def shallow_simplification(self, *, must_evaluate=False, **defaults_config):
'''
For the simple binary case Max(a, b), returns a proven
simplification equation for this Max expression assuming the
operands 'a' and 'b' have been simplified and that we know or
have assumed that either a >= b or b > a. If such relational
knowledge is not to be had, we simply return the equation of
the Max expression with itself.
Cases with more than 2 operands are not yet handled.
'''
from proveit.logic import Equals
from proveit.numbers import greater_eq
# We're only set up to deal with binary operator version
if not self.operands.is_double():
# Default is no simplification if not a binary operation.
return Equals(self, self).prove()
# If binary and we know how operands 'a' and 'b' are related ...
op_01, op_02 = self.operands[0], self.operands[1]
if (greater_eq(op_01, op_02).proven()
or greater_eq(op_02, op_01).proven()):
from proveit import x, y
from proveit.numbers.ordering import max_def_bin
return max_def_bin.instantiate({x: op_01, y: op_02})
# Otherwise still no simplification.
return Equals(self, self).prove()
def left_mult_both_sides(self,
multiplier,
*,
simplify=True,
assumptions=USE_DEFAULTS):
'''
Multiply both sides of the relation by the 'multiplier'
on the left.
'''
from proveit.numbers import greater_eq, zero
from proveit.numbers.multiplication import (left_mult_pos_lesseq,
left_mult_neg_lesseq)
if greater_eq(multiplier, zero).proven(assumptions):
new_rel = left_mult_pos_lesseq.instantiate(
{
a: multiplier,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif LessEq(multiplier, zero).proven(assumptions):
new_rel = left_mult_neg_lesseq.instantiate(
{
a: multiplier,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
else:
raise Exception(
"Cannot 'left_mult_both_sides' a LessEq relation without "
"knowing the multiplier's relation with zero.")
return new_rel.with_mimicked_style(self)
def conclude(self, assumptions):
from proveit.logic import InSet, NotEquals
from proveit.numbers import (Rational, RationalNonZero, RationalPos,
RationalNeg, RationalNonNeg, Less,
greater, greater_eq, zero)
# If we known the element is in Q, we may be able to
# prove that is in RationalNonZero, RationalPos, RationalNeg, or
# RationalNonNeg if we know its relation to zero.
if (self.number_set != Rational
and InSet(self.element, Rational).proven(assumptions)):
if self.number_set == RationalNonZero:
if NotEquals(self.element, zero).proven(assumptions):
from . import non_zero_rational_is_rational_non_zero
return non_zero_rational_is_rational_non_zero.instantiate(
{q: self.element}, assumptions=assumptions)
if self.number_set == RationalPos:
if greater(self.element, zero).proven(assumptions):
from . import positive_rational_is_rational_pos
return positive_rational_is_rational_pos.instantiate(
{q: self.element}, assumptions=assumptions)
if self.number_set == RationalNeg:
if Less(self.element, zero).proven():
from . import negative_rational_is_rational_neg
return negative_rational_is_rational_neg.instantiate(
{q: self.element}, assumptions=assumptions)
if self.number_set == RationalNonNeg:
if greater_eq(self.element, zero).proven():
from . import non_neg_rational_in_rational_neg
return non_neg_rational_in_rational_neg.instantiate(
{q: self.element}, assumptions=assumptions)
# Resort to the default NumberMembership.conclude strategies.
return NumberMembership.conclude(self, assumptions)
def derive_element_in_restricted_number_set(self, **defaults_config):
'''
From (member in Interval(x, y)), where x ≥ 0 or y ≤ 0,
deduce that the element is in Natural, NaturalPos, IntegerNeg,
or IntegerNonPos as appropriate.
