def simplification(self, *, must_evaluate=False, **defaults_config):
'''
Proves that this ExprTuple is equal to an ExprTuple
with all of its entries simplified (and ExprRanges reduced).
'''
from proveit.relation import TransRelUpdater
from proveit import ExprRange
expr = self
eq = TransRelUpdater(expr)
_k = 0
for entry in self.entries:
if isinstance(entry, ExprRange):
entry_simp = entry.reduction(preserve_all=True)
num_entries = entry_simp.rhs.num_entries()
else:
if must_evaluate:
entry_simp = entry.evaluation()
else:
entry_simp = entry.simplification()
num_entries = 1
if entry_simp.lhs != entry_simp.rhs:
expr = eq.update(expr.substitution(entry_simp,
start_idx=_k))
_k += num_entries
return eq.relation
def simplification_of_operands(self, **defaults_config):
'''
Prove this Operation equal to a form in which its operands
have been simplified.
'''
from proveit.relation import TransRelUpdater
from proveit import ExprRange, NamedExprs
from proveit.logic import is_irreducible_value
if any(isinstance(operand, ExprRange) for operand in self.operands):
# If there is any ExprRange in the operands, simplify the
# operands together as an ExprTuple.
return self.inner_expr().operands[:].simplification()
else:
expr = self
eq = TransRelUpdater(expr)
with defaults.temporary() as temp_defaults:
# No auto-simplification or replacements here;
# just simplify operands one at a time.
temp_defaults.preserve_all = True
operands = self.operands
if isinstance(operands, NamedExprs):
# operands as NamedExprs
for key in operands.keys():
operand = operands[key]
if not is_irreducible_value(operand):
inner_operand = getattr(expr.inner_expr(), key)
expr = eq.update(inner_operand.simplification())
else:
# operands as ExprTuple
for k, operand in enumerate(operands):
if not is_irreducible_value(operand):
inner_operand = expr.inner_expr().operands[k]
expr = eq.update(inner_operand.simplification())
return eq.relation
def _redundant_mod_elimination(
expr, mod_elimination_thm, mod_elimination_in_sum_thm):
'''
For use by Mod and ModAbs for shallow_simplification.
'''
dividend = expr.dividend
divisor = expr.divisor
if isinstance(dividend, Mod) and dividend.divisor==divisor:
# [(a mod b) mod b] = [a mod b]
return mod_elimination_thm.instantiate(
{a:dividend.dividend, b:divisor})
elif isinstance(dividend, Add):
# Eliminate 'mod L' from each term.
eq = TransRelUpdater(expr)
_L = divisor
mod_terms = []
for _k, term in enumerate(dividend.terms):
if isinstance(term, Mod) and term.divisor==_L:
mod_terms.append(_k)
for _k in mod_terms:
# Use preserve_all=True for all but the last term.
preserve_all = (_k != mod_terms[-1])
_a = expr.dividend.terms[:_k]
_b = expr.dividend.terms[_k].dividend
_c = expr.dividend.terms[_k+1:]
_i = _a.num_elements()
_j = _c.num_elements()
expr = eq.update(
mod_elimination_in_sum_thm
.instantiate(
{i:_i, j:_j, a:_a, b:_b, c:_c, L:_L},
preserve_all=preserve_all))
return eq.relation
return Equals(expr, expr).conclude_via_reflexivity()
def simplification(self, assumptions=USE_DEFAULTS, automation=True):
if not automation:
return Expression.simplification(self,
assumptions,
automation=False)
from proveit.relation import TransRelUpdater
eq = TransRelUpdater(self, assumptions)
expr = eq.update(self.computation(assumptions))
expr = eq.update(expr.simplification(assumptions))
return eq.relation
def shallow_simplification(self, *, must_evaluate=False,
**defaults_config):
'''
Returns a proven simplification equation for this NumKet
expression assuming the operands have been simplified.
Currently deals only with:
(1) simplifying a NumKet with register size = 1 to a
simple Ket. It's not immediately clear that we always want
to do such a thing, but here goes.
'''
if self.size == one:
# from . import single_qubit_register_ket
from . import single_qubit_num_ket
return single_qubit_num_ket.instantiate(
{b: self.num}, preserve_all=True)
# Else simply return self=self.
# Establishing some minimal infrastructure
# for future development
expr = self
# for convenience updating our equation:
eq = TransRelUpdater(expr)
# Future processing possible here.
return eq.relation
def do_reduced_simplification(self, assumptions=USE_DEFAULTS):
'''
For trivial cases, a zero or one exponent or zero or one base,
derive and return this exponential expression equated with a
simplified form. Assumptions may be necessary to deduce
necessary conditions for the simplification.
'''
from proveit.logic import Equals, InSet
from proveit.numbers import one, two, Rational, Real, Abs
from proveit.relation import TransRelUpdater
from . import complex_x_to_first_power_is_x
if self.exponent == one:
return complex_x_to_first_power_is_x.instantiate({a: self.base})
if (isinstance(self.base, Exp) and isinstance(self.base.exponent, Div)
and self.base.exponent.numerator == one
and self.base.exponent.denominator == self.exponent):
from . import nth_power_of_nth_root
_n, _x = nth_power_of_nth_root.instance_params
return nth_power_of_nth_root.instantiate(
{
_n: self.exponent,
_x: self.base.base
},
assumptions=assumptions)
expr = self
# for convenience updating our equation:
eq = TransRelUpdater(expr, assumptions)
if self.exponent == two and isinstance(self.base, Abs):
from . import (square_abs_rational_simp, square_abs_real_simp)
# |a|^2 = a if a is real
rational_base = InSet(self.base, Rational).proven(assumptions)
real_base = InSet(self.base, Real).proven(assumptions)
thm = None
if rational_base:
thm = square_abs_rational_simp
elif real_base:
thm = square_abs_real_simp
if thm is not None:
simp = thm.instantiate({a: self.base.operand},
assumptions=assumptions)
expr = eq.update(simp)
# A further simplification may be possible after
# eliminating the absolute value.
expr = eq.update(expr.simplification(assumptions))
return eq.relation
def doReducedSimplification(self, assumptions=USE_DEFAULTS, **kwargs):
'''
Derive and return this negation expression equated with a simpler form.
Deals with double negation specifically.
