_INT hwi_nit;

  if (!estimated_loop_iterations (loop, conservative, &nit))
    return -1;

  if (!double_int_fits_in_shwi_p (nit))
    return -1;
  hwi_nit = double_int_to_shwi (nit);

  return hwi_nit < 0 ? -1 : hwi_nit;
}
    
/* Similar to estimated_loop_iterations, but returns the estimate as a tree,
   and only if it fits to the int type.  If this is not the case, or the
   estimate on the number of iterations of LOOP could not be derived, returns
   chrec_dont_know.  */

static tree
estimated_loop_iterations_tree (struct loop *loop, bool conservative)
{
  double_int nit;
  tree type;

  if (!estimated_loop_iterations (loop, conservative, &nit))
    return chrec_dont_know;

  type = lang_hooks.types.type_for_size (INT_TYPE_SIZE, true);
  if (!double_int_fits_to_tree_p (type, nit))
    return chrec_dont_know;

  return double_int_to_tree (type, nit);
}

/* Analyze a SIV (Single Index Variable) subscript where CHREC_A is a
   constant, and CHREC_B is an affine function.  *OVERLAPS_A and
   *OVERLAPS_B are initialized to the functions that describe the
   relation between the elements accessed twice by CHREC_A and
   CHREC_B.  For k >= 0, the following property is verified:

   CHREC_A (*OVERLAPS_A (k)) = CHREC_B (*OVERLAPS_B (k)).  */

static void
analyze_siv_subscript_cst_affine (tree chrec_a, 
                                  tree chrec_b,
                                  conflict_function **overlaps_a, 
                                  conflict_function **overlaps_b, 
                                  tree *last_conflicts)
{
  bool value0, value1, value2;
  tree difference, tmp;

  chrec_a = chrec_convert (integer_type_node, chrec_a, NULL_TREE);
  chrec_b = chrec_convert (integer_type_node, chrec_b, NULL_TREE);
  difference = chrec_fold_minus 
    (integer_type_node, initial_condition (chrec_b), chrec_a);
  
  if (!chrec_is_positive (initial_condition (difference), &value0))
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "siv test failed: chrec is not positive.\n"); 

      dependence_stats.num_siv_unimplemented++;
      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      return;
    }
  else
    {
      if (value0 == false)
        {
          if (!chrec_is_positive (CHREC_RIGHT (chrec_b), &value1))
            {
              if (dump_file && (dump_flags & TDF_DETAILS))
                fprintf (dump_file, "siv test failed: chrec not positive.\n");

              *overlaps_a = conflict_fn_not_known ();
              *overlaps_b = conflict_fn_not_known ();      
              *last_conflicts = chrec_dont_know;
              dependence_stats.num_siv_unimplemented++;
              return;
            }
          else
            {
              if (value1 == true)
                {
                  /* Example:  
                     chrec_a = 12
                     chrec_b = {10, +, 1}
                  */
                  
                  if (tree_fold_divides_p (CHREC_RIGHT (chrec_b), difference))
                    {
                      HOST_WIDE_INT numiter;
                      struct loop *loop = get_chrec_loop (chrec_b);

                      *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
                      tmp = fold_build2 (EXACT_DIV_EXPR, integer_type_node,
                                         fold_build1 (ABS_EXPR,
                                                      integer_type_node,
                                                      difference),
                                         CHREC_RIGHT (chrec_b));
                      *overlaps_b = conflict_fn (1, affine_fn_cst (tmp));
                      *last_conflicts = integer_one_node;
                      

                      /* Perform weak-zero siv test to see if overlap is
                         outside the loop bounds.  */
                      numiter = estimated_loop_iterations_int (loop, true);

                      if (numiter >= 0
                          && compare_tree_int (tmp, numiter) > 0)
                        {
                          free_conflict_function (*overlaps_a);
                          free_conflict_function (*overlaps_b);
                          *overlaps_a = conflict_fn_no_dependence ();
                          *overlaps_b = conflict_fn_no_dependence ();
                          *last_conflicts = integer_zero_node;
                          dependence_stats.num_siv_independent++;
                          return;
                        }               
                      dependence_stats.num_siv_dependent++;
                      return;
                    }
                  
                  /* When the step does not divide the difference, there are
                     no overlaps.  */
                  else
                    {
                      *overlaps_a = conflict_fn_no_dependence ();
                      *overlaps_b = conflict_fn_no_dependence ();      
                      *last_conflicts = integer_zero_node;
                      dependence_stats.num_siv_independent++;
                      return;
                    }
                }
              
              else
                {
                  /* Example:  
                     chrec_a = 12
                     chrec_b = {10, +, -1}
                     
                     In this case, chrec_a will not overlap with chrec_b.  */
                  *overlaps_a = conflict_fn_no_dependence ();
                  *overlaps_b = conflict_fn_no_dependence ();
                  *last_conflicts = integer_zero_node;
                  dependence_stats.num_siv_independent++;
                  return;
                }
            }
        }
      else 
        {
          if (!chrec_is_positive (CHREC_RIGHT (chrec_b), &value2))
            {
              if (dump_file && (dump_flags & TDF_DETAILS))
                fprintf (dump_file, "siv test failed: chrec not positive.\n");

              *overlaps_a = conflict_fn_not_known ();
              *overlaps_b = conflict_fn_not_known ();      
              *last_conflicts = chrec_dont_know;
              dependence_stats.num_siv_unimplemented++;
              return;
            }
          else
            {
              if (value2 == false)
                {
                  /* Example:  
                     chrec_a = 3
                     chrec_b = {10, +, -1}
                  */
                  if (tree_fold_divides_p (CHREC_RIGHT (chrec_b), difference))
                    {
                      HOST_WIDE_INT numiter;
                      struct loop *loop = get_chrec_loop (chrec_b);

                      *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
                      tmp = fold_build2 (EXACT_DIV_EXPR,
                                         integer_type_node, difference, 
                                         CHREC_RIGHT (chrec_b));
                      *overlaps_b = conflict_fn (1, affine_fn_cst (tmp));
                      *last_conflicts = integer_one_node;

                      /* Perform weak-zero siv test to see if overlap is
                         outside the loop bounds.  */
                      numiter = estimated_loop_iterations_int (loop, true);

                      if (numiter >= 0
                          && compare_tree_int (tmp, numiter) > 0)
                        {
                          free_conflict_function (*overlaps_a);
                          free_conflict_function (*overlaps_b);
                          *overlaps_a = conflict_fn_no_dependence ();
                          *overlaps_b = conflict_fn_no_dependence ();
                          *last_conflicts = integer_zero_node;
                          dependence_stats.num_siv_independent++;
                          return;
                        }       
                      dependence_stats.num_siv_dependent++;
                      return;
                    }
                  
                  /* When the step does not divide the difference, there
                     are no overlaps.  */
                  else
                    {
                      *overlaps_a = conflict_fn_no_dependence ();
                      *overlaps_b = conflict_fn_no_dependence ();      
                      *last_conflicts = integer_zero_node;
                      dependence_stats.num_siv_independent++;
                      return;
                    }
                }
              else
                {
                  /* Example:  
                     chrec_a = 3  
                     chrec_b = {4, +, 1}
                 
                     In this case, chrec_a will not overlap with chrec_b.  */
                  *overlaps_a = conflict_fn_no_dependence ();
                  *overlaps_b = conflict_fn_no_dependence ();
                  *last_conflicts = integer_zero_node;
                  dependence_stats.num_siv_independent++;
                  return;
                }
            }
        }
    }
}

/* Helper recursive function for initializing the matrix A.  Returns
   the initial value of CHREC.  */

static int
initialize_matrix_A (lambda_matrix A, tree chrec, unsigned index, int mult)
{
  gcc_assert (chrec);

  if (TREE_CODE (chrec) != POLYNOMIAL_CHREC)
    return int_cst_value (chrec);

  A[index][0] = mult * int_cst_value (CHREC_RIGHT (chrec));
  return initialize_matrix_A (A, CHREC_LEFT (chrec), index + 1, mult);
}

#define FLOOR_DIV(x,y) ((x) / (y))

/* Solves the special case of the Diophantine equation: 
   | {0, +, STEP_A}_x (OVERLAPS_A) = {0, +, STEP_B}_y (OVERLAPS_B)

   Computes the descriptions OVERLAPS_A and OVERLAPS_B.  NITER is the
   number of iterations that loops X and Y run.  The overlaps will be
   constructed as evolutions in dimension DIM.  */

static void
compute_overlap_steps_for_affine_univar (int niter, int step_a, int step_b, 
                                         affine_fn *overlaps_a,
                                         affine_fn *overlaps_b, 
                                         tree *last_conflicts, int dim)
{
  if (((step_a > 0 && step_b > 0)
       || (step_a < 0 && step_b < 0)))
    {
      int step_overlaps_a, step_overlaps_b;
      int gcd_steps_a_b, last_conflict, tau2;

      gcd_steps_a_b = gcd (step_a, step_b);
      step_overlaps_a = step_b / gcd_steps_a_b;
      step_overlaps_b = step_a / gcd_steps_a_b;

      tau2 = FLOOR_DIV (niter, step_overlaps_a);
      tau2 = MIN (tau2, FLOOR_DIV (niter, step_overlaps_b));
      last_conflict = tau2;

      *overlaps_a = affine_fn_univar (integer_zero_node, dim, 
                                      build_int_cst (NULL_TREE,
                                                     step_overlaps_a));
      *overlaps_b = affine_fn_univar (integer_zero_node, dim, 
                                      build_int_cst (NULL_TREE, 
                                                     step_overlaps_b));
      *last_conflicts = build_int_cst (NULL_TREE, last_conflict);
    }

  else
    {
      *overlaps_a = affine_fn_cst (integer_zero_node);
      *overlaps_b = affine_fn_cst (integer_zero_node);
      *last_conflicts = integer_zero_node;
    }
}

/* Solves the special case of a Diophantine equation where CHREC_A is
   an affine bivariate function, and CHREC_B is an affine univariate
   function.  For example, 

   | {{0, +, 1}_x, +, 1335}_y = {0, +, 1336}_z
   
   has the following overlapping functions: 

   | x (t, u, v) = {{0, +, 1336}_t, +, 1}_v
   | y (t, u, v) = {{0, +, 1336}_u, +, 1}_v
   | z (t, u, v) = {{{0, +, 1}_t, +, 1335}_u, +, 1}_v

   FORNOW: This is a specialized implementation for a case occurring in
   a common benchmark.  Implement the general algorithm.  */

static void
compute_overlap_steps_for_affine_1_2 (tree chrec_a, tree chrec_b, 
                                      conflict_function **overlaps_a,
                                      conflict_function **overlaps_b, 
                                      tree *last_conflicts)
{
  bool xz_p, yz_p, xyz_p;
  int step_x, step_y, step_z;
  HOST_WIDE_INT niter_x, niter_y, niter_z, niter;
  affine_fn overlaps_a_xz, overlaps_b_xz;
  affine_fn overlaps_a_yz, overlaps_b_yz;
  affine_fn overlaps_a_xyz, overlaps_b_xyz;
  affine_fn ova1, ova2, ovb;
  tree last_conflicts_xz, last_conflicts_yz, last_conflicts_xyz;

  step_x = int_cst_value (CHREC_RIGHT (CHREC_LEFT (chrec_a)));
  step_y = int_cst_value (CHREC_RIGHT (chrec_a));
  step_z = int_cst_value (CHREC_RIGHT (chrec_b));

  niter_x = estimated_loop_iterations_int
                (get_chrec_loop (CHREC_LEFT (chrec_a)), true);
  niter_y = estimated_loop_iterations_int (get_chrec_loop (chrec_a), true);
  niter_z = estimated_loop_iterations_int (get_chrec_loop (chrec_b), true);
  
  if (niter_x < 0 || niter_y < 0 || niter_z < 0)
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "overlap steps test failed: no iteration counts.\n");
           