'''
_a = self.domain.lower_bound
_b = self.domain.upper_bound
_n = self.element
# We wish to deduce a fact based upon the following
# membership fact:
self.expr.prove()
if (not InSet(_a, Natural).proven()
and not InSet(_b, IntegerNonPos).proven()):
# If we don't know that a ≥ 0 or b ≤ 0, we can't prove
# the element is in either restricted number set
# (NaturalPos or IntegerNeg). So, try to sort a, b, 0
# to work this out.
LessEq.sort([_a, _b, zero])
if InSet(_a, Natural).proven():
try:
_a.deduce_in_number_set(NaturalPos, automation=False)
except Exception:
pass
if InSet(_a, NaturalPos).proven():
# member in N^{>0}
lower_bounding = self.derive_element_lower_bound()
a_bounding = greater(_a, zero)
lower_bounding.apply_transitivity(a_bounding)
return InSet(_n, NaturalPos).prove()
else:
# member in N
lower_bounding = self.derive_element_lower_bound()
a_bounding = greater_eq(_a, zero)
lower_bounding.apply_transitivity(a_bounding)
return InSet(_n, Natural).prove()
if InSet(_b, IntegerNonPos).proven():
try:
_b.deduce_in_number_set(IntegerNeg, automation=False)
except Exception:
pass
if InSet(_b, IntegerNeg).proven():
# member in Z^{<0}
upper_bounding = self.derive_element_upper_bound()
b_bounding = Less(_b, zero)
upper_bounding.apply_transitivity(b_bounding)
return InSet(_n, IntegerNeg).prove()
else:
# member in Z^{≤0}
upper_bounding = self.derive_element_upper_bound()
b_bounding = LessEq(_b, zero)
upper_bounding.apply_transitivity(b_bounding)
return InSet(_n, IntegerNonPos).prove()
def left_mult_both_sides(self, multiplier, **defaults_config):
'''
Multiply both sides of the relation by the 'multiplier'
on the left.
'''
from proveit.numbers import greater_eq, zero, deduce_number_set
from proveit.numbers.multiplication import (
weak_bound_via_right_factor_bound,
reversed_weak_bound_via_right_factor_bound)
was_reversed = False
deduce_number_set(multiplier)
if greater_eq(multiplier, zero).proven():
new_rel = weak_bound_via_right_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
elif LessEq(multiplier, zero).proven():
new_rel = reversed_weak_bound_via_right_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
was_reversed = True
else:
raise Exception(
"Cannot left-multiply both sides of %s by %s "
"without knowing the multiplier's relation with zero." %
(self, multiplier))
new_rel = new_rel.with_mimicked_style(self)
if was_reversed:
new_rel = new_rel.with_direction_reversed()
return new_rel
def do_reduced_simplification(self, assumptions=USE_DEFAULTS):
'''
For the case Abs(x) where the operand x is already known to
be or assumed to be a non-negative real, derive and return
this Abs expression equated with the operand itself:
|- Abs(x) = x. For the case where x is already known or assumed
to be a negative real, return the Abs expression equated with
the negative of the operand: |- Abs(x) = -x.
Assumptions may be necessary to deduce necessary conditions for
the simplification.
'''
from proveit.numbers import greater, greater_eq, Mult, Neg
from proveit.numbers import (zero, Natural, NaturalPos, RealNeg,
RealNonNeg, RealPos)
# among other things, convert any assumptions=None
# to assumptions=() (thus averting len(None) errors)
assumptions = defaults.checked_assumptions(assumptions)
#-- -------------------------------------------------------- --#
#-- Case (1): Abs(x) where entire operand x is known or --#
#-- assumed to be non-negative Real. --#
#-- -------------------------------------------------------- --#
if InSet(self.operand, RealNonNeg).proven(assumptions=assumptions):
# Entire operand is known to be or assumed to be a
# non-negative real, so we can return Abs(x) = x
return self.abs_elimination(operand_type='non-negative',
assumptions=assumptions)
#-- -------------------------------------------------------- --#
#-- Case (2): Abs(x) where entire operand x is known or --#
#-- assumed to be a negative Real. --#
#-- -------------------------------------------------------- --#
if InSet(self.operand, RealNeg).proven(assumptions=assumptions):
# Entire operand is known to be or assumed to be a
# negative real, so we can return Abs(x) = -x
return self.abs_elimination(operand_type='negative',
assumptions=assumptions)