'''
from proveit.relation import TransRelUpdater
expr = self
# For convenience updating our equation:
eq = TransRelUpdater(expr, assumptions)
# Handle double negation:
if isinstance(self.operand, Neg):
# simplify double negation
expr = eq.update(self.doubleNegSimplification(assumptions))
# simplify what is inside the double-negation.
expr = eq.update(expr.simplification(assumptions))
return eq.relation
def simplification(self, **defaults_config):
from proveit.relation import TransRelUpdater
from proveit.logic import And
expr = self
eq = TransRelUpdater(expr)
if True: #not self.value.is_simplified():
# Simplify the 'value'.
expr = eq.update(
expr.value_substitution(self.value.simplification()))
if isinstance(self.condition, And):
# Simplify the conditions.
if True: #any(not condition.is_simplified() for condition
# in self.condition.operands.entries):
inner_condition = expr.inner_expr().condition
expr = eq.update(inner_condition.simplification_of_operands())
elif True: #not self.condition.is_simplified():
# Simplify the condition.
expr = eq.update(
expr.condition_substitution(self.condition.simplification()))
# Perform a shallow simplifation on this Conditional.
# If the expression has not been reduced yet, no need for an
# "evaluation check" by the @prover decorator.
_no_eval_check = (expr == self)
eq.update(expr.shallow_simplification(_no_eval_check=_no_eval_check))
return eq.relation
def sub_expr_substitution(self, new_sub_exprs, **defaults_config):
'''
Given new sub-expressions to replace existing sub-expressions,
return the equality between this Expression and the new
one with the new sub-expressions.
'''
from proveit.logic import Equals
from proveit.relation import TransRelUpdater
assert len(new_sub_exprs) == 2, (
"Expecting 2 sub-expressions: operator and operands")
eq = TransRelUpdater(self)
expr = self
if new_sub_exprs[0] != self.sub_expr(0):
expr = eq.update(
expr.operator_substitution(
Equals(self.sub_expr(0), new_sub_exprs[0])))
if new_sub_exprs[1] != self.sub_expr(1):
expr = eq.update(
expr.operands_substitution(
Equals(self.sub_expr(1), new_sub_exprs[1])))
return eq.relation
def distribution(self, assumptions=USE_DEFAULTS):
'''
Distribute negation through a sum, deducing and returning
the equality between the original and distributed forms.
'''
from ._theorems_ import distributeNegThroughBinarySum
from ._theorems_ import distributeNegThroughSubtract, distributeNegThroughSum
from proveit.number import Add, num
from proveit.relation import TransRelUpdater
expr = self
eq = TransRelUpdater(
expr, assumptions) # for convenience updating our equation
if isinstance(self.operand, Add):
# Distribute negation through a sum.
add_expr = self.operand
if len(add_expr.operands) == 2:
# special case of 2 operands
if isinstance(add_expr.operands[1], Neg):
expr = eq.update(
distributeNegThroughSubtract.specialize(
{
a: add_expr.operands[0],
b: add_expr.operands[1].operand
},
assumptions=assumptions))
else:
expr = eq.update(
distributeNegThroughBinarySum.specialize(
{
a: add_expr.operands[0],
b: add_expr.operands[1]
},
assumptions=assumptions))
else:
# distribute the negation over the sum
expr = eq.update(distributeNegThroughSum.specialize({
n:
num(len(add_expr.operands)),
xx:
add_expr.operands
}),
assumptions=assumptions)
assert isinstance(
expr, Add
), "distributeNeg theorems are expected to yield an Add expression"
# check for double negation
for k, operand in enumerate(expr.operands):
assert isinstance(
operand, Neg
), "Each term from distributeNegThroughSum is expected to be negated"
if isinstance(operand.operand, Neg):
expr = eq.update(
expr.innerExpr().operands[k].doubleNegSimplification())
return eq.relation
else:
raise Exception(
'Only negation distribution through a sum or subtract is implemented'
)
def shallow_simplification(self, *, must_evaluate=False,
**defaults_config):
'''
Proves that this ExprTuple is equal to an ExprTuple
with ExprRanges reduced unless these are "preserved"
expressions.
'''
from proveit.relation import TransRelUpdater
from proveit import ExprRange
from proveit.logic import is_irreducible_value, EvaluationError
expr = self
eq = TransRelUpdater(expr)
if defaults.preserve_all:
# Preserve all sub-expressions -- don't simplify.
return eq.relation
_k = 0
for entry in self.entries:
if isinstance(entry, ExprRange):
if entry in defaults.preserved_exprs:
if must_evaluate:
# An ExprRange is not irreducible.
raise EvaluationError(self)
# Preserve this entry -- don't simplify it.
_k += ExprTuple(entry).num_entries()
continue
entry_simp = entry.reduction()
if must_evaluate and not is_irreducible_value(entry_simp.rhs):
raise EvaluationError(self)
if entry_simp.lhs != entry_simp.rhs:
substitution = expr.substitution(
entry_simp, start_idx=_k, preserve_all=True)
expr = eq.update(substitution)
_k += entry_simp.rhs.num_entries()
else:
if must_evaluate and not is_irreducible_value(entry):
raise EvaluationError(self)
_k += 1
return eq.relation
def conclude_via_inout_consolidation(self, **defaults_config):
'''
Prove this circuit equivalence by consolidating the input
(at the beginning) and/or output (at the end).
'''
lhs_cols, rhs_cols = (self.lhs.vert_expr_array.entries,
self.rhs.vert_expr_array.entries)
if len(lhs_cols) != len(rhs_cols):
raise ValueError(
"Cannot perform 'conclude_via_inout_consolidation' "
"if the number of columns is different: %s ≠ %s." %
(len(lhs_cols), len(rhs_cols)))
if not all(
lhs_col == rhs_col
for lhs_col, rhs_col in zip(lhs_cols[1:-1], rhs_cols[1:-1])):
raise ValueError(
"Cannot perform 'conclude_via_inout_consolidation' "
"if the inner columns do not all match.")