      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      return;
    }

  niter = MIN (niter_x, niter_z);
  compute_overlap_steps_for_affine_univar (niter, step_x, step_z,
                                           &overlaps_a_xz,
                                           &overlaps_b_xz,
                                           &last_conflicts_xz, 1);
  niter = MIN (niter_y, niter_z);
  compute_overlap_steps_for_affine_univar (niter, step_y, step_z,
                                           &overlaps_a_yz,
                                           &overlaps_b_yz,
                                           &last_conflicts_yz, 2);
  niter = MIN (niter_x, niter_z);
  niter = MIN (niter_y, niter);
  compute_overlap_steps_for_affine_univar (niter, step_x + step_y, step_z,
                                           &overlaps_a_xyz,
                                           &overlaps_b_xyz,
                                           &last_conflicts_xyz, 3);

  xz_p = !integer_zerop (last_conflicts_xz);
  yz_p = !integer_zerop (last_conflicts_yz);
  xyz_p = !integer_zerop (last_conflicts_xyz);

  if (xz_p || yz_p || xyz_p)
    {
      ova1 = affine_fn_cst (integer_zero_node);
      ova2 = affine_fn_cst (integer_zero_node);
      ovb = affine_fn_cst (integer_zero_node);
      if (xz_p)
        {
          affine_fn t0 = ova1;
          affine_fn t2 = ovb;

          ova1 = affine_fn_plus (ova1, overlaps_a_xz);
          ovb = affine_fn_plus (ovb, overlaps_b_xz);
          affine_fn_free (t0);
          affine_fn_free (t2);
          *last_conflicts = last_conflicts_xz;
        }
      if (yz_p)
        {
          affine_fn t0 = ova2;
          affine_fn t2 = ovb;

          ova2 = affine_fn_plus (ova2, overlaps_a_yz);
          ovb = affine_fn_plus (ovb, overlaps_b_yz);
          affine_fn_free (t0);
          affine_fn_free (t2);
          *last_conflicts = last_conflicts_yz;
        }
      if (xyz_p)
        {
          affine_fn t0 = ova1;
          affine_fn t2 = ova2;
          affine_fn t4 = ovb;

          ova1 = affine_fn_plus (ova1, overlaps_a_xyz);
          ova2 = affine_fn_plus (ova2, overlaps_a_xyz);
          ovb = affine_fn_plus (ovb, overlaps_b_xyz);
          affine_fn_free (t0);
          affine_fn_free (t2);
          affine_fn_free (t4);
          *last_conflicts = last_conflicts_xyz;
        }
      *overlaps_a = conflict_fn (2, ova1, ova2);
      *overlaps_b = conflict_fn (1, ovb);
    }
  else
    {
      *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *overlaps_b = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *last_conflicts = integer_zero_node;
    }

  affine_fn_free (overlaps_a_xz);
  affine_fn_free (overlaps_b_xz);
  affine_fn_free (overlaps_a_yz);
  affine_fn_free (overlaps_b_yz);
  affine_fn_free (overlaps_a_xyz);
  affine_fn_free (overlaps_b_xyz);
}

/* Determines the overlapping elements due to accesses CHREC_A and
   CHREC_B, that are affine functions.  This function cannot handle
   symbolic evolution functions, ie. when initial conditions are
   parameters, because it uses lambda matrices of integers.  */

static void
analyze_subscript_affine_affine (tree chrec_a, 
                                 tree chrec_b,
                                 conflict_function **overlaps_a, 
                                 conflict_function **overlaps_b, 
                                 tree *last_conflicts)
{
  unsigned nb_vars_a, nb_vars_b, dim;
  int init_a, init_b, gamma, gcd_alpha_beta;
  int tau1, tau2;
  lambda_matrix A, U, S;

  if (eq_evolutions_p (chrec_a, chrec_b))
    {
      /* The accessed index overlaps for each iteration in the
         loop.  */
      *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *overlaps_b = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *last_conflicts = chrec_dont_know;
      return;
    }
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(analyze_subscript_affine_affine \n");
  
  /* For determining the initial intersection, we have to solve a
     Diophantine equation.  This is the most time consuming part.
     
     For answering to the question: "Is there a dependence?" we have
     to prove that there exists a solution to the Diophantine
     equation, and that the solution is in the iteration domain,
     i.e. the solution is positive or zero, and that the solution
     happens before the upper bound loop.nb_iterations.  Otherwise
     there is no dependence.  This function outputs a description of
     the iterations that hold the intersections.  */

  nb_vars_a = nb_vars_in_chrec (chrec_a);
  nb_vars_b = nb_vars_in_chrec (chrec_b);

  dim = nb_vars_a + nb_vars_b;
  U = lambda_matrix_new (dim, dim);
  A = lambda_matrix_new (dim, 1);
  S = lambda_matrix_new (dim, 1);

  init_a = initialize_matrix_A (A, chrec_a, 0, 1);
  init_b = initialize_matrix_A (A, chrec_b, nb_vars_a, -1);
  gamma = init_b - init_a;

  /* Don't do all the hard work of solving the Diophantine equation
     when we already know the solution: for example, 
     | {3, +, 1}_1
     | {3, +, 4}_2
     | gamma = 3 - 3 = 0.
     Then the first overlap occurs during the first iterations: 
     | {3, +, 1}_1 ({0, +, 4}_x) = {3, +, 4}_2 ({0, +, 1}_x)
  */
  if (gamma == 0)
    {
      if (nb_vars_a == 1 && nb_vars_b == 1)
        {
          int step_a, step_b;
          HOST_WIDE_INT niter, niter_a, niter_b;
          affine_fn ova, ovb;

          niter_a = estimated_loop_iterations_int
                        (get_chrec_loop (chrec_a), true);
          niter_b = estimated_loop_iterations_int
                        (get_chrec_loop (chrec_b), true);
          if (niter_a < 0 || niter_b < 0)
            {
              if (dump_file && (dump_flags & TDF_DETAILS))
                fprintf (dump_file, "affine-affine test failed: missing iteration counts.\n");
              *overlaps_a = conflict_fn_not_known ();
              *overlaps_b = conflict_fn_not_known ();
              *last_conflicts = chrec_dont_know;
              goto end_analyze_subs_aa;
            }

          niter = MIN (niter_a, niter_b);

          step_a = int_cst_value (CHREC_RIGHT (chrec_a));
          step_b = int_cst_value (CHREC_RIGHT (chrec_b));

          compute_overlap_steps_for_affine_univar (niter, step_a, step_b, 
                                                   &ova, &ovb, 
                                                   last_conflicts, 1);
          *overlaps_a = conflict_fn (1, ova);
          *overlaps_b = conflict_fn (1, ovb);
        }

      else if (nb_vars_a == 2 && nb_vars_b == 1)
        compute_overlap_steps_for_affine_1_2
          (chrec_a, chrec_b, overlaps_a, overlaps_b, last_conflicts);

      else if (nb_vars_a == 1 && nb_vars_b == 2)
        compute_overlap_steps_for_affine_1_2
          (chrec_b, chrec_a, overlaps_b, overlaps_a, last_conflicts);

      else
        {
          if (dump_file && (dump_flags & TDF_DETAILS))
            fprintf (dump_file, "affine-affine test failed: too many variables.\n");
          *overlaps_a = conflict_fn_not_known ();
          *overlaps_b = conflict_fn_not_known ();
          *last_conflicts = chrec_dont_know;
        }
      goto end_analyze_subs_aa;
    }

  /* U.A = S */
  lambda_matrix_right_hermite (A, dim, 1, S, U);

  if (S[0][0] < 0)
    {
      S[0][0] *= -1;
      lambda_matrix_row_negate (U, dim, 0);
    }
  gcd_alpha_beta = S[0][0];

  /* Something went wrong: for example in {1, +, 0}_5 vs. {0, +, 0}_5,
     but that is a quite strange case.  Instead of ICEing, answer
     don't know.  */
  if (gcd_alpha_beta == 0)
    {
      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      goto end_analyze_subs_aa;
    }

  /* The classic "gcd-test".  */
  if (!int_divides_p (gcd_alpha_beta, gamma))
    {
      /* The "gcd-test" has determined that there is no integer
         solution, i.e. there is no dependence.  */
      *overlaps_a = conflict_fn_no_dependence ();
      *overlaps_b = conflict_fn_no_dependence ();
      *last_conflicts = integer_zero_node;
    }

  /* Both access functions are univariate.  This includes SIV and MIV cases.  */
  else if (nb_vars_a == 1 && nb_vars_b == 1)
    {
      /* Both functions should have the same evolution sign.  */
      if (((A[0][0] > 0 && -A[1][0] > 0)
           || (A[0][0] < 0 && -A[1][0] < 0)))
        {
          /* The solutions are given by:
             | 
             | [GAMMA/GCD_ALPHA_BETA  t].[u11 u12]  = [x0]
             |                           [u21 u22]    [y0]
         
             For a given integer t.  Using the following variables,
         
             | i0 = u11 * gamma / gcd_alpha_beta
             | j0 = u12 * gamma / gcd_alpha_beta
             | i1 = u21
             | j1 = u22
         
             the solutions are:
         
             | x0 = i0 + i1 * t, 
             | y0 = j0 + j1 * t.  */
      
          int i0, j0, i1, j1;

          /* X0 and Y0 are the first iterations for which there is a
             dependence.  X0, Y0 are two solutions of the Diophantine
             equation: chrec_a (X0) = chrec_b (Y0).  */
          int x0, y0;
          int niter, niter_a, niter_b;

          niter_a = estimated_loop_iterations_int
                        (get_chrec_loop (chrec_a), true);
          niter_b = estimated_loop_iterations_int
                        (get_chrec_loop (chrec_b), true);

          if (niter_a < 0 || niter_b < 0)
            {
              if (dump_file && (dump_flags & TDF_DETAILS))
                fprintf (dump_file, "affine-affine test failed: missing iteration counts.\n");
              *overlaps_a = conflict_fn_not_known ();
              *overlaps_b = conflict_fn_not_known ();
              *last_conflicts = chrec_dont_know;
              goto end_analyze_subs_aa;
            }

          niter = MIN (niter_a, niter_b);

          i0 = U[0][0] * gamma / gcd_alpha_beta;
          j0 = U[0][1] * gamma / gcd_alpha_beta;
          i1 = U[1][0];
          j1 = U[1][1];

          if ((i1 == 0 && i0 < 0)
              || (j1 == 0 && j0 < 0))
            {
              /* There is no solution.  
                 FIXME: The case "i0 > nb_iterations, j0 > nb_iterations" 
                 falls in here, but for the moment we don't look at the 
                 upper bound of the iteration domain.  */
              *overlaps_a = conflict_fn_no_dependence ();
              *overlaps_b = conflict_fn_no_dependence ();
              *last_conflicts = integer_zero_node;
            }

          else 
            {
              if (i1 > 0)
                {
                  tau1 = CEIL (-i0, i1);
                  tau2 = FLOOR_DIV (niter - i0, i1);

                  if (j1 > 0)
                    {
                      int last_conflict, min_multiple;
                      tau1 = MAX (tau1, CEIL (-j0, j1));
                      tau2 = MIN (tau2, FLOOR_DIV (niter - j0, j1));

                      x0 = i1 * tau1 + i0;
                      y0 = j1 * tau1 + j0;

                      /* At this point (x0, y0) is one of the
                         solutions to the Diophantine equation.  The
                         next step has to compute the smallest
                         positive solution: the first conflicts.  */
                      min_multiple = MIN (x0 / i1, y0 / j1);
                      x0 -= i1 * min_multiple;
                      y0 -= j1 * min_multiple;

                      tau1 = (x0 - i0)/i1;
                      last_conflict = tau2 - tau1;