#-- -------------------------------------------------------- --#
# -- Case (3): Abs(x) where entire operand x is not yet known --*
#-- to be a non-negative Real, but can easily be --#
#-- proven to be a non-negative Real because it is --#
# -- (a) known or assumed to be ≥ 0 or
#-- (b) known or assumed to be in a subset of the --#
#-- non-negative Real numbers, or --#
#-- (c) the addition or product of operands, all --#
#-- of which are known or assumed to be non- --#
# -- negative real numbers. TBA!
#-- -------------------------------------------------------- --#
if (greater(self.operand, zero).proven(assumptions=assumptions)
and not greater_eq(self.operand,
zero).proven(assumptions=assumptions)):
greater_eq(self.operand, zero).prove(assumptions=assumptions)
# and then it will get picked up in the next if() below
if greater_eq(self.operand, zero).proven(assumptions=assumptions):
from proveit.numbers.number_sets.real_numbers import (
in_real_non_neg_if_greater_eq_zero)
in_real_non_neg_if_greater_eq_zero.instantiate(
{a: self.operand}, assumptions=assumptions)
return self.abs_elimination(operand_type='non-negative',
assumptions=assumptions)
if self.operand in InSet.known_memberships.keys():
for kt in InSet.known_memberships[self.operand]:
if kt.is_sufficient(assumptions):
if is_equal_to_or_subset_eq_of(
kt.expr.operands[1],
equal_sets=[RealNonNeg, RealPos],
subset_eq_sets=[Natural, NaturalPos, RealPos],
assumptions=assumptions):
InSet(self.operand,
RealNonNeg).prove(assumptions=assumptions)
return self.abs_elimination(
operand_type='non-negative',
assumptions=assumptions)
if isinstance(self.operand, Add) or isinstance(self.operand, Mult):
count_of_known_memberships = 0
count_of_known_relevant_memberships = 0
for op in self.operand.operands:
if op in InSet.known_memberships.keys():
count_of_known_memberships += 1
if (count_of_known_memberships ==
self.operand.operands.num_entries()):
for op in self.operand.operands:
op_temp_known_memberships = InSet.known_memberships[op]
for kt in op_temp_known_memberships:
if (kt.is_sufficient(assumptions)
and is_equal_to_or_subset_eq_of(
kt.expr.operands[1],
equal_sets=[RealNonNeg, RealPos],
subset_eq_sets=[
Natural, NaturalPos, RealPos,
RealNonNeg
],
assumptions=assumptions)):
count_of_known_relevant_memberships += 1
break
if (count_of_known_relevant_memberships ==
self.operand.operands.num_entries()):
# Prove that the sum or product is in
# RealNonNeg and then instantiate abs_elimination.
for op in self.operand.operands:
InSet(op, RealNonNeg).prove(assumptions=assumptions)
return self.abs_elimination(assumptions=assumptions)
#-- -------------------------------------------------------- --#
#-- Case (4): Abs(x) where operand x can easily be proven --#
#-- to be a negative Real number because -x is --#
#-- known to be in a subset of the positive Real --#
#-- numbers. --#
#-- -------------------------------------------------------- --#
negated_op = None
if isinstance(self.operand, Neg):
negated_op = self.operand.operand
else:
negated_op = Neg(self.operand)
negated_op_simp = negated_op.simplification(
assumptions=assumptions).rhs
if negated_op_simp in InSet.known_memberships.keys():
from proveit.numbers.number_sets.real_numbers import (
neg_is_real_neg_if_pos_is_real_pos)
for kt in InSet.known_memberships[negated_op_simp]:
if kt.is_sufficient(assumptions):
if is_equal_to_or_subset_eq_of(
kt.expr.operands[1],
equal_sets=[RealNonNeg, RealPos],
subset_sets=[NaturalPos, RealPos],
subset_eq_sets=[NaturalPos, RealPos],
assumptions=assumptions):
InSet(negated_op_simp,
RealPos).prove(assumptions=assumptions)
neg_is_real_neg_if_pos_is_real_pos.instantiate(
{a: negated_op_simp}, assumptions=assumptions)
return self.abs_elimination(operand_type='negative',
assumptions=assumptions)
# for updating our equivalence claim(s) for the
# remaining possibilities
from proveit import TransRelUpdater
eq = TransRelUpdater(self, assumptions)
return eq.relation
def _generateCoordOrderAssumptions(coords):
from proveit.numbers import LessEq, greater_eq
for prev_coord, next_coord in zip(coords[:-1], coords[1:]):
yield LessEq(prev_coord, next_coord)
yield greater_eq(next_coord, prev_coord)
def conclude(self, **defaults_config):
from proveit.numbers import zero, greater_eq
if (InSet(self.element, Real).proven() and
greater_eq(self.element, zero).proven()):
return self.conclude_as_last_resort()
return NumberMembership.conclude(self)
def bound_via_denominator_bound(self, relation, **defaults_config):
'''
Given a relation applicable to the numerator, bound this
division accordingly. For example,
if self is "a / b" and the relation is b > y
return (a / b) < (a / y), provided a, b, and y are positive.
