def double_scaling_reduction(self, **defaults_config):
from . import doubly_scaled_as_singly_scaled
if not isinstance(self.scaled, ScalarMult):
raise ValueError("'double_scaling_reduction' is only applicable "
"for a doubly nested ScalarMult")
# Reduce doubly-nested ScalarMult
_x = self.scaled.scaled
# _V = VecSpaces.known_vec_space(_x)
# the following is a little klunky, but trying to avoid the
# use of a default field=Real if we're actually dealing with
# complex scalars somewhere in the vector
from proveit import free_vars
if any([InSet(elem, Complex).proven() for elem in free_vars(self)]):
_V = VecSpaces.known_vec_space(self, field=Complex)
else:
_V = VecSpaces.known_vec_space(self)
_K = VecSpaces.known_field(_V)
_alpha = self.scalar
_beta = self.scaled.scalar
return doubly_scaled_as_singly_scaled.instantiate({
K: _K,
V: _V,
x: _x,
alpha: _alpha,
beta: _beta
})
def vec_sum_elimination(self, field=None, **defaults_config):
'''
For a VecSum in which the summand does not depend on the
summation index, return an equality between this VecSum and
the equivalent expression in which the VecSum is eliminated.
For example, suppose self = VecSum(i, v, Interval(2, 4)).
Then self.vec_sum_elimination() would return
|- self = 3*v
where the 3*v is actually ScalarMult(3, v).
The method works only for a VecSum over a single summation
index, and simply returns self = self if the VecSum elimination
is not possible due to the summand being dependent on the
index of summation.
'''
expr = self
summation_index = expr.index
eq = TransRelUpdater(expr)
if summation_index not in free_vars(expr.summand):
vec_space_membership = expr.summand.deduce_in_vec_space(
field=field,
assumptions = defaults.assumptions + expr.conditions.entries)
_V_sub = vec_space_membership.domain
_K_sub = VecSpaces.known_field(_V_sub)
_j_sub = expr.condition.domain.lower_bound
_k_sub = expr.condition.domain.upper_bound
_v_sub = expr.summand
from proveit.linear_algebra.addition import vec_sum_of_constant_vec
eq.update(vec_sum_of_constant_vec.instantiate(
{V: _V_sub, K: _K_sub, j: _j_sub, k: _k_sub, v: _v_sub}))
else:
print("VecSum cannot be eliminated. The summand {0} appears "
"to depend on the index of summation {1}".
format(expr.summand, summation_index))
return eq.relation
def shallow_simplification(self,
*,
must_evaluate=False,
**defaults_config):
'''
Returns a proven simplification equation for this Sum
expression assuming the operands have been simplified.
For the trivial case of summing over only one item (currently
implemented just for a Interval where the endpoints are equal),
derive and return this summation expression equated with the
simplified form of the single term.
Assumptions may be necessary to deduce necessary conditions
for the simplification.
NEEDS UPDATING
'''
from proveit.logic import TRUE, SimplificationError
from . import sum_single, trivial_sum
if (isinstance(self.domain, Interval)
and self.domain.lower_bound == self.domain.upper_bound):
if hasattr(self, 'index'):
return sum_single.instantiate({
Function(f, self.index): self.summand,
a: self.domain.lower_bound
})
if (isinstance(self.domain, Interval)
and self.instance_param not in free_vars(self.summand)
and self.non_domain_condition() == TRUE):
# Trivial sum: summand independent of parameter.
_a = self.domain.lower_bound
_b = self.domain.upper_bound
_x = self.summand
return trivial_sum.instantiate({a: _a, b: _b, x: _x})
raise SimplificationError(
"Sum simplification only implemented for a summation over an "
"integer Interval of one instance variable where the upper "
"and lower bounds are the same.")
def factor(self,
theFactor,
pull="left",
groupFactor=False,
groupRemainder=None,
assumptions=frozenset()):
'''
Pull out a common factor from a summation, pulling it either to the "left" or "right".