                      /* If the overlap occurs outside of the bounds of the
                         loop, there is no dependence.  */
                      if (x0 > niter || y0  > niter)
                        {
                          *overlaps_a = conflict_fn_no_dependence ();
                          *overlaps_b = conflict_fn_no_dependence ();
                          *last_conflicts = integer_zero_node;
                        }
                      else
                        {
                          *overlaps_a
                            = conflict_fn (1,
                                affine_fn_univar (build_int_cst (NULL_TREE, x0),
                                                  1,
                                                  build_int_cst (NULL_TREE, i1)));
                          *overlaps_b
                            = conflict_fn (1,
                                affine_fn_univar (build_int_cst (NULL_TREE, y0),
                                                  1,
                                                  build_int_cst (NULL_TREE, j1)));
                          *last_conflicts = build_int_cst (NULL_TREE, last_conflict);
                        }
                    }
                  else
                    {
                      /* FIXME: For the moment, the upper bound of the
                         iteration domain for j is not checked.  */
                      if (dump_file && (dump_flags & TDF_DETAILS))
                        fprintf (dump_file, "affine-affine test failed: unimplemented.\n");
                      *overlaps_a = conflict_fn_not_known ();
                      *overlaps_b = conflict_fn_not_known ();
                      *last_conflicts = chrec_dont_know;
                    }
                }
          
              else
                {
                  /* FIXME: For the moment, the upper bound of the
                     iteration domain for i is not checked.  */
                  if (dump_file && (dump_flags & TDF_DETAILS))
                    fprintf (dump_file, "affine-affine test failed: unimplemented.\n");
                  *overlaps_a = conflict_fn_not_known ();
                  *overlaps_b = conflict_fn_not_known ();
                  *last_conflicts = chrec_dont_know;
                }
            }
        }
      else
        {
          if (dump_file && (dump_flags & TDF_DETAILS))
            fprintf (dump_file, "affine-affine test failed: unimplemented.\n");
          *overlaps_a = conflict_fn_not_known ();
          *overlaps_b = conflict_fn_not_known ();
          *last_conflicts = chrec_dont_know;
        }
    }

  else
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "affine-affine test failed: unimplemented.\n");
      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
    }

end_analyze_subs_aa:  
  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "  (overlaps_a = ");
      dump_conflict_function (dump_file, *overlaps_a);
      fprintf (dump_file, ")\n  (overlaps_b = ");
      dump_conflict_function (dump_file, *overlaps_b);
      fprintf (dump_file, ")\n");
      fprintf (dump_file, ")\n");
    }
}

/* Returns true when analyze_subscript_affine_affine can be used for
   determining the dependence relation between chrec_a and chrec_b,
   that contain symbols.  This function modifies chrec_a and chrec_b
   such that the analysis result is the same, and such that they don't
   contain symbols, and then can safely be passed to the analyzer.  

   Example: The analysis of the following tuples of evolutions produce
   the same results: {x+1, +, 1}_1 vs. {x+3, +, 1}_1, and {-2, +, 1}_1
   vs. {0, +, 1}_1
   
   {x+1, +, 1}_1 ({2, +, 1}_1) = {x+3, +, 1}_1 ({0, +, 1}_1)
   {-2, +, 1}_1 ({2, +, 1}_1) = {0, +, 1}_1 ({0, +, 1}_1)
*/

static bool
can_use_analyze_subscript_affine_affine (tree *chrec_a, tree *chrec_b)
{
  tree diff, type, left_a, left_b, right_b;

  if (chrec_contains_symbols (CHREC_RIGHT (*chrec_a))
      || chrec_contains_symbols (CHREC_RIGHT (*chrec_b)))
    /* FIXME: For the moment not handled.  Might be refined later.  */
    return false;

  type = chrec_type (*chrec_a);
  left_a = CHREC_LEFT (*chrec_a);
  left_b = chrec_convert (type, CHREC_LEFT (*chrec_b), NULL_TREE);
  diff = chrec_fold_minus (type, left_a, left_b);

  if (!evolution_function_is_constant_p (diff))
    return false;

  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "can_use_subscript_aff_aff_for_symbolic \n");

  *chrec_a = build_polynomial_chrec (CHREC_VARIABLE (*chrec_a), 
                                     diff, CHREC_RIGHT (*chrec_a));
  right_b = chrec_convert (type, CHREC_RIGHT (*chrec_b), NULL_TREE);
  *chrec_b = build_polynomial_chrec (CHREC_VARIABLE (*chrec_b),
                                     build_int_cst (type, 0),
                                     right_b);
  return true;
}

/* Analyze a SIV (Single Index Variable) subscript.  *OVERLAPS_A and
   *OVERLAPS_B are initialized to the functions that describe the
   relation between the elements accessed twice by CHREC_A and
   CHREC_B.  For k >= 0, the following property is verified:

   CHREC_A (*OVERLAPS_A (k)) = CHREC_B (*OVERLAPS_B (k)).  */

static void
analyze_siv_subscript (tree chrec_a, 
                       tree chrec_b,
                       conflict_function **overlaps_a, 
                       conflict_function **overlaps_b, 
                       tree *last_conflicts)
{
  dependence_stats.num_siv++;
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(analyze_siv_subscript \n");
  
  if (evolution_function_is_constant_p (chrec_a)
      && evolution_function_is_affine_p (chrec_b))
    analyze_siv_subscript_cst_affine (chrec_a, chrec_b, 
                                      overlaps_a, overlaps_b, last_conflicts);
  
  else if (evolution_function_is_affine_p (chrec_a)
           && evolution_function_is_constant_p (chrec_b))
    analyze_siv_subscript_cst_affine (chrec_b, chrec_a, 
                                      overlaps_b, overlaps_a, last_conflicts);
  
  else if (evolution_function_is_affine_p (chrec_a)
           && evolution_function_is_affine_p (chrec_b))
    {
      if (!chrec_contains_symbols (chrec_a)
          && !chrec_contains_symbols (chrec_b))
        {
          analyze_subscript_affine_affine (chrec_a, chrec_b, 
                                           overlaps_a, overlaps_b, 
                                           last_conflicts);

          if (CF_NOT_KNOWN_P (*overlaps_a)
              || CF_NOT_KNOWN_P (*overlaps_b))
            dependence_stats.num_siv_unimplemented++;
          else if (CF_NO_DEPENDENCE_P (*overlaps_a)
                   || CF_NO_DEPENDENCE_P (*overlaps_b))
            dependence_stats.num_siv_independent++;
          else
            dependence_stats.num_siv_dependent++;
        }
      else if (can_use_analyze_subscript_affine_affine (&chrec_a, 
                                                        &chrec_b))
        {
          analyze_subscript_affine_affine (chrec_a, chrec_b, 
                                           overlaps_a, overlaps_b, 
                                           last_conflicts);
          /* FIXME: The number of iterations is a symbolic expression.
             Compute it properly.  */
          *last_conflicts = chrec_dont_know;

          if (CF_NOT_KNOWN_P (*overlaps_a)
              || CF_NOT_KNOWN_P (*overlaps_b))
            dependence_stats.num_siv_unimplemented++;
          else if (CF_NO_DEPENDENCE_P (*overlaps_a)
                   || CF_NO_DEPENDENCE_P (*overlaps_b))
            dependence_stats.num_siv_independent++;
          else
            dependence_stats.num_siv_dependent++;
        }
      else
        goto siv_subscript_dontknow;
    }

  else
    {
    siv_subscript_dontknow:;
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "siv test failed: unimplemented.\n");
      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      dependence_stats.num_siv_unimplemented++;
    }
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, ")\n");
}

/* Returns false if we can prove that the greatest common divisor of the steps
   of CHREC does not divide CST, false otherwise.  */

static bool
gcd_of_steps_may_divide_p (tree chrec, tree cst)
{
  HOST_WIDE_INT cd = 0, val;
  tree step;

  if (!host_integerp (cst, 0))
    return true;
  val = tree_low_cst (cst, 0);

  while (TREE_CODE (chrec) == POLYNOMIAL_CHREC)
    {
      step = CHREC_RIGHT (chrec);
      if (!host_integerp (step, 0))
        return true;
      cd = gcd (cd, tree_low_cst (step, 0));
      chrec = CHREC_LEFT (chrec);
    }

  return val % cd == 0;
}

/* Analyze a MIV (Multiple Index Variable) subscript.  *OVERLAPS_A and
   *OVERLAPS_B are initialized to the functions that describe the
   relation between the elements accessed twice by CHREC_A and
   CHREC_B.  For k >= 0, the following property is verified:

   CHREC_A (*OVERLAPS_A (k)) = CHREC_B (*OVERLAPS_B (k)).  */

static void
analyze_miv_subscript (tree chrec_a, 
                       tree chrec_b, 
                       conflict_function **overlaps_a, 
                       conflict_function **overlaps_b, 
                       tree *last_conflicts)
{
  /* FIXME:  This is a MIV subscript, not yet handled.
     Example: (A[{1, +, 1}_1] vs. A[{1, +, 1}_2]) that comes from 
     (A[i] vs. A[j]).  
     
     In the SIV test we had to solve a Diophantine equation with two
     variables.  In the MIV case we have to solve a Diophantine
     equation with 2*n variables (if the subscript uses n IVs).
  */
  tree difference;
  dependence_stats.num_miv++;
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(analyze_miv_subscript \n");

  chrec_a = chrec_convert (integer_type_node, chrec_a, NULL_TREE);
  chrec_b = chrec_convert (integer_type_node, chrec_b, NULL_TREE);
  difference = chrec_fold_minus (integer_type_node, chrec_a, chrec_b);
  
  if (eq_evolutions_p (chrec_a, chrec_b))
    {
      /* Access functions are the same: all the elements are accessed
         in the same order.  */
      *overlaps_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *overlaps_b = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *last_conflicts = estimated_loop_iterations_tree
                                (get_chrec_loop (chrec_a), true);
      dependence_stats.num_miv_dependent++;
    }
  
  else if (evolution_function_is_constant_p (difference)
           /* For the moment, the following is verified:
              evolution_function_is_affine_multivariate_p (chrec_a, 0) */
           && !gcd_of_steps_may_divide_p (chrec_a, difference))
    {
      /* testsuite/.../ssa-chrec-33.c
         {{21, +, 2}_1, +, -2}_2  vs.  {{20, +, 2}_1, +, -2}_2 
         
         The difference is 1, and all the evolution steps are multiples
         of 2, consequently there are no overlapping elements.  */
      *overlaps_a = conflict_fn_no_dependence ();
      *overlaps_b = conflict_fn_no_dependence ();
      *last_conflicts = integer_zero_node;
      dependence_stats.num_miv_independent++;
    }
  
  else if (evolution_function_is_affine_multivariate_p (chrec_a, 0)
           && !chrec_contains_symbols (chrec_a)
           && evolution_function_is_affine_multivariate_p (chrec_b, 0)
           && !chrec_contains_symbols (chrec_b))
    {
      /* testsuite/.../ssa-chrec-35.c
         {0, +, 1}_2  vs.  {0, +, 1}_3
         the overlapping elements are respectively located at iterations:
         {0, +, 1}_x and {0, +, 1}_x, 
         in other words, we have the equality: 
         {0, +, 1}_2 ({0, +, 1}_x) = {0, +, 1}_3 ({0, +, 1}_x)
         
         Other examples: 
         {{0, +, 1}_1, +, 2}_2 ({0, +, 1}_x, {0, +, 1}_y) = 
         {0, +, 1}_1 ({{0, +, 1}_x, +, 2}_y)

         {{0, +, 2}_1, +, 3}_2 ({0, +, 1}_y, {0, +, 1}_x) = 
         {{0, +, 3}_1, +, 2}_2 ({0, +, 1}_x, {0, +, 1}_y)
      */
      analyze_subscript_affine_affine (chrec_a, chrec_b, 
                                       overlaps_a, overlaps_b, last_conflicts);

      if (CF_NOT_KNOWN_P (*overlaps_a)
          || CF_NOT_KNOWN_P (*overlaps_b))
        dependence_stats.num_miv_unimplemented++;
      else if (CF_NO_DEPENDENCE_P (*overlaps_a)
               || CF_NO_DEPENDENCE_P (*overlaps_b))
        dependence_stats.num_miv_independent++;
      else
        dependence_stats.num_miv_dependent++;
    }
  
  else
    {
      /* When the analysis is too difficult, answer "don't know".  */
      if (dump_file && (dump_flags & TDF_DETAILS))
        fprintf (dump_file, "analyze_miv_subscript test failed: unimplemented.\n");

      *overlaps_a = conflict_fn_not_known ();
      *overlaps_b = conflict_fn_not_known ();
      *last_conflicts = chrec_dont_know;
      dependence_stats.num_miv_unimplemented++;
    }
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, ")\n");
}

/* Determines the iterations for which CHREC_A is equal to CHREC_B.
   OVERLAP_ITERATIONS_A and OVERLAP_ITERATIONS_B are initialized with
   two functions that describe the iterations that contain conflicting
   elements.
   