Also see NumberOperation.deduce_bound.
'''
from proveit.numbers import zero, Less, LessEq, greater, greater_eq
from . import (strong_div_from_denom_bound__all_pos,
weak_div_from_denom_bound__all_pos,
strong_div_from_denom_bound__all_neg,
weak_div_from_denom_bound__all_neg,
strong_div_from_denom_bound__neg_over_pos,
weak_div_from_denom_bound__neg_over_pos,
strong_div_from_denom_bound__pos_over_neg,
weak_div_from_denom_bound__pos_over_neg)
if isinstance(relation, Judgment):
relation = relation.expr
if not (isinstance(relation, Less) or isinstance(relation, LessEq)):
raise TypeError("relation is expected to be Less "
"or LessEq number relations, not %s" % relation)
if self.denominator not in relation.operands:
raise ValueError("relation is expected to involve the "
"denominator of %s. %s does not." %
(self, relation))
_a = self.numerator
_x = relation.normal_lhs
_y = relation.normal_rhs
try:
# Ensure that we relate both _x and _y to zero by knowing
# one of these.
ordering = LessEq.sort((_x, _y, zero))
ordering.operands[0].apply_transitivity(ordering.operands[1])
except:
pass # We'll generate an appropriate error below.
try:
deduce_number_set(self.numerator)
deduce_number_set(self.denominator)
except UnsatisfiedPrerequisites:
pass
pos_numer = greater_eq(self.numerator, zero).proven()
neg_numer = LessEq(self.numerator, zero).proven()
pos_denom = greater(self.denominator, zero).proven()
neg_denom = Less(self.denominator, zero).proven()
if not (pos_numer or neg_numer) or not (pos_denom or neg_denom):
raise UnsatisfiedPrerequisites(
"We must know the sign of the numerator and "
"denominator of %s before we can use "
"'bound_via_denominator_bound'." % self)
if pos_numer and pos_denom:
if isinstance(relation, Less):
bound = strong_div_from_denom_bound__all_pos.instantiate({
a: _a,
x: _x,
y: _y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_denom_bound__all_pos.instantiate({
a: _a,
x: _x,
y: _y
})
elif neg_numer and neg_denom:
if isinstance(relation, Less):
bound = strong_div_from_denom_bound__all_neg.instantiate({
a: _a,
x: _x,
y: _y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_denom_bound__all_neg.instantiate({
a: _a,
x: _x,
y: _y
})
elif pos_numer and neg_denom:
if isinstance(relation, Less):
bound = strong_div_from_denom_bound__pos_over_neg.instantiate({
a:
_a,
x:
_x,
y:
_y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_denom_bound__pos_over_neg.instantiate({
a:
_a,
x:
_x,
y:
_y
})
elif neg_numer and pos_denom:
if isinstance(relation, Less):
bound = strong_div_from_denom_bound__neg_over_pos.instantiate({
a:
_a,
x:
_x,
y:
_y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_denom_bound__neg_over_pos.instantiate({
a:
_a,
x:
_x,
y:
_y
})
else:
raise UnsatisfiedPrerequisites(
"We must know whether or not the denominator of %s "
"is positive or negative before we can use "
"'bound_via_denominator_bound'." % self)
if bound.rhs == self:
return bound.with_direction_reversed()
return bound
def conclude(self, **defaults_config):
if (InSet(self.element, Integer).proven()
and greater_eq(self.element, zero).proven()):
return self.conclude_as_last_resort()
return NumberMembership.conclude(self)
def derive_element_in_restricted_number_set(self, **defaults_config):
'''
From (element in IntervalXX(x, y)), where x ≥ 0 or y ≤ 0,
derive that the element is in RealPos, RealNeg, RealNonPos, or
RealNonNeg as appropriate.
