#
# Copyright (c) 2017 Intel Corporation
# SPDX-License-Identifier: BSD-2-Clause
#

from __future__ import print_function, division, absolute_import
import types as pytypes  # avoid confusion with numba.types
import numpy
import operator
from numba import ir, analysis, types, config, cgutils, typing
from numba.ir_utils import (
    mk_unique_var,
    replace_vars_inner,
    find_topo_order,
    dprint_func_ir,
    get_global_func_typ,
    guard,
    require,
    get_definition,
    find_callname,
    find_build_sequence,
    find_const,
    is_namedtuple_class,
    build_definitions)
from numba.analysis import (compute_cfg_from_blocks)
from numba.typing import npydecl, signature
import collections
import copy
from numba.extending import intrinsic
import llvmlite.llvmpy.core as lc
import llvmlite

UNKNOWN_CLASS = -1
CONST_CLASS = 0
MAP_TYPES = [numpy.ufunc]

array_analysis_extensions = {}

# declaring call classes
array_creation = ['empty', 'zeros', 'ones', 'full']

random_int_args = ['rand', 'randn']

random_1arg_size = ['ranf', 'random_sample', 'sample',
                    'random', 'standard_normal']

random_2arg_sizelast = ['chisquare', 'weibull', 'power',
                        'geometric', 'exponential',
                        'poisson', 'rayleigh']

random_3arg_sizelast = ['normal', 'uniform', 'beta',
                        'binomial', 'f', 'gamma',
                        'lognormal', 'laplace']

random_calls = (random_int_args +
                random_1arg_size +
                random_2arg_sizelast +
                random_3arg_sizelast +
                ['randint', 'triangular'])

@intrinsic
def wrap_index(typingctx, idx, size):
    """
    Calculate index value "idx" relative to a size "size" value as
    (idx % size), where "size" is known to be positive.
    Note that we use the mod(%) operation here instead of
    (idx < 0 ? idx + size : idx) because we may have situations
    where idx > size due to the way indices are calculated
    during slice/range analysis.
    """
    unified_ty = typingctx.unify_types(idx, size)
    if not unified_ty:
        raise ValueError("Argument types for wrap_index must match")

    def codegen(context, builder, sig, args):
        ll_unified_ty = context.get_data_type(unified_ty)
        idx = builder.sext(args[0], ll_unified_ty)
        size = builder.sext(args[1], ll_unified_ty)
        neg_size = builder.neg(size)
        zero = llvmlite.ir.Constant(ll_unified_ty, 0)
        idx_negative = builder.icmp_signed('<', idx, zero)
        pos_oversize = builder.icmp_signed('>=', idx, size)
        neg_oversize = builder.icmp_signed('<=', idx, neg_size)
        pos_res = builder.select(pos_oversize, size, idx)
        neg_res = builder.select(neg_oversize, zero, builder.add(idx, size))
        mod = builder.select(idx_negative, neg_res, pos_res)
        return mod

    return signature(unified_ty, idx, size), codegen

def wrap_index_literal(idx, size):
    if idx < 0:
        if idx <= -size:
            return 0
        else:
            return idx + size
    else:
        if idx >= size:
            return size
        else:
            return idx

@intrinsic
def assert_equiv(typingctx, *val):
    """
    A function that asserts the inputs are of equivalent size,
    and throws runtime error when they are not. The input is
    a vararg that contains an error message, followed by a set
    of objects of either array, tuple or integer.
    """
    if len(val) > 1:
        # Make sure argument is a single tuple type. Note that this only
        # happens when IR containing assert_equiv call is being compiled
        # (and going through type inference) again.
        val = (types.Tuple(val),)

    assert(len(val[0]) > 1)
    # Arguments must be either array, tuple, or integer
    assert all(map(lambda a: (isinstance(a, types.ArrayCompatible) or
                              isinstance(a, types.BaseTuple) or
                              isinstance(a, types.SliceType) or
                              isinstance(a, types.Integer)), val[0][1:]))

    def codegen(context, builder, sig, args):
        assert(len(args) == 1)  # it is a vararg tuple
        tup = cgutils.unpack_tuple(builder, args[0])
        tup_type = sig.args[0]
        msg = sig.args[0][0].literal_value

        def unpack_shapes(a, aty):
            if isinstance(aty, types.ArrayCompatible):
                ary = context.make_array(aty)(context, builder, a)
                return cgutils.unpack_tuple(builder, ary.shape)
            elif isinstance(aty, types.BaseTuple):
                return cgutils.unpack_tuple(builder, a)
            else:  # otherwise it is a single integer
                return [a]

        def pairwise(a, aty, b, bty):
            ashapes = unpack_shapes(a, aty)
            bshapes = unpack_shapes(b, bty)
            assert len(ashapes) == len(bshapes)
            for (m, n) in zip(ashapes, bshapes):
                m_eq_n = builder.icmp(lc.ICMP_EQ, m, n)
                with builder.if_else(m_eq_n) as (then, orelse):
                    with then:
                        pass
                    with orelse:
                        context.call_conv.return_user_exc(
                            builder, AssertionError, (msg,))

        for i in range(1, len(tup_type) - 1):
            pairwise(tup[i], tup_type[i], tup[i + 1], tup_type[i + 1])
        r = context.get_constant_generic(builder, types.NoneType, None)
        return r
    return signature(types.none, *val), codegen


class EquivSet(object):

    """EquivSet keeps track of equivalence relations between
    a set of objects.
    """

    def __init__(self, obj_to_ind=None, ind_to_obj=None, next_ind=0):
        """Create a new EquivSet object. Optional keyword arguments are for
        internal use only.
        """
        # obj_to_ind maps object to equivalence index (sometimes also called
        # equivalence class) is a non-negative number that uniquely identifies
        # a set of objects that are equivalent.
        self.obj_to_ind = obj_to_ind if obj_to_ind else {}
        # ind_to_obj maps equivalence index to a list of objects.
        self.ind_to_obj = ind_to_obj if ind_to_obj else {}
        # next index number that is incremented each time a new equivalence
        # relation is created.
        self.next_ind = next_ind

    def empty(self):
        """Return an empty EquivSet object.
        """
        return EquivSet()

    def clone(self):
        """Return a new copy.
        """
        return EquivSet(obj_to_ind=copy.deepcopy(self.obj_to_ind),
                        ind_to_obj=copy.deepcopy(self.ind_to_obj),
                        next_id=self.next_ind)

    def __repr__(self):
        return "EquivSet({})".format(self.ind_to_obj)

    def is_empty(self):
        """Return true if the set is empty, or false otherwise.
        """
        return self.obj_to_ind == {}

    def _get_ind(self, x):
        """Return the internal index (greater or equal to 0) of the given
        object, or -1 if not found.
        """
        return self.obj_to_ind.get(x, -1)

    def _get_or_add_ind(self, x):
        """Return the internal index (greater or equal to 0) of the given
        object, or create a new one if not found.
        """
        if x in self.obj_to_ind:
            i = self.obj_to_ind[x]
        else:
            i = self.next_ind
            self.next_ind += 1
        return i

    def _insert(self, objs):
        """Base method that inserts a set of equivalent objects by modifying
        self.
        """
        assert len(objs) > 1

        inds = tuple(self._get_or_add_ind(x) for x in objs)
        ind = min(inds)

        if config.DEBUG_ARRAY_OPT >= 2:
            print("_insert:", objs, inds)

        if not (ind in self.ind_to_obj):
            self.ind_to_obj[ind] = []

        for i, obj in zip(inds, objs):
            if i == ind:
                if not (obj in self.ind_to_obj[ind]):
                    self.ind_to_obj[ind].append(obj)
                    self.obj_to_ind[obj] = ind
            else:
                if i in self.ind_to_obj:
                    # those already existing are reassigned
                    for x in self.ind_to_obj[i]:
                        self.obj_to_ind[x] = ind
                        self.ind_to_obj[ind].append(x)
                    del self.ind_to_obj[i]
                else:
                    # those that are new are assigned.
                    self.obj_to_ind[obj] = ind
                    self.ind_to_obj[ind].append(obj)

    def is_equiv(self, *objs):
        """Try to derive if given objects are equivalent, return true
        if so, or false otherwise.
        """
        inds = [self._get_ind(x) for x in objs]
        ind = max(inds)
        if ind != -1:
            return all(i == ind for i in inds)
        else:
            return all([x == objs[0] for x in objs])

    def get_equiv_const(self, obj):
        """Check if obj is equivalent to some int constant, and return
        the constant if found, or None otherwise.
        """
        ind = self._get_ind(obj)
        if ind >= 0:
            objs = self.ind_to_obj[ind]
            for x in objs:
                if isinstance(x, int):
                    return x
        return None

    def get_equiv_set(self, obj):
        """Return the set of equivalent objects.
        """
        ind = self._get_ind(obj)
        if ind >= 0:
            return set(self.ind_to_obj[ind])
        return set()

    def insert_equiv(self, *objs):
        """Insert a set of equivalent objects by modifying self. This
        method can be overloaded to transform object type before insertion.
        """
        return self._insert(objs)

    def intersect(self, equiv_set):
        """ Return the intersection of self and the given equiv_set,
        without modifying either of them. The result will also keep
        old equivalence indices unchanged.
        """
        new_set = self.empty()
        new_set.next_ind = self.next_ind

        for objs in equiv_set.ind_to_obj.values():
            inds = tuple(self._get_ind(x) for x in objs)
            ind_to_obj = {}

            for i, x in zip(inds, objs):
                if i in ind_to_obj:
                    ind_to_obj[i].append(x)
                elif i >= 0:
                    ind_to_obj[i] = [x]

            for v in ind_to_obj.values():
                if len(v) > 1:
                    new_set._insert(v)

        return new_set


class ShapeEquivSet(EquivSet):

    """Just like EquivSet, except that it accepts only numba IR variables
    and constants as objects, guided by their types. Arrays are considered
    equivalent as long as their shapes are equivalent. Scalars are
    equivalent only when they are equal in value. Tuples are equivalent
    when they are of the same size, and their elements are equivalent.
    """

    def __init__(self, typemap, defs=None, ind_to_var=None,
                 obj_to_ind=None, ind_to_obj=None, next_id=0, ind_to_const=None):
        """Create a new ShapeEquivSet object, where typemap is a dictionary
        that maps variable names to their types, and it will not be modified.
        Optional keyword arguments are for internal use only.
        """
        self.typemap = typemap
        # defs maps variable name to an int, where
        # 1 means the variable is defined only once, and numbers greater
        # than 1 means defined more than onces.
        self.defs = defs if defs else {}
        # ind_to_var maps index number to a list of variables (of ir.Var type).
        # It is used to retrieve defined shape variables given an equivalence
        # index.
        self.ind_to_var = ind_to_var if ind_to_var else {}
        # ind_to_const maps index number to a constant, if known.
        self.ind_to_const = ind_to_const if ind_to_const else {}

        super(ShapeEquivSet, self).__init__(obj_to_ind, ind_to_obj, next_id)

    def empty(self):
        """Return an empty ShapeEquivSet.
        """
        return ShapeEquivSet(self.typemap, {})

    def clone(self):
        """Return a new copy.
        """
        return ShapeEquivSet(
            self.typemap,
            defs=copy.copy(self.defs),
            ind_to_var=copy.copy(self.ind_to_var),
            obj_to_ind=copy.deepcopy(self.obj_to_ind),
            ind_to_obj=copy.deepcopy(self.ind_to_obj),
            next_id=self.next_ind,
            ind_to_const=copy.deepcopy(self.ind_toconst))