If groupFactor is True and theFactor is a product, it will be grouped together as a
sub-product. groupRemainder is not relevant kept for compatibility with other factor
methods. Returns the equality that equates self to this new version.
Give any assumptions necessary to prove that the operands are in Complexes so that
the associative and commutation theorems are applicable.
'''
from proveit.number.multiplication.theorems import distributeThroughSummationRev
from proveit.number import Mult
if not free_vars(theFactor).isdisjoint(self.indices):
raise Exception(
'Cannot factor anything involving summation indices out of a summation'
)
# We may need to factor the summand within the summation
summand_assumptions = assumptions | {
InSet(index, self.domain)
for index in self.indices
}
summandFactorEq = self.summand.factor(theFactor,
pull,
groupFactor=False,
groupRemainder=True,
assumptions=summand_assumptions)
summandInstanceEquivalence = summandFactorEq.generalize(
self.indices, domain=self.domain).checked(assumptions)
eq = Equation(
self.instanceSubstitution(summandInstanceEquivalence).checked(
assumptions))
factorOperands = theFactor.operands if isinstance(theFactor,
Mult) else theFactor
xDummy, zDummy = self.safeDummyVars(2)
# Now do the actual factoring by reversing distribution
if pull == 'left':
Pop, Pop_sub = Operation(
P, self.indices), summandFactorEq.rhs.operands[-1]
xSub = factorOperands
zSub = []
elif pull == 'right':
Pop, Pop_sub = Operation(
P, self.indices), summandFactorEq.rhs.operands[0]
xSub = []
zSub = factorOperands
# We need to deduce that theFactor is in Complexes and that all instances of Pop_sup are in Complexes.
deduceInComplexes(factorOperands, assumptions=assumptions)
deduceInComplexes(Pop_sub,
assumptions=assumptions
| {InSet(idx, self.domain)
for idx in self.indices}).generalize(
self.indices,
domain=self.domain).checked(assumptions)
# Now we specialize distributThroughSummationRev
spec1 = distributeThroughSummationRev.specialize({
Pop:
Pop_sub,
S:
self.domain,
yEtc:
self.indices,
xEtc:
Etcetera(MultiVariable(xDummy)),
zEtc:
Etcetera(MultiVariable(zDummy))
}).checked()
eq.update(spec1.deriveConclusion().specialize({
Etcetera(MultiVariable(xDummy)):
xSub,
Etcetera(MultiVariable(zDummy)):
zSub
}))
if groupFactor and len(factorOperands) > 1:
eq.update(
eq.eqExpr.rhs.group(endIdx=len(factorOperands),
assumptions=assumptions))
return eq.eqExpr #.checked(assumptions)
def tensor_prod_factoring(self, idx=None, idx_beg=None, idx_end=None,
field=None, **defaults_config):
'''
For a VecSum with a TensorProd summand or ScalarMult summand
with a scaled attribute being a TensorProd, factor out from
the VecSum the TensorProd vectors other than the ones indicated
by the (0-based) idx, or idx_beg and idx_end pair and return
an equality between the original VecSum and the new TensorProd.
For example, we could take the VecSum defined by
vec_sum = VecSum(TensorProd(x, f(i), y, z))
and call vec_sum.tensor_prod_factoring(idx_beg=1, idx_end=2)
to obtain:
|- VecSum(TensorProd(x, f(i), y, z)) =
TensorProd(x, VecSum(TensorProd(f(i), y)), z)
This method should work even if the summand is a nested
ScalarMult. Note that any vectors inside the TensorProd that
depend on the index of summation cannot be pulled out of the
VecSum and thus will cause the method to fail if not chosen
to remain inside the VecSum. If all idx args are 'None',
method will factor out all possible vector factors, including
the case where all factors could be removed and the VecSum
eliminated entirely.