   Remark: For an integer k >= 0, the following equality is true:
   
   CHREC_A (OVERLAP_ITERATIONS_A (k)) == CHREC_B (OVERLAP_ITERATIONS_B (k)).
*/

static void 
analyze_overlapping_iterations (tree chrec_a, 
                                tree chrec_b, 
                                conflict_function **overlap_iterations_a, 
                                conflict_function **overlap_iterations_b, 
                                tree *last_conflicts)
{
  dependence_stats.num_subscript_tests++;
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "(analyze_overlapping_iterations \n");
      fprintf (dump_file, "  (chrec_a = ");
      print_generic_expr (dump_file, chrec_a, 0);
      fprintf (dump_file, ")\n  (chrec_b = ");
      print_generic_expr (dump_file, chrec_b, 0);
      fprintf (dump_file, ")\n");
    }

  if (chrec_a == NULL_TREE
      || chrec_b == NULL_TREE
      || chrec_contains_undetermined (chrec_a)
      || chrec_contains_undetermined (chrec_b))
    {
      dependence_stats.num_subscript_undetermined++;
      
      *overlap_iterations_a = conflict_fn_not_known ();
      *overlap_iterations_b = conflict_fn_not_known ();
    }

  /* If they are the same chrec, and are affine, they overlap 
     on every iteration.  */
  else if (eq_evolutions_p (chrec_a, chrec_b)
           && evolution_function_is_affine_multivariate_p (chrec_a, 0))
    {
      dependence_stats.num_same_subscript_function++;
      *overlap_iterations_a = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *overlap_iterations_b = conflict_fn (1, affine_fn_cst (integer_zero_node));
      *last_conflicts = chrec_dont_know;
    }

  /* If they aren't the same, and aren't affine, we can't do anything
     yet. */
  else if ((chrec_contains_symbols (chrec_a) 
            || chrec_contains_symbols (chrec_b))
           && (!evolution_function_is_affine_multivariate_p (chrec_a, 0)
               || !evolution_function_is_affine_multivariate_p (chrec_b, 0)))
    {
      dependence_stats.num_subscript_undetermined++;
      *overlap_iterations_a = conflict_fn_not_known ();
      *overlap_iterations_b = conflict_fn_not_known ();
    }

  else if (ziv_subscript_p (chrec_a, chrec_b))
    analyze_ziv_subscript (chrec_a, chrec_b, 
                           overlap_iterations_a, overlap_iterations_b,
                           last_conflicts);
  
  else if (siv_subscript_p (chrec_a, chrec_b))
    analyze_siv_subscript (chrec_a, chrec_b, 
                           overlap_iterations_a, overlap_iterations_b, 
                           last_conflicts);
  
  else
    analyze_miv_subscript (chrec_a, chrec_b, 
                           overlap_iterations_a, overlap_iterations_b,
                           last_conflicts);
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "  (overlap_iterations_a = ");
      dump_conflict_function (dump_file, *overlap_iterations_a);
      fprintf (dump_file, ")\n  (overlap_iterations_b = ");
      dump_conflict_function (dump_file, *overlap_iterations_b);
      fprintf (dump_file, ")\n");
      fprintf (dump_file, ")\n");
    }
}

/* Helper function for uniquely inserting distance vectors.  */

static void
save_dist_v (struct data_dependence_relation *ddr, lambda_vector dist_v)
{
  unsigned i;
  lambda_vector v;

  for (i = 0; VEC_iterate (lambda_vector, DDR_DIST_VECTS (ddr), i, v); i++)
    if (lambda_vector_equal (v, dist_v, DDR_NB_LOOPS (ddr)))
      return;

  VEC_safe_push (lambda_vector, heap, DDR_DIST_VECTS (ddr), dist_v);
}

/* Helper function for uniquely inserting direction vectors.  */

static void
save_dir_v (struct data_dependence_relation *ddr, lambda_vector dir_v)
{
  unsigned i;
  lambda_vector v;

  for (i = 0; VEC_iterate (lambda_vector, DDR_DIR_VECTS (ddr), i, v); i++)
    if (lambda_vector_equal (v, dir_v, DDR_NB_LOOPS (ddr)))
      return;

  VEC_safe_push (lambda_vector, heap, DDR_DIR_VECTS (ddr), dir_v);
}

/* Add a distance of 1 on all the loops outer than INDEX.  If we
   haven't yet determined a distance for this outer loop, push a new
   distance vector composed of the previous distance, and a distance
   of 1 for this outer loop.  Example:

   | loop_1
   |   loop_2
   |     A[10]
   |   endloop_2
   | endloop_1

   Saved vectors are of the form (dist_in_1, dist_in_2).  First, we
   save (0, 1), then we have to save (1, 0).  */

static void
add_outer_distances (struct data_dependence_relation *ddr,
                     lambda_vector dist_v, int index)
{
  /* For each outer loop where init_v is not set, the accesses are
     in dependence of distance 1 in the loop.  */
  while (--index >= 0)
    {
      lambda_vector save_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
      lambda_vector_copy (dist_v, save_v, DDR_NB_LOOPS (ddr));
      save_v[index] = 1;
      save_dist_v (ddr, save_v);
    }
}

/* Return false when fail to represent the data dependence as a
   distance vector.  INIT_B is set to true when a component has been
   added to the distance vector DIST_V.  INDEX_CARRY is then set to
   the index in DIST_V that carries the dependence.  */

static bool
build_classic_dist_vector_1 (struct data_dependence_relation *ddr,
                             struct data_reference *ddr_a,
                             struct data_reference *ddr_b,
                             lambda_vector dist_v, bool *init_b,
                             int *index_carry)
{
  unsigned i;
  lambda_vector init_v = lambda_vector_new (DDR_NB_LOOPS (ddr));

  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    {
      tree access_fn_a, access_fn_b;
      struct subscript *subscript = DDR_SUBSCRIPT (ddr, i);

      if (chrec_contains_undetermined (SUB_DISTANCE (subscript)))
        {
          non_affine_dependence_relation (ddr);
          return false;
        }

      access_fn_a = DR_ACCESS_FN (ddr_a, i);
      access_fn_b = DR_ACCESS_FN (ddr_b, i);

      if (TREE_CODE (access_fn_a) == POLYNOMIAL_CHREC 
          && TREE_CODE (access_fn_b) == POLYNOMIAL_CHREC)
        {
          int dist, index;
          int index_a = index_in_loop_nest (CHREC_VARIABLE (access_fn_a),
                                            DDR_LOOP_NEST (ddr));
          int index_b = index_in_loop_nest (CHREC_VARIABLE (access_fn_b),
                                            DDR_LOOP_NEST (ddr));

          /* The dependence is carried by the outermost loop.  Example:
             | loop_1
             |   A[{4, +, 1}_1]
             |   loop_2
             |     A[{5, +, 1}_2]
             |   endloop_2
             | endloop_1
             In this case, the dependence is carried by loop_1.  */
          index = index_a < index_b ? index_a : index_b;
          *index_carry = MIN (index, *index_carry);

          if (chrec_contains_undetermined (SUB_DISTANCE (subscript)))
            {
              non_affine_dependence_relation (ddr);
              return false;
            }
          
          dist = int_cst_value (SUB_DISTANCE (subscript));

          /* This is the subscript coupling test.  If we have already
             recorded a distance for this loop (a distance coming from
             another subscript), it should be the same.  For example,
             in the following code, there is no dependence:

             | loop i = 0, N, 1
             |   T[i+1][i] = ...
             |   ... = T[i][i]
             | endloop
          */
          if (init_v[index] != 0 && dist_v[index] != dist)
            {
              finalize_ddr_dependent (ddr, chrec_known);
              return false;
            }

          dist_v[index] = dist;
          init_v[index] = 1;
          *init_b = true;
        }
      else if (!operand_equal_p (access_fn_a, access_fn_b, 0))
        {
          /* This can be for example an affine vs. constant dependence
             (T[i] vs. T[3]) that is not an affine dependence and is
             not representable as a distance vector.  */
          non_affine_dependence_relation (ddr);
          return false;
        }
    }

  return true;
}

/* Return true when the DDR contains two data references that have the
   same access functions.  */

static bool
same_access_functions (struct data_dependence_relation *ddr)
{
  unsigned i;

  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    if (!eq_evolutions_p (DR_ACCESS_FN (DDR_A (ddr), i),
                          DR_ACCESS_FN (DDR_B (ddr), i)))
      return false;

  return true;
}

/* Return true when the DDR contains only constant access functions.  */

static bool
constant_access_functions (struct data_dependence_relation *ddr)
{
  unsigned i;

  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    if (!evolution_function_is_constant_p (DR_ACCESS_FN (DDR_A (ddr), i))
        || !evolution_function_is_constant_p (DR_ACCESS_FN (DDR_B (ddr), i)))
      return false;

  return true;
}


/* Helper function for the case where DDR_A and DDR_B are the same
   multivariate access function.  */

static void
add_multivariate_self_dist (struct data_dependence_relation *ddr, tree c_2)
{
  int x_1, x_2;
  tree c_1 = CHREC_LEFT (c_2);
  tree c_0 = CHREC_LEFT (c_1);
  lambda_vector dist_v;
  int v1, v2, cd;

  /* Polynomials with more than 2 variables are not handled yet.  */
  if (TREE_CODE (c_0) != INTEGER_CST)
    {
      DDR_ARE_DEPENDENT (ddr) = chrec_dont_know;
      return;
    }

  x_2 = index_in_loop_nest (CHREC_VARIABLE (c_2), DDR_LOOP_NEST (ddr));
  x_1 = index_in_loop_nest (CHREC_VARIABLE (c_1), DDR_LOOP_NEST (ddr));

  /* For "{{0, +, 2}_1, +, 3}_2" the distance vector is (3, -2).  */
  dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
  v1 = int_cst_value (CHREC_RIGHT (c_1));
  v2 = int_cst_value (CHREC_RIGHT (c_2));
  cd = gcd (v1, v2);
  v1 /= cd;
  v2 /= cd;

  if (v2 < 0)
    {
      v2 = -v2;
      v1 = -v1;
    }

  dist_v[x_1] = v2;
  dist_v[x_2] = -v1;
  save_dist_v (ddr, dist_v);

  add_outer_distances (ddr, dist_v, x_1);
}

/* Helper function for the case where DDR_A and DDR_B are the same
   access functions.  */

static void
add_other_self_distances (struct data_dependence_relation *ddr)
{
  lambda_vector dist_v;
  unsigned i;
  int index_carry = DDR_NB_LOOPS (ddr);

  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    {
      tree access_fun = DR_ACCESS_FN (DDR_A (ddr), i);

      if (TREE_CODE (access_fun) == POLYNOMIAL_CHREC)
        {
          if (!evolution_function_is_univariate_p (access_fun))
            {
              if (DDR_NUM_SUBSCRIPTS (ddr) != 1)
                {
                  DDR_ARE_DEPENDENT (ddr) = chrec_dont_know;
                  return;
                }

              add_multivariate_self_dist (ddr, DR_ACCESS_FN (DDR_A (ddr), 0));
              return;
            }

          index_carry = MIN (index_carry,
                             index_in_loop_nest (CHREC_VARIABLE (access_fun),
                                                 DDR_LOOP_NEST (ddr)));
        }
    }

  dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
  add_outer_distances (ddr, dist_v, index_carry);
}

static void
insert_innermost_unit_dist_vector (struct data_dependence_relation *ddr)
{
  lambda_vector dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));

  dist_v[DDR_INNER_LOOP (ddr)] = 1;
  save_dist_v (ddr, dist_v);
}

/* Adds a unit distance vector to DDR when there is a 0 overlap.  This
   is the case for example when access functions are the same and
   equal to a constant, as in:

   | loop_1
   |   A[3] = ...
   |   ... = A[3]
   | endloop_1

   in which case the distance vectors are (0) and (1).  */

static void
add_distance_for_zero_overlaps (struct data_dependence_relation *ddr)
{
  unsigned i, j;

