def empty_range(_i, _j, _f, assumptions):
# If the start and end are literal ints and form an
# empty range, then it should be straightforward to
# prove that the range is empty.
from proveit.numbers import is_literal_int, Add
from proveit.logic import Equals
from proveit import m
_m = entries[0].start_index
_n = entries[0].end_index
empty_req = Equals(Add(_n, one), _m)
if is_literal_int(_m) and is_literal_int(_n):
if _n.as_int() + 1 == _m.as_int():
empty_req.prove()
if empty_req.proven(assumptions):
_f = Lambda(
(entries[0].parameter, entries[0].body.parameter),
entries[0].body.body)
_i = entry_map(_i)
_j = entry_map(_j)
return len_of_empty_range_of_ranges.instantiate(
{
m: _m,
n: _n,
f: _f,
i: _i,
j: _j
},
assumptions=assumptions)
def conclude(self, **defaults_config):
'''
Attempt to conclude that the element is in the domain. First,
see if it is known to be contained in a known subset of the
domain. Next, check if the element has a known simplification;
if so, try to derive membership via this simplification.
If there isn't a known simplification, next try to call
the 'self.domain.membership_object.conclude(..)' method to prove
the membership. If that fails, try simplifying the element
again, this time using automation to push the simplification
through if possible.
'''
from proveit.logic import Equals, SubsetEq, evaluation_or_simplification
# See if the element, or something known to be equal to
# the element, is known to be a member of the domain or a subset
# of the domain.
for elem_sub in Equals.yield_known_equal_expressions(self.element):
same_membership = None # membership in self.domain
eq_membership = None # membership in an equal domain
subset_membership = None # membership in a subset
for known_membership in InClass.yield_known_memberships(elem_sub):
eq_rel = Equals(known_membership.domain, self.domain)
sub_rel = SubsetEq(known_membership.domain, self.domain)
if known_membership.domain == self.domain:
same_membership = known_membership
break # this is the best to use; we are done
elif eq_rel.proven():
eq_membership = known_membership
elif sub_rel.proven():
subset_membership = known_membership
elem_sub_in_domain = None
if same_membership is not None:
elem_sub_in_domain = same_membership
elif eq_membership is not None:
# domains are equal -- just substitute to domain.
eq_rel = Equals(eq_membership.domain, self.domain)
elem_sub_in_domain = eq_rel.sub_right_side_into(
eq_membership.inner_expr().domain)
elif subset_membership is not None:
# S is a superset of R, so now we can prove x in S.
sub_rel = SubsetEq(subset_membership.domain, self.domain)
elem_sub_in_domain = sub_rel.derive_superset_membership(
elem_sub)
if elem_sub_in_domain is not None:
# We found what we are looking for.
if elem_sub == self.element:
return elem_sub_in_domain # done
# Just need to sub in the element for _elem_sub.
Equals(elem_sub, self.element).conclude_via_transitivity()
return elem_sub_in_domain.inner_expr().element.substitute(
self.element)
# Try the standard Relation strategies -- evaluate or
# simplify both sides.
try:
return Relation.conclude(self)
except ProofFailure:
# Both sides are already irreducible or simplified
# or we were unable to simplify either side.
pass
# Unable to simplify the element. Try to conclude via
# the 'membership_object' if there is one.
if hasattr(self, 'membership_object'):
return self.membership_object.conclude()
raise ProofFailure(
self, defaults.assumptions, "Unable to conclude automatically; "
"the domain, %s, has no 'membership_object' "
"method with a strategy for proving "
"membership." % self.domain)
def conclude(self, assumptions):
'''
Attempt to conclude that the element is in the domain.
First, see if it is known to be contained in a known subset of the
domain. Next, check if the element has a known simplification; if so,
try to derive membership via this simplification.
If there isn't a known simplification, next try to call
the 'self.domain.membershipObject.conclude(..)' method to prove
the membership. If that fails, try simplifying the element
again, this time using automation to push the simplification through
if possible.
'''
from proveit.logic import Equals, SubsetEq
from proveit import ProofFailure
from proveit.logic import SimplificationError
# See if the membership is already known.
if self.element in InSet.knownMemberships:
for knownMembership in InSet.knownMemberships[self.element]:
if knownMembership.isSufficient(assumptions):
# x in R is a known truth; if we know that R subseteq S,
# or R = S we are done.
eqRel = Equals(knownMembership.domain, self.domain)
if eqRel.proven(assumptions):
return eqRel.subRightSideInto(
knownMembership.innerExpr().domain)
subRel = SubsetEq(knownMembership.domain, self.domain)
if subRel.proven(assumptions):
# S is a superset of R, so now we can prove x in S.
return subRel.deriveSupersetMembership(
self.element, assumptions)
# No known membership works. Let's see if there is a known
# simplification of the element before trying anything else.
try:
elem_simplification = self.element.simplification(assumptions,
automation=False)
if elem_simplification.lhs == elem_simplification.rhs:
elem_simplification = None # reflection doesn't count
except SimplificationError:
elem_simplification = None
if elem_simplification is None:
# Let's try harder to simplify the element.
try:
elem_simplification = self.element.simplification(assumptions)
if elem_simplification.lhs == elem_simplification.rhs:
elem_simplification = None # reflection doesn't count
except SimplificationError:
pass
# If the element simplification succeeded, prove the membership
# via the simplified form of the element.
if elem_simplification is not None:
simple_elem = elem_simplification.rhs
simple_membership = InSet(simple_elem,
self.domain).prove(assumptions)
inner_expr = simple_membership.innerExpr().element
return elem_simplification.subLeftSideInto(inner_expr, assumptions)
else:
# Unable to simplify the element. Try to conclude via
# the 'membershipObject' if there is one.
if hasattr(self, 'membershipObject'):
return self.membershipObject.conclude(assumptions)
raise ProofFailure(
self, assumptions, "Unable to conclude automatically; "
"the domain, %s, has no 'membershipObject' "
"method with a strategy for proving "
"membership." % self.domain)