Note that this method only works when self has a single
index of summation.
'''
expr = self
the_summand = self.summand
eq = TransRelUpdater(expr)
# Check that
# (1) the_summand is a TensorProd
# or (2) the_summand is a ScalarMult;
# otherwise, this method does not apply
from proveit.linear_algebra import ScalarMult, TensorProd
if isinstance(the_summand, ScalarMult):
# try shallow simplification first to remove nested
# ScalarMults and multiplicative identities
expr = eq.update(expr.inner_expr().summand.shallow_simplification())
the_summand = expr.summand
if isinstance(the_summand, TensorProd):
tensor_prod_expr = the_summand
tensor_prod_summand = True
tensor_prod_factors_list = list(
the_summand.operands.entries)
elif (isinstance(the_summand, ScalarMult)
and isinstance(the_summand.scaled, TensorProd)):
tensor_prod_expr = the_summand.scaled
tensor_prod_summand = False
tensor_prod_factors_list = list(
the_summand.scaled.operands.entries)
else:
raise ValueError(
"tensor_prod_factoring() requires the VecSum summand "
"to be a TensorProd or a ScalarMult (with its 'scaled' "
"attribute a TensorProd); instead the "
"summand is {}".format(self.instance_expr))
if idx is None and idx_beg is None and idx_end is None:
# prepare to take out all possible factors, including
# the complete elimination of the VecSum if possible
if expr.index not in free_vars(expr.summand):
# summand does not depend on index of summation
# so we can eliminate the VecSum entirely
return expr.vec_sum_elimination(field=field)
if expr.index in free_vars(tensor_prod_expr):
# identify the extractable vs. non-extractable
# TensorProd factors (and there must be at least
# one such non-extractable factor)
idx_beg = -1
idx_end = -1
for i in range(len(expr.summand.operands.entries)):
if expr.index in free_vars(tensor_prod_expr.operands[i]):
if idx_beg == -1:
idx_beg = i
idx_end = idx_beg
else:
idx_end = i
else:
# The alternative is that the summand is
# a ScalarMult with the scalar (but not the scaled)
# being dependent on the index of summation. It's not
# obvious what's best to do in this case, but we set
# things up to factor out all but the last of the
# TensorProd factors (so we'll factor out at least
# 1 factor)
idx_beg = len(tensor_prod_expr.operands.entries) - 1
idx_end = idx_beg
# Check that the provided idxs are within bounds
# (it should refer to an actual TensorProd operand)
num_vec_factors = len(tensor_prod_factors_list)
if idx is not None and idx >= num_vec_factors:
raise ValueError(
"idx value {0} provided for tensor_prod_factoring() "
"method is out-of-bounds; the TensorProd summand has "
"{1} factors: {2}, and thus possibly indices 0-{3}".
format(idx, len(tensor_prod_factors_list),
tensor_prod_factors_list,
len(tensor_prod_factors_list)-1))
if idx_beg is not None and idx_end is not None:
if (idx_end < idx_beg or idx_beg >= num_vec_factors or
idx_end >= num_vec_factors):
raise ValueError(
"idx_beg value {0} or idx_end value {1} (or both) "
"provided for tensor_prod_factoring() "
"method is/are out-of-bounds; the TensorProd summand "
"has {2} factors: {3}, and thus possibly indices 0-{3}".
format(idx_beg, idx_end, num_vec_factors,
tensor_prod_factors_list,num_vec_factors-1))
if idx is not None:
# take single idx as the default
idx_beg = idx
idx_end = idx
# Check that the TensorProd factors to be factored out do not
# rely on the VecSum index of summation
summation_index = expr.index
for i in range(num_vec_factors):
if i < idx_beg or i > idx_end:
the_factor = tensor_prod_factors_list[i]
if summation_index in free_vars(the_factor):
raise ValueError(
"TensorProd factor {0} cannot be factored "
"out of the given VecSum summation because "
"it is a function of the summation index {1}.".