    def __repr__(self):
        return "ShapeEquivSet({}, ind_to_var={}, ind_to_const={})".format(
            self.ind_to_obj, self.ind_to_var, self.ind_to_const)

    def _get_names(self, obj):
        """Return a set of names for the given obj, where array and tuples
        are broken down to their individual shapes or elements. This is
        safe because both Numba array shapes and Python tuples are immutable.
        """
        if isinstance(obj, ir.Var) or isinstance(obj, str):
            name = obj if isinstance(obj, str) else obj.name
            if name not in self.typemap:
                return (name,)

            typ = self.typemap[name]
            if (isinstance(typ, types.BaseTuple) or
                    isinstance(typ, types.ArrayCompatible)):
                ndim = (typ.ndim if isinstance(typ, types.ArrayCompatible)
                        else len(typ))
                # Treat 0d array as if it were a scalar.
                if ndim == 0:
                    return (name,)
                else:
                    return tuple("{}#{}".format(name, i) for i in range(ndim))
            else:
                return (name,)
        elif isinstance(obj, ir.Const):
            if isinstance(obj.value, tuple):
                return obj.value
            else:
                return (obj.value,)
        elif isinstance(obj, tuple):
            return tuple(self._get_names(x)[0] for x in obj)
        elif isinstance(obj, int):
            return (obj,)
        else:
            raise NotImplementedError(
                "ShapeEquivSet does not support {}".format(obj))

    def is_equiv(self, *objs):
        """Overload EquivSet.is_equiv to handle Numba IR variables and
        constants.
        """
        assert(len(objs) > 1)
        obj_names = [self._get_names(x) for x in objs]
        obj_names = [x for x in obj_names if x != ()] # rule out 0d shape
        if len(obj_names) <= 1:
            return False;
        ndims = [len(names) for names in obj_names]
        ndim = ndims[0]
        if not all(ndim == x for x in ndims):
            if config.DEBUG_ARRAY_OPT >= 1:
                print("is_equiv: Dimension mismatch for {}".format(objs))
            return False
        for i in range(ndim):
            names = [obj_name[i] for obj_name in obj_names]
            if not super(ShapeEquivSet, self).is_equiv(*names):
                return False
        return True

    def get_equiv_const(self, obj):
        """If the given object is equivalent to a constant scalar,
        return the scalar value, or None otherwise.
        """
        names = self._get_names(obj)
        if len(names) > 1:
            return None
        return super(ShapeEquivSet, self).get_equiv_const(names[0])

    def get_equiv_var(self, obj):
        """If the given object is equivalent to some defined variable,
        return the variable, or None otherwise.
        """
        names = self._get_names(obj)
        if len(names) != 1:
            return None
        ind = self._get_ind(names[0])
        vs = self.ind_to_var.get(ind, [])
        return vs[0] if vs != [] else None

    def get_equiv_set(self, obj):
        """Return the set of equivalent objects.
        """
        names = self._get_names(obj)
        if len(names) > 1:
            return None
        return super(ShapeEquivSet, self).get_equiv_set(names[0])

    def _insert(self, objs):
        """Overload EquivSet._insert to manage ind_to_var dictionary.
        """
        inds = []
        for obj in objs:
            if obj in self.obj_to_ind:
                inds.append(self.obj_to_ind[obj])
        varlist = []
        constval = None
        names = set()
        for i in sorted(inds):
            if i in self.ind_to_var:
                for x in self.ind_to_var[i]:
                    if not (x.name in names):
                        varlist.append(x)
                        names.add(x.name)
            if i in self.ind_to_const:
                assert(constval is None)
                constval = self.ind_to_const[i]
        super(ShapeEquivSet, self)._insert(objs)
        new_ind = self.obj_to_ind[objs[0]]
        for i in set(inds):
            if i in self.ind_to_var:
                del self.ind_to_var[i]
        self.ind_to_var[new_ind] = varlist
        if constval is not None:
            self.ind_to_const[new_ind] = constval

    def insert_equiv(self, *objs):
        """Overload EquivSet.insert_equiv to handle Numba IR variables and
        constants. Input objs are either variable or constant, and at least
        one of them must be variable.
        """
        assert(len(objs) > 1)
        obj_names = [self._get_names(x) for x in objs]
        obj_names = [x for x in obj_names if x != ()] # rule out 0d shape
        if len(obj_names) <= 1:
            return;
        names = sum([list(x) for x in obj_names], [])
        ndims = [len(x) for x in obj_names]
        ndim = ndims[0]
        assert all(ndim == x for x in ndims), (
            "Dimension mismatch for {}".format(objs))
        varlist = []
        constlist = []
        for obj in objs:
            if not isinstance(obj, tuple):
                obj = (obj,)
            for var in obj:
                if isinstance(var, ir.Var) and not (var.name in varlist):
                    # favor those already defined, move to front of varlist
                    if var.name in self.defs:
                        varlist.insert(0, var)
                    else:
                        varlist.append(var)
                if isinstance(var, ir.Const) and not (var.value in constlist):
                    constlist.append(var.value)

        # try to populate ind_to_var if variables are present
        for obj in varlist:
            name = obj.name
            if name in names and not (name in self.obj_to_ind):
                self.ind_to_obj[self.next_ind] = [name]
                self.obj_to_ind[name] = self.next_ind
                self.ind_to_var[self.next_ind] = [obj]
                self.next_ind += 1

        # create equivalence classes for previously unseen constants
        for const in constlist:
            if const in names and not (const in self.obj_to_ind):
                self.ind_to_obj[self.next_ind] = [const]
                self.obj_to_ind[const] = self.next_ind
                self.ind_to_const[self.next_ind] = const
                self.next_ind += 1

        some_change = False

        for i in range(ndim):
            names = [obj_name[i] for obj_name in obj_names]
            ie_res = super(ShapeEquivSet, self).insert_equiv(*names)
            some_change = some_change or ie_res

        return some_change

    def has_shape(self, name):
        """Return true if the shape of the given variable is available.
        """
        return self.get_shape(name) != None

    def get_shape(self, name):
        """Return a tuple of variables that corresponds to the shape
        of the given array, or None if not found.
        """
        return guard(self._get_shape, name)

    def _get_shape(self, name):
        """Return a tuple of variables that corresponds to the shape
        of the given array, or raise GuardException if not found.
        """
        inds = self.get_shape_classes(name)
        require (inds != ())
        shape = []
        for i in inds:
            require(i in self.ind_to_var)
            vs = self.ind_to_var[i]
            if vs != []:
                shape.append(vs[0])
            else:
                require(i in self.ind_to_const)
                vs = self.ind_to_const[i]
                shape.append(vs)
        return tuple(shape)

    def get_shape_classes(self, name):
        """Instead of the shape tuple, return tuple of int, where
        each int is the corresponding class index of the size object.
        Unknown shapes are given class index -1. Return empty tuple
        if the input name is a scalar variable.
        """
        if isinstance(name, ir.Var):
            name = name.name
        typ = self.typemap[name] if name in self.typemap else None
        if not (isinstance(typ, types.BaseTuple) or
                isinstance(typ, types.SliceType) or
                isinstance(typ, types.ArrayCompatible)):
            return []
        # Treat 0d arrays like scalars.
        if isinstance(typ, types.ArrayCompatible) and typ.ndim == 0:
            return []
        names = self._get_names(name)
        inds = tuple(self._get_ind(name) for name in names)
        return inds

    def intersect(self, equiv_set):
        """Overload the intersect method to handle ind_to_var.
        """
        newset = super(ShapeEquivSet, self).intersect(equiv_set)
        ind_to_var = {}
        for i, objs in newset.ind_to_obj.items():
            assert(len(objs) > 0)
            obj = objs[0]
            assert(obj in self.obj_to_ind)
            assert(obj in equiv_set.obj_to_ind)
            j = self.obj_to_ind[obj]
            k = equiv_set.obj_to_ind[obj]
            assert(j in self.ind_to_var)
            assert(k in equiv_set.ind_to_var)
            varlist = []
            names = [x.name for x in equiv_set.ind_to_var[k]]
            for x in self.ind_to_var[j]:
                if x.name in names:
                    varlist.append(x)
            ind_to_var[i] = varlist
        newset.ind_to_var = ind_to_var
        return newset

    def define(self, name, redefined):
        """Increment the internal count of how many times a variable is being
        defined. Most variables in Numba IR are SSA, i.e., defined only once,
        but not all of them. When a variable is being re-defined, it must
        be removed from the equivalence relation and added to the redefined
        set but only if that redefinition is not known to have the same
        equivalence classes. Those variables redefined are removed from all
        the blocks' equivalence sets later.

        Arrays passed to define() use their whole name but these do not
        appear in the equivalence sets since they are stored there per
        dimension. Calling _get_names() here converts array names to
        dimensional names.

        This function would previously invalidate if there were any multiple
        definitions of a variable.  However, we realized that this behavior
        is overly restrictive.  You need only invalidate on multiple
        definitions if they are not known to be equivalent. So, the
        equivalence insertion functions now return True if some change was
        made (meaning the definition was not equivalent) and False
        otherwise. If no change was made, then define() need not be
        called. For no change to have been made, the variable must
        already be present. If the new definition of the var has the
        case where lhs and rhs are in the same equivalence class then
        again, no change will be made and define() need not be called
        or the variable invalidated.
        """
        if isinstance(name, ir.Var):
            name = name.name
        if name in self.defs:
            self.defs[name] += 1
            name_res = list(self._get_names(name))
            for one_name in name_res:
                # NOTE: variable being redefined, must invalidate previous
                # equivalences. Believe it is a rare case, and only happens to
                # scalar accumuators.
                if one_name in self.obj_to_ind:
                    redefined.add(one_name) # remove this var from all equiv sets
                    i = self.obj_to_ind[one_name]
                    del self.obj_to_ind[one_name]
                    self.ind_to_obj[i].remove(one_name)
                    if self.ind_to_obj[i] == []:
                        del self.ind_to_obj[i]
                    assert(i in self.ind_to_var)
                    names = [x.name for x in self.ind_to_var[i]]
                    if name in names:
                        j = names.index(name)
                        del self.ind_to_var[i][j]
                        if self.ind_to_var[i] == []:
                            del self.ind_to_var[i]
                            # no more size variables, remove equivalence too
                            if i in self.ind_to_obj:
                                for obj in self.ind_to_obj[i]:
                                    del self.obj_to_ind[obj]
                                del self.ind_to_obj[i]
        else:
            self.defs[name] = 1

    def union_defs(self, defs, redefined):
        """Union with the given defs dictionary. This is meant to handle
        branch join-point, where a variable may have been defined in more
        than one branches.
        """
        for k, v in defs.items():
            if v > 0:
                self.define(k, redefined)

class SymbolicEquivSet(ShapeEquivSet):