expr = self.lhs
# Use this updater to go from the lhs to the rhs via
# transitive equivalences.
equiv = TransRelUpdater(expr)
if lhs_cols[0] != rhs_cols[0]:
# The first column does not match, so try to consolidate
# inputs.
lhs_consolidation = Qcircuit(VertExprArray(
lhs_cols[0])).input_consolidation()
expr = equiv.update(lhs_consolidation.substitution(expr))
rhs_consolidation = Qcircuit(VertExprArray(
rhs_cols[0])).input_consolidation()
input_equiv = QcircuitEquiv(lhs_consolidation.rhs,
rhs_consolidation.rhs).prove()
expr = equiv.update(input_equiv.substitution(expr))
expr = equiv.update(rhs_consolidation.substitution(expr))
if lhs_cols[-1] != rhs_cols[-1]:
# The first column does not match, so try to consolidate
# inputs.
lhs_consolidation = Qcircuit(VertExprArray(
lhs_cols[-1])).output_consolidation()
expr = equiv.update(lhs_consolidation.substitution(expr))
rhs_consolidation = Qcircuit(VertExprArray(
rhs_cols[-1])).output_consolidation()
output_equiv = QcircuitEquiv(lhs_consolidation.rhs,
rhs_consolidation.rhs).prove()
expr = equiv.update(output_equiv.substitution(expr))
expr = equiv.update(rhs_consolidation.substitution(expr))
return equiv.relation
def simplification(self, assumptions=USE_DEFAULTS):
'''
Derive and return this negation expression equated with a simpler form.
Deals with double negation specifically.
'''
from proveit.number import zero
from proveit.logic import Equals
from ._theorems_ import doubleNegation
from proveit.relation import TransRelUpdater
# Handle double negation:
if isinstance(self.operand, Neg):
expr = self
eq = TransRelUpdater(
expr, assumptions) # for convenience updating our equation
# simplify double negation
expr = eq.update(self.doubleNegSimplification(assumptions))
# simplify what is inside the double-negation.
expr = eq.update(expr.simplification(assumptions))
return eq.relation
# otherwise, just use the default simplification
return Operation.simplification(self, assumptions)
def distribution(self, assumptions=USE_DEFAULTS):
'''
Distribute negation through a sum, deducing and returning
the equality between the original and distributed forms.
'''
from . import distribute_neg_through_binary_sum
from . import distribute_neg_through_subtract, distribute_neg_through_sum
from proveit.numbers import Add
from proveit.relation import TransRelUpdater
expr = self
# for convenience updating our equation
eq = TransRelUpdater(expr, assumptions)
if isinstance(self.operand, Add):
# Distribute negation through a sum.
add_expr = self.operand
if add_expr.operands.is_double():
# special case of 2 operands
if isinstance(add_expr.operands[1], Neg):
expr = eq.update(distribute_neg_through_subtract.instantiate(
{a: add_expr.operands[0], b: add_expr.operands[1].operand}, assumptions=assumptions))
else:
expr = eq.update(distribute_neg_through_binary_sum.instantiate(
{a: add_expr.operands[0], b: add_expr.operands[1]}, assumptions=assumptions))
else:
# distribute the negation over the sum
_x = add_expr.operands
_n = _x.num_elements(assumptions)
expr = eq.update(distribute_neg_through_sum.instantiate(
{n: _n, x: _x}), assumptions=assumptions)
assert isinstance(
expr, Add), "distribute_neg theorems are expected to yield an Add expression"
# check for double negation
for k, operand in enumerate(expr.operands):
assert isinstance(
operand, Neg), "Each term from distribute_neg_through_sum is expected to be negated"
if isinstance(operand.operand, Neg):
expr = eq.update(
expr.inner_expr().operands[k].double_neg_simplification())
return eq.relation
else:
raise Exception(
'Only negation distribution through a sum or subtract is implemented')
def factorization_of_scalars(self, **defaults_config):
'''
Prove equality with a Qmult in which the complex
numbers are pulled to the front andassociated together via Mult.
For example,
(a A b B c C d D e) = ((a*b*c*d*e) A B C D)
where a, b, c, d, and e are complex numbers, '*' denotes
number multiplication, and spaces, here, denote the Qmult
operation.
Also see scalar_mult_factorization.
'''
from . import (QmultCodomain, qmult_pulling_scalar_out_front,
qmult_pulling_scalars_out_front,
qmult_scalar_association)
expr = self
eq = TransRelUpdater(expr)
# First, lets prove the Qmult is well-formed and, in the
# process, ensure to know which operands are Complex.
if not InClass(self, QmultCodomain).proven():
QmultCodomain.membership_object(self).conclude()
# Go through the operands in reverse order so the complex
# factors will be in the original order out front in the end.
n_complex_entries = 0
for _k, operand in enumerate(reversed(self.operands.entries)):
_idx = self.operands.num_entries() - _k - 1 + n_complex_entries
if InSet(operand, Complex).proven():
# We have a complex number to pull out in front.
_A = expr.operands[:_idx]
_b = operand
_C = expr.operands[_idx + 1:]
_l = _A.num_elements()
_n = _C.num_elements()
expr = eq.update(
qmult_pulling_scalar_out_front.instantiate(
{
l: _l,
n: _n,
b: _b,
A: _A,
C: _C
}, preserve_all=True))
n_complex_entries += 1
elif isinstance(operand, ExprRange):
if ExprRange(operand.parameter, InSet(operand.body, Complex),
operand.true_start_index,
operand.true_end_index).proven():
# We have a range of complex numbers to pull out in
# front.
_A = expr.operands[:_idx]
_b = ExprTuple(operand)
_C = expr.operands[_idx + 1:]
_j = _b.num_elements()
_l = _A.num_elements()
_n = _C.num_elements()
thm = qmult_pulling_scalars_out_front
expr = eq.update(
thm.instantiate(
{
j: _j,
l: _l,
n: _n,
b: _b,
A: _A,
C: _C
},
preserve_all=True))