  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    {
      subscript_p sub = DDR_SUBSCRIPT (ddr, i);
      conflict_function *ca = SUB_CONFLICTS_IN_A (sub);
      conflict_function *cb = SUB_CONFLICTS_IN_B (sub);

      for (j = 0; j < ca->n; j++)
        if (affine_function_zero_p (ca->fns[j]))
          {
            insert_innermost_unit_dist_vector (ddr);
            return;
          }

      for (j = 0; j < cb->n; j++)
        if (affine_function_zero_p (cb->fns[j]))
          {
            insert_innermost_unit_dist_vector (ddr);
            return;
          }
    }
}

/* Compute the classic per loop distance vector.  DDR is the data
   dependence relation to build a vector from.  Return false when fail
   to represent the data dependence as a distance vector.  */

static bool
build_classic_dist_vector (struct data_dependence_relation *ddr)
{
  bool init_b = false;
  int index_carry = DDR_NB_LOOPS (ddr);
  lambda_vector dist_v;

  if (DDR_ARE_DEPENDENT (ddr) != NULL_TREE)
    return true;

  if (same_access_functions (ddr))
    {
      /* Save the 0 vector.  */
      dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
      save_dist_v (ddr, dist_v);

      if (constant_access_functions (ddr))
        add_distance_for_zero_overlaps (ddr);

      if (DDR_NB_LOOPS (ddr) > 1)
        add_other_self_distances (ddr);

      return true;
    }

  dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
  if (!build_classic_dist_vector_1 (ddr, DDR_A (ddr), DDR_B (ddr),
                                    dist_v, &init_b, &index_carry))
    return false;

  /* Save the distance vector if we initialized one.  */
  if (init_b)
    {
      /* Verify a basic constraint: classic distance vectors should
         always be lexicographically positive.

         Data references are collected in the order of execution of
         the program, thus for the following loop

         | for (i = 1; i < 100; i++)
         |   for (j = 1; j < 100; j++)
         |     {
         |       t = T[j+1][i-1];  // A
         |       T[j][i] = t + 2;  // B
         |     }

         references are collected following the direction of the wind:
         A then B.  The data dependence tests are performed also
         following this order, such that we're looking at the distance
         separating the elements accessed by A from the elements later
         accessed by B.  But in this example, the distance returned by
         test_dep (A, B) is lexicographically negative (-1, 1), that
         means that the access A occurs later than B with respect to
         the outer loop, ie. we're actually looking upwind.  In this
         case we solve test_dep (B, A) looking downwind to the
         lexicographically positive solution, that returns the
         distance vector (1, -1).  */
      if (!lambda_vector_lexico_pos (dist_v, DDR_NB_LOOPS (ddr)))
        {
          lambda_vector save_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
          subscript_dependence_tester_1 (ddr, DDR_B (ddr), DDR_A (ddr));
          compute_subscript_distance (ddr);
          build_classic_dist_vector_1 (ddr, DDR_B (ddr), DDR_A (ddr),
                                       save_v, &init_b, &index_carry);
          save_dist_v (ddr, save_v);

          /* In this case there is a dependence forward for all the
             outer loops:

             | for (k = 1; k < 100; k++)
             |  for (i = 1; i < 100; i++)
             |   for (j = 1; j < 100; j++)
             |     {
             |       t = T[j+1][i-1];  // A
             |       T[j][i] = t + 2;  // B
             |     }

             the vectors are: 
             (0,  1, -1)
             (1,  1, -1)
             (1, -1,  1)
          */
          if (DDR_NB_LOOPS (ddr) > 1)
            {
              add_outer_distances (ddr, save_v, index_carry);
              add_outer_distances (ddr, dist_v, index_carry);
            }
        }
      else
        {
          lambda_vector save_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
          lambda_vector_copy (dist_v, save_v, DDR_NB_LOOPS (ddr));
          save_dist_v (ddr, save_v);

          if (DDR_NB_LOOPS (ddr) > 1)
            {
              lambda_vector opposite_v = lambda_vector_new (DDR_NB_LOOPS (ddr));

              subscript_dependence_tester_1 (ddr, DDR_B (ddr), DDR_A (ddr));
              compute_subscript_distance (ddr);
              build_classic_dist_vector_1 (ddr, DDR_B (ddr), DDR_A (ddr),
                                           opposite_v, &init_b, &index_carry);

              add_outer_distances (ddr, dist_v, index_carry);
              add_outer_distances (ddr, opposite_v, index_carry);
            }
        }
    }
  else
    {
      /* There is a distance of 1 on all the outer loops: Example:
         there is a dependence of distance 1 on loop_1 for the array A.

         | loop_1
         |   A[5] = ...
         | endloop
      */
      add_outer_distances (ddr, dist_v,
                           lambda_vector_first_nz (dist_v,
                                                   DDR_NB_LOOPS (ddr), 0));
    }

  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      unsigned i;

      fprintf (dump_file, "(build_classic_dist_vector\n");
      for (i = 0; i < DDR_NUM_DIST_VECTS (ddr); i++)
        {
          fprintf (dump_file, "  dist_vector = (");
          print_lambda_vector (dump_file, DDR_DIST_VECT (ddr, i),
                               DDR_NB_LOOPS (ddr));
          fprintf (dump_file, "  )\n");
        }
      fprintf (dump_file, ")\n");
    }

  return true;
}

/* Return the direction for a given distance.
   FIXME: Computing dir this way is suboptimal, since dir can catch
   cases that dist is unable to represent.  */

static inline enum data_dependence_direction
dir_from_dist (int dist)
{
  if (dist > 0)
    return dir_positive;
  else if (dist < 0)
    return dir_negative;
  else
    return dir_equal;
}

/* Compute the classic per loop direction vector.  DDR is the data
   dependence relation to build a vector from.  */

static void
build_classic_dir_vector (struct data_dependence_relation *ddr)
{
  unsigned i, j;
  lambda_vector dist_v;

  for (i = 0; VEC_iterate (lambda_vector, DDR_DIST_VECTS (ddr), i, dist_v); i++)
    {
      lambda_vector dir_v = lambda_vector_new (DDR_NB_LOOPS (ddr));

      for (j = 0; j < DDR_NB_LOOPS (ddr); j++)
        dir_v[j] = dir_from_dist (dist_v[j]);

      save_dir_v (ddr, dir_v);
    }
}

/* Helper function.  Returns true when there is a dependence between
   data references DRA and DRB.  */

static bool
subscript_dependence_tester_1 (struct data_dependence_relation *ddr,
                               struct data_reference *dra,
                               struct data_reference *drb)
{
  unsigned int i;
  tree last_conflicts;
  struct subscript *subscript;

  for (i = 0; VEC_iterate (subscript_p, DDR_SUBSCRIPTS (ddr), i, subscript);
       i++)
    {
      conflict_function *overlaps_a, *overlaps_b;

      analyze_overlapping_iterations (DR_ACCESS_FN (dra, i), 
                                      DR_ACCESS_FN (drb, i),
                                      &overlaps_a, &overlaps_b, 
                                      &last_conflicts);

      if (CF_NOT_KNOWN_P (overlaps_a)
          || CF_NOT_KNOWN_P (overlaps_b))
        {
          finalize_ddr_dependent (ddr, chrec_dont_know);
          dependence_stats.num_dependence_undetermined++;
          free_conflict_function (overlaps_a);
          free_conflict_function (overlaps_b);
          return false;
        }

      else if (CF_NO_DEPENDENCE_P (overlaps_a)
               || CF_NO_DEPENDENCE_P (overlaps_b))
        {
          finalize_ddr_dependent (ddr, chrec_known);
          dependence_stats.num_dependence_independent++;
          free_conflict_function (overlaps_a);
          free_conflict_function (overlaps_b);
          return false;
        }

      else
        {
          SUB_CONFLICTS_IN_A (subscript) = overlaps_a;
          SUB_CONFLICTS_IN_B (subscript) = overlaps_b;
          SUB_LAST_CONFLICT (subscript) = last_conflicts;
        }
    }

  return true;
}

/* Computes the conflicting iterations, and initialize DDR.  */

static void
subscript_dependence_tester (struct data_dependence_relation *ddr)
{
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, "(subscript_dependence_tester \n");
  
  if (subscript_dependence_tester_1 (ddr, DDR_A (ddr), DDR_B (ddr)))
    dependence_stats.num_dependence_dependent++;

  compute_subscript_distance (ddr);
  if (build_classic_dist_vector (ddr))
    build_classic_dir_vector (ddr);

  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, ")\n");
}

/* Returns true when all the access functions of A are affine or
   constant.  */

static bool 
access_functions_are_affine_or_constant_p (struct data_reference *a)
{
  unsigned int i;
  VEC(tree,heap) *fns = DR_ACCESS_FNS (a);
  tree t;

  for (i = 0; VEC_iterate (tree, fns, i, t); i++)
    if (!evolution_function_is_constant_p (t)
        && !evolution_function_is_affine_multivariate_p (t, 0))
      return false;
  
  return true;
}

/* Initializes an equation for an OMEGA problem using the information
   contained in the ACCESS_FUN.  Returns true when the operation
   succeeded.

   PB is the omega constraint system.
   EQ is the number of the equation to be initialized.
   OFFSET is used for shifting the variables names in the constraints:
   a constrain is composed of 2 * the number of variables surrounding
   dependence accesses.  OFFSET is set either to 0 for the first n variables,
   then it is set to n.
   ACCESS_FUN is expected to be an affine chrec.  */

static bool
init_omega_eq_with_af (omega_pb pb, unsigned eq, 
                       unsigned int offset, tree access_fun, 
                       struct data_dependence_relation *ddr)
{
  switch (TREE_CODE (access_fun))
    {
    case POLYNOMIAL_CHREC:
      {
        tree left = CHREC_LEFT (access_fun);
        tree right = CHREC_RIGHT (access_fun);
        int var = CHREC_VARIABLE (access_fun);
        unsigned var_idx;

        if (TREE_CODE (right) != INTEGER_CST)
          return false;

        var_idx = index_in_loop_nest (var, DDR_LOOP_NEST (ddr));
        pb->eqs[eq].coef[offset + var_idx + 1] = int_cst_value (right);

        /* Compute the innermost loop index.  */
        DDR_INNER_LOOP (ddr) = MAX (DDR_INNER_LOOP (ddr), var_idx);

        if (offset == 0)
          pb->eqs[eq].coef[var_idx + DDR_NB_LOOPS (ddr) + 1] 
            += int_cst_value (right);

        switch (TREE_CODE (left))
          {
          case POLYNOMIAL_CHREC:
            return init_omega_eq_with_af (pb, eq, offset, left, ddr);

          case INTEGER_CST:
            pb->eqs[eq].coef[0] += int_cst_value (left);
            return true;

          default:
            return false;
          }
      }

    case INTEGER_CST:
      pb->eqs[eq].coef[0] += int_cst_value (access_fun);
      return true;

    default:
      return false;
    }
}

/* As explained in the comments preceding init_omega_for_ddr, we have
   to set up a system for each loop level, setting outer loops
   variation to zero, and current loop variation to positive or zero.
   Save each lexico positive distance vector.  */

static void
omega_extract_distance_vectors (omega_pb pb,
                                struct data_dependence_relation *ddr)
{
  int eq, geq;
  unsigned i, j;
  struct loop *loopi, *loopj;
  enum omega_result res;

  /* Set a new problem for each loop in the nest.  The basis is the
     problem that we have initialized until now.  On top of this we
     add new constraints.  */
  for (i = 0; i <= DDR_INNER_LOOP (ddr) 
         && VEC_iterate (loop_p, DDR_LOOP_NEST (ddr), i, loopi); i++)
    {
      int dist = 0;
      omega_pb copy = omega_alloc_problem (2 * DDR_NB_LOOPS (ddr),
                                           DDR_NB_LOOPS (ddr));

      omega_copy_problem (copy, pb);