'''
from proveit.numbers import (zero, RealPos, RealNonNeg,
RealNeg, RealNonPos)
from proveit.numbers import Less, LessEq, greater, greater_eq
_a = self.domain.lower_bound
_b = self.domain.upper_bound
_n = self.element
# We wish to deduce a fact based upon the following
# membership fact:
self.expr.prove()
if (not InSet(_a, RealNonNeg).proven() and
not InSet(_b, RealNonPos).proven()):
try:
_a.deduce_in_number_set(RealNonNeg, automation=False)
except Exception:
pass
try:
_b.deduce_in_number_set(RealNonPos, automation=False)
except Exception:
pass
if (not InSet(_a, RealNonNeg).proven() and
not InSet(_b, RealNonPos).proven()):
# If we don't know that a ≥ 0 or b ≤ 0, we can't prove
# the element is in either restricted number set
# (NaturalPos or IntegerNeg). So, try to sort a, b, 0
# to work this out.
LessEq.sort([_a, _b, zero])
if InSet(_a, RealNonNeg).proven():
try:
_a.deduce_in_number_set(RealPos, automation=False)
except Exception:
pass
lower_bound = self.derive_element_lower_bound()
a_bound = greater_eq(_a, zero)
if InSet(_a, RealPos).proven():
a_bound = greater(_a, zero)
lower_bound.apply_transitivity(a_bound)
if (isinstance(self, IntervalOO)
or isinstance(self, IntervalOC)
or InSet(_a, RealPos).proven()):
# member in R^{>0}
return InSet(_n, RealPos).prove()
else:
# member in R^{≥0}
return InSet(_n, RealNonNeg).prove()
if InSet(_b, RealNonPos).proven():
try:
_b.deduce_in_number_set(RealNeg, automation=False)
except Exception:
pass
upper_bound = self.derive_element_upper_bound()
b_bound = LessEq(_b, zero)
if InSet(_b, RealNeg).proven():
b_bound = Less(_b, zero)
upper_bound.apply_transitivity(b_bound)
if (isinstance(self, IntervalOO)
or isinstance(self, IntervalCO)
or InSet(_b, RealNeg).proven()):
# member in R^{<0}
return InSet(_n, RealNeg).prove()
else:
# member in R^{≤0}
return InSet(_n, RealNonPos).prove()
def bound_via_operand_bound(self, operand_relation, **defaults_config):
'''
For simple cases, deduce a bound on this Exp object given a
bound on its operand. For example, suppose x = Exp(y, 2) and
we know that y >= 2. Then x.bound_via_operand_bound(y >= 2)
returns x >= 2^2 = 4.
This method currently works MAINLY for expressions
of the form Exp(x, a) for non-negative real x and real exponent
'a', where we know something of the form x < y (or x ≤ y, x > y,
x ≥ y) involving the base of the exponential expression.
The result also depends on knowing the relationship between the
exponent 'a' and zero, which might need to be pre-proven or
provided as an assumption (e.g. in the form 'a > 0' or
InSet(a, RealNeg), etc).
A special case also deals with a negative base raised to the
power of 2.
This method also works for special cases of the form Exp(a, x),
where a > 1 and the operand_relation involve the exponent x.
Future development will address operand_relations
involving the exponent x in expressions of the form a^x when
the base 0 < a < 1, and expand the special negative
base case to include all even and odd exponent cases.
Also see NumberOperation.deduce_bound and compare to the
bound_via_operand_bound() method found in the Div and Neg
classes.
'''
from proveit import Judgment
from proveit.numbers import (
two, greater, greater_eq, Less, LessEq,
NumberOrderingRelation, RealNonNeg)
if isinstance(operand_relation, Judgment):
operand_relation = operand_relation.expr
if not isinstance(operand_relation, NumberOrderingRelation):
raise TypeError(
"In Exp.bound_via_operand_bound(), the "
"'operand_relation' argument is expected to be a number "
"relation (<, >, ≤, or ≥), but instead was {}.".
format(operand_relation))
lhs = operand_relation.lhs
# should be able to generalize this later
# no need to limit to just lhs, right?
if lhs != self.base and lhs != self.exponent:
raise ValueError(
"In Exp.bound_via_operand_bound(), the left side of "
"the 'operand_relation' argument {0} is expected to "
"match either the Exp base operand {1} or the "
"Exp exponent operand {2}.".