    """Just like ShapeEquivSet, except that it also reasons about variable
    equivalence symbolically by using their arithmetic definitions.
    The goal is to automatically derive the equivalence of array ranges
    (slicing). For instance, a[1:m] and a[0:m-1] shall be considered
    size-equivalence.
    """

    def __init__(self, typemap, def_by=None, ref_by=None, ext_shapes=None,
                 defs=None, ind_to_var=None, obj_to_ind=None,
                 ind_to_obj=None, next_id=0):
        """Create a new SymbolicEquivSet object, where typemap is a dictionary
        that maps variable names to their types, and it will not be modified.
        Optional keyword arguments are for internal use only.
        """
        # A "defined-by" table that maps A to a tuple of (B, i), which
        # means A is defined as: A = B + i, where A,B are variable names,
        # and i is an integer constants.
        self.def_by = def_by if def_by else {}
        # A "refered-by" table that maps A to a list of [(B, i), (C, j) ...],
        # which implies a sequence of definitions: B = A - i, C = A - j, and
        # so on, where A,B,C,... are variable names, and i,j,... are
        # integer constants.
        self.ref_by = ref_by if ref_by else {}
        # A extended shape table that can map an arbitrary object to a shape,
        # currently used to remember shapes for SetItem IR node, and wrapped
        # indices for Slice objects.
        self.ext_shapes = ext_shapes if ext_shapes else {}
        # rel_map keeps a map of relative sizes that we have seen so
        # that if we compute the same relative sizes different times
        # in different ways we can associate those two instances
        # of the same relative size to the same equivalence class.
        self.rel_map = {}
        # wrap_index() computes the effectual index given a slice and a
        # dimension's size.  We need to be able to know that two wrap_index
        # calls are equivalent.  They are known to be equivalent if the slice
        # and dimension sizes of the two wrap_index calls are equivalent.
        # wrap_map maps from a tuple of equivalence class ids for a slice and
        # a dimension size to some new equivalence class id for the output size.
        self.wrap_map = {}
        super(SymbolicEquivSet, self).__init__(
            typemap, defs, ind_to_var, obj_to_ind, ind_to_obj, next_id)

    def empty(self):
        """Return an empty SymbolicEquivSet.
        """
        return SymbolicEquivSet(self.typemap)

    def __repr__(self):
        return ("SymbolicEquivSet({}, ind_to_var={}, def_by={}, "
                "ref_by={}, ext_shapes={})".format(self.ind_to_obj,
                self.ind_to_var, self.def_by, self.ref_by, self.ext_shapes))

    def clone(self):
        """Return a new copy.
        """
        return SymbolicEquivSet(
            self.typemap,
            def_by=copy.copy(self.def_by),
            ref_by=copy.copy(self.ref_by),
            ext_shapes=copy.copy(self.ext_shapes),
            defs=copy.copy(self.defs),
            ind_to_var=copy.copy(self.ind_to_var),
            obj_to_ind=copy.deepcopy(self.obj_to_ind),
            ind_to_obj=copy.deepcopy(self.ind_to_obj),
            next_id=self.next_ind)

    def get_rel(self, name):
        """Retrieve a definition pair for the given variable,
        or return None if it is not available.
        """
        return guard(self._get_or_set_rel, name)

    def _get_or_set_rel(self, name, func_ir=None):
        """Retrieve a definition pair for the given variable,
        and if it is not already available, try to look it up
        in the given func_ir, and remember it for future use.
        """
        if isinstance(name, ir.Var):
            name = name.name
        require(self.defs.get(name, 0) == 1)
        if name in self.def_by:
            return self.def_by[name]
        else:
            require(func_ir != None)
            def plus(x, y):
                x_is_const = isinstance(x, int)
                y_is_const = isinstance(y, int)
                if x_is_const:
                    if y_is_const:
                        return x + y
                    else:
                        (var, offset) = y
                        return (var, x + offset)
                else:
                    (var, offset) = x
                    if y_is_const:
                        return (var, y + offset)
                    else:
                        return None
            def minus(x, y):
                if isinstance(y, int):
                    return plus(x, -y)
                elif (isinstance(x, tuple) and isinstance(y, tuple) and
                      x[0] == y[0]):
                    return minus(x[1], y[1])
                else:
                    return None
            expr = get_definition(func_ir, name)
            value = (name, 0) # default to its own name
            if isinstance(expr, ir.Expr):
                if expr.op == 'call':
                    fname, mod_name = find_callname(
                            func_ir, expr, typemap=self.typemap)
                    if fname == 'wrap_index' and mod_name == 'numba.array_analysis':
                        index = tuple(self.obj_to_ind.get(x.name, -1)
                                      for x in expr.args)
                        if -1 in index:
                            return None
                        names = self.ext_shapes.get(index, [])
                        names.append(name)
                        if len(names) > 0:
                            self._insert(names)
                        self.ext_shapes[index] = names
                elif expr.op == 'binop':
                    lhs = self._get_or_set_rel(expr.lhs, func_ir)
                    rhs = self._get_or_set_rel(expr.rhs, func_ir)
                    if expr.fn == operator.add:
                        value = plus(lhs, rhs)
                    elif expr.fn == operator.sub:
                        value = minus(lhs, rhs)
            elif isinstance(expr, ir.Const) and isinstance(expr.value, int):
                value = expr.value
            require(value != None)
            # update def_by table
            self.def_by[name] = value
            if isinstance(value, int) or (isinstance(value, tuple) and
                (value[0] != name or value[1] != 0)):
                # update ref_by table too
                if isinstance(value, tuple):
                    (var, offset) = value
                    if not (var in self.ref_by):
                        self.ref_by[var] = []
                    self.ref_by[var].append((name, -offset))
                    # insert new equivalence if found
                    ind = self._get_ind(var)
                    if ind >= 0:
                        objs = self.ind_to_obj[ind]
                        names = []
                        for obj in objs:
                            if obj in self.ref_by:
                                names += [ x for (x, i) in self.ref_by[obj]
                                           if i == -offset ]
                        if len(names) > 1:
                            super(SymbolicEquivSet, self)._insert(names)
            return value

    def define(self, var, redefined, func_ir=None, typ=None):
        """Besides incrementing the definition count of the given variable
        name, it will also retrieve and simplify its definition from func_ir,
        and remember the result for later equivalence comparison. Supported
        operations are:
          1. arithmetic plus and minus with constants
          2. wrap_index (relative to some given size)
        """
        if isinstance(var, ir.Var):
            name = var.name
        else:
            name = var
        super(SymbolicEquivSet, self).define(name, redefined)
        if (func_ir and self.defs.get(name, 0) == 1 and
                isinstance(typ, types.Number)):
            value = guard(self._get_or_set_rel, name, func_ir)
            # turn constant definition into equivalence
            if isinstance(value, int):
                self._insert([name, value])
            if isinstance(var, ir.Var):
                ind = self._get_or_add_ind(name)
                if not (ind in self.ind_to_obj):
                    self.ind_to_obj[ind] = [name]
                    self.obj_to_ind[name] = ind
                if ind in self.ind_to_var:
                    self.ind_to_var[ind].append(var)
                else:
                    self.ind_to_var[ind] = [var]

    def _insert(self, objs):
        """Overload _insert method to handle ind changes between relative
        objects.  Returns True if some change is made, false otherwise.
        """
        indset = set()
        uniqs = set()
        for obj in objs:
            ind = self._get_ind(obj)
            if ind == -1:
                uniqs.add(obj)
            elif not (ind in indset):
                uniqs.add(obj)
                indset.add(ind)
        if len(uniqs) <= 1:
            return False
        uniqs = list(uniqs)
        super(SymbolicEquivSet, self)._insert(uniqs)
        objs = self.ind_to_obj[self._get_ind(uniqs[0])]

        # New equivalence guided by def_by and ref_by
        offset_dict = {}
        def get_or_set(d, k):
            if k in d:
                v = d[k]
            else:
                v = []
                d[k] = v
            return v
        for obj in objs:
            if obj in self.def_by:
                value = self.def_by[obj]
                if isinstance(value, tuple):
                    (name, offset) = value
                    get_or_set(offset_dict, -offset).append(name)
                    if name in self.ref_by: # relative to name
                        for (v, i) in self.ref_by[name]:
                            get_or_set(offset_dict, -(offset+i)).append(v)
            if obj in self.ref_by:
                for (name, offset) in self.ref_by[obj]:
                    get_or_set(offset_dict, offset).append(name)
        for names in offset_dict.values():
            self._insert(names)
        return True

    def set_shape_setitem(self, obj, shape):
        """remember shapes of SetItem IR nodes.
        """
        assert isinstance(obj, ir.StaticSetItem) or isinstance(obj, ir.SetItem)
        self.ext_shapes[obj] = shape

    def _get_shape(self, obj):
        """Overload _get_shape to retrieve the shape of SetItem IR nodes.
        """
        if isinstance(obj, ir.StaticSetItem) or isinstance(obj, ir.SetItem):
            require(obj in self.ext_shapes)
            return self.ext_shapes[obj]
        else:
            assert(isinstance(obj, ir.Var))
            typ = self.typemap[obj.name]
            # for slice type, return the shape variable itself
            if isinstance(typ, types.SliceType):
                return (obj,)
            else:
                return super(SymbolicEquivSet, self)._get_shape(obj)

class WrapIndexMeta(object):
    """
      Array analysis should be able to analyze all the function
      calls that it adds to the IR.  That way, array analysis can
      be run as often as needed and you should get the same
      equivalencies.  One modification to the IR that array analysis
      makes is the insertion of wrap_index calls.  Thus, repeated
      array analysis passes should be able to analyze these wrap_index
      calls.  The difficulty of these calls is that the equivalence
      class of the left-hand side of the assignment is not present in
      the arguments to wrap_index in the right-hand side.  Instead,
      the equivalence class of the wrap_index output is a combination
      of the wrap_index args.  The important thing to
      note is that if the equivalence classes of the slice size
      and the dimension's size are the same for two wrap index
      calls then we can be assured of the answer being the same.
      So, we maintain the wrap_map dict that maps from a tuple
      of equivalence class ids for the slice and dimension size
      to some new equivalence class id for the output size.
      However, when we are analyzing the first such wrap_index
      call we don't have a variable there to associate to the
      size since we're in the process of analyzing the instruction
      that creates that mapping.  So, instead we return an object
      of this special class and analyze_inst will establish the
      connection between a tuple of the parts of this object
      below and the left-hand side variable.
    """
    def __init__(self, slice_size, dim_size):
        self.slice_size = slice_size
        self.dim_size = dim_size

class ArrayAnalysis(object):
    aa_count = 0

    """Analyzes Numpy array computations for properties such as
    shape/size equivalence, and keeps track of them on a per-block
    basis. The analysis should only be run once because it modifies
    the incoming IR by inserting assertion statements that safeguard
    parfor optimizations.
    """

    def __init__(self, context, func_ir, typemap, calltypes):
        self.context = context
        self.func_ir = func_ir
        self.typemap = typemap
        self.calltypes = calltypes

        # EquivSet of variables, indexed by block number
        self.equiv_sets = {}
        # keep attr calls to arrays like t=A.sum() as {t:('sum',A)}
        self.array_attr_calls = {}
        # keep prepended instructions from conditional branch
        self.prepends = {}
        # keep track of pruned precessors when branch degenerates to jump
        self.pruned_predecessors = {}

    def get_equiv_set(self, block_label):
        """Return the equiv_set object of an block given its label.
        """
        return self.equiv_sets[block_label]

    def remove_redefineds(self, redefineds):
        """Take a set of variables in redefineds and go through all
        the currently existing equivalence sets (created in topo order)
        and remove that variable from all of them since it is multiply
        defined within the function.
        """
        unused = set()
        for r in redefineds:
            for eslabel in self.equiv_sets:
                es = self.equiv_sets[eslabel]
                es.define(r, unused)

    def run(self, blocks=None, equiv_set=None):
        """run array shape analysis on the given IR blocks, resulting in
        modified IR and finalized EquivSet for each block.
        """
        if blocks == None:
            blocks = self.func_ir.blocks

        self.func_ir._definitions = build_definitions(self.func_ir.blocks)

        if equiv_set == None:
            init_equiv_set = SymbolicEquivSet(self.typemap)
        else:
            init_equiv_set = equiv_set

        aa_count_save = ArrayAnalysis.aa_count
        ArrayAnalysis.aa_count += 1
        if config.DEBUG_ARRAY_OPT >= 1:
            print("Starting ArrayAnalysis:", aa_count_save)
        dprint_func_ir(self.func_ir, "before array analysis", blocks)

        if config.DEBUG_ARRAY_OPT >= 1:
            print("ArrayAnalysis variable types: ", sorted(self.typemap.items()))
            print("ArrayAnalysis call types: ", self.calltypes)

        cfg = compute_cfg_from_blocks(blocks)
        topo_order = find_topo_order(blocks, cfg=cfg)
        # Traverse blocks in topological order
        for label in topo_order:
            if config.DEBUG_ARRAY_OPT >= 2:
                print("Processing block:", label)
            block = blocks[label]
            scope = block.scope
            new_body = []
            equiv_set = None

