def right_mult_both_sides(self, multiplier, **defaults_config):
'''
Multiply both sides of the relation by the 'multiplier'
on the right.
'''
from proveit.numbers import LessEq, zero, deduce_number_set
from proveit.numbers.multiplication import (
strong_bound_via_left_factor_bound,
weak_bound_via_left_factor_bound,
reversed_strong_bound_via_left_factor_bound,
reversed_weak_bound_via_left_factor_bound)
was_reversed = False
deduce_number_set(multiplier)
if Less(zero, multiplier).proven():
new_rel = strong_bound_via_left_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
elif Less(multiplier, zero).proven():
new_rel = reversed_strong_bound_via_left_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
was_reversed = True
elif LessEq(zero, multiplier).proven():
new_rel = weak_bound_via_left_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
elif LessEq(multiplier, zero).proven():
new_rel = reversed_weak_bound_via_left_factor_bound.instantiate({
a:
multiplier,
x:
self.lower,
y:
self.upper
})
was_reversed = True
else:
raise Exception(
"Cannot right-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 divide_both_sides(self,
divisor,
*,
simplify=True,
assumptions=USE_DEFAULTS):
'''
Divide both sides of the relation by the 'divisor'.
'''
from proveit.numbers import Less, zero
from proveit.numbers.division import (div_pos_lesseq, div_neg_lesseq)
if Less(zero, divisor).proven(assumptions):
new_rel = div_pos_lesseq.instantiate(
{
a: divisor,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif Less(divisor, zero).proven(assumptions):
new_rel = div_neg_lesseq.instantiate(
{
a: divisor,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
else:
raise Exception("Cannot 'divide' a LessEq relation without "
"knowing whether the divisor is greater than "
"or less than zero.")
return new_rel.with_mimicked_style(self)
def right_mult_both_sides(self,
multiplier,
*,
simplify=True,
assumptions=USE_DEFAULTS):
'''
Multiply both sides of the relation by the 'multiplier'
on the right.
'''
from proveit.numbers import LessEq, zero
from proveit.numbers.multiplication import (right_mult_pos_less,
right_mult_nonneg_less,
right_mult_neg_less,
right_mult_nonpos_less)
if Less(zero, multiplier).proven(assumptions):
new_rel = right_mult_pos_less.instantiate(
{
a: multiplier,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif Less(multiplier, zero).proven(assumptions):
new_rel = right_mult_neg_less.instantiate(
{
a: multiplier,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif LessEq(zero, multiplier).proven(assumptions):
new_rel = right_mult_nonneg_less.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 = right_mult_nonpos_less.instantiate(
{
a: multiplier,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
else:
raise Exception(
"Cannot 'right_mult_both_sides' a Less relation without "
"knowing the multiplier's relation with zero.")
return new_rel.with_mimicked_style(self)
def exponentiate_both_sides(self,
exponent,
*,
simplify=True,
assumptions=USE_DEFAULTS):
'''
Exponentiate both sides of the relation by the 'exponent'.
'''
from proveit.numbers import Less, zero
from proveit.numbers.exponentiation import (exp_pos_lesseq,
exp_nonneg_lesseq,
exp_neg_lesseq,
exp_nonpos_lesseq)
if Less(exponent, zero).proven(assumptions):
new_rel = exp_pos_lesseq.instantiate(
{
a: exponent,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif Less(exponent, zero).proven(assumptions):
new_rel = exp_neg_lesseq.instantiate(
{
a: exponent,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif LessEq(exponent, zero).proven(assumptions):
new_rel = exp_nonneg_lesseq.instantiate(
{
a: exponent,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
elif LessEq(exponent, zero).proven(assumptions):
new_rel = exp_nonpos_lesseq.instantiate(
{
a: exponent,
x: self.lower,
y: self.upper
},
assumptions=assumptions)._simplify_both_sides(
simplify=simplify, assumptions=assumptions)
else:
raise Exception("Cannot 'exponentiate' a Less relation without "
"knowing the exponent'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 not_equal(self, other, **defaults_config):
from proveit.numbers import Less
from proveit.numbers.ordering import less_is_not_eq
_a, _b = Less.sorted_items([self, other])
not_eq_stmt = less_is_not_eq.instantiate({a: _a, b: _b})
if not_eq_stmt.lhs != self:
# We need to reverse the statement.
return not_eq_stmt.derive_reversed()
return not_eq_stmt
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 divide_both_sides(self, divisor, **defaults_config):
'''
Divide both sides of the relation by the 'divisor'.