format(the_factor, summation_index))
# Everything checks out as best we can tell, so prepare to
# import and instantiate the appropriate theorem,
# depending on whether:
# (1) the_summand is a TensorProd, or
# (2) the_summand is a ScalarMult (with a TensorProd 'scaled')
if tensor_prod_summand:
from proveit.linear_algebra.tensors import (
tensor_prod_distribution_over_summation)
else:
from proveit.linear_algebra.tensors import (
tensor_prod_distribution_over_summation_with_scalar_mult)
if idx_beg != idx_end:
# need to associate the elements and change idx value
# but process is slightly different in the two cases
if tensor_prod_summand:
expr = eq.update(expr.inner_expr().summand.association(
idx_beg, idx_end-idx_beg+1))
tensor_prod_expr = expr.summand
else:
expr = eq.update(expr.inner_expr().summand.scaled.association(
idx_beg, idx_end-idx_beg+1))
tensor_prod_expr = expr.summand.scaled
idx = idx_beg
from proveit import K, f, Q, i, j, k, V, a, b, c, s
# actually, maybe it doesn't matter and we can deduce the
# vector space regardless: (Adding this temp 12/26/21)
vec_space_membership = expr.summand.deduce_in_vec_space(
field=field,
assumptions = defaults.assumptions + expr.conditions.entries)
_V_sub = vec_space_membership.domain
# Substitutions regardless of Case
_K_sub = VecSpaces.known_field(_V_sub)
_b_sub = expr.indices
_j_sub = _b_sub.num_elements()
_Q_sub = Lambda(expr.indices, expr.condition)
# Case-specific substitutions, using updated tensor_prod_expr:
_a_sub = tensor_prod_expr.operands[:idx]
_c_sub = tensor_prod_expr.operands[idx+1:]
_f_sub = Lambda(expr.indices, tensor_prod_expr.operands[idx])
if not tensor_prod_summand:
_s_sub = Lambda(expr.indices, expr.summand.scalar)
# Case-dependent substitutions:
_i_sub = _a_sub.num_elements()
_k_sub = _c_sub.num_elements()
if tensor_prod_summand:
impl = tensor_prod_distribution_over_summation.instantiate(
{K:_K_sub, f:_f_sub, Q:_Q_sub, i:_i_sub, j:_j_sub,
k:_k_sub, V:_V_sub, a:_a_sub, b:_b_sub, c:_c_sub},
preserve_expr=expr)
else:
impl = (tensor_prod_distribution_over_summation_with_scalar_mult.
instantiate(
{K:_K_sub, f:_f_sub, Q:_Q_sub, i:_i_sub, j:_j_sub,
k:_k_sub, V:_V_sub, a:_a_sub, b:_b_sub, c:_c_sub,
s: _s_sub}, preserve_expr=expr))
expr = eq.update(impl.derive_consequent(
assumptions = defaults.assumptions + expr.conditions.entries).
derive_reversed())
return eq.relation
def factors_extraction(self, field=None, **defaults_config):
'''
Derive an equality between this VecSum and the result
when all possible leading scalar factors have been extracted
and moved to the front of the VecSum (for example, in the
case where the summand of the VecSum is a ScalarMult) and
all possible tensor product factors have been moved outside
the VecSum (in front if possible, or afterward if necessary).
For example, we could take the VecSum
vec_sum = VecSum(ScalarMult(a, TensorProd(x, f(i), y))),
where the index of summation is i, and call
vec_sum.factor_extraction() to obtain:
|- vec_sum =
ScalarMult(a, TensorProd(x, VecSum(f(i)), y))
Note that any factors inside the summand that depend on the
index of summation cannot be pulled out from inside the VecSum,
and thus pose limitations on the result.
Note that this method only works when self has a single
index of summation, and only when self has a summand that is
a ScalarMult or TensorProd.