      /* For all the outer loops "loop_j", add "dj = 0".  */
      for (j = 0;
           j < i && VEC_iterate (loop_p, DDR_LOOP_NEST (ddr), j, loopj); j++)
        {
          eq = omega_add_zero_eq (copy, omega_black);
          copy->eqs[eq].coef[j + 1] = 1;
        }

      /* For "loop_i", add "0 <= di".  */
      geq = omega_add_zero_geq (copy, omega_black);
      copy->geqs[geq].coef[i + 1] = 1;

      /* Reduce the constraint system, and test that the current
         problem is feasible.  */
      res = omega_simplify_problem (copy);
      if (res == omega_false 
          || res == omega_unknown
          || copy->num_geqs > (int) DDR_NB_LOOPS (ddr))
        goto next_problem;

      for (eq = 0; eq < copy->num_subs; eq++)
        if (copy->subs[eq].key == (int) i + 1)
          {
            dist = copy->subs[eq].coef[0];
            goto found_dist;
          }

      if (dist == 0)
        {
          /* Reinitialize problem...  */
          omega_copy_problem (copy, pb);
          for (j = 0;
               j < i && VEC_iterate (loop_p, DDR_LOOP_NEST (ddr), j, loopj); j++)
            {
              eq = omega_add_zero_eq (copy, omega_black);
              copy->eqs[eq].coef[j + 1] = 1;
            }

          /* ..., but this time "di = 1".  */
          eq = omega_add_zero_eq (copy, omega_black);
          copy->eqs[eq].coef[i + 1] = 1;
          copy->eqs[eq].coef[0] = -1;

          res = omega_simplify_problem (copy);
          if (res == omega_false 
              || res == omega_unknown
              || copy->num_geqs > (int) DDR_NB_LOOPS (ddr))
            goto next_problem;

          for (eq = 0; eq < copy->num_subs; eq++)
            if (copy->subs[eq].key == (int) i + 1)
              {
                dist = copy->subs[eq].coef[0];
                goto found_dist;
              }
        }

    found_dist:;
      /* Save the lexicographically positive distance vector.  */
      if (dist >= 0)
        {
          lambda_vector dist_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
          lambda_vector dir_v = lambda_vector_new (DDR_NB_LOOPS (ddr));

          dist_v[i] = dist;

          for (eq = 0; eq < copy->num_subs; eq++)
            if (copy->subs[eq].key > 0)
              {
                dist = copy->subs[eq].coef[0];
                dist_v[copy->subs[eq].key - 1] = dist;
              }

          for (j = 0; j < DDR_NB_LOOPS (ddr); j++)
            dir_v[j] = dir_from_dist (dist_v[j]);

          save_dist_v (ddr, dist_v);
          save_dir_v (ddr, dir_v);
        }

    next_problem:;
      omega_free_problem (copy);
    }
}

/* This is called for each subscript of a tuple of data references:
   insert an equality for representing the conflicts.  */

static bool
omega_setup_subscript (tree access_fun_a, tree access_fun_b,
                       struct data_dependence_relation *ddr,
                       omega_pb pb, bool *maybe_dependent)
{
  int eq;
  tree fun_a = chrec_convert (integer_type_node, access_fun_a, NULL_TREE);
  tree fun_b = chrec_convert (integer_type_node, access_fun_b, NULL_TREE);
  tree difference = chrec_fold_minus (integer_type_node, fun_a, fun_b);

  /* When the fun_a - fun_b is not constant, the dependence is not
     captured by the classic distance vector representation.  */
  if (TREE_CODE (difference) != INTEGER_CST)
    return false;

  /* ZIV test.  */
  if (ziv_subscript_p (fun_a, fun_b) && !integer_zerop (difference))
    {
      /* There is no dependence.  */
      *maybe_dependent = false;
      return true;
    }

  fun_b = chrec_fold_multiply (integer_type_node, fun_b, 
                               integer_minus_one_node);

  eq = omega_add_zero_eq (pb, omega_black);
  if (!init_omega_eq_with_af (pb, eq, DDR_NB_LOOPS (ddr), fun_a, ddr)
      || !init_omega_eq_with_af (pb, eq, 0, fun_b, ddr))
    /* There is probably a dependence, but the system of
       constraints cannot be built: answer "don't know".  */
    return false;

  /* GCD test.  */
  if (DDR_NB_LOOPS (ddr) != 0 && pb->eqs[eq].coef[0]
      && !int_divides_p (lambda_vector_gcd 
                         ((lambda_vector) &(pb->eqs[eq].coef[1]),
                          2 * DDR_NB_LOOPS (ddr)),
                         pb->eqs[eq].coef[0]))
    {
      /* There is no dependence.  */
      *maybe_dependent = false;
      return true;
    }

  return true;
}

/* Helper function, same as init_omega_for_ddr but specialized for
   data references A and B.  */

static bool
init_omega_for_ddr_1 (struct data_reference *dra, struct data_reference *drb,
                      struct data_dependence_relation *ddr,
                      omega_pb pb, bool *maybe_dependent)
{
  unsigned i;
  int ineq;
  struct loop *loopi;
  unsigned nb_loops = DDR_NB_LOOPS (ddr);

  /* Insert an equality per subscript.  */
  for (i = 0; i < DDR_NUM_SUBSCRIPTS (ddr); i++)
    {
      if (!omega_setup_subscript (DR_ACCESS_FN (dra, i), DR_ACCESS_FN (drb, i),
                                  ddr, pb, maybe_dependent))
        return false;
      else if (*maybe_dependent == false)
        {
          /* There is no dependence.  */
          DDR_ARE_DEPENDENT (ddr) = chrec_known;
          return true;
        }
    }

  /* Insert inequalities: constraints corresponding to the iteration
     domain, i.e. the loops surrounding the references "loop_x" and
     the distance variables "dx".  The layout of the OMEGA
     representation is as follows:
     - coef[0] is the constant
     - coef[1..nb_loops] are the protected variables that will not be
     removed by the solver: the "dx"
     - coef[nb_loops + 1, 2*nb_loops] are the loop variables: "loop_x".
  */
  for (i = 0; i <= DDR_INNER_LOOP (ddr) 
         && VEC_iterate (loop_p, DDR_LOOP_NEST (ddr), i, loopi); i++)
    {
      HOST_WIDE_INT nbi = estimated_loop_iterations_int (loopi, true);

      /* 0 <= loop_x */
      ineq = omega_add_zero_geq (pb, omega_black);
      pb->geqs[ineq].coef[i + nb_loops + 1] = 1;

      /* 0 <= loop_x + dx */
      ineq = omega_add_zero_geq (pb, omega_black);
      pb->geqs[ineq].coef[i + nb_loops + 1] = 1;
      pb->geqs[ineq].coef[i + 1] = 1;

      if (nbi != -1)
        {
          /* loop_x <= nb_iters */
          ineq = omega_add_zero_geq (pb, omega_black);
          pb->geqs[ineq].coef[i + nb_loops + 1] = -1;
          pb->geqs[ineq].coef[0] = nbi;

          /* loop_x + dx <= nb_iters */
          ineq = omega_add_zero_geq (pb, omega_black);
          pb->geqs[ineq].coef[i + nb_loops + 1] = -1;
          pb->geqs[ineq].coef[i + 1] = -1;
          pb->geqs[ineq].coef[0] = nbi;

          /* A step "dx" bigger than nb_iters is not feasible, so
             add "0 <= nb_iters + dx",  */
          ineq = omega_add_zero_geq (pb, omega_black);
          pb->geqs[ineq].coef[i + 1] = 1;
          pb->geqs[ineq].coef[0] = nbi;
          /* and "dx <= nb_iters".  */
          ineq = omega_add_zero_geq (pb, omega_black);
          pb->geqs[ineq].coef[i + 1] = -1;
          pb->geqs[ineq].coef[0] = nbi;
        }
    }

  omega_extract_distance_vectors (pb, ddr);

  return true;
}

/* Sets up the Omega dependence problem for the data dependence
   relation DDR.  Returns false when the constraint system cannot be
   built, ie. when the test answers "don't know".  Returns true
   otherwise, and when independence has been proved (using one of the
   trivial dependence test), set MAYBE_DEPENDENT to false, otherwise
   set MAYBE_DEPENDENT to true.

   Example: for setting up the dependence system corresponding to the
   conflicting accesses 

   | loop_i
   |   loop_j
   |     A[i, i+1] = ...
   |     ... A[2*j, 2*(i + j)]
   |   endloop_j
   | endloop_i
   
   the following constraints come from the iteration domain:

   0 <= i <= Ni
   0 <= i + di <= Ni
   0 <= j <= Nj
   0 <= j + dj <= Nj

   where di, dj are the distance variables.  The constraints
   representing the conflicting elements are:

   i = 2 * (j + dj)
   i + 1 = 2 * (i + di + j + dj)

   For asking that the resulting distance vector (di, dj) be
   lexicographically positive, we insert the constraint "di >= 0".  If
   "di = 0" in the solution, we fix that component to zero, and we
   look at the inner loops: we set a new problem where all the outer
   loop distances are zero, and fix this inner component to be
   positive.  When one of the components is positive, we save that
   distance, and set a new problem where the distance on this loop is
   zero, searching for other distances in the inner loops.  Here is
   the classic example that illustrates that we have to set for each
   inner loop a new problem:

   | loop_1
   |   loop_2
   |     A[10]
   |   endloop_2
   | endloop_1

   we have to save two distances (1, 0) and (0, 1).

   Given two array references, refA and refB, we have to set the
   dependence problem twice, refA vs. refB and refB vs. refA, and we
   cannot do a single test, as refB might occur before refA in the
   inner loops, and the contrary when considering outer loops: ex.

   | loop_0
   |   loop_1
   |     loop_2
   |       T[{1,+,1}_2][{1,+,1}_1]  // refA
   |       T[{2,+,1}_2][{0,+,1}_1]  // refB
   |     endloop_2
   |   endloop_1
   | endloop_0

   refB touches the elements in T before refA, and thus for the same
   loop_0 refB precedes refA: ie. the distance vector (0, 1, -1)
   but for successive loop_0 iterations, we have (1, -1, 1)

   The Omega solver expects the distance variables ("di" in the
   previous example) to come first in the constraint system (as
   variables to be protected, or "safe" variables), the constraint
   system is built using the following layout:

   "cst | distance vars | index vars".
*/

static bool
init_omega_for_ddr (struct data_dependence_relation *ddr,
                    bool *maybe_dependent)
{
  omega_pb pb;
  bool res = false;

  *maybe_dependent = true;

  if (same_access_functions (ddr))
    {
      unsigned j;
      lambda_vector dir_v;

      /* Save the 0 vector.  */
      save_dist_v (ddr, lambda_vector_new (DDR_NB_LOOPS (ddr)));
      dir_v = lambda_vector_new (DDR_NB_LOOPS (ddr));
      for (j = 0; j < DDR_NB_LOOPS (ddr); j++)
        dir_v[j] = dir_equal;
      save_dir_v (ddr, dir_v);

      /* Save the dependences carried by outer loops.  */
      pb = omega_alloc_problem (2 * DDR_NB_LOOPS (ddr), DDR_NB_LOOPS (ddr));
      res = init_omega_for_ddr_1 (DDR_A (ddr), DDR_B (ddr), ddr, pb,
                                  maybe_dependent);
      omega_free_problem (pb);
      return res;
    }

  /* Omega expects the protected variables (those that have to be kept
     after elimination) to appear first in the constraint system.
     These variables are the distance variables.  In the following
     initialization we declare NB_LOOPS safe variables, and the total
     number of variables for the constraint system is 2*NB_LOOPS.  */
  pb = omega_alloc_problem (2 * DDR_NB_LOOPS (ddr), DDR_NB_LOOPS (ddr));
  res = init_omega_for_ddr_1 (DDR_A (ddr), DDR_B (ddr), ddr, pb,
                              maybe_dependent);
  omega_free_problem (pb);