n_complex_entries += 1
# Associate the complex numbers, now out in front.
_b = expr.operands[:n_complex_entries]
_A = expr.operands[n_complex_entries:]
_j = _b.num_elements()
_l = _A.num_elements()
if (_b.num_entries() > 0 and not _b.is_single()
and _A.num_entries() > 0):
expr = eq.update(
qmult_scalar_association.instantiate(
{
j: _j,
l: _l,
b: _b,
A: _A
}, preserve_expr=expr))
# The multiplication of complex numbers is complex.
expr.operands[0].deduce_in_number_set(Complex)
return eq.relation
def merger(self, **defaults_config):
'''
If this is an tuple of expressions that can be directly merged
together into a single ExprRange, return this proven
equivalence. For example,
{j \in Natural, k-(j+1) \in Natural}
|- (x_1, .., x_j, x_{j+1}, x_{j+2}, ..., x_k) = (x_1, ..., x_k)
'''
from proveit._core_.expression.lambda_expr import (
Lambda, ArgumentExtractionError)
from .expr_range import ExprRange, simplified_index
from proveit.relation import TransRelUpdater
from proveit.core_expr_types.tuples import (
merge, merge_front, merge_back, merge_extension,
merge_pair, merge_series)
from proveit import f, i, j, k, l, x
from proveit.numbers import one, Add, subtract
# A convenience to allow successive update to the equation via
# transitivities (starting with self=self).
eq = TransRelUpdater(self)
# Determine the position of the first ExprRange item and get the
# lambda map.
first_range_pos = len(self.entries)
lambda_map = None
for _k, item in enumerate(self):
if isinstance(item, ExprRange):
lambda_map = Lambda(item.lambda_map.parameter,
item.lambda_map.body)
first_range_pos = _k
break
if 1 < first_range_pos:
if lambda_map is None:
raise NotImplementedError("Means of extracting a lambda "
"map has not been implemented")
pass # need the lambda map
# Collapse singular items at the beginning.
front_singles = ExprTuple(eq.expr[:first_range_pos])
i_sub = lambda_map.extract_argument(front_singles[0])
j_sub = lambda_map.extract_argument(front_singles[-1])
if len(front_singles.entries) == 2:
# Merge a pair of singular items.
front_merger = merge_pair.instantiate(
{f: lambda_map, i: i_sub, j: j_sub})
else:
# Merge a series of singular items in one shot.
front_merger = merge_series.instantiate(
{f: lambda_map, x: front_singles, i: i_sub, j: j_sub})
eq.update(
front_merger.substitution(
self.inner_expr()[:first_range_pos]))
if eq.expr.num_entries() == 1:
# We have accomplished a merger down to one item.
return eq.relation
if eq.expr.num_entries() == 2:
# Merge a pair.
if isinstance(eq.expr[0], ExprRange):
if isinstance(eq.expr[1], ExprRange):
# Merge a pair of ExprRanges.
item = eq.expr[1]
other_lambda_map = Lambda(item.lambda_map.parameter,
item.lambda_map.body)
if other_lambda_map != lambda_map:
raise ExprTupleError(
"Cannot merge together ExprRanges "
"with different lambda maps: %s vs %s" %
(lambda_map, other_lambda_map))
_i, _j = eq.expr[0].true_start_index, eq.expr[0].true_end_index
_k, _l = eq.expr[1].true_start_index, eq.expr[1].true_end_index
merger = \
merge.instantiate(
{f: lambda_map, i: _i, j: _j, k: _k, l: _l})
else:
# Merge an ExprRange and a singular item.
_i, _j = eq.expr[0].true_start_index, eq.expr[0].true_end_index
try:
_k = lambda_map.extract_argument(eq.expr[1])
except ArgumentExtractionError:
_k = simplified_index(Add(_j, one))
if _k == Add(_j, one):
merger = merge_extension.instantiate(
{f: lambda_map, i: _i, j: _j})
else:
merger = merge_back.instantiate(
{f: lambda_map, i: _i, j: _j, k: _k})
else:
# Merge a singular item and ExprRange.
_i = simplified_index(
subtract(eq.expr[1].true_start_index, one))
_j, _k = eq.expr[1].true_start_index, eq.expr[1].true_end_index
merger = \
merge_front.instantiate({f: lambda_map, i: _i,
j: _j, k: _k})
all_decreasing = all(expr.is_decreasing() for expr in eq.expr
if isinstance(expr, ExprRange))
if all_decreasing:
# Apply the 'decreasing' order style to match what we
# had originally.
for _i in (0, 1):
if isinstance(eq.expr[_i], ExprRange):
merger = (merger.inner_expr().lhs[_i]
.with_decreasing_order())
merger = merger.inner_expr().rhs[0].with_decreasing_order()
eq.update(merger)
return eq.relation
while eq.expr.num_entries() > 1:
front_merger = ExprTuple(*eq.expr[:2].entries).merger()
eq.update(front_merger.substitution(
eq.expr.inner_expr()[:2]))
return eq.relation
def deduce_equality(self, equality, *,
eq_via_elem_eq_thm=None, **defaults_config):
from proveit import ExprRange
from proveit import a, b, i
from proveit.logic import Equals
from proveit.core_expr_types.tuples import tuple_eq_via_elem_eq
from proveit.relation import TransRelUpdater
if not isinstance(equality, Equals):
raise ValueError("The 'equality' should be an Equals expression")
if equality.lhs != self:
raise ValueError("The left side of 'equality' should be 'self'")
from proveit.numbers import num, one
# Handle the special counting cases. For example,
# (1, 2, 3, 4) = (1, ..., 4)
_n = len(self.entries)
if all(self[_k] == num(_k + 1) for _k in range(_n)):
if (isinstance(equality.rhs, ExprTuple)
and equality.rhs.num_entries() == 1
and isinstance(equality.rhs[0], ExprRange)):
expr_range = equality.rhs[0]
if (expr_range.true_start_index == one and
expr_range.true_end_index == num(_n)):
if len(self.entries) >= 10:
raise NotImplementedError("counting range equality "
"not implemented for more "
"then 10 elements")
import proveit.numbers.numerals.decimals
equiv_thm = proveit.numbers.numerals.decimals\
.__getattr__('count_to_%d_range' % _n)
return equiv_thm
lhs, rhs = equality.lhs, equality.rhs
if (lhs.num_entries() == rhs.num_entries() == 1
and isinstance(lhs[0], ExprRange)
and isinstance(rhs[0], ExprRange)):
# Prove the equality of two ExprRanges.
r_range = rhs[0]
expr = lhs
if expr[0].is_decreasing():
# We could handle different styles later, but
# let's be consistent with increasing order for now
# to make this easier to implement.
expr = expr.inner_expr()[0].with_increasing_order()
eq = TransRelUpdater(expr)
if expr[0].true_start_index != r_range.true_start_index:
# Shift indices so they have the same start.
expr = eq.update(expr[0].shift_equivalence(
new_start=r_range.true_start_index))
if expr[0].lambda_map != r_range.lambda_map:
# Change the lambda map.
expr = eq.update(expr[0].range_fn_transformation(
r_range.lambda_map))
if expr[0].true_end_index != r_range.true_end_index:
# Make the end indices be the same:
end_eq = Equals(expr[0].true_end_index, r_range.true_end_index).prove()
expr = eq.update(end_eq.substitution(
expr.inner_expr()[0].true_end_index))
return eq.relation
# Try tuple_eq_via_elem_eq as the last resort.
_i = lhs.num_elements()
_a = lhs
_b = rhs
if eq_via_elem_eq_thm is None:
eq_via_elem_eq_thm = tuple_eq_via_elem_eq
return eq_via_elem_eq_thm.instantiate({i:_i, a:_a, b:_b})
def bundle(expr, bundle_thm, num_levels=2, *, assumptions=USE_DEFAULTS):
'''
Given a nested OperationOverInstances, derive or equate an
equivalent form in which a given number of nested levels is
bundled together. Use the given theorem specific to the particular
OperationOverInstances.