Later versions of this method should provide mechanisms to
specify factors to extract from, and/or leave behind in, the
VecSum.
'''
expr = self
summation_index = expr.index
assumptions = defaults.assumptions + expr.conditions.entries
assumptions_with_conditions = (
defaults.assumptions + expr.conditions.entries)
# for convenience in updating our equation:
# this begins with eq.relation as expr = expr
eq = TransRelUpdater(expr)
# If the summand is a ScalarMult, perform a
# shallow_simplification(), which will remove nested
# ScalarMults and multiplicative identities. This is
# intended to simplify without changing too much the
# intent of the user. This might even transform the
# ScalarMult object into something else.
from proveit.linear_algebra import ScalarMult, TensorProd
if isinstance(expr.summand, ScalarMult):
expr = eq.update(
expr.inner_expr().summand.shallow_simplification())
if isinstance(expr.summand, ScalarMult):
# had to re-check, b/c the shallow_simplification might
# have transformed the ScalarMult into the scaled object
tensor_prod_summand = False # not clearly useful; review please
the_scalar = expr.summand.scalar
elif isinstance(expr.summand, TensorProd):
tensor_prod_summand = True # not clearly useful; review please
if isinstance(expr.summand, ScalarMult):
if summation_index not in free_vars(expr.summand.scalar):
# it doesn't matter what the scalar is; the whole thing
# can be pulled out in front of the VecSum
from proveit.linear_algebra.scalar_multiplication import (
distribution_over_vec_sum)
summand_in_vec_space = expr.summand.deduce_in_vec_space(
field=field, assumptions=assumptions_with_conditions)
_V_sub = summand_in_vec_space.domain
_K_sub = VecSpaces.known_field(_V_sub)
_b_sub = expr.indices
_j_sub = _b_sub.num_elements()
_f_sub = Lambda(expr.indices, expr.summand.scaled)
_Q_sub = Lambda(expr.indices, expr.condition)
_k_sub = expr.summand.scalar
imp = distribution_over_vec_sum.instantiate(
{V: _V_sub, K: _K_sub, b: _b_sub, j: _j_sub,
f: _f_sub, Q: _Q_sub, k: _k_sub},
assumptions=assumptions_with_conditions)
expr = eq.update(imp.derive_consequent(
assumptions=assumptions_with_conditions).derive_reversed())
else:
# The scalar portion is dependent on summation index.
# If the scalar itself is a Mult of things, go through
# and pull to the front of the Mult all individual
# factors that are not dependent on the summation index.
if isinstance(expr.summand.scalar, Mult):
# Repeatedly pull index-independent factors #
# to the front of the Mult factors #
# prepare to count the extractable and
# unextractable factors
_num_factored = 0
_num_unfactored = len(expr.summand.scalar.operands.entries)
# go through factors from back to front
for the_factor in reversed(
expr.summand.scalar.operands.entries):
if summation_index not in free_vars(the_factor):
expr = eq.update(
expr.inner_expr().summand.scalar.factorization(
the_factor,
assumptions=assumptions_with_conditions,
preserve_all=True))
_num_factored += 1
_num_unfactored -= 1
# group the factorable factors
if _num_factored > 0:
expr = eq.update(
expr.inner_expr().summand.scalar.association(
0, _num_factored,
assumptions=assumptions_with_conditions,
preserve_all=True))
# group the unfactorable factors
if _num_unfactored > 1:
expr = eq.update(
expr.inner_expr().summand.scalar.association(
1, _num_unfactored,
assumptions=assumptions_with_conditions,
preserve_all=True))
# finally, extract any factorable scalar factors
if _num_factored > 0:
from proveit.linear_algebra.scalar_multiplication import (
distribution_over_vec_sum_with_scalar_mult)
# Mult._simplification_directives_.ungroup = False
# _V_sub = VecSpaces.known_vec_space(expr, field=field)
summand_in_vec_space = (
expr.summand.deduce_in_vec_space(
field = field,
assumptions =
assumptions_with_conditions))
_V_sub = summand_in_vec_space.domain
_K_sub = VecSpaces.known_field(_V_sub)
_b_sub = expr.indices
_j_sub = _b_sub.num_elements()
_f_sub = Lambda(expr.indices, expr.summand.scaled)
_Q_sub = Lambda(expr.indices, expr.condition)
_c_sub = Lambda(expr.indices,
expr.summand.scalar.operands[1])
_k_sub = expr.summand.scalar.operands[0]