            # equiv_set is the intersection of predecessors
            preds = cfg.predecessors(label)
            # some incoming edge may be pruned due to prior analysis
            if label in self.pruned_predecessors:
                pruned = self.pruned_predecessors[label]
            else:
                pruned = []
            # Go through each incoming edge, process prepended instructions and
            # calculate beginning equiv_set of current block as an intersection
            # of incoming ones.
            if config.DEBUG_ARRAY_OPT >= 2:
                print("preds:", preds)
            for (p, q) in preds:
                if config.DEBUG_ARRAY_OPT >= 2:
                    print("p, q:", p, q)
                if p in pruned:
                    continue
                if p in self.equiv_sets:
                    from_set = self.equiv_sets[p].clone()
                    if config.DEBUG_ARRAY_OPT >= 2:
                        print("p in equiv_sets", from_set)
                    if (p, label) in self.prepends:
                        instrs = self.prepends[(p, label)]
                        for inst in instrs:
                            redefined = set()
                            self._analyze_inst(label, scope, from_set, inst, redefined)
                            # Remove anything multiply defined in this block 
                            # from every block equivs.
                            self.remove_redefineds(redefined)
                    if equiv_set == None:
                        equiv_set = from_set
                    else:
                        equiv_set = equiv_set.intersect(from_set)
                        redefined = set()
                        equiv_set.union_defs(from_set.defs, redefined)
                        # Remove anything multiply defined in this block 
                        # from every block equivs.
                        self.remove_redefineds(redefined)

            # Start with a new equiv_set if none is computed
            if equiv_set == None:
                equiv_set = init_equiv_set
            self.equiv_sets[label] = equiv_set
            # Go through instructions in a block, and insert pre/post
            # instructions as we analyze them.
            for inst in block.body:
                redefined = set()
                if isinstance(inst, ir.StaticSetItem):
                    orig_calltype = self.calltypes[inst]
                    inst = ir.SetItem(inst.target, inst.index_var, inst.value, inst.loc)
                    self.calltypes[inst] = orig_calltype
                pre, post = self._analyze_inst(label, scope, equiv_set, inst, redefined)
                # Remove anything multiply defined in this block from every block equivs.
                if len(redefined) > 0:
                    self.remove_redefineds(redefined)
                for instr in pre:
                    new_body.append(instr)
                new_body.append(inst)
                for instr in post:
                    new_body.append(instr)
            block.body = new_body

        if config.DEBUG_ARRAY_OPT >= 1:
            self.dump()
            print("ArrayAnalysis post variable types: ", sorted(self.typemap.items()))
            print("ArrayAnalysis post call types: ", self.calltypes)

        dprint_func_ir(self.func_ir, "after array analysis", blocks)
        if config.DEBUG_ARRAY_OPT >= 1:
            print("Ending ArrayAnalysis:", aa_count_save)

    def dump(self):
        """dump per-block equivalence sets for debugging purposes.
        """
        print("Array Analysis: ", self.equiv_sets)

    def _define(self, equiv_set, var, typ, value):
        self.typemap[var.name] = typ
        self.func_ir._definitions[var.name] = [value]
        redefineds = set()
        equiv_set.define(var, redefineds, self.func_ir, typ)

    def _analyze_inst(self, label, scope, equiv_set, inst, redefined):
        pre = []
        post = []
        if config.DEBUG_ARRAY_OPT >= 2:
            print("analyze_inst:", inst)
        if isinstance(inst, ir.Assign):
            lhs = inst.target
            typ = self.typemap[lhs.name]
            shape = None
            if isinstance(typ, types.ArrayCompatible) and typ.ndim == 0:
                shape = ()
            elif isinstance(inst.value, ir.Expr):
                result = self._analyze_expr(scope, equiv_set, inst.value)
                if result:
                    shape = result[0]
                    pre = result[1]
                    if len(result) > 2:
                        rhs = result[2]
                        inst.value = rhs
            elif (isinstance(inst.value, ir.Var) or
                  isinstance(inst.value, ir.Const)):
                shape = inst.value

            if isinstance(shape, ir.Const):
                if isinstance(shape.value, tuple):
                    loc = shape.loc
                    shape = tuple(ir.Const(x, loc) for x in shape.value)
                elif isinstance(shape.value, int):
                    shape = (shape,)
                else:
                    shape = None
            elif (isinstance(shape, ir.Var) and
                  isinstance(self.typemap[shape.name], types.Integer)):
                shape = (shape,)
            elif isinstance(shape, WrapIndexMeta):
                """ Here we've got the special WrapIndexMeta object
                    back from analyzing a wrap_index call.  We define
                    the lhs and then get it's equivalence class then
                    add the mapping from the tuple of slice size and
                    dimensional size equivalence ids to the lhs
                    equivalence id.
                """
                equiv_set.define(lhs, redefined, self.func_ir, typ)
                lhs_ind = equiv_set._get_ind(lhs.name)
                if lhs_ind != -1:
                    equiv_set.wrap_map[(shape.slice_size, shape.dim_size)] = lhs_ind
                return pre, post

            if isinstance(typ, types.ArrayCompatible):
                if (shape is not None and
                    isinstance(shape, ir.Var) and
                    isinstance(self.typemap[shape.name], types.containers.Tuple)):
                    pass
                elif (shape == None or isinstance(shape, tuple) or
                    (isinstance(shape, ir.Var) and
                     not equiv_set.has_shape(shape))):
                    (shape, post) = self._gen_shape_call(equiv_set, lhs,
                                                         typ.ndim, shape)
            elif isinstance(typ, types.UniTuple):
                if shape and isinstance(typ.dtype, types.Integer):
                    (shape, post) = self._gen_shape_call(equiv_set, lhs,
                                                         len(typ), shape)

            """ See the comment on the define() function.

                We need only call define(), which will invalidate a variable
                from being in the equivalence sets on multiple definitions,
                if the variable was not previously defined or if the new
                definition would be in a conflicting equivalence class to the
                original equivalence class for the variable.

                insert_equiv() returns True if either of these conditions are
                True and then we call define() in those cases.  If insert_equiv()
                returns False then no changes were made and all equivalence
                classes are consistent upon a redefinition so no invalidation
                is needed and we don't call define().
            """
            needs_define = True
            if shape != None:
                needs_define = equiv_set.insert_equiv(lhs, shape)
            if needs_define:
                equiv_set.define(lhs, redefined, self.func_ir, typ)
        elif isinstance(inst, ir.StaticSetItem) or isinstance(inst, ir.SetItem):
            index = inst.index if isinstance(inst, ir.SetItem) else inst.index_var
            result = guard(self._index_to_shape,
                scope, equiv_set, inst.target, index)
            if not result:
                return [], []
            if result[0] is not None:
                assert(isinstance(inst, ir.SetItem))
                inst.index = result[0]
                #inst.index_var = result[0]
            result = result[1]
            (target_shape, pre) = result
            value_shape = equiv_set.get_shape(inst.value)
            if value_shape == (): # constant
                equiv_set.set_shape_setitem(inst, target_shape)
                return pre, []
            elif value_shape != None:
                target_typ = self.typemap[inst.target.name]
                require(isinstance(target_typ, types.ArrayCompatible))
                target_ndim = target_typ.ndim
                shapes = [target_shape, value_shape]
                names = [inst.target.name, inst.value.name]
                shape, asserts = self._broadcast_assert_shapes(
                                scope, equiv_set, inst.loc, shapes, names)
                n = len(shape)
                # shape dimension must be within target dimension
                assert(target_ndim >= n)
                equiv_set.set_shape_setitem(inst, shape)
                return pre + asserts, []
            else:
                return pre, []
        elif isinstance(inst, ir.Branch):
            cond_var = inst.cond
            cond_def = guard(get_definition, self.func_ir, cond_var)
            if not cond_def:  # phi variable has no single definition
                # We'll use equiv_set to try to find a cond_def instead
                equivs = equiv_set.get_equiv_set(cond_var)
                defs = []
                for name in equivs:
                    if isinstance(name, str) and name in self.typemap:
                        var_def = guard(get_definition, self.func_ir, name,
                                        lhs_only=True)
                        if isinstance(var_def, ir.Var):
                            var_def = var_def.name
                        if var_def:
                            defs.append(var_def)
                    else:
                        defs.append(name)
                defvars = set(filter(lambda x: isinstance(x, str), defs))
                defconsts = set(defs).difference(defvars)
                if len(defconsts) == 1:
                    cond_def = list(defconsts)[0]
                elif len(defvars) == 1:
                    cond_def = guard(get_definition, self.func_ir,
                                     list(defvars)[0])
            if isinstance(cond_def, ir.Expr) and cond_def.op == 'binop':
                br = None
                if cond_def.fn == operator.eq:
                    br = inst.truebr
                    otherbr = inst.falsebr
                    cond_val = 1
                elif cond_def.fn == operator.ne:
                    br = inst.falsebr
                    otherbr = inst.truebr
                    cond_val = 0
                lhs_typ = self.typemap[cond_def.lhs.name]
                rhs_typ = self.typemap[cond_def.rhs.name]
                if (br != None and
                    ((isinstance(lhs_typ, types.Integer) and
                      isinstance(rhs_typ, types.Integer)) or
                     (isinstance(lhs_typ, types.BaseTuple) and
                      isinstance(rhs_typ, types.BaseTuple)))):
                    loc = inst.loc
                    args = (cond_def.lhs, cond_def.rhs)
                    asserts = self._make_assert_equiv(
                        scope, loc, equiv_set, args)
                    asserts.append(
                        ir.Assign(ir.Const(cond_val, loc), cond_var, loc))
                    self.prepends[(label, br)] = asserts
                    self.prepends[(label, otherbr)] = [
                        ir.Assign(ir.Const(1 - cond_val, loc), cond_var, loc)]
            else:
                if isinstance(cond_def, ir.Const):
                    cond_def = cond_def.value
                if isinstance(cond_def, int) or isinstance(cond_def, bool):
                    # condition is always true/false, prune the outgoing edge
                    pruned_br = inst.falsebr if cond_def else inst.truebr
                    if pruned_br in self.pruned_predecessors:
                        self.pruned_predecessors[pruned_br].append(label)
                    else:
                        self.pruned_predecessors[pruned_br] = [label]

        elif type(inst) in array_analysis_extensions:
            # let external calls handle stmt if type matches
            f = array_analysis_extensions[type(inst)]
            pre, post = f(inst, equiv_set, self.typemap, self)

        return pre, post

    def _analyze_expr(self, scope, equiv_set, expr):
        fname = "_analyze_op_{}".format(expr.op)
        try:
            fn = getattr(self, fname)
        except AttributeError:
            return None
        return guard(fn, scope, equiv_set, expr)

    def _analyze_op_getattr(self, scope, equiv_set, expr):
        # TODO: getattr of npytypes.Record
        if expr.attr == 'T' and self._isarray(expr.value.name):
            return self._analyze_op_call_numpy_transpose(scope, equiv_set, [expr.value], {})
        elif expr.attr == 'shape':
            shape = equiv_set.get_shape(expr.value)
            return shape, []
        return None