For example,
\forall_{x, y | Q(x, y)} \forall_{z | R(z)} P(x, y, z)
can become
\forall_{x, y, z | Q(x, y), R(z)} P(x, y, z)
via bundle with num_levels=2.
For example of the form of the theorem required, see
proveit.logic.boolean.quantification.bundling or
proveit.logic.boolean.quantification.bundling_equality.
'''
from proveit.relation import TransRelUpdater
from proveit.logic import Implies, Equals
# Make a TransRelUpdater only if the bundle_thm yield an
# equation, in which case we'll want the result to be an equation.
eq = None
bundled = expr
while num_levels >= 2:
if (not isinstance(bundled, OperationOverInstances) or
not isinstance(bundled.instanceExpr, OperationOverInstances)):
raise ValueError(
"May only 'bundle' nested OperationOverInstances, "
"not %s" % bundled)
_m = bundled.instanceParams.length()
_n = bundled.instanceExpr.instanceParams.length()
_P = bundled.instanceExpr.instanceExpr
_Q = bundled.effectiveCondition()
_R = bundled.instanceExpr.effectiveCondition()
m, n = bundle_thm.instanceVars
P, Q, R = bundle_thm.instanceExpr.instanceVars
correspondence = bundle_thm.instanceExpr.instanceExpr
if isinstance(correspondence, Implies):
if (not isinstance(correspondence.antecedent,
OperationOverInstances)
or not len(correspondence.consequent.instanceParams) == 2):
raise ValueError("'bundle_thm', %s, does not have the "
"expected form with the bundled form as "
"the consequent of the implication, %s" %
(bundle_thm, correspondence))
x_1_to_m, y_1_to_n = correspondence.consequent.instanceParams
elif isinstance(correspondence, Equals):
if not isinstance(
correspondence.rhs, OperationOverInstances
or not len(correspondence.antecedent.instanceParams) == 2):
raise ValueError("'bundle_thm', %s, does not have the "
"expected form with the bundled form on "
"right of the an equality, %s" %
(bundle_thm, correspondence))
x_1_to_m, y_1_to_n = correspondence.rhs.instanceParams
all_params = bundled.instanceParams + bundled.instanceExpr.instanceParams
Pxy = Function(P, all_params)
Qx = Function(Q, bundled.instanceParams)
Rxy = Function(R, all_params)
x_1_to_m = x_1_to_m.replaced({m: _m})
y_1_to_n = y_1_to_n.replaced({n: _n})
instantiation = bundle_thm.instantiate(
{
m: _m,
n: _n,
ExprTuple(x_1_to_m): bundled.instanceParams,
ExprTuple(y_1_to_n): bundled.instanceExpr.instanceParams,
Pxy: _P,
Qx: _Q,
Rxy: _R
},
assumptions=assumptions)
if isinstance(instantiation.expr, Implies):
bundled = instantiation.deriveConsequent()
elif isinstance(instantiation.expr, Equals):
if eq is None:
eq = TransRelUpdater(bundled)
try:
bundled = eq.update(instantiation)
except ValueError:
raise ValueError(
"Instantiation of bundle_thm %s is %s but "
"should match %s on one side of the equation." %
(bundle_thm, instantiation, bundled))
else:
raise ValueError("Instantiation of bundle_thm %s is %s but "
"should be an Implies or Equals expression." %
(bundle_thm, instantiation))
num_levels -= 1
if eq is None:
# Return the bundled result.
return bundled
else:
# Return the equality between the original expression and
# the bundled result.
return eq.relation
def merger(self, assumptions=USE_DEFAULTS):
'''
If this is an tuple of expressions that can be directly merged
together into a single ExprRange, return this proven
equivalence. For example,
{j \in Naturals, k-(j+1) \in Naturals}
|- (x_1, .., x_j, x_{j+1}, x_{j+2}, ..., x_k) = (x_1, ..., x_k)
'''
from proveit._core_.expression.lambda_expr import Lambda
from .expr_range import ExprRange
from proveit.relation import TransRelUpdater
from proveit.core_expr_types.tuples._theorems_ import (
merge, merge_front, merge_back, merge_extension, merge_pair,
merge_series)
from proveit._common_ import f, i, j, k, l, x
from proveit.number import Add, one
# A convenience to allow successive update to the equation via
# transitivities (starting with self=self).
eq = TransRelUpdater(self, assumptions)
# Determine the position of the first ExprRange item and get the
# lambda map.
first_range_pos = len(self)
lambda_map = None
for _k, item in enumerate(self):
if isinstance(item, ExprRange):
lambda_map = Lambda(item.lambda_map.parameter,
item.lambda_map.body)
first_range_pos = _k
break
if 1 < first_range_pos:
if lambda_map is None:
raise NotImplementedError("Means of extracting a lambda "
"map has not been implemented")
pass # need the lambda map
# Collapse singular items at the beginning.
front_singles = ExprTuple(eq.expr[:first_range_pos])
i_sub = lambda_map.extractArgument(front_singles[0])
j_sub = lambda_map.extractArgument(front_singles[-1])
if len(front_singles) == 2:
# Merge a pair of singular items.
front_merger = merge_pair.specialize(
{
f: lambda_map,
i: i_sub,
j: j_sub
},
assumptions=assumptions)
else:
# Merge a series of singular items in one shot.
front_merger = merge_series.specialize(
{
f: lambda_map,
x: front_singles,
i: i_sub,
j: j_sub
},
assumptions=assumptions)
eq.update(
front_merger.substitution(self.innerExpr()[:first_range_pos],
assumptions=assumptions))
if len(eq.expr) == 1:
# We have accomplished a merger down to one item.
return eq.relation
if len(eq.expr) == 2:
# Merge a pair.
if isinstance(eq.expr[0], ExprRange):
if isinstance(eq.expr[1], ExprRange):
# Merge a pair of ExprRanges.
item = eq.expr[1]
other_lambda_map = Lambda(item.lambda_map.parameter,
item.lambda_map.body)
if other_lambda_map != lambda_map:
raise ExprTupleError(
"Cannot merge together ExprRanges "
"with different lambda maps: %s vs %s" %
(lambda_map, other_lambda_map))
_i, _j = eq.expr[0].start_index, eq.expr[0].end_index
_k, _l = eq.expr[1].start_index, eq.expr[1].end_index
merger = \
merge.specialize({f:lambda_map, i:_i, j:_j, k:_k, l:_l},
assumptions=assumptions)
else:
# Merge an ExprRange and a singular item.
_i, _j = eq.expr[0].start_index, eq.expr[0].end_index
_k = lambda_map.extractArgument(eq.expr[1])
if _k == Add(_j, one):
merger = merge_extension.specialize(
{
f: lambda_map,
i: _i,
j: _j
},
assumptions=assumptions)
else:
merger = merge_back.specialize(
{
f: lambda_map,
i: _i,
j: _j,
k: _k
},
assumptions=assumptions)
else:
# Merge a singular item and ExprRange.
iSub = lambda_map.extractArgument(eq.expr[0])
jSub, kSub = eq.expr[1].start_index, eq.expr[1].end_index
merger = \
merge_front.specialize({f:lambda_map, i:iSub, j:jSub,
k:kSub}, assumptions=assumptions)
eq.update(merger)
return eq.relation
while len(eq.expr) > 1:
front_merger = ExprTuple(*eq.expr[:2]).merger(assumptions)
eq.update(
front_merger.substitution(eq.expr.innerExpr(assumptions)[:2],
assumptions=assumptions))
return eq.relation
def shallow_simplification(self, *, must_evaluate=False,
**defaults_config):
'''
Returns a proven simplification equation for this Exp
expression assuming the operands have been simplified.
Handles the following evaluations:
a^0 = 1 for any complex a
0^x = 0 for any positive x
1^x = 1 for any complex x
a^(Log(a, x)) = x for RealPos a and x, a != 1.
x^n = x*x*...*x = ? for a natural n and irreducible x.
Handles a zero or one exponent or zero or one base as
simplifications.
'''
from proveit.relation import TransRelUpdater
from proveit.logic import EvaluationError, is_irreducible_value
from proveit.logic import InSet
from proveit.numbers import (zero, one, two, is_literal_int,
is_literal_rational,
Log, Rational, Abs)
from . import (exp_zero_eq_one, exponentiated_zero,
exponentiated_one, exp_nat_pos_expansion)
if self.is_irreducible_value():
# already irreducible
return Equals(self, self).conclude_via_reflexivity()
if must_evaluate:
if not all(is_irreducible_value(operand) for
operand in self.operands):
for operand in self.operands:
if not is_irreducible_value(operand):
# The simplification of the operands may not have
# worked hard enough. Let's work harder if we
# must evaluate.
operand.evaluation()
return self.evaluation()
if self.exponent == zero:
return exp_zero_eq_one.instantiate({a: self.base}) # =1
elif self.base == zero:
# Will fail if the exponent is not positive, but this
# is the only sensible thing to try.
return exponentiated_zero.instantiate({x: self.exponent}) # =0
elif self.exponent == one:
return self.power_of_one_reduction()
elif self.base == one:
return exponentiated_one.instantiate({x: self.exponent}) # =1
elif (isinstance(self.base, Exp) and
isinstance(self.base.exponent, Div) and
self.base.exponent.numerator == one and
self.base.exponent.denominator == self.exponent):
from . import nth_power_of_nth_root
_n, _x = nth_power_of_nth_root.instance_params
return nth_power_of_nth_root.instantiate(
{_n: self.exponent, _x: self.base.base})
elif (isinstance(self.base, Exp) and
isinstance(self.exponent, Div) and
self.exponent.numerator == one and
self.exponent.denominator == self.base.exponent):
from . import nth_root_of_nth_power, sqrt_of_square
_n = self.base.exponent
_x = self.base.base
if _n == two:
return sqrt_of_square.instantiate({x: _x})
return nth_root_of_nth_power.instantiate({n: _n, x: _x})
elif (is_literal_rational(self.base) and
is_literal_int(self.exponent) and
self.exponent.as_int() > 1):
expr = self
eq = TransRelUpdater(expr)
expr = eq.update(exp_nat_pos_expansion.instantiate(
{x:self.base, n:self.exponent}, preserve_all=True))
# We should come up with a better way of reducing
# ExprRanges representing repetitions:
_n = self.exponent.as_int()
if _n <= 0 or _n > 9:
raise NotImplementedError("Currently only implemented for 1-9")
repetition_thm = proveit.numbers.numerals.decimals \
.__getattr__('reduce_%s_repeats' % _n)
rep_reduction = repetition_thm.instantiate({x: self.base})
expr = eq.update(expr.inner_expr().operands.substitution(
rep_reduction.rhs, preserve_all=True))
expr = eq.update(expr.evaluation())
return eq.relation
elif (isinstance(self.exponent, Log)
and self.base == self.exponent.base):
# base_ns = self.base.deduce_number_set()
# antilog_ns = self.exponent.antilog.deduce_number_set()
if (InSet(self.base, RealPos).proven()
and InSet(self.exponent.antilog, RealPos).proven()
and NotEquals(self.base, one).proven()):
return self.power_of_log_reduction()
expr = self
# for convenience updating our equation:
eq = TransRelUpdater(expr)
if self.exponent == two and isinstance(self.base, Abs):
from . import (square_abs_rational_simp,
square_abs_real_simp)
# |a|^2 = a if a is real
try:
deduce_number_set(self.base)
except UnsatisfiedPrerequisites:
pass
rational_base = InSet(self.base, Rational).proven()
real_base = InSet(self.base, Real).proven()
thm = None
if rational_base:
thm = square_abs_rational_simp
elif real_base:
thm = square_abs_real_simp
if thm is not None:
simp = thm.instantiate({a: self.base.operand})
expr = eq.update(simp)
# A further simplification may be possible after
# eliminating the absolute value.
expr = eq.update(expr.simplification())
return eq.relation
def deduce_bound(self,
inner_expr_bound_or_bounds,
inner_exprs_to_bound=None,
**defaults_config):
'''
Return a bound of this arithmetic expression based upon
the bounds of any number of inner expressions. The inner
expression should appear on the left side of the corresponding
bound which should be a number ordering relation (< or <=).
