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 abs_elimination(self, operand_type=None, **defaults_config):
'''
For some |x| expression, deduce either |x| = x (the default) OR
|x| = -x (for operand_type = 'negative'). Assumptions may be
needed to deduce x >= 0 or x < 0, respectively.
'''
from proveit.numbers import LessEq, zero
from . import abs_non_neg_elim, abs_neg_elim
if operand_type is None:
# LessEq.sort uses a bidirectional search which should
# be fairly efficient, as long as there aren't too
# many known relationship directly or indirectly involving
# self.operand or zero.
relation_with_zero = LessEq.sort([zero, self.operand])
if relation_with_zero.normal_lhs == zero:
operand_type = 'non-negative'
else:
operand_type = 'negative'
# deduce_non_neg(self.operand, assumptions) # NOT YET IMPLEMENTED
if operand_type is None or operand_type == 'non-negative':
return abs_non_neg_elim.instantiate({x: self.operand})
elif operand_type == 'negative':
return abs_neg_elim.instantiate({x: self.operand})
else:
raise ValueError(
"Unsupported operand type for Abs.abs_elimination() "
"method; operand type should be omitted or specified "
"as 'negative' or 'non-negative', but instead was "
"given as operand_type = {}.".format(operand_type))
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 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 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()