format(operand_relation, self.base, self.exponent))
# assign x and y subs according to std Less or LessEq relations
_x_sub = operand_relation.normal_lhs
_y_sub = operand_relation.normal_rhs
if lhs == self.base:
_a_sub = self.exponent
else:
_a_sub = self.base
# I. Default case: the user-supplied operand relation involves
# the BASE of the Exp expression x^a
if lhs == self.base:
# Several cases to consider:
# (1) a > 0, 0 ≤ x < y
# (2) a > 0, 0 ≤ x ≤ y
# (3) a ≥ 0, 0 < x < y
# (4) a ≥ 0, 0 < x ≤ y
# (5) a < 0, 0 < x < y
# (6) a < 0, 0 < x ≤ y
# (7) a ≤ 0, 0 < x < y
# (8) a ≤ 0, 0 < x ≤ y
# =====================
# (9) a = 2, y < x < 0
# (10) a = 2, y ≤ x < 0
# Cases (1) and (2): exponent a > 0
if (greater(_a_sub, zero).proven() and
greater_eq(_x_sub, zero).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import exp_pos_less
bound = exp_pos_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import exp_pos_lesseq
bound = exp_pos_lesseq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
# Cases (3) and (4): exponent a ≥ 0
elif (greater_eq(_a_sub, zero).proven() and
greater(_x_sub, zero).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import exp_nonneg_less
bound = exp_nonneg_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import exp_nonneg_lesseq
bound = exp_nonneg_lesseq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
# Cases (5) and (6): exponent a < 0
elif (Less(_a_sub, zero).proven() and
greater(_x_sub, zero).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import exp_neg_less
bound = exp_neg_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import exp_neg_lesseq
bound = exp_neg_lesseq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
# Cases (7) and (8): exponent a ≤ 0
elif (LessEq(_a_sub, zero).proven() and
greater(_x_sub, zero).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import exp_nonpos_less
bound = exp_nonpos_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import exp_nonpos_lesseq
bound = exp_nonpos_lesseq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
# Cases (9) and (10): exponent a = 2
# with x < y < 0 or x ≤ y < 0
elif (_a_sub == two and
Less(_y_sub, zero).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import (
exp_even_neg_base_less)
bound = exp_even_neg_base_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import (
exp_even_neg_base_lesseq)
bound = exp_even_neg_base_lesseq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
else:
raise ValueError(
"In calling Exp.bound_via_operand_bound(), a "
"specific matching case was not found for {}.".
format(self))
# II. 2nd main case: the user-supplied operand relation involves
# the EXPONENT of the Exp expression a^x
elif lhs == self.exponent:
# Several cases to consider (others to be developed)
# considering the Exp expression a^x with a, x in Real:
# (1) a > 1, x < y
# (2) a > 1, x ≤ y
# (3) a > 1, y < x
# (4) a > 1, y ≤ x
# Other cases to be developed involving base a < 1,
# which produces a monotonically-decreasing function.
# Cases (1)-(4): base a > 1, a^x monotonically increasing
if (greater(_a_sub, one).proven() and
InSet(_x_sub, Real).proven()):
if isinstance(operand_relation, Less):
from proveit.numbers.exponentiation import (
exp_monotonicity_large_base_less)
bound = exp_monotonicity_large_base_less.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
elif isinstance(operand_relation, LessEq):
from proveit.numbers.exponentiation import (
exp_monotonicity_large_base_less_eq)
bound = exp_monotonicity_large_base_less_eq.instantiate(
{x: _x_sub, y: _y_sub, a: _a_sub})
else:
raise TypeError(
"In Exp.bound_via_operand_bound(), the 'operand_relation' "
"argument is expected to be a 'Less', 'LessEq', 'greater', "
"or 'greater_eq' relation. Instead we have {}.".
format(operand_relation))
else:
raise ValueError(
"In Exp.bound_via_operand_bound(), either the "
"base {0} is not known to be greater than 1 and/or "
"the operand {1} is not known to be Real.".
format(_a_sub, _x_sub))
else:
raise ValueError("OOOPS!")
if bound.rhs == self:
return bound.with_direction_reversed()
return bound