    def _analyze_op_cast(self, scope, equiv_set, expr):
        return expr.value, []

    def _analyze_op_exhaust_iter(self, scope, equiv_set, expr):
        var = expr.value
        typ = self.typemap[var.name]
        if isinstance(typ, types.BaseTuple):
            require(len(typ) == expr.count)
            require(equiv_set.has_shape(var))
            return var, []
        return None

    def gen_literal_slice_part(self, arg_val, loc, scope, stmts, equiv_set,
                               name="static_literal_slice_part"):
        # Create var to hold the calculated slice size.
        static_literal_slice_part_var = ir.Var(scope, mk_unique_var(name), loc)
        static_literal_slice_part_val = ir.Const(arg_val, loc)
        static_literal_slice_part_typ = types.IntegerLiteral(arg_val)
        # We'll prepend this slice size calculation to the get/setitem.
        stmts.append(ir.Assign(value=static_literal_slice_part_val,
                               target=static_literal_slice_part_var,
                               loc=loc))
        self._define(equiv_set,
                     static_literal_slice_part_var,
                     static_literal_slice_part_typ,
                     static_literal_slice_part_val)
        return static_literal_slice_part_var, static_literal_slice_part_typ

    def gen_static_slice_size(self, lhs_rel, rhs_rel, loc, scope, stmts, equiv_set):
        the_var, the_typ = self.gen_literal_slice_part(rhs_rel - lhs_rel,
                                                       loc,
                                                       scope,
                                                       stmts,
                                                       equiv_set,
                                                       name="static_slice_size")
        return the_var

    def gen_explicit_neg(self, arg, arg_rel, arg_typ, size_typ, loc, scope,
                         dsize, stmts, equiv_set):
        assert(not isinstance(size_typ, int))
        # Create var to hold the calculated slice size.
        explicit_neg_var = ir.Var(scope, mk_unique_var("explicit_neg"), loc)
        explicit_neg_val = ir.Expr.binop(operator.add, dsize, arg, loc=loc)
        # Determine the type of that var.  Can be literal if we know the
        # literal size of the dimension.
        explicit_neg_typ = types.intp
        self.calltypes[explicit_neg_val] = signature(explicit_neg_typ,
                                                     size_typ, arg_typ)
        # We'll prepend this slice size calculation to the get/setitem.
        stmts.append(ir.Assign(value=explicit_neg_val,
                               target=explicit_neg_var, loc=loc))
        self._define(equiv_set, explicit_neg_var,
                     explicit_neg_typ, explicit_neg_val)
        return explicit_neg_var, explicit_neg_typ

    def update_replacement_slice(self, lhs, lhs_typ, lhs_rel, dsize_rel,
                                 replacement_slice, slice_index,
                                 need_replacement, loc, scope, stmts,
                                 equiv_set, size_typ, dsize):
        # Do compile-time calculation of real index value if both the given
        # index value and the array length are known at compile time.
        known = False
        if isinstance(lhs_rel, int):
            # If the index and the array size are known then the real index
            # can be calculated at compile time.
            if lhs_rel == 0:
                # Special-case 0 as nothing needing to be done.
                known = True
            elif isinstance(dsize_rel, int):
                known = True
                # Calculate the real index.
                wil = wrap_index_literal(lhs_rel, dsize_rel)
                # If the given index value is between 0 and dsize then
                # there's no need to rewrite anything.
                if wil != lhs_rel:
                    if config.DEBUG_ARRAY_OPT >= 2:
                        print("Replacing slice to hard-code known slice size.")
                    # Indicate we will need to replace the slice var.
                    need_replacement = True
                    literal_var, literal_typ = self.gen_literal_slice_part(
                        wil, loc, scope, stmts, equiv_set)
                    assert(slice_index == 0 or slice_index == 1)
                    if slice_index == 0:
                        replacement_slice.args = (literal_var,
                                                  replacement_slice.args[1])
                    else:
                        replacement_slice.args = (replacement_slice.args[0],
                                                  literal_var)
                    # Update lhs information with the negative removed.
                    lhs = replacement_slice.args[slice_index]
                    lhs_typ = literal_typ
                    lhs_rel = equiv_set.get_rel(lhs)
            elif lhs_rel < 0:
                # Indicate we will need to replace the slice var.
                need_replacement = True
                if config.DEBUG_ARRAY_OPT >= 2:
                    print("Replacing slice due to known negative index.")
                explicit_neg_var, explicit_neg_typ = self.gen_explicit_neg(lhs,
                    lhs_rel, lhs_typ, size_typ, loc, scope, dsize, stmts, equiv_set)
                if slice_index == 0:
                    replacement_slice.args = (explicit_neg_var,
                                              replacement_slice.args[1])
                else:
                    replacement_slice.args = (replacement_slice.args[0],
                                              explicit_neg_var)
                # Update lhs information with the negative removed.
                lhs = replacement_slice.args[slice_index]
                lhs_typ = explicit_neg_typ
                lhs_rel = equiv_set.get_rel(lhs)
        return lhs, lhs_typ, lhs_rel, replacement_slice, need_replacement, known

    def slice_size(self, index, dsize, equiv_set, scope, stmts):
        """Reason about the size of a slice represented by the "index"
        variable, and return a variable that has this size data, or
        raise GuardException if it cannot reason about it.

        The computation takes care of negative values used in the slice
        with respect to the given dimensional size ("dsize").

        Extra statments required to produce the result are appended
        to parent function's stmts list.
        """
        loc = index.loc
        # Get the definition of the index variable.
        index_def = get_definition(self.func_ir, index)
        fname, mod_name = find_callname(
            self.func_ir, index_def, typemap=self.typemap)
        require(fname == 'slice' and mod_name in ('__builtin__', 'builtins'))
        require(len(index_def.args) == 2)
        lhs = index_def.args[0]
        rhs = index_def.args[1]
        size_typ = self.typemap[dsize.name]
        lhs_typ = self.typemap[lhs.name]
        rhs_typ = self.typemap[rhs.name]

        if config.DEBUG_ARRAY_OPT >= 2:
            print("slice_size", "index=", index, "dsize=", dsize,
                  "index_def=", index_def, "lhs=", lhs, "rhs=", rhs,
                  "size_typ=", size_typ, "lhs_typ=", lhs_typ, "rhs_typ=", rhs_typ)

        # Make a deepcopy of the original slice to use as the
        # replacement slice, which we will modify as necessary
        # below to convert all negative constants in the slice
        # to be relative to the dimension size.
        replacement_slice = copy.deepcopy(index_def)
        need_replacement = False

        # Fill in the left side of the slice's ":" with 0 if it wasn't specified.
        if isinstance(lhs_typ, types.NoneType):
            zero_var = ir.Var(scope, mk_unique_var("zero"), loc)
            zero = ir.Const(0, loc)
            stmts.append(ir.Assign(value=zero, target=zero_var, loc=loc))
            self._define(equiv_set, zero_var, types.IntegerLiteral(0), zero)
            lhs = zero_var
            lhs_typ = types.IntegerLiteral(0)
            replacement_slice.args = (lhs, replacement_slice.args[1])
            need_replacement = True
            if config.DEBUG_ARRAY_OPT >= 2:
                print("Replacing slice because lhs is None.")

        # Fill in the right side of the slice's ":" with the array
        # length if it wasn't specified.
        if isinstance(rhs_typ, types.NoneType):
            rhs = dsize
            rhs_typ = size_typ
            replacement_slice.args = (replacement_slice.args[0], rhs)
            need_replacement = True
            if config.DEBUG_ARRAY_OPT >= 2:
                print("Replacing slice because lhs is None.")

        lhs_rel = equiv_set.get_rel(lhs)
        rhs_rel = equiv_set.get_rel(rhs)
        dsize_rel = equiv_set.get_rel(dsize)
        if config.DEBUG_ARRAY_OPT >= 2:
            print("lhs_rel", lhs_rel, "rhs_rel", rhs_rel, "dsize_rel", dsize_rel)

        # Update replacement slice with the real index value if we can
        # compute it at compile time.
        lhs, lhs_typ, lhs_rel, replacement_slice, need_replacement, lhs_known = self.update_replacement_slice(
              lhs, lhs_typ, lhs_rel, dsize_rel, replacement_slice, 0,
              need_replacement, loc, scope, stmts, equiv_set, size_typ, dsize)
        rhs, rhs_typ, rhs_rel, replacement_slice, need_replacement, rhs_known = self.update_replacement_slice(
              rhs, rhs_typ, rhs_rel, dsize_rel, replacement_slice, 1,
              need_replacement, loc, scope, stmts, equiv_set, size_typ, dsize)
        if config.DEBUG_ARRAY_OPT >= 2:
            print("lhs_known:", lhs_known)
            print("rhs_known:", rhs_known)

        # If neither of the parts of the slice were negative constants
        # then we don't need to do slice replacement in the IR.
        if not need_replacement:
            replacement_slice_var = None
        else:
            # Create a new var for the replacement slice.
            replacement_slice_var = ir.Var(scope, mk_unique_var("replacement_slice"), loc)
            # Create a deepcopy of slice calltype so that when we change it below
            # the original isn't changed.  Make the types of the parts of the slice
            # intp.
            new_arg_typs = (types.intp, types.intp)
            rs_calltype = self.typemap[index_def.func.name].get_call_type(self.context, new_arg_typs, {})
            self.calltypes[replacement_slice] = rs_calltype
            stmts.append(ir.Assign(value=replacement_slice, target=replacement_slice_var, loc=loc))
            # The type of the replacement slice is the same type as the original.
            self.typemap[replacement_slice_var.name] = self.typemap[index.name]

        if config.DEBUG_ARRAY_OPT >= 2:
            print("after rewriting negatives", "lhs_rel", lhs_rel, "rhs_rel", rhs_rel)

        if lhs_known and rhs_known:
            if config.DEBUG_ARRAY_OPT >= 2:
                print("lhs and rhs known so return static size")
            return self.gen_static_slice_size(lhs_rel, rhs_rel, loc, scope, stmts, equiv_set), replacement_slice_var

        if (lhs_rel == 0 and isinstance(rhs_rel, tuple) and
            equiv_set.is_equiv(dsize, rhs_rel[0]) and
            rhs_rel[1] == 0):
            return dsize, None

        slice_typ = types.intp
        orig_slice_typ = slice_typ

        size_var = ir.Var(scope, mk_unique_var("slice_size"), loc)
        size_val = ir.Expr.binop(operator.sub, rhs, lhs, loc=loc)
        self.calltypes[size_val] = signature(slice_typ, rhs_typ, lhs_typ)
        self._define(equiv_set, size_var, slice_typ, size_val)
        size_rel = equiv_set.get_rel(size_var)
        if config.DEBUG_ARRAY_OPT >= 2:
            print("size_rel", size_rel, type(size_rel))

        wrap_var = ir.Var(scope, mk_unique_var("wrap"), loc)
        wrap_def = ir.Global('wrap_index', wrap_index, loc=loc)
        fnty = get_global_func_typ(wrap_index)
        sig = self.context.resolve_function_type(fnty, (orig_slice_typ, size_typ,), {})
        self._define(equiv_set, wrap_var, fnty, wrap_def)

        def gen_wrap_if_not_known(val, val_typ, known):
            if not known:
                var = ir.Var(scope, mk_unique_var("var"), loc)
                var_typ = types.intp
                new_value = ir.Expr.call(wrap_var, [val, dsize], {}, loc)
                self._define(equiv_set, var, var_typ, new_value)
                self.calltypes[new_value] = sig
                return (var, var_typ, new_value)
            else:
                return (val, val_typ, None)

        var1, var1_typ, value1 = gen_wrap_if_not_known(lhs, lhs_typ, lhs_known)
        var2, var2_typ, value2 = gen_wrap_if_not_known(rhs, rhs_typ, rhs_known)

        post_wrap_size_var = ir.Var(scope, mk_unique_var("post_wrap_slice_size"), loc)
        post_wrap_size_val = ir.Expr.binop(operator.sub, var2, var1, loc=loc)
        self.calltypes[post_wrap_size_val] = signature(slice_typ, var2_typ, var1_typ)
        self._define(equiv_set, post_wrap_size_var, slice_typ, post_wrap_size_val)

        stmts.append(ir.Assign(value=size_val, target=size_var, loc=loc))
        stmts.append(ir.Assign(value=wrap_def, target=wrap_var, loc=loc))
        if value1 is not None:
            stmts.append(ir.Assign(value=value1, target=var1, loc=loc))
        if value2 is not None:
            stmts.append(ir.Assign(value=value2, target=var2, loc=loc))
        stmts.append(ir.Assign(value=post_wrap_size_val, target=post_wrap_size_var, loc=loc))