The returned, proven bound will have this expression on the
left-hand side. The bounds of the inner expressions will be
processed in the order they are provided.
If inner_exprs_to_bound is provided, restrict the bounding
to these particular InnerExpr objects. Otherwise, all inner
expressions are fair game.
'''
if isinstance(inner_expr_bound_or_bounds, Judgment):
inner_expr_bound_or_bounds = inner_expr_bound_or_bounds.expr
if isinstance(inner_expr_bound_or_bounds, ExprTuple):
inner_expr_bounds = inner_expr_bound_or_bounds.entries
elif isinstance(inner_expr_bound_or_bounds, Expression):
inner_expr_bounds = [inner_expr_bound_or_bounds]
else:
inner_expr_bounds = inner_expr_bound_or_bounds
inner_expr_bounds = deque(inner_expr_bounds)
inner_relations = dict()
if len(inner_expr_bounds) == 0:
raise ValueError("Expecting one or more 'inner_expr_bounds'")
while len(inner_expr_bounds) > 0:
inner_expr_bound = inner_expr_bounds.popleft()
print('inner_expr_bound', inner_expr_bound)
if isinstance(inner_expr_bound, TransRelUpdater):
# May be one of the internally generated
# TransRelUpdater for percolating bounds up through
# the expression hierarchy to the root.
inner_expr_bound = inner_expr_bound.relation
elif isinstance(inner_expr_bound, Judgment):
inner_expr_bound = inner_expr_bound.expr
inner = inner_expr_bound.lhs
if inner == self:
raise ValueError(
"Why supply a bound for the full expression when "
"calling 'deduce_bound'? There is nothing to deduce.")
no_such_inner_expr = True
no_such_number_op_inner_expr = True
# Apply bound to each inner expression as applicable.
if inner_exprs_to_bound is None:
inner_exprs = generate_inner_expressions(self, inner)
else:
inner_exprs = inner_exprs_to_bound
for inner_expr in inner_exprs:
no_such_inner_expr = False
inner_expr_depth = len(inner_expr.expr_hierarchy)
assert inner_expr_depth > 1, (
"We already checked that the inner expression was not "
"equal to the full expression. What's the deal?")
# Create/update the relation for the container of this
# inner expression.
if inner_expr_depth >= 3:
container = inner_expr.expr_hierarchy[-2]
if isinstance(container, ExprTuple):
# Skip an ExprTuple layer.
if inner_expr_depth >= 4:
container = inner_expr.expr_hierarchy[-3]
else:
container = self
else:
container = self
if not isinstance(container, NumberOperation):
# Skip over any 'container' that is not a
# NumberOperation.
continue
no_such_number_op_inner_expr = False
container_relation = inner_relations.setdefault(
container, TransRelUpdater(container))
expr = container_relation.expr
# Don't simplify or make replacements if there
# is more to go:
preserve_all = (len(inner_expr_bounds) > 0)
container_relation.update(
expr.bound_via_operand_bound(inner_expr_bound,
preserve_all=preserve_all))
# Append the relation for processing
if container is self:
# No further processing needed when the container
continue # is self.
if (len(inner_expr_bounds) == 0
or inner_expr_bounds[-1] != container_relation):
inner_expr_bounds.append(container_relation)
if no_such_inner_expr:
raise ValueError(
"The left side of %s does not appear within %s" %
(inner_expr_bound, self))
if no_such_number_op_inner_expr:
raise ValueError(
"The left side of %s is not contained within a "
"NumberOperation expression" % (inner_expr_bound, self))
assert self in inner_relations, (
"If there are more than one inner bounds and they are "
"valid, they should have percolated to the top")
return inner_relations[self].relation
def unbundle(expr,
unbundle_thm,
num_param_entries=(1, ),
*,
assumptions=USE_DEFAULTS):
'''
Given a nested OperationOverInstances, derive or equate an
equivalent form in which the parameter entries are split in
number according to 'num_param_entries'. Use the given theorem
specific to the particular OperationOverInstances.
For example,
\forall_{x, y, z | Q(x, y), R(z)} P(x, y, z)
can become
\forall_{x, y | Q(x, y)} \forall_{z | R(z)} P(x, y, z)
via bundle with num_param_entries=(2, 1) or
num_param_entries=(2,) -- the last number can be implied
by the remaining number of parameters.
For example of the form of the theorem required, see
proveit.logic.boolean.quantification.unbundling or
proveit.logic.boolean.quantification.bundling_equality.
'''