'''
from proveit.numbers import Less, zero, deduce_number_set
from proveit.numbers.division import (
strong_div_from_numer_bound__pos_denom,
strong_div_from_numer_bound__neg_denom)
deduce_number_set(divisor)
if Less(zero, divisor).proven():
thm = strong_div_from_numer_bound__pos_denom
elif Less(divisor, zero).proven():
thm = strong_div_from_numer_bound__neg_denom
else:
raise Exception("Cannot divide both sides of %s by %s "
"without knowing the divisors "
"relation to zero." % (self, divisor))
new_rel = thm.instantiate({a: divisor, x: self.lower, y: self.upper})
return new_rel.with_mimicked_style(self)
def exponentiate_both_sides(self, exponent, **defaults_config):
'''
Exponentiate both sides of the relation by the 'exponent'.
'''
from proveit.numbers import Less, LessEq, zero, deduce_number_set
from proveit.numbers.exponentiation import (exp_pos_less,
exp_nonneg_less,
exp_neg_less,
exp_nonpos_less)
# We need to know how the exponent relates to zero.
deduce_number_set(exponent)
LessEq.sort([zero, exponent])
if Less(zero, exponent).proven():
new_rel = exp_pos_less.instantiate({
a: exponent,
x: self.lower,
y: self.upper
})
elif Less(exponent, zero).proven():
new_rel = exp_neg_less.instantiate({
a: exponent,
x: self.lower,
y: self.upper
})
elif LessEq(zero, exponent).proven():
new_rel = exp_nonneg_less.instantiate({
a: exponent,
x: self.lower,
y: self.upper
})
elif LessEq(exponent, zero).proven():
new_rel = exp_nonpos_less.instantiate({
a: exponent,
x: self.lower,
y: self.upper
})
else:
raise Exception("Cannot exponentiate both sides of %s by %s "
"without knowing the exponent's relation with "
"zero" % (self, exponent))
return new_rel.with_mimicked_style(self)
def not_equal(self, other, assumptions=USE_DEFAULTS):
from proveit.numbers import Less
from proveit.numbers.ordering import less_is_not_eq
_a, _b = Less.sorted_items([self, other], assumptions=assumptions)
not_eq_stmt = less_is_not_eq.instantiate({
a: _a,
b: _b
},
assumptions=assumptions)
if not_eq_stmt.lhs != self:
# We need to reverse the statement.
return not_eq_stmt.derive_reversed(assumptions)
return not_eq_stmt
def deduce_in_interval(self, **defaults_config):
'''
Bound the interval of the Sin function evaluation depending
upon what is known about the angle.
'''
from . import (sine_interval, sine_nonneg_interval, sine_pos_interval,
sine_nonpos_interval, sine_neg_interval)
from proveit.numbers import (zero, pi, Less, LessEq, Neg)
_theta = self.angle
deduce_number_set(_theta)
if (Less(zero, _theta).proven() and Less(_theta, pi).proven()):
return sine_pos_interval.instantiate({theta: _theta})
elif (LessEq(zero, _theta).proven() and LessEq(_theta, pi).proven()):
return sine_nonneg_interval.instantiate({theta: _theta})
elif (Less(Neg(pi), _theta).proven() and Less(_theta, zero).proven()):
return sine_neg_interval.instantiate({theta: _theta})
elif (Less(Neg(pi), _theta).proven() and Less(_theta, zero).proven()):
return sine_neg_interval.instantiate({theta: _theta})
elif (LessEq(Neg(pi), _theta).proven()
and LessEq(_theta, zero).proven()):
return sine_nonpos_interval.instantiate({theta: _theta})
else:
# Without knowing more, we can only bound it by +/-1.
return sine_interval.instantiate({theta: _theta})
def bound_via_numerator_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 a < x
return (a / b) < (x / b), provided b > 0.