        # rel_map keeps a map of relative sizes that we have seen so
        # that if we compute the same relative sizes different times
        # in different ways we can associate those two instances
        # of the same relative size to the same equivalence class.
        if isinstance(size_rel, tuple):
            if config.DEBUG_ARRAY_OPT >= 2:
                print("size_rel is tuple", equiv_set.rel_map)
            rel_map_entry = None
            for rme, rme_tuple in equiv_set.rel_map.items():
                if rme[1] == size_rel[1] and equiv_set.is_equiv(rme[0], size_rel[0]):
                    rel_map_entry = rme_tuple
                    break

            if rel_map_entry is not None:
                # We have seen this relative size before so establish
                # equivalence to the previous variable.
                if config.DEBUG_ARRAY_OPT >= 2:
                    print("establishing equivalence to", rel_map_entry)
                equiv_set.insert_equiv(size_var, rel_map_entry[0])
                equiv_set.insert_equiv(post_wrap_size_var, rel_map_entry[1])
            else:
                # The first time we've seen this relative size so
                # remember the variable defining that size.
                equiv_set.rel_map[size_rel] = (size_var, post_wrap_size_var)

        return post_wrap_size_var, replacement_slice_var

    def _index_to_shape(self, scope, equiv_set, var, ind_var):
        """For indexing like var[index] (either write or read), see if
        the index corresponds to a range/slice shape.
        Returns a 2-tuple where the first item is either None or a ir.Var
        to be used to replace the index variable in the outer getitem or
        setitem instruction.  The second item is also a tuple returning
        the shape and prepending instructions.
        """
        typ = self.typemap[var.name]
        require(isinstance(typ, types.ArrayCompatible))
        ind_typ = self.typemap[ind_var.name]
        ind_shape = equiv_set._get_shape(ind_var)
        var_shape = equiv_set._get_shape(var)
        if isinstance(ind_typ, types.SliceType):
            seq_typs = (ind_typ,)
        else:
            require(isinstance(ind_typ, types.BaseTuple))
            seq, op = find_build_sequence(self.func_ir, ind_var)
            require(op == 'build_tuple')
            seq_typs = tuple(self.typemap[x.name] for x in seq)
        require(len(ind_shape)==len(seq_typs)==len(var_shape))
        stmts = []

        def to_shape(typ, index, dsize):
            if isinstance(typ, types.SliceType):
                return self.slice_size(index, dsize, equiv_set, scope, stmts)
            elif isinstance(typ, types.Number):
                return None, None
            else:
                # unknown dimension size for this index,
                # so we'll raise GuardException
                require(False)

        shape_list = []
        index_var_list = []
        replace_index = False
        for (typ, size, dsize) in zip(seq_typs, ind_shape, var_shape):
            # Convert the given dimension of the get/setitem index expr.
            shape_part, index_var_part = to_shape(typ, size, dsize)
            shape_list.append(shape_part)

            # to_shape will return index_var_part as not None if a
            # replacement of the slice is required to convert from
            # negative indices to positive relative indices.
            if index_var_part is not None:
                # Remember that we need to replace the build_tuple.
                replace_index = True
                index_var_list.append(index_var_part)
            else:
                index_var_list.append(size)

        # If at least one of the dimensions required a new slice variable
        # then we'll need to replace the build_tuple for this get/setitem.
        if replace_index:
            # Multi-dimensional array access needs a replacement tuple built.
            if len(index_var_list) > 1:
                # Make a variable to hold the new build_tuple.
                replacement_build_tuple_var = ir.Var(scope,
                                              mk_unique_var("replacement_build_tuple"),
                                              ind_shape[0].loc)
                # Create the build tuple from the accumulated index vars above.
                new_build_tuple = ir.Expr.build_tuple(index_var_list, ind_shape[0].loc)
                stmts.append(ir.Assign(value=new_build_tuple,
                                       target=replacement_build_tuple_var,
                                       loc=ind_shape[0].loc))
                # New build_tuple has same type as the original one.
                self.typemap[replacement_build_tuple_var.name] = ind_typ
            else:
                replacement_build_tuple_var = index_var_list[0]
        else:
            replacement_build_tuple_var = None

        shape = tuple(shape_list)
        require(not all(x == None for x in shape))
        shape = tuple(x for x in shape if x != None)
        return (replacement_build_tuple_var, (shape, stmts))

    def _analyze_op_getitem(self, scope, equiv_set, expr):
        result = self._index_to_shape(scope, equiv_set, expr.value, expr.index)
        if result[0] is not None:
            expr.index = result[0]
        return result[1]

    def _analyze_op_static_getitem(self, scope, equiv_set, expr):
        var = expr.value
        typ = self.typemap[var.name]
        if not isinstance(typ, types.BaseTuple):
            result = self._index_to_shape(scope, equiv_set, expr.value, expr.index_var)
            if result[0] is not None:
                expr.index_var = result[0]
            return result[1]
        shape = equiv_set._get_shape(var)
        require(isinstance(expr.index, int) and expr.index < len(shape))
        return shape[expr.index], []

    def _analyze_op_unary(self, scope, equiv_set, expr):
        require(expr.fn in UNARY_MAP_OP)
        # for scalars, only + operator results in equivalence
        # for example, if "m = -n", m and n are not equivalent
        if self._isarray(expr.value.name) or expr.fn == operator.add:
            return expr.value, []
        return None

    def _analyze_op_binop(self, scope, equiv_set, expr):
        require(expr.fn in BINARY_MAP_OP)
        return self._analyze_broadcast(scope, equiv_set, expr.loc, [expr.lhs, expr.rhs])

    def _analyze_op_inplace_binop(self, scope, equiv_set, expr):
        require(expr.fn in INPLACE_BINARY_MAP_OP)
        return self._analyze_broadcast(scope, equiv_set, expr.loc, [expr.lhs, expr.rhs])

    def _analyze_op_arrayexpr(self, scope, equiv_set, expr):
        return self._analyze_broadcast(scope, equiv_set, expr.loc, expr.list_vars())

    def _analyze_op_build_tuple(self, scope, equiv_set, expr):
        consts = []
        for var in expr.items:
            x = guard(find_const, self.func_ir, var)
            if x is not None:
                consts.append(x)
            else:
                break
        else:
            out = tuple([ir.Const(x, expr.loc) for x in consts])
            return out, [], ir.Const(tuple(consts), expr.loc)
        # default return for non-const
        return tuple(expr.items), []


    def _analyze_op_call(self, scope, equiv_set, expr):
        from numba.stencil import StencilFunc

        callee = expr.func
        callee_def = get_definition(self.func_ir, callee)
        if (isinstance(callee_def, (ir.Global, ir.FreeVar))
                and is_namedtuple_class(callee_def.value)):
            return tuple(expr.args), []
        if (isinstance(callee_def, (ir.Global, ir.FreeVar))
                and isinstance(callee_def.value, StencilFunc)):
            args = expr.args
            return self._analyze_stencil(scope, equiv_set, callee_def.value,
                                         expr.loc, args, dict(expr.kws))

        fname, mod_name = find_callname(
            self.func_ir, expr, typemap=self.typemap)
        added_mod_name = False
        # call via attribute (i.e. array.func)
        if (isinstance(mod_name, ir.Var)
                and isinstance(self.typemap[mod_name.name],
                                types.ArrayCompatible)):
            args = [mod_name] + expr.args
            mod_name = 'numpy'
            # Remember that args and expr.args don't alias.
            added_mod_name = True
        else:
            args = expr.args
        fname = "_analyze_op_call_{}_{}".format(
            mod_name, fname).replace('.', '_')
        if fname in UFUNC_MAP_OP:  # known numpy ufuncs
            return self._analyze_broadcast(scope, equiv_set, expr.loc, args)
        else:
            try:
                fn = getattr(self, fname)
            except AttributeError:
                return None
            result = guard(fn, scope, equiv_set, args, dict(expr.kws))
            # We want the ability for function fn to modify arguments.
            # If args and expr.args don't alias then we need the extra
            # step of assigning back into expr.args from the args that
            # was passed to fn.
            if added_mod_name:
                expr.args = args[1:]
            return result

    def _analyze_op_call___builtin___len(self, scope, equiv_set, args, kws):
        # python 2 version of len()
        return self._analyze_op_call_builtins_len(scope, equiv_set, args, kws)

    def _analyze_op_call_builtins_len(self, scope, equiv_set, args, kws):
        # python 3 version of len()
        require(len(args) == 1)
        var = args[0]
        typ = self.typemap[var.name]
        require(isinstance(typ, types.ArrayCompatible))
        shape = equiv_set._get_shape(var)
        return shape[0], [], shape[0]

    def _analyze_op_call_numba_array_analysis_assert_equiv(self, scope,
                                                        equiv_set, args, kws):
        equiv_set.insert_equiv(*args[1:])
        return None

    def _analyze_op_call_numba_array_analysis_wrap_index(self, scope,
                                                        equiv_set, args, kws):
        """ Analyze wrap_index calls added by a previous run of
            Array Analysis
        """
        require(len(args) == 2)
        # Two parts to wrap index, the specified slice size...
        slice_size = args[0].name
        # ...and the size of the dimension.
        dim_size = args[1].name
        # Get the equivalence class ids for both.
        slice_eq = equiv_set._get_or_add_ind(slice_size)
        dim_eq = equiv_set._get_or_add_ind(dim_size)
        # See if a previous wrap_index calls we've analyzed maps from
        # the same pair of equivalence class ids for slice and dim size.
        if (slice_eq, dim_eq) in equiv_set.wrap_map:
            wrap_ind = equiv_set.wrap_map[(slice_eq, dim_eq)]
            require(wrap_ind in equiv_set.ind_to_var)
            vs = equiv_set.ind_to_var[wrap_ind]
            require(vs != [])
            # Return the shape of the variable from the previous wrap_index.
            return ((vs[0],),[])
        else:
            # We haven't seen this combination of slice and dim
            # equivalence class ids so return a WrapIndexMeta so that
            # _analyze_inst can establish the connection to the lhs var.
            return (WrapIndexMeta(slice_eq, dim_eq),[])

    def _analyze_numpy_create_array(self, scope, equiv_set, args, kws):
        shape_var = None
        if len(args) > 0:
            shape_var = args[0]
        elif 'shape' in kws:
            shape_var = kws['shape']
        if shape_var:
            return shape_var, []
        raise NotImplementedError("Must specify a shape for array creation")