from proveit.relation import TransRelUpdater
from proveit.logic import Implies, Equals, And
# Make a TransRelUpdater only if the bundle_thm yield an
# equation, in which case we'll want the result to be an equation.
eq = None
unbundled = expr
net_indicated_param_entries = sum(num_param_entries)
num_actual_param_entries = len(expr.instanceParams)
for n in num_param_entries:
if not isinstance(n, int) or n <= 0:
raise ValueError(
"Each of 'num_param_entries', must be an "
"integer greater than 0. %s fails this requirement." %
(num_param_entries))
if net_indicated_param_entries > num_actual_param_entries:
raise ValueError(
"Sum of 'num_param_entries', %s=%d should not "
"be greater than the number of parameter entries "
"of %s for unbundling." %
(num_param_entries, net_indicated_param_entries, expr))
if net_indicated_param_entries < num_actual_param_entries:
diff = num_actual_param_entries - net_indicated_param_entries
num_param_entries = list(num_param_entries) + [diff]
else:
num_param_entries = list(num_param_entries)
while len(num_param_entries) > 1:
n_last_entries = num_param_entries.pop(-1)
first_params = ExprTuple(*unbundled.instanceParams[:-n_last_entries])
first_param_vars = {getParamVar(param) for param in first_params}
remaining_params = \
ExprTuple(*unbundled.instanceParams[-n_last_entries:])
_m = first_params.length()
_n = remaining_params.length()
_P = unbundled.instanceExpr
# Split up the conditions between the outer
# OperationOverInstances and inner OperationOverInstances
condition = unbundled.effectiveCondition()
if isinstance(condition, And):
_nQ = 0
for cond in condition.operands:
cond_vars = free_vars(cond, err_inclusively=True)
if first_param_vars.isdisjoint(cond_vars): break
_nQ += 1
if _nQ == 0:
_Q = And()
elif _nQ == 1:
_Q = condition.operands[0]
else:
_Q = And(*condition.operands[:_nQ])
_nR = len(condition.operands) - _nQ
if _nR == 0:
_R = And()
elif _nR == 1:
_R = condition.operands[-1]
else:
_R = And(*condition.operands[_nQ:])
elif first_param_vars.isdisjoint(
free_vars(condition, err_inclusively=True)):
_Q = condition
_R = And()
else:
_Q = And()
_R = condition
m, n = unbundle_thm.instanceVars
P, Q, R = unbundle_thm.instanceExpr.instanceVars
correspondence = unbundle_thm.instanceExpr.instanceExpr
if isinstance(correspondence, Implies):
if (not isinstance(correspondence.antecedent,
OperationOverInstances)
or not len(correspondence.antecedent.instanceParams) == 2):
raise ValueError("'unbundle_thm', %s, does not have the "
"expected form with the bundled form as "
"the antecedent of the implication, %s" %
(unbundle_thm, correspondence))
x_1_to_m, y_1_to_n = correspondence.antecedent.instanceParams
elif isinstance(correspondence, Equals):
if not isinstance(
correspondence.rhs, OperationOverInstances
or not len(correspondence.antecedent.instanceParams) == 2):
raise ValueError("'unbundle_thm', %s, does not have the "
"expected form with the bundled form on "
"right of the an equality, %s" %
(unbundle_thm, correspondence))
x_1_to_m, y_1_to_n = correspondence.rhs.instanceParams
else:
raise ValueError("'unbundle_thm', %s, does not have the expected "
"form with an equality or implication "
"correspondence, %s" %
(unbundle_thm, correspondence))
Qx = Function(Q, first_params)
Rxy = Function(R, unbundled.instanceParams)
Pxy = Function(P, unbundled.instanceParams)
x_1_to_m = x_1_to_m.replaced({m: _m})
y_1_to_n = y_1_to_n.replaced({n: _n})
instantiation = unbundle_thm.instantiate(
{
m: _m,
n: _n,
ExprTuple(x_1_to_m): first_params,
ExprTuple(y_1_to_n): remaining_params,
Pxy: _P,
Qx: _Q,
Rxy: _R
},
assumptions=assumptions)
if isinstance(instantiation.expr, Implies):
unbundled = instantiation.deriveConsequent()
elif isinstance(instantiation.expr, Equals):
if eq is None:
eq = TransRelUpdater(unbundled)
try:
unbundled = eq.update(instantiation)
except ValueError:
raise ValueError(
"Instantiation of bundle_thm %s is %s but "
"should match %s on one side of the equation." %
(unbundle_thm, instantiation, unbundled))
else:
raise ValueError("Instantiation of bundle_thm %s is %s but "
"should be an Implies or Equals expression." %
(unbundle_thm, instantiation))
if eq is None:
# Return the unbundled result.
return unbundled
else:
# Return the equality between the original expression and
# the unbundled result.
return eq.relation
def shallow_simplification(self,
*,
must_evaluate=False,
**defaults_config):
'''
Returns a proven simplification equation for this Qmult
expression assuming the operands have been simplified.
Currently deals only with:
(1) simplify unary case
(2) Ungrouping nested tensor products.
(3) Factoring out scalars.
'''
from proveit.linear_algebra import ScalarMult
from proveit.physics.quantum import (Bra, Ket, HilbertSpaces, varphi,
var_ket_psi)
from proveit.physics.quantum.algebra import Hspace
yield_known_hilbert_spaces = HilbertSpaces.yield_known_hilbert_spaces
if self.operands.is_single():
# Handle unary cases
from . import (qmult_of_ket, qmult_of_bra, qmult_of_complex,
qmult_of_linmap)
operand = self.operands[0]
if InSet(operand, Complex).proven():
# Qmult of a complex number is just the complex number
return qmult_of_complex.instantiate({c: operand})
elif isinstance(operand, Bra):
# Qmult of a bra is the bra (equal to the
# linear map it represents).
for _Hspace in yield_known_hilbert_spaces(Ket(
operand.operand)):
return qmult_of_bra.instantiate({
Hspace: _Hspace,
varphi: operand.operand
})
elif operand in MatrixSpace.known_memberships:
# Qmult of a matrix. It equates to a lambda
# map, but this isn't considered a simplification.
# Use linmap_reduction instead if this is desired.
return Equals(self, self).conclude_via_reflexivity()
else:
for _Hspace in yield_known_hilbert_spaces(operand):
# Qmult of a ket is just the ket
return qmult_of_ket.instantiate({
Hspace: _Hspace,
var_ket_psi: operand
})
for linmap in containing_hilbert_space_linmap_sets(operand):
# Qmult of a linear map is just the linear map
_Hspace, _X = linmap.from_vspace, linmap.to_vspace
return qmult_of_linmap.instantiate({
Hspace: _Hspace,
X: _X,
A: operand
})
return Equals(self, self).conclude_via_reflexivity()
# for convenience updating our equation:
expr = self
eq = TransRelUpdater(expr)
if Qmult._simplification_directives_.ungroup:
# ungroup the expression (disassociate nested additions).
_n = 0
length = expr.operands.num_entries() - 1
# loop through all operands
while _n < length:
operand = expr.operands[_n]
# print("n, length", n, length)
if isinstance(operand, Qmult):
# if it is grouped, ungroup it
expr = eq.update(expr.disassociation(_n,
preserve_all=True))
elif isinstance(operand, ScalarMult):
# ungroup contained ScalarMult's
expr = eq.update(
expr.scalar_mult_absorption(_n, preserve_all=True))
length = expr.operands.num_entries()
_n += 1
if Qmult._simplification_directives_.factor_scalars:
# Next, pull out scalar factors
expr = eq.update(expr.factorization_of_scalars())
if Qmult._simplification_directives_.use_scalar_mult:
# Finally, use ScalarMult for any scalar operands.
if (isinstance(expr, Qmult) and expr.operands.num_entries() > 1
and InSet(expr.operands[0], Complex).proven()):
expr = eq.update(expr.scalar_mult_factorization())
return eq.relation