# when instantiating, we set preserve_expr=expr;
# otherwise auto_simplification disassociates inside
# the Mult.
impl = distribution_over_vec_sum_with_scalar_mult.instantiate(
{V:_V_sub, K:_K_sub, b: _b_sub, j: _j_sub,
f: _f_sub, Q: _Q_sub, c:_c_sub, k: _k_sub},
preserve_expr=expr,
assumptions=assumptions_with_conditions)
expr = eq.update(impl.derive_consequent(
assumptions=assumptions_with_conditions).
derive_reversed())
else:
# The scalar component is dependent on summation
# index but is not a Mult.
# Revert everything and return self = self.
print("Found summation index {0} in the scalar {1} "
"and the scalar is not a Mult object.".
format(summation_index, expr.summand.scalar))
eq = TransRelUpdater(self)
# ============================================================ #
# VECTOR FACTORS #
# ============================================================ #
# After the scalar factors (if any) have been dealt with,
# proceed with the vector factors in any remaining TensorProd
# in the summand.
# Notice that we are not guaranteed at this point that we even
# have a TensorProd to factor, and if we do have a TensorProd
# we have not identified the non-index-dependent factors to
# extract.
# After processing above for scalar factors, we might now have
# (1) expr = VecSum (we didn't find scalar factors to extract),
# inside of which we might have a ScalarMult or a TensorProd;
# or (2) expr = ScalarMult (we found some scalar factors to
# extract), with a VecSum as the scaled component.
if isinstance(expr, VecSum):
expr = eq.update(expr.tensor_prod_factoring())
elif isinstance(expr, ScalarMult) and isinstance(expr.scaled, VecSum):
expr = eq.update(expr.inner_expr().scaled.tensor_prod_factoring())
return eq.relation
def factorization(self,
the_factor,
pull="left",
group_factor=True,
group_remainder=None,
assumptions=USE_DEFAULTS):
'''
If group_factor is True and the_factor is a product, it will be grouped together as a
sub-product. group_remainder is not relevant kept for compatibility with other factor
methods. Returns the equality that equates self to this new version.
Give any assumptions necessary to prove that the operands are in Complex so that
the associative and commutation theorems are applicable.
'''
from proveit.numbers.multiplication import distribute_through_summation
from proveit.numbers import Mult
if not free_vars(the_factor).isdisjoint(self.indices):
raise Exception(
'Cannot factor anything involving summation indices out of a summation'
)
expr = self
# for convenience updating our equation
eq = TransRelUpdater(expr, assumptions)
assumptions = defaults.checked_assumptions(assumptions)
# We may need to factor the summand within the summation
summand_assumptions = assumptions + self.condition
summand_factorization = self.summand.factorization(
the_factor,
pull,
group_factor=False,
group_remainder=True,
assumptions=summand_assumptions)
summand_instance_equivalence = summand_factor_eq.generalize(
self.indices, domain=self.domain).checked(assumptions)
eq = Equation(
self.instance_substitution(summand_instance_equivalence).checked(
assumptions))
factor_operands = the_factor.operands if isinstance(
the_factor, Mult) else the_factor
x_dummy, z_dummy = self.safe_dummy_vars(2)
# Now do the actual factoring by reversing distribution
if pull == 'left':
Pop, Pop_sub = Operation(
P, self.indices), summand_factor_eq.rhs.operands[-1]
x_sub = factor_operands
z_sub = []
elif pull == 'right':
Pop, Pop_sub = Operation(
P, self.indices), summand_factor_eq.rhs.operands[0]
x_sub = []