    def _analyze_op_call_numpy_empty(self, scope, equiv_set, args, kws):
        return self._analyze_numpy_create_array(scope, equiv_set, args, kws)

    def _analyze_op_call_numba_unsafe_ndarray_empty_inferred(self, scope,
                                                         equiv_set, args, kws):
        return self._analyze_numpy_create_array(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_zeros(self, scope, equiv_set, args, kws):
        return self._analyze_numpy_create_array(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_ones(self, scope, equiv_set, args, kws):
        return self._analyze_numpy_create_array(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_eye(self, scope, equiv_set, args, kws):
        if len(args) > 0:
            N = args[0]
        elif 'N' in kws:
            N = kws['N']
        else:
            raise NotImplementedError(
                "Expect one argument (or 'N') to eye function")
        if 'M' in kws:
            M = kws['M']
        else:
            M = N
        return (N, M), []

    def _analyze_op_call_numpy_identity(self, scope, equiv_set, args, kws):
        assert len(args) > 0
        N = args[0]
        return (N, N), []

    def _analyze_op_call_numpy_diag(self, scope, equiv_set, args, kws):
        # We can only reason about the output shape when the input is 1D or
        # square 2D.
        assert len(args) > 0
        a = args[0]
        assert(isinstance(a, ir.Var))
        atyp = self.typemap[a.name]
        if isinstance(atyp, types.ArrayCompatible):
            if atyp.ndim == 2:
                if 'k' in kws:  # will proceed only when k = 0 or absent
                    k = kws['k']
                    if not equiv_set.is_equiv(k, 0):
                        return None
                (m, n) = equiv_set._get_shape(a)
                if equiv_set.is_equiv(m, n):
                    return (m,), []
            elif atyp.ndim == 1:
                (m,) = equiv_set._get_shape(a)
                return (m, m), []
        return None

    def _analyze_numpy_array_like(self, scope, equiv_set, args, kws):
        assert(len(args) > 0)
        var = args[0]
        typ = self.typemap[var.name]
        if isinstance(typ, types.Integer):
            return (1,), []
        elif (isinstance(typ, types.ArrayCompatible) and
              equiv_set.has_shape(var)):
            return var, []
        return None

    def _analyze_op_call_numpy_ravel(self, scope, equiv_set, args, kws):
        assert(len(args) == 1)
        var = args[0]
        typ = self.typemap[var.name]
        assert isinstance(typ, types.ArrayCompatible)
        # output array is same shape as input if input is 1D
        if typ.ndim == 1 and equiv_set.has_shape(var):
            if typ.layout == 'C':
                # output is the same as input (no copy) for 'C' layout
                # optimize out the call
                return var, [], var
            else:
                return var, []
        # TODO: handle multi-D input arrays (calc array size)
        return None

    def _analyze_op_call_numpy_copy(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_empty_like(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_zeros_like(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_ones_like(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_full_like(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_asfortranarray(self, *args):
        return self._analyze_numpy_array_like(*args)

    def _analyze_op_call_numpy_reshape(self, scope, equiv_set, args, kws):
        n = len(args)
        assert(n > 1)
        if n == 2:
            typ = self.typemap[args[1].name]
            if isinstance(typ, types.BaseTuple):
                return args[1], []

        # Reshape is allowed to take one argument that has the value <0.
        # This means that the size of that dimension should be inferred from
        # the size of the array being reshaped and the other dimensions
        # specified.  Our general approach here is to see if the reshape
        # has any <0 arguments.  If it has more than one then throw a
        # ValueError.  If exactly one <0 argument is found, remember its
        # argument index.
        stmts = []
        neg_one_index = -1
        for arg_index in range(1, len(args)):
            reshape_arg = args[arg_index]
            reshape_arg_def = guard(get_definition, self.func_ir, reshape_arg)
            if isinstance(reshape_arg_def, ir.Const):
                if reshape_arg_def.value < 0:
                    if neg_one_index == -1:
                        neg_one_index = arg_index
                    else:
                        msg = ("The reshape API may only include one negative"
                               " argument. %s" % str(reshape_arg.loc))
                        raise ValueError(msg)

        if neg_one_index >= 0:
            # If exactly one <0 argument to reshape was found, then we are
            # going to insert code to calculate the missing dimension and then
            # replace the negative with the calculated size.  We do this because we
            # can't let array equivalence analysis think that some array has
            # a negative dimension size.
            loc = args[0].loc
            # Create a variable to hold the size of the array being reshaped.
            calc_size_var = ir.Var(scope, mk_unique_var("calc_size_var"), loc)
            self.typemap[calc_size_var.name] = types.intp
            # Assign the size of the array calc_size_var.
            init_calc_var = ir.Assign(ir.Expr.getattr(args[0], "size", loc), calc_size_var, loc)
            stmts.append(init_calc_var)
            # For each other dimension, divide the current size by the specified
            # dimension size.  Once all such dimensions have been done then what is
            # left is the size of the negative dimension.
            for arg_index in range(1, len(args)):
                # Skip the negative dimension.
                if arg_index == neg_one_index:
                    continue
                div_calc_size_var = ir.Var(scope, mk_unique_var("calc_size_var"), loc)
                self.typemap[div_calc_size_var.name] = types.intp
                # Calculate the next size as current size // the current arg's dimension size.
                new_binop = ir.Expr.binop(operator.floordiv, calc_size_var, args[arg_index], loc)
                div_calc = ir.Assign(new_binop, div_calc_size_var, loc)
                self.calltypes[new_binop] = signature(types.intp, types.intp, types.intp)
                stmts.append(div_calc)
                calc_size_var = div_calc_size_var
            # Put the calculated value back into the reshape arguments, replacing the negative.
            args[neg_one_index] = calc_size_var

        return tuple(args[1:]), stmts

    def _analyze_op_call_numpy_transpose(self, scope, equiv_set, args, kws):
        in_arr = args[0]
        typ = self.typemap[in_arr.name]
        assert isinstance(typ, types.ArrayCompatible), \
            "Invalid np.transpose argument"
        shape = equiv_set._get_shape(in_arr)
        if len(args) == 1:
            return tuple(reversed(shape)), []
        axes = [guard(find_const, self.func_ir, a) for a in args[1:]]
        if isinstance(axes[0], tuple):
            axes = list(axes[0])
        if None in axes:
            return None
        ret = [shape[i] for i in axes]
        return tuple(ret), []

    def _analyze_op_call_numpy_random_rand(self, scope, equiv_set, args, kws):
        if len(args) > 0:
            return tuple(args), []
        return None

    def _analyze_op_call_numpy_random_randn(self, *args):
        return self._analyze_op_call_numpy_random_rand(*args)

    def _analyze_op_numpy_random_with_size(self, pos, scope, equiv_set, args, kws):
        if 'size' in kws:
            return kws['size'], []
        if len(args) > pos:
            return args[pos], []
        return None

    def _analyze_op_call_numpy_random_ranf(self, *args):
        return self._analyze_op_numpy_random_with_size(0, *args)

    def _analyze_op_call_numpy_random_random_sample(self, *args):
        return self._analyze_op_numpy_random_with_size(0, *args)

    def _analyze_op_call_numpy_random_sample(self, *args):
        return self._analyze_op_numpy_random_with_size(0, *args)

    def _analyze_op_call_numpy_random_random(self, *args):
        return self._analyze_op_numpy_random_with_size(0, *args)

    def _analyze_op_call_numpy_random_standard_normal(self, *args):
        return self._analyze_op_numpy_random_with_size(0, *args)

    def _analyze_op_call_numpy_random_chisquare(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_weibull(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_power(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_geometric(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_exponential(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_poisson(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_rayleigh(self, *args):
        return self._analyze_op_numpy_random_with_size(1, *args)

    def _analyze_op_call_numpy_random_normal(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_uniform(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_beta(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_binomial(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_f(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_gamma(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_lognormal(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_laplace(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_randint(self, *args):
        return self._analyze_op_numpy_random_with_size(2, *args)

    def _analyze_op_call_numpy_random_triangular(self, *args):
        return self._analyze_op_numpy_random_with_size(3, *args)

    def _analyze_op_call_numpy_concatenate(self, scope, equiv_set, args, kws):
        assert(len(args) > 0)
        loc = args[0].loc
        seq, op = find_build_sequence(self.func_ir, args[0])
        n = len(seq)
        require(n > 0)
        axis = 0
        if 'axis' in kws:
            if isinstance(kws['axis'], int):  # internal use only
                axis = kws['axis']
            else:
                axis = find_const(self.func_ir, kws['axis'])
        elif len(args) > 1:
            axis = find_const(self.func_ir, args[1])
        require(isinstance(axis, int))
        require(op == 'build_tuple')
        shapes = [equiv_set._get_shape(x) for x in seq]
        if axis < 0:
            axis = len(shapes[0]) + axis
        require(0 <= axis < len(shapes[0]))
        asserts = []
        new_shape = []
        if n == 1:  # from one array N-dimension to (N-1)-dimension
            shape = shapes[0]
            # first size is the count, pop it out of shapes
            n = equiv_set.get_equiv_const(shapes[0])
            shape.pop(0)
            for i in range(len(shape)):
                if i == axis:
                    m = equiv_set.get_equiv_const(shape[i])
                    size = m * n if (m and n) else None
                else:
                    size = self._sum_size(equiv_set, shapes[0])
            new_shape.append(size)
        else:  # from n arrays N-dimension to N-dimension
            for i in range(len(shapes[0])):
                if i == axis:
                    size = self._sum_size(
                        equiv_set, [shape[i] for shape in shapes])
                else:
                    sizes = [shape[i] for shape in shapes]
                    asserts.append(
                        self._call_assert_equiv(scope, loc, equiv_set, sizes))
                    size = sizes[0]
                new_shape.append(size)
        return tuple(new_shape), sum(asserts, [])

    def _analyze_op_call_numpy_stack(self, scope, equiv_set, args, kws):
        assert(len(args) > 0)
        loc = args[0].loc
        seq, op = find_build_sequence(self.func_ir, args[0])
        n = len(seq)
        require(n > 0)
        axis = 0
        if 'axis' in kws:
            if isinstance(kws['axis'], int):  # internal use only
                axis = kws['axis']
            else:
                axis = find_const(self.func_ir, kws['axis'])
        elif len(args) > 1:
            axis = find_const(self.func_ir, args[1])
        require(isinstance(axis, int))
        # only build_tuple can give reliable count
        require(op == 'build_tuple')
        shapes = [equiv_set._get_shape(x) for x in seq]
        asserts = self._call_assert_equiv(scope, loc, equiv_set, seq)
        shape = shapes[0]
        if axis < 0:
            axis = len(shape) + axis + 1
        require(0 <= axis <= len(shape))
        new_shape = list(shape[0:axis]) + [n] + list(shape[axis:])
        return tuple(new_shape), asserts

