def scalar_one_elimination(self, **defaults_config):
'''
Equivalence method that derives a simplification in which
a single scalar multiplier of 1 is eliminated.
For example, letting v denote a vector element of some VecSpace,
then ScalarMult(one, v).scalar_one_elimination() would return
|- ScalarMult(one, v) = v
Will need to know that v is in VecSpaces(K) and that the
multiplicative identity 1 is in the field K. This might require
assumptions or pre-proving of a VecSpace that contains the
vector appearing in the ScalarMult expression.
'''
from proveit.numbers import one
# from . import elim_one_left, elim_one_right
from . import one_as_scalar_mult_id
# The following seems silly -- if the scalar is not 1, just
# return with no change instead?
if self.scalar != one:
raise ValueError(
"For ScalarMult.scalar_one_elimination(), the scalar "
"multiplier in {0} was expected to be 1 but instead "
"was {1}.".format(self, self.scalar))
# obtain the instance params:
# _K is the field; _V is the vector space over field _K;
# and _v is the vector in _V being scaled
_K, _v, _V = one_as_scalar_mult_id.all_instance_params()
# isolate the vector portion
_v_sub = self.scaled
# Find a containing vector space V in the theorem.
# This may fail!
_V_sub = list(VecSpaces.yield_known_vec_spaces(_v_sub))[0]
# Find a vec space field, hopefully one that contains the mult
# identity 1. This may fail!
_K_sub = list(VecSpaces.yield_known_fields(_V_sub))[0]
# If we made it this far, can probably instantiate the theorem
# (although it could still fail if 1 is not in the field _K_sub)
return one_as_scalar_mult_id.instantiate({
_K: _K_sub,
_v: _v_sub,
_V: _V_sub
})
def yield_known_hilbert_spaces(vec):
'''
Given a vector expression, vec, yield any Hilbert spaces
known to contain vec.
'''
for vec_space in VecSpaces.yield_known_vec_spaces(vec, field=Complex):
if vec_space in HilbertSpacesLiteral.known_spaces_memberships:
# This vector space is already known to be an inner
# product space.
yield vec_space
else:
try:
deduce_as_hilbert_space(vec_space)
yield vec_space
except NotImplementedError:
# Not known you to prove 'vec_space' is an inner
# product space.
pass