z_sub = factor_operands
# We need to deduce that the_factor is in Complex and that all
# instances of Pop_sup are in Complex.
deduce_in_complex(factor_operands, assumptions=assumptions)
deduce_in_complex(Pop_sub,
assumptions=assumptions
| {InSet(idx, self.domain)
for idx in self.indices}).generalize(
self.indices,
domain=self.domain).checked(assumptions)
# Now we instantiate distribut_through_summation_rev
spec1 = distribute_through_summation_rev.instantiate({
Pop:
Pop_sub,
S:
self.domain,
y_etc:
self.indices,
x_etc:
Etcetera(Multi_variable(x_dummy)),
z_etc:
Etcetera(Multi_variable(z_dummy))
}).checked()
eq.update(spec1.derive_conclusion().instantiate({
Etcetera(Multi_variable(x_dummy)):
x_sub,
Etcetera(Multi_variable(z_dummy)):
z_sub
}))
if group_factor and factor_operands.num_entries() > 1:
eq.update(
eq.eq_expr.rhs.group(end_idx=factor_operands.num_entries(),
assumptions=assumptions))
return eq.eq_expr # .checked(assumptions)
def factorization(self,
the_factors,
pull="left",
group_factors=True,
group_remainder=None,
**defaults_config):
'''
Return the proven factorization (equality with the factored
form) from pulling the factor(s) from this summation to the
"left" or "right".
If group_factors is True, the factors will be grouped together
as a sub-product. group_remainder is not relevant kept for
compatibility with other factor methods.
'''
from proveit import ExprTuple, var_range, IndexedVar
from proveit.numbers.multiplication import distribute_through_summation
from proveit.numbers import Mult, one
if not isinstance(the_factors, Expression):
# If 'the_factors' is not an Expression, assume it is
# an iterable and make it a Mult.
the_factors = Mult(*the_factors)
if not free_vars(the_factors).isdisjoint(self.instance_params):
raise ValueError(
'Cannot factor anything involving summation indices '
'out of a summation')
expr = self
# for convenience updating our equation
eq = TransRelUpdater(expr)
# We may need to factor the summand within the summation
summand_assumptions = defaults.assumptions + self.conditions.entries
summand_factorization = self.summand.factorization(
the_factors,
pull,
group_factors=group_factors,
group_remainder=True,
assumptions=summand_assumptions)
if summand_factorization.lhs != summand_factorization.rhs:
gen_summand_factorization = summand_factorization.generalize(
self.instance_params, conditions=self.conditions)
expr = eq.update(
expr.instance_substitution(gen_summand_factorization,
preserve_all=True))
if not group_factors and isinstance(the_factors, Mult):
factors = the_factors.factors
else:
factors = ExprTuple(the_factors)
if pull == 'left':
_a = factors
_c = ExprTuple()
summand_remainder = expr.summand.factors[-1]
elif pull == 'right':
_a = ExprTuple()
_c = factors
summand_remainder = expr.summand.factors[0]
else:
raise ValueError("'pull' must be 'left' or 'right', not %s" % pull)
_b = self.instance_params
_i = _a.num_elements()
_j = _b.num_elements()
_k = _c.num_elements()
_f = Lambda(expr.instance_params, summand_remainder)
_Q = Lambda(expr.instance_params, expr.condition)
_impl = distribute_through_summation.instantiate(
{
i: _i,
j: _j,
k: _k,
f: _f,
Q: _Q,
b: _b
}, preserve_all=True)
quantified_eq = _impl.derive_consequent(preserve_all=True)
eq.update(quantified_eq.instantiate({a: _a, c: _c}, preserve_all=True))
return eq.relation