    def _analyze_op_call_numpy_vstack(self, scope, equiv_set, args, kws):
        assert(len(args) == 1)
        seq, op = find_build_sequence(self.func_ir, args[0])
        n = len(seq)
        require(n > 0)
        typ = self.typemap[seq[0].name]
        require(isinstance(typ, types.ArrayCompatible))
        if typ.ndim < 2:
            return self._analyze_op_call_numpy_stack(scope, equiv_set, args, kws)
        else:
            kws['axis'] = 0
            return self._analyze_op_call_numpy_concatenate(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_hstack(self, scope, equiv_set, args, kws):
        assert(len(args) == 1)
        seq, op = find_build_sequence(self.func_ir, args[0])
        n = len(seq)
        require(n > 0)
        typ = self.typemap[seq[0].name]
        require(isinstance(typ, types.ArrayCompatible))
        if typ.ndim < 2:
            kws['axis'] = 0
        else:
            kws['axis'] = 1
        return self._analyze_op_call_numpy_concatenate(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_dstack(self, scope, equiv_set, args, kws):
        assert(len(args) == 1)
        seq, op = find_build_sequence(self.func_ir, args[0])
        n = len(seq)
        require(n > 0)
        typ = self.typemap[seq[0].name]
        require(isinstance(typ, types.ArrayCompatible))
        if typ.ndim == 1:
            kws['axis'] = 1
            result = self._analyze_op_call_numpy_stack(
                scope, equiv_set, args, kws)
            require(result)
            (shape, pre) = result
            shape = tuple([1] + list(shape))
            return shape, pre
        elif typ.ndim == 2:
            kws['axis'] = 2
            return self._analyze_op_call_numpy_stack(scope, equiv_set, args, kws)
        else:
            kws['axis'] = 2
            return self._analyze_op_call_numpy_concatenate(scope, equiv_set, args, kws)

    def _analyze_op_call_numpy_cumsum(self, scope, equiv_set, args, kws):
        # TODO
        return None

    def _analyze_op_call_numpy_cumprod(self, scope, equiv_set, args, kws):
        # TODO
        return None

    def _analyze_op_call_numpy_linspace(self, scope, equiv_set, args, kws):
        n = len(args)
        num = 50
        if n > 2:
            num = args[2]
        elif 'num' in kws:
            num = kws['num']
        return (num,), []

    def _analyze_op_call_numpy_dot(self, scope, equiv_set, args, kws):
        n = len(args)
        assert(n >= 2)
        loc = args[0].loc
        require(all([self._isarray(x.name) for x in args]))
        typs = [self.typemap[x.name] for x in args]
        dims = [ty.ndim for ty in typs]
        require(all(x > 0 for x in dims))
        if dims[0] == 1 and dims[1] == 1:
            return None
        shapes = [equiv_set._get_shape(x) for x in args]
        if dims[0] == 1:
            asserts = self._call_assert_equiv(
                scope, loc, equiv_set, [shapes[0][0], shapes[1][-2]])
            return tuple(shapes[1][0:-2] + shapes[1][-1:]), asserts
        if dims[1] == 1:
            asserts = self._call_assert_equiv(
                scope, loc, equiv_set, [shapes[0][-1], shapes[1][0]])
            return tuple(shapes[0][0:-1]), asserts
        if dims[0] == 2 and dims[1] == 2:
            asserts = self._call_assert_equiv(
                scope, loc, equiv_set, [shapes[0][1], shapes[1][0]])
            return (shapes[0][0], shapes[1][1]), asserts
        if dims[0] > 2:  # TODO: handle higher dimension cases
            pass
        return None

    def _analyze_stencil(self, scope, equiv_set, stencil_func, loc, args, kws):
        # stencil requires that all relatively indexed array arguments are
        # of same size
        std_idx_arrs = stencil_func.options.get('standard_indexing', ())
        kernel_arg_names = stencil_func.kernel_ir.arg_names
        if isinstance(std_idx_arrs, str):
            std_idx_arrs = (std_idx_arrs,)
        rel_idx_arrs = []
        assert(len(args) > 0 and len(args) == len(kernel_arg_names))
        for arg, var in zip(kernel_arg_names, args):
            typ = self.typemap[var.name]
            if (isinstance(typ, types.ArrayCompatible) and
                not(arg in std_idx_arrs)):
                rel_idx_arrs.append(var)
        n = len(rel_idx_arrs)
        require(n > 0)
        asserts = self._call_assert_equiv(scope, loc, equiv_set, rel_idx_arrs)
        shape = equiv_set.get_shape(rel_idx_arrs[0])
        return shape, asserts

    def _analyze_op_call_numpy_linalg_inv(self, scope, equiv_set, args, kws):
        require(len(args) >= 1)
        return equiv_set._get_shape(args[0]), []

    def _analyze_broadcast(self, scope, equiv_set, loc, args):
        """Infer shape equivalence of arguments based on Numpy broadcast rules
        and return shape of output
        https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html
        """
        arrs = list(filter(lambda a: self._isarray(a.name), args))
        require(len(arrs) > 0)
        names = [x.name for x in arrs]
        dims = [self.typemap[x.name].ndim for x in arrs]
        max_dim = max(dims)
        require(max_dim > 0)
        try:
            shapes = [equiv_set.get_shape(x) for x in arrs]
        except GuardException:
            return arrs[0], self._call_assert_equiv(scope, loc, equiv_set, arrs)
        return self._broadcast_assert_shapes(scope, equiv_set, loc, shapes, names)

    def _broadcast_assert_shapes(self, scope, equiv_set, loc, shapes, names):
        """Produce assert_equiv for sizes in each dimension, taking into account
        of dimension coercion and constant size of 1.
        """
        asserts = []
        new_shape = []
        max_dim = max([len(shape) for shape in shapes])
        const_size_one = None
        for i in range(max_dim):
            sizes = []
            size_names = []
            for name, shape in zip(names, shapes):
                if i < len(shape):
                    size = shape[len(shape) - 1 - i]
                    const_size = equiv_set.get_equiv_const(size)
                    if const_size == 1:
                        const_size_one = size
                    else:
                        sizes.append(size)  # non-1 size to front
                        size_names.append(name)
            if sizes == []:
                assert(const_size_one != None)
                sizes.append(const_size_one)
                size_names.append("1")
            asserts.append(self._call_assert_equiv(scope, loc, equiv_set,
                                                   sizes, names=size_names))
            new_shape.append(sizes[0])
        return tuple(reversed(new_shape)), sum(asserts, [])

    def _call_assert_equiv(self, scope, loc, equiv_set, args, names=None):
        insts = self._make_assert_equiv(
            scope, loc, equiv_set, args, names=names)
        if len(args) > 1:
            equiv_set.insert_equiv(*args)
        return insts

    def _make_assert_equiv(self, scope, loc, equiv_set, _args, names=None):
        # filter out those that are already equivalent
        if config.DEBUG_ARRAY_OPT >= 2:
            print("make_assert_equiv:", _args, names)
        if names == None:
            names = [x.name for x in _args]
        args = []
        arg_names = []
        for name, x in zip(names, _args):
            if config.DEBUG_ARRAY_OPT >= 2:
                print("name, x:", name, x)
            seen = False
            for y in args:
                if config.DEBUG_ARRAY_OPT >= 2:
                    print("is equiv to?", y, equiv_set.is_equiv(x, y))
                if equiv_set.is_equiv(x, y):
                    seen = True
                    break
            if not seen:
                args.append(x)
                arg_names.append(name)

        # no assertion necessary if there are less than two
        if len(args) < 2:
            if config.DEBUG_ARRAY_OPT >= 2:
                print("Will not insert assert_equiv as args are known to be equivalent.")
            return []

        msg = "Sizes of {} do not match on {}".format(', '.join(arg_names), loc)
        msg_val = ir.Const(msg, loc)
        msg_typ = types.StringLiteral(msg)
        msg_var = ir.Var(scope, mk_unique_var("msg"), loc)
        self.typemap[msg_var.name] = msg_typ
        argtyps = tuple([msg_typ] + [self.typemap[x.name] for x in args])

        # assert_equiv takes vararg, which requires a tuple as argument type
        tup_typ = types.BaseTuple.from_types(argtyps)

        # prepare function variable whose type may vary since it takes vararg
        assert_var = ir.Var(scope, mk_unique_var("assert"), loc)
        assert_def = ir.Global('assert_equiv', assert_equiv, loc=loc)
        fnty = get_global_func_typ(assert_equiv)
        sig = self.context.resolve_function_type(fnty, (tup_typ,), {})
        self._define(equiv_set, assert_var, fnty, assert_def)

        # The return value from assert_equiv is always of none type.
        var = ir.Var(scope, mk_unique_var("ret"), loc)
        value = ir.Expr.call(assert_var, [msg_var] + args, {}, loc=loc)
        self._define(equiv_set, var, types.none, value)
        self.calltypes[value] = sig

        return [ir.Assign(value=msg_val, target=msg_var, loc=loc),
                ir.Assign(value=assert_def, target=assert_var, loc=loc),
                ir.Assign(value=value, target=var, loc=loc),
                ]

    def _gen_shape_call(self, equiv_set, var, ndims, shape):
        out = []
        # attr call: A_sh_attr = getattr(A, shape)
        if isinstance(shape, ir.Var):
            shape = equiv_set.get_shape(shape)
        # already a tuple variable that contains size
        if isinstance(shape, ir.Var):
            attr_var = shape
            shape_attr_call = None
            shape = None
        else:
            shape_attr_call = ir.Expr.getattr(var, "shape", var.loc)
            attr_var = ir.Var(var.scope, mk_unique_var(
                              "{}_shape".format(var.name)), var.loc)
            shape_attr_typ = types.containers.UniTuple(types.intp, ndims)
        size_vars = []
        use_attr_var = False
        # trim shape tuple if it is more than ndim
        if shape:
            nshapes = len(shape)
            if ndims < nshapes:
                shape = shape[(nshapes-ndims):]
        for i in range(ndims):
            skip = False
            if shape and shape[i]:
                if isinstance(shape[i], ir.Var):
                    typ = self.typemap[shape[i].name]
                    if (isinstance(typ, types.Number) or
                        isinstance(typ, types.SliceType)):
                        size_var = shape[i]
                        skip = True
                else:
                    if isinstance(shape[i], int):
                        size_val = ir.Const(shape[i], var.loc)
                    else:
                        size_val = shape[i]
                    assert(isinstance(size_val, ir.Const))
                    size_var = ir.Var(var.scope, mk_unique_var(
                        "{}_size{}".format(var.name, i)), var.loc)
                    out.append(ir.Assign(size_val, size_var, var.loc))
                    self._define(equiv_set, size_var, types.intp, size_val)
                    skip = True
            if not skip:
                # get size: Asize0 = A_sh_attr[0]
                size_var = ir.Var(var.scope, mk_unique_var(
                                  "{}_size{}".format(var.name, i)), var.loc)
                getitem = ir.Expr.static_getitem(attr_var, i, None, var.loc)
                use_attr_var = True
                self.calltypes[getitem] = None
                out.append(ir.Assign(getitem, size_var, var.loc))
                self._define(equiv_set, size_var, types.intp, getitem)
            size_vars.append(size_var)
        if use_attr_var and shape_attr_call:
            # only insert shape call if there is any getitem call
            out.insert(0, ir.Assign(shape_attr_call, attr_var, var.loc))
            self._define(equiv_set, attr_var, shape_attr_typ, shape_attr_call)
        return tuple(size_vars), out

    def _isarray(self, varname):
        typ = self.typemap[varname]
        return (isinstance(typ, types.npytypes.Array) and typ.ndim > 0)

    def _sum_size(self, equiv_set, sizes):
        """Return the sum of the given list of sizes if they are all equivalent
        to some constant, or None otherwise.
        """
        s = 0
        for size in sizes:
            n = equiv_set.get_equiv_const(size)
            if n == None:
                return None
            else:
                s += n
        return s

UNARY_MAP_OP = list(
    npydecl.NumpyRulesUnaryArrayOperator._op_map.keys()) + [operator.pos]
BINARY_MAP_OP = npydecl.NumpyRulesArrayOperator._op_map.keys()
INPLACE_BINARY_MAP_OP = npydecl.NumpyRulesInplaceArrayOperator._op_map.keys()
UFUNC_MAP_OP = [f.__name__ for f in npydecl.supported_ufuncs]