  /* Stop computation if not decidable, or no dependence.  */
  if (res == false || *maybe_dependent == false)
    return res;

  pb = omega_alloc_problem (2 * DDR_NB_LOOPS (ddr), DDR_NB_LOOPS (ddr));
  res = init_omega_for_ddr_1 (DDR_B (ddr), DDR_A (ddr), ddr, pb,
                              maybe_dependent);
  omega_free_problem (pb);

  return res;
}

/* Return true when DDR contains the same information as that stored
   in DIR_VECTS and in DIST_VECTS, return false otherwise.   */

static bool
ddr_consistent_p (FILE *file,
                  struct data_dependence_relation *ddr,
                  VEC (lambda_vector, heap) *dist_vects,
                  VEC (lambda_vector, heap) *dir_vects)
{
  unsigned int i, j;

  /* If dump_file is set, output there.  */
  if (dump_file && (dump_flags & TDF_DETAILS))
    file = dump_file;

  if (VEC_length (lambda_vector, dist_vects) != DDR_NUM_DIST_VECTS (ddr))
    {
      lambda_vector b_dist_v;
      fprintf (file, "\n(Number of distance vectors differ: Banerjee has %d, Omega has %d.\n",
               VEC_length (lambda_vector, dist_vects),
               DDR_NUM_DIST_VECTS (ddr));

      fprintf (file, "Banerjee dist vectors:\n");
      for (i = 0; VEC_iterate (lambda_vector, dist_vects, i, b_dist_v); i++)
        print_lambda_vector (file, b_dist_v, DDR_NB_LOOPS (ddr));

      fprintf (file, "Omega dist vectors:\n");
      for (i = 0; i < DDR_NUM_DIST_VECTS (ddr); i++)
        print_lambda_vector (file, DDR_DIST_VECT (ddr, i), DDR_NB_LOOPS (ddr));

      fprintf (file, "data dependence relation:\n");
      dump_data_dependence_relation (file, ddr);

      fprintf (file, ")\n");
      return false;
    }

  if (VEC_length (lambda_vector, dir_vects) != DDR_NUM_DIR_VECTS (ddr))
    {
      fprintf (file, "\n(Number of direction vectors differ: Banerjee has %d, Omega has %d.)\n",
               VEC_length (lambda_vector, dir_vects),
               DDR_NUM_DIR_VECTS (ddr));
      return false;
    }

  for (i = 0; i < DDR_NUM_DIST_VECTS (ddr); i++)
    {
      lambda_vector a_dist_v;
      lambda_vector b_dist_v = DDR_DIST_VECT (ddr, i);

      /* Distance vectors are not ordered in the same way in the DDR
         and in the DIST_VECTS: search for a matching vector.  */
      for (j = 0; VEC_iterate (lambda_vector, dist_vects, j, a_dist_v); j++)
        if (lambda_vector_equal (a_dist_v, b_dist_v, DDR_NB_LOOPS (ddr)))
          break;

      if (j == VEC_length (lambda_vector, dist_vects))
        {
          fprintf (file, "\n(Dist vectors from the first dependence analyzer:\n");
          print_dist_vectors (file, dist_vects, DDR_NB_LOOPS (ddr));
          fprintf (file, "not found in Omega dist vectors:\n");
          print_dist_vectors (file, DDR_DIST_VECTS (ddr), DDR_NB_LOOPS (ddr));
          fprintf (file, "data dependence relation:\n");
          dump_data_dependence_relation (file, ddr);
          fprintf (file, ")\n");
        }
    }

  for (i = 0; i < DDR_NUM_DIR_VECTS (ddr); i++)
    {
      lambda_vector a_dir_v;
      lambda_vector b_dir_v = DDR_DIR_VECT (ddr, i);

      /* Direction vectors are not ordered in the same way in the DDR
         and in the DIR_VECTS: search for a matching vector.  */
      for (j = 0; VEC_iterate (lambda_vector, dir_vects, j, a_dir_v); j++)
        if (lambda_vector_equal (a_dir_v, b_dir_v, DDR_NB_LOOPS (ddr)))
          break;

      if (j == VEC_length (lambda_vector, dist_vects))
        {
          fprintf (file, "\n(Dir vectors from the first dependence analyzer:\n");
          print_dir_vectors (file, dir_vects, DDR_NB_LOOPS (ddr));
          fprintf (file, "not found in Omega dir vectors:\n");
          print_dir_vectors (file, DDR_DIR_VECTS (ddr), DDR_NB_LOOPS (ddr));
          fprintf (file, "data dependence relation:\n");
          dump_data_dependence_relation (file, ddr);
          fprintf (file, ")\n");
        }
    }

  return true;  
}

/* This computes the affine dependence relation between A and B.
   CHREC_KNOWN is used for representing the independence between two
   accesses, while CHREC_DONT_KNOW is used for representing the unknown
   relation.
   
   Note that it is possible to stop the computation of the dependence
   relation the first time we detect a CHREC_KNOWN element for a given
   subscript.  */

static void
compute_affine_dependence (struct data_dependence_relation *ddr)
{
  struct data_reference *dra = DDR_A (ddr);
  struct data_reference *drb = DDR_B (ddr);
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    {
      fprintf (dump_file, "(compute_affine_dependence\n");
      fprintf (dump_file, "  (stmt_a = \n");
      print_generic_expr (dump_file, DR_STMT (dra), 0);
      fprintf (dump_file, ")\n  (stmt_b = \n");
      print_generic_expr (dump_file, DR_STMT (drb), 0);
      fprintf (dump_file, ")\n");
    }

  /* Analyze only when the dependence relation is not yet known.  */
  if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE)
    {
      dependence_stats.num_dependence_tests++;

      if (access_functions_are_affine_or_constant_p (dra)
          && access_functions_are_affine_or_constant_p (drb))
        {
          if (flag_check_data_deps)
            {
              /* Compute the dependences using the first algorithm.  */
              subscript_dependence_tester (ddr);

              if (dump_file && (dump_flags & TDF_DETAILS))
                {
                  fprintf (dump_file, "\n\nBanerjee Analyzer\n");
                  dump_data_dependence_relation (dump_file, ddr);
                }

              if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE)
                {
                  bool maybe_dependent;
                  VEC (lambda_vector, heap) *dir_vects, *dist_vects;

                  /* Save the result of the first DD analyzer.  */
                  dist_vects = DDR_DIST_VECTS (ddr);
                  dir_vects = DDR_DIR_VECTS (ddr);

                  /* Reset the information.  */
                  DDR_DIST_VECTS (ddr) = NULL;
                  DDR_DIR_VECTS (ddr) = NULL;

                  /* Compute the same information using Omega.  */
                  if (!init_omega_for_ddr (ddr, &maybe_dependent))
                    goto csys_dont_know;

                  if (dump_file && (dump_flags & TDF_DETAILS))
                    {
                      fprintf (dump_file, "Omega Analyzer\n");
                      dump_data_dependence_relation (dump_file, ddr);
                    }

                  /* Check that we get the same information.  */
                  if (maybe_dependent)
                    gcc_assert (ddr_consistent_p (stderr, ddr, dist_vects,
                                                  dir_vects));
                }
            }
          else
            subscript_dependence_tester (ddr);
        }
     
      /* As a last case, if the dependence cannot be determined, or if
         the dependence is considered too difficult to determine, answer
         "don't know".  */
      else
        {
        csys_dont_know:;
          dependence_stats.num_dependence_undetermined++;

          if (dump_file && (dump_flags & TDF_DETAILS))
            {
              fprintf (dump_file, "Data ref a:\n");
              dump_data_reference (dump_file, dra);
              fprintf (dump_file, "Data ref b:\n");
              dump_data_reference (dump_file, drb);
              fprintf (dump_file, "affine dependence test not usable: access function not affine or constant.\n");
            }
          finalize_ddr_dependent (ddr, chrec_dont_know);
        }
    }
  
  if (dump_file && (dump_flags & TDF_DETAILS))
    fprintf (dump_file, ")\n");
}

/* This computes the dependence relation for the same data
   reference into DDR.  */

static void
compute_self_dependence (struct data_dependence_relation *ddr)
{
  unsigned int i;
  struct subscript *subscript;

  if (DDR_ARE_DEPENDENT (ddr) != NULL_TREE)
    return;

  for (i = 0; VEC_iterate (subscript_p, DDR_SUBSCRIPTS (ddr), i, subscript);
       i++)
    {
      /* The accessed index overlaps for each iteration.  */
      SUB_CONFLICTS_IN_A (subscript)
              = conflict_fn (1, affine_fn_cst (integer_zero_node));
      SUB_CONFLICTS_IN_B (subscript)
              = conflict_fn (1, affine_fn_cst (integer_zero_node));
      SUB_LAST_CONFLICT (subscript) = chrec_dont_know;
    }

  /* The distance vector is the zero vector.  */
  save_dist_v (ddr, lambda_vector_new (DDR_NB_LOOPS (ddr)));
  save_dir_v (ddr, lambda_vector_new (DDR_NB_LOOPS (ddr)));
}

/* Compute in DEPENDENCE_RELATIONS the data dependence graph for all
   the data references in DATAREFS, in the LOOP_NEST.  When
   COMPUTE_SELF_AND_RR is FALSE, don't compute read-read and self
   relations.  */

void 
compute_all_dependences (VEC (data_reference_p, heap) *datarefs,
                         VEC (ddr_p, heap) **dependence_relations,
                         VEC (loop_p, heap) *loop_nest,
                         bool compute_self_and_rr)
{
  struct data_dependence_relation *ddr;
  struct data_reference *a, *b;
  unsigned int i, j;

  for (i = 0; VEC_iterate (data_reference_p, datarefs, i, a); i++)
    for (j = i + 1; VEC_iterate (data_reference_p, datarefs, j, b); j++)
      if (!DR_IS_READ (a) || !DR_IS_READ (b) || compute_self_and_rr)
        {
          ddr = initialize_data_dependence_relation (a, b, loop_nest);
          VEC_safe_push (ddr_p, heap, *dependence_relations, ddr);
          compute_affine_dependence (ddr);
        }

  if (compute_self_and_rr)
    for (i = 0; VEC_iterate (data_reference_p, datarefs, i, a); i++)
      {
        ddr = initialize_data_dependence_relation (a, a, loop_nest);
        VEC_safe_push (ddr_p, heap, *dependence_relations, ddr);
        compute_self_dependence (ddr);
      }
}

/* Stores the locations of memory references in STMT to REFERENCES.  Returns
   true if STMT clobbers memory, false otherwise.  */

bool
get_references_in_stmt (tree stmt, VEC (data_ref_loc, heap) **references)
{
  bool clobbers_memory = false;
  data_ref_loc *ref;
  tree *op0, *op1, call;

  *references = NULL;

  /* ASM_EXPR and CALL_EXPR may embed arbitrary side effects.
     Calls have side-effects, except those to const or pure
     functions.  */
  call = get_call_expr_in (stmt);
  if ((call
       && !(call_expr_flags (call) & (ECF_CONST | ECF_PURE)))
      || (TREE_CODE (stmt) == ASM_EXPR
          && ASM_VOLATILE_P (stmt)))
    clobbers_memory = true;

  if (ZERO_SSA_OPERANDS (stmt, SSA_OP_ALL_VIRTUALS))
    return clobbers_memory;

  if (TREE_CODE (stmt) ==  GIMPLE_MODIFY_STMT)
    {
      op0 = &GIMPLE_STMT_OPERAND (stmt, 0);
      op1 = &GIMPLE_STMT_OPERAND (stmt, 1);
                
      if (DECL_P (*op1)
          || REFERENCE_CLASS_P (*op1))
        {
          ref = VEC_safe_push (data_ref_loc, heap, *references, NULL);
          ref->pos = op1;
          ref->is_read = true;
        }

      if (DECL_P (*op0)
          || REFERENCE_CLASS_P (*op0))
        {
          ref = VEC_safe_push (data_ref_loc, heap, *references, NULL);
          ref->pos = op0;
          ref->is_read = false;
        }
    }

  if (call)
    {
      unsigned i, n = call_expr_nargs (call);

      for (i = 0; i < n; i++)
        {
          op0 = &CALL_EXPR_ARG (call, i);

          if (DECL_P (*op0)
              || REFERENCE_CLASS_P (*op0))
            {
              ref = VEC_safe_push (data_ref_loc, heap, *references, NULL);
              ref->pos = op0;
              ref->is_read = true;
            }
        }
    }

  return clobbers_memory;
}

/* Stores the data references in STMT to DATAREFS.  If there is an unanalyzable
   reference, returns false, otherwise returns true.  NEST is the outermost
   loop of the loop nest in that the references should be analyzed.  */

static bool
find_data_references_in_stmt (struct loop *nest, tree stmt,
                              VEC (data_reference_p, heap) **datarefs)
{
  unsigned i;
  VEC (data_ref_loc, heap) *references;
  data_ref_loc *ref;
  bool ret = true;
  data_reference_p dr;

  if (get_references_in_stmt (stmt, &references))
    {
      VEC_free (data_ref_loc, heap, references);
      return false;
    }

  for (i = 0; VEC_iterate (data_ref_loc, references, i, ref); i++)
    {
      dr = create_data_ref (nest, *ref->pos, stmt, ref->is_read);
      gcc_assert (dr != NULL);
  
      /* FIXME -- data dependence analysis does not work correctly for objects with
         invariant addresses.  Let us fail here until the problem is fixed.  */
      if (dr_address_invariant_p (dr))
        {
          free_data_ref (dr);
          if (dump_file && (dump_flags & TDF_DETAILS))
            fprintf (dump_file, "\tFAILED as dr address is invariant\n");
          ret = false;
          break;
        }

      VEC_safe_push (data_reference_p, heap, *datarefs, dr);
    }
  VEC_free (data_ref_loc, heap, references);
  return ret;
}

/* Search the data references in LOOP, and record the information into
   DATAREFS.  Returns chrec_dont_know when failing to analyze a
   difficult case, returns NULL_TREE otherwise.