Also see NumberOperation.deduce_bound.
'''
from proveit.numbers import zero, Less, LessEq, greater
from . import (strong_div_from_numer_bound__pos_denom,
weak_div_from_numer_bound__pos_denom,
strong_div_from_numer_bound__neg_denom,
weak_div_from_numer_bound__neg_denom)
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.numerator not in relation.operands:
raise ValueError("relation is expected to involve the "
"numerator of %s. %s does not." %
(self, relation))
_a = self.denominator
_x = relation.normal_lhs
_y = relation.normal_rhs
try:
deduce_number_set(self.denominator)
except UnsatisfiedPrerequisites:
pass
if greater(self.denominator, zero).proven():
if isinstance(relation, Less):
bound = strong_div_from_numer_bound__pos_denom.instantiate({
a:
_a,
x:
_x,
y:
_y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_numer_bound__pos_denom.instantiate({
a: _a,
x: _x,
y: _y
})
elif Less(self.denominator, zero).proven():
if isinstance(relation, Less):
bound = strong_div_from_numer_bound__neg_denom.instantiate({
a:
_a,
x:
_x,
y:
_y
})
elif isinstance(relation, LessEq):
bound = weak_div_from_numer_bound__neg_denom.instantiate({
a: _a,
x: _x,
y: _y
})
else:
raise UnsatisfiedPrerequisites(
"We must know whether the denominator of %s "
"is positive or negative before we can use "
"'bound_via_numerator_bound'." % self)
if bound.rhs == self:
return bound.with_direction_reversed()
return bound
def conclude(self, **defaults_config):
from proveit.numbers import zero, Less
if (InSet(self.element, Real).proven() and
Less(self.element, zero).proven()):
return self.conclude_as_last_resort()
return NumberMembership.conclude(self)
def greater(a, b):
'''
Return an expression representing a > b, internally represented
as b < a but with a style that reverses the direction.
'''
return Less(b, a).with_styles(direction='reversed')
def deduce_number_set(expr, **defaults_config):
'''
Prove that 'expr' is an Expression that represents a number
in a standard number set that is as restrictive as we can
readily know.
'''
from proveit.logic import And, InSet, Equals, NotEquals
from proveit.numbers import Less, LessEq, zero
# Find the first (most restrictive) number set that
# contains 'expr' or something equal to it.
for number_set in sorted_number_sets:
membership = None
for eq_expr in Equals.yield_known_equal_expressions(expr):
if isinstance(eq_expr, ExprRange):
membership = And(
ExprRange(eq_expr.parameter,
InSet(eq_expr.body, number_set),
eq_expr.true_start_index,
eq_expr.true_end_index,
styles=eq_expr.get_styles()))
else:
membership = InSet(eq_expr, number_set)
if membership.proven():
break # found a known number set membership
else:
membership = None
if membership is not None:
membership = InSet(expr, number_set).prove()
break
if hasattr(expr, 'deduce_number_set'):
# Use 'deduce_number_set' method.
try:
deduced_membership = expr.deduce_number_set()
except (UnsatisfiedPrerequisites, ProofFailure):
deduced_membership = None
if deduced_membership is not None:
assert isinstance(deduced_membership, Judgment)
if not isinstance(deduced_membership.expr, InSet):
raise TypeError("'deduce_number_set' expected to prove an "
"InSet type expression")
if deduced_membership.expr.element != expr:
raise TypeError("'deduce_number_set' was expected to prove "
"that %s is in some number set" % expr)