   TODO: This function should be made smarter so that it can handle address
   arithmetic as if they were array accesses, etc.  */

static tree 
find_data_references_in_loop (struct loop *loop,
                              VEC (data_reference_p, heap) **datarefs)
{
  basic_block bb, *bbs;
  unsigned int i;
  block_stmt_iterator bsi;

  bbs = get_loop_body_in_dom_order (loop);

  for (i = 0; i < loop->num_nodes; i++)
    {
      bb = bbs[i];

      for (bsi = bsi_start (bb); !bsi_end_p (bsi); bsi_next (&bsi))
        {
          tree stmt = bsi_stmt (bsi);

          if (!find_data_references_in_stmt (loop, stmt, datarefs))
            {
              struct data_reference *res;
              res = XCNEW (struct data_reference);
              VEC_safe_push (data_reference_p, heap, *datarefs, res);

              free (bbs);
              return chrec_dont_know;
            }
        }
    }
  free (bbs);

  return NULL_TREE;
}

/* Recursive helper function.  */

static bool
find_loop_nest_1 (struct loop *loop, VEC (loop_p, heap) **loop_nest)
{
  /* Inner loops of the nest should not contain siblings.  Example:
     when there are two consecutive loops,

     | loop_0
     |   loop_1
     |     A[{0, +, 1}_1]
     |   endloop_1
     |   loop_2
     |     A[{0, +, 1}_2]
     |   endloop_2
     | endloop_0

     the dependence relation cannot be captured by the distance
     abstraction.  */
  if (loop->next)
    return false;

  VEC_safe_push (loop_p, heap, *loop_nest, loop);
  if (loop->inner)
    return find_loop_nest_1 (loop->inner, loop_nest);
  return true;
}

/* Return false when the LOOP is not well nested.  Otherwise return
   true and insert in LOOP_NEST the loops of the nest.  LOOP_NEST will
   contain the loops from the outermost to the innermost, as they will
   appear in the classic distance vector.  */

bool
find_loop_nest (struct loop *loop, VEC (loop_p, heap) **loop_nest)
{
  VEC_safe_push (loop_p, heap, *loop_nest, loop);
  if (loop->inner)
    return find_loop_nest_1 (loop->inner, loop_nest);
  return true;
}

/* Given a loop nest LOOP, the following vectors are returned:
   DATAREFS is initialized to all the array elements contained in this loop, 
   DEPENDENCE_RELATIONS contains the relations between the data references.  
   Compute read-read and self relations if 
   COMPUTE_SELF_AND_READ_READ_DEPENDENCES is TRUE.  */

void
compute_data_dependences_for_loop (struct loop *loop, 
                                   bool compute_self_and_read_read_dependences,
                                   VEC (data_reference_p, heap) **datarefs,
                                   VEC (ddr_p, heap) **dependence_relations)
{
  VEC (loop_p, heap) *vloops = VEC_alloc (loop_p, heap, 3);

  memset (&dependence_stats, 0, sizeof (dependence_stats));

  /* If the loop nest is not well formed, or one of the data references 
     is not computable, give up without spending time to compute other
     dependences.  */
  if (!loop
      || !find_loop_nest (loop, &vloops)
      || find_data_references_in_loop (loop, datarefs) == chrec_dont_know)
    {
      struct data_dependence_relation *ddr;

      /* Insert a single relation into dependence_relations:
         chrec_dont_know.  */
      ddr = initialize_data_dependence_relation (NULL, NULL, vloops);
      VEC_safe_push (ddr_p, heap, *dependence_relations, ddr);
    }
  else
    compute_all_dependences (*datarefs, dependence_relations, vloops,
                             compute_self_and_read_read_dependences);

  if (dump_file && (dump_flags & TDF_STATS))
    {
      fprintf (dump_file, "Dependence tester statistics:\n");

      fprintf (dump_file, "Number of dependence tests: %d\n", 
               dependence_stats.num_dependence_tests);
      fprintf (dump_file, "Number of dependence tests classified dependent: %d\n", 
               dependence_stats.num_dependence_dependent);
      fprintf (dump_file, "Number of dependence tests classified independent: %d\n", 
               dependence_stats.num_dependence_independent);
      fprintf (dump_file, "Number of undetermined dependence tests: %d\n", 
               dependence_stats.num_dependence_undetermined);

      fprintf (dump_file, "Number of subscript tests: %d\n", 
               dependence_stats.num_subscript_tests);
      fprintf (dump_file, "Number of undetermined subscript tests: %d\n", 
               dependence_stats.num_subscript_undetermined);
      fprintf (dump_file, "Number of same subscript function: %d\n", 
               dependence_stats.num_same_subscript_function);

      fprintf (dump_file, "Number of ziv tests: %d\n",
               dependence_stats.num_ziv);
      fprintf (dump_file, "Number of ziv tests returning dependent: %d\n",
               dependence_stats.num_ziv_dependent);
      fprintf (dump_file, "Number of ziv tests returning independent: %d\n",
               dependence_stats.num_ziv_independent);
      fprintf (dump_file, "Number of ziv tests unimplemented: %d\n",
               dependence_stats.num_ziv_unimplemented);      

      fprintf (dump_file, "Number of siv tests: %d\n", 
               dependence_stats.num_siv);
      fprintf (dump_file, "Number of siv tests returning dependent: %d\n",
               dependence_stats.num_siv_dependent);
      fprintf (dump_file, "Number of siv tests returning independent: %d\n",
               dependence_stats.num_siv_independent);
      fprintf (dump_file, "Number of siv tests unimplemented: %d\n",
               dependence_stats.num_siv_unimplemented);

      fprintf (dump_file, "Number of miv tests: %d\n", 
               dependence_stats.num_miv);
      fprintf (dump_file, "Number of miv tests returning dependent: %d\n",
               dependence_stats.num_miv_dependent);
      fprintf (dump_file, "Number of miv tests returning independent: %d\n",
               dependence_stats.num_miv_independent);
      fprintf (dump_file, "Number of miv tests unimplemented: %d\n",
               dependence_stats.num_miv_unimplemented);
    }    
}

/* Entry point (for testing only).  Analyze all the data references
   and the dependence relations in LOOP.

   The data references are computed first.  
   
   A relation on these nodes is represented by a complete graph.  Some
   of the relations could be of no interest, thus the relations can be
   computed on demand.
   
   In the following function we compute all the relations.  This is
   just a first implementation that is here for:
   - for showing how to ask for the dependence relations, 
   - for the debugging the whole dependence graph,
   - for the dejagnu testcases and maintenance.
   
   It is possible to ask only for a part of the graph, avoiding to
   compute the whole dependence graph.  The computed dependences are
   stored in a knowledge base (KB) such that later queries don't
   recompute the same information.  The implementation of this KB is
   transparent to the optimizer, and thus the KB can be changed with a
   more efficient implementation, or the KB could be disabled.  */
static void 
analyze_all_data_dependences (struct loop *loop)
{
  unsigned int i;
  int nb_data_refs = 10;
  VEC (data_reference_p, heap) *datarefs = 
    VEC_alloc (data_reference_p, heap, nb_data_refs);
  VEC (ddr_p, heap) *dependence_relations = 
    VEC_alloc (ddr_p, heap, nb_data_refs * nb_data_refs);

  /* Compute DDs on the whole function.  */
  compute_data_dependences_for_loop (loop, false, &datarefs,
                                     &dependence_relations);

  if (dump_file)
    {
      dump_data_dependence_relations (dump_file, dependence_relations);
      fprintf (dump_file, "\n\n");

      if (dump_flags & TDF_DETAILS)
        dump_dist_dir_vectors (dump_file, dependence_relations);

      if (dump_flags & TDF_STATS)
        {
          unsigned nb_top_relations = 0;
          unsigned nb_bot_relations = 0;
          unsigned nb_basename_differ = 0;
          unsigned nb_chrec_relations = 0;
          struct data_dependence_relation *ddr;

          for (i = 0; VEC_iterate (ddr_p, dependence_relations, i, ddr); i++)
            {
              if (chrec_contains_undetermined (DDR_ARE_DEPENDENT (ddr)))
                nb_top_relations++;
          
              else if (DDR_ARE_DEPENDENT (ddr) == chrec_known)
                {
                  struct data_reference *a = DDR_A (ddr);
                  struct data_reference *b = DDR_B (ddr);

                  if (!bitmap_intersect_p (DR_VOPS (a), DR_VOPS (b)))
                    nb_basename_differ++;
                  else
                    nb_bot_relations++;
                }
          
              else 
                nb_chrec_relations++;
            }
      
          gather_stats_on_scev_database ();
        }
    }

  free_dependence_relations (dependence_relations);
  free_data_refs (datarefs);
}

/* Computes all the data dependences and check that the results of
   several analyzers are the same.  */

void
tree_check_data_deps (void)
{
  loop_iterator li;
  struct loop *loop_nest;

  FOR_EACH_LOOP (li, loop_nest, 0)
    analyze_all_data_dependences (loop_nest);
}

/* Free the memory used by a data dependence relation DDR.  */

void
free_dependence_relation (struct data_dependence_relation *ddr)
{
  if (ddr == NULL)
    return;

  if (DDR_ARE_DEPENDENT (ddr) == NULL_TREE && DDR_SUBSCRIPTS (ddr))
    free_subscripts (DDR_SUBSCRIPTS (ddr));

  free (ddr);
}

/* Free the memory used by the data dependence relations from
   DEPENDENCE_RELATIONS.  */

void 
free_dependence_relations (VEC (ddr_p, heap) *dependence_relations)
{
  unsigned int i;
  struct data_dependence_relation *ddr;
  VEC (loop_p, heap) *loop_nest = NULL;

  for (i = 0; VEC_iterate (ddr_p, dependence_relations, i, ddr); i++)
    {
      if (ddr == NULL)
        continue;
      if (loop_nest == NULL)
        loop_nest = DDR_LOOP_NEST (ddr);
      else
        gcc_assert (DDR_LOOP_NEST (ddr) == NULL
                    || DDR_LOOP_NEST (ddr) == loop_nest);
      free_dependence_relation (ddr);
    }

  if (loop_nest)
    VEC_free (loop_p, heap, loop_nest);
  VEC_free (ddr_p, heap, dependence_relations);
}

/* Free the memory used by the data references from DATAREFS.  */

void
free_data_refs (VEC (data_reference_p, heap) *datarefs)
{
  unsigned int i;
  struct data_reference *dr;

  for (i = 0; VEC_iterate (data_reference_p, datarefs, i, dr); i++)
    free_data_ref (dr);
  VEC_free (data_reference_p, heap, datarefs);
}