# See if this deduced number set is more restrictive than
# what we had surmised already.
deduced_number_set = deduced_membership.domain
if membership is None:
membership = deduced_membership
number_set = deduced_number_set
elif (deduced_number_set != number_set
and number_set.includes(deduced_number_set)):
number_set = deduced_number_set
membership = deduced_membership
if membership is None:
from proveit import defaults
raise UnsatisfiedPrerequisites(
"Unable to prove any number membership for %s" % expr)
# Already proven to be in some number set,
# Let's see if we can restrict it further.
if Less(zero, expr).proven(): # positive
number_set = pos_number_set.get(number_set, None)
elif Less(expr, zero).proven(): # negative
number_set = neg_number_set.get(number_set, None)
elif LessEq(zero, expr).proven(): # non-negative
number_set = nonneg_number_set.get(number_set, None)
elif LessEq(expr, zero).proven(): # non-positive
number_set = nonpos_number_set.get(number_set, None)
elif NotEquals(expr, zero).proven():
number_set = nonzero_number_set.get(number_set, None)
if number_set is None:
# Just use what we have already proven.
return membership.prove()
return InSet(expr, number_set).prove()
def deduce_in_number_set(self, number_set, **defaults_config):
'''
Given a number set number_set, attempt to prove that the given
expression is in that number set using the appropriate closure
theorem.
'''
from proveit import a, b
from proveit.numbers import Less, zero
from proveit.numbers.division import (
div_rational_closure, div_rational_nonzero_closure,
div_rational_pos_closure, div_rational_pos_from_double_neg,
div_rational_neg_from_neg_denom, div_rational_neg_from_neg_numer,
div_rational_nonneg_closure, div_rational_nonneg_from_double_neg,
div_rational_nonpos_from_neg_denom,
div_rational_nonpos_from_nonpos_numer, div_real_closure,
div_real_nonzero_closure, div_real_pos_closure,
div_real_pos_from_double_neg, div_real_neg_from_neg_denom,
div_real_neg_from_neg_numer, div_real_nonneg_closure,
div_real_nonneg_from_double_neg, div_real_nonpos_from_neg_denom,
div_real_nonpos_from_nonpos_numer, div_complex_nonzero_closure,
div_complex_closure)
thm = None
if number_set == Rational:
thm = div_rational_closure
elif number_set == RationalNonZero:
thm = div_rational_nonzero_closure
elif number_set == RationalPos:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_rational_pos_from_double_neg
else:
thm = div_rational_pos_closure
elif number_set == RationalNeg:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_rational_neg_from_neg_denom
else:
thm = div_rational_neg_from_neg_numer
elif number_set == RationalNonNeg:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_rational_nonneg_from_double_neg
else:
thm = div_rational_nonneg_closure
elif number_set == RationalNonPos:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_rational_nonpos_from_neg_denom
else:
thm = div_rational_nonpos_from_neg_numer
elif number_set == Real:
thm = div_real_closure
elif number_set == RealNonZero:
thm = div_real_nonzero_closure
elif number_set == RealPos:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_real_pos_from_double_neg
else:
thm = div_real_pos_closure
elif number_set == RealNeg:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_real_neg_from_neg_denom
else:
thm = div_real_neg_from_neg_numer
elif number_set == RealNonNeg:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_real_nonneg_from_double_neg
else:
thm = div_real_nonneg_closure
elif number_set == RealNonPos:
deduce_number_set(self.denominator)
if Less(self.denominator, zero).proven():
thm = div_real_nonpos_from_neg_denom
else:
thm = div_real_nonpos_from_nonpos_numer
elif number_set == ComplexNonZero:
thm = div_complex_nonzero_closure
elif number_set == Complex:
thm = div_complex_closure
if thm is not None:
return thm.instantiate({a: self.numerator, b: self.denominator})
raise NotImplementedError(
"'Div.deduce_in_number_set()' not implemented for the %s set" %
str(number_set))
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
def conclude(self, **defaults_config):
if (InSet(self.element, Integer).proven()
and Less(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