def reduceOperands(innerExpr,
inPlace=True,
mustEvaluate=False,
assumptions=USE_DEFAULTS):
'''
Attempt to return an InnerExpr object that is provably equivalent to
the given innerExpr but with simplified operands at the inner-expression
level.
If inPlace is True, the top-level expression must be a KnownTruth
and the simplified KnownTruth is derived instead of an equivalence
relation.
If mustEvaluate is True, the simplified
operands must be irreducible values (see isIrreducibleValue).
'''
# Any of the operands that can be simplified must be replaced with their evaluation
from proveit import InnerExpr
assert isinstance(
innerExpr,
InnerExpr), "Expecting 'innerExpr' to be of type 'InnerExpr'"
inner = innerExpr.exprHierarchy[-1]
while True:
allReduced = True
for operand in inner.operands:
if not mustEvaluate or not isIrreducibleValue(operand):
# the operand is not an irreducible value so it must be evaluated
operandEval = operand.evaluation(
assumptions=assumptions
) if mustEvaluate else operand.simplification(
assumptions=assumptions)
if mustEvaluate and not isIrreducibleValue(operandEval.rhs):
raise EvaluationError(
'Evaluations expected to be irreducible values')
if operandEval.lhs != operandEval.rhs:
# compose map to replace all instances of the operand within the inner expression
lambdaMap = innerExpr.replMap().compose(
Lambda.globalRepl(inner, operand))
# substitute in the evaluated value
if inPlace:
innerExpr = InnerExpr(
operandEval.subRightSideInto(lambdaMap),
innerExpr.innerExprPath)
else:
innerExpr = InnerExpr(
operandEval.substitution(lambdaMap).rhs,
innerExpr.innerExprPath)
allReduced = False
break # start over (there may have been multiple substitutions)
if allReduced: return innerExpr
inner = innerExpr.exprHierarchy[-1]
def evaluation(self, assumptions=USE_DEFAULTS):
'''
Given operands that may be evaluated to irreducible values that
may be compared, or if there is a known evaluation of this
equality, derive and return this expression equated to
TRUE or FALSE.
'''
if self.lhs == self.rhs:
# prove equality is true by reflexivity
return evaluateTruth(self.prove().expr, assumptions=[])
if isIrreducibleValue(self.lhs) and isIrreducibleValue(self.rhs):
# Irreducible values must know how to evaluate the equality
# between each other, where appropriate.
return self.lhs.evalEquality(self.rhs)
return TransitiveRelation.evaluation(self, assumptions)
def sideEffects(self, knownTruth):
'''
Record the knownTruth in Equals.knownEqualities, associated from
the left hand side and the right hand side. This information may
be useful for concluding new equations via transitivity.
If the right hand side is an "irreducible value" (see
isIrreducibleValue), also record it in Equals.evaluations for use
when the evaluation method is called. Some side-effects
derivations are also attempted depending upon the form of
this equality.
'''
from proveit.logic.boolean._common_ import TRUE, FALSE
Equals.knownEqualities.setdefault(self.lhs, set()).add(knownTruth)
Equals.knownEqualities.setdefault(self.rhs, set()).add(knownTruth)
if isIrreducibleValue(self.rhs):
Equals.simplifications.setdefault(self.lhs, set()).add(knownTruth)
Equals.evaluations.setdefault(self.lhs, set()).add(knownTruth)
if (self.lhs != self.rhs):
# automatically derive the reversed form which is equivalent
yield self.deriveReversed
if self.rhs == FALSE:
# derive lhs => FALSE from lhs = FALSE
yield self.deriveContradiction
# derive lhs from Not(lhs) = FALSE, if self is in this form
#yield self.deriveViaFalsifiedNegation
if self.rhs in (TRUE, FALSE):
# automatically derive A from A=TRUE or Not(A) from A=FALSE
yield self.deriveViaBooleanEquality
if hasattr(self.lhs, 'equalitySideEffects'):
for sideEffect in self.lhs.equalitySideEffects(knownTruth):
yield sideEffect
def conclude(self, assumptions):
'''
Attempt to conclude the equality various ways:
simple reflexivity (x=x), via an evaluation (if one side is an irreducible),
or via transitivity.
'''
from proveit.logic import TRUE, FALSE, Implies, Iff
if self.lhs == self.rhs:
# Trivial x=x
return self.concludeViaReflexivity()
if self.lhs or self.rhs in (TRUE, FALSE):
try:
# Try to conclude as TRUE or FALSE.
return self.concludeBooleanEquality(assumptions)
except ProofFailure:
pass
if isIrreducibleValue(self.rhs):
return self.lhs.evaluation()
elif isIrreducibleValue(self.lhs):
return self.rhs.evaluation().deriveReversed()
try:
Implies(self.lhs, self.rhs).prove(assumptions, automation=False)
Implies(self.rhs, self.lhs).prove(assumptions, automation=False)
# lhs => rhs and rhs => lhs, so attempt to prove lhs = rhs via lhs <=> rhs
# which works when they can both be proven to be Booleans.
try:
return Iff(self.lhs, self.rhs).deriveEquality(assumptions)
except:
from proveit.logic.boolean.implication._theorems_ import eqFromMutualImpl
return eqFromMutualImpl.specialize({
A: self.lhs,
B: self.rhs
},
assumptions=assumptions)
except ProofFailure:
pass
"""
# Use concludeEquality if available
if hasattr(self.lhs, 'concludeEquality'):
return self.lhs.concludeEquality(assumptions)
if hasattr(self.rhs, 'concludeEquality'):
return self.rhs.concludeEquality(assumptions)
"""
# Use a breadth-first search approach to find the shortest
# path to get from one end-point to the other.
return TransitiveRelation.conclude(self, assumptions)
def sideEffects(self, knownTruth):
'''
Record the knownTruth in Equals.knownEqualities, associated from
the left hand side and the right hand side. This information
may be useful for concluding new equations via transitivity.
If the right hand side is an "irreducible value" (see
isIrreducibleValue), also record it in
Equals.known_evaluation_sets for use when the evaluation
method is called. Some side-effects derivations are also
attempted depending upon the form of this equality.
If the rhs is an "irreducible value" (see isIrreducibleValue),
also record the knownTruth in the Equals.known_simplifications
and Equals.known_evaluation_sets dictionaries, for use when the
simplification or evaluation method is called. The key for the
known_simplifications dictionary is the specific *combination*
of the lhs expression along with the assumptions in the form
(expr, tuple(sorted(assumptions))); the key for the
known_evaluation_sets dictionary is just the lhs expression
without the specific assumptions. Some side-effects
derivations are also attempted depending upon the form of this
equality.
'''
from proveit.logic.boolean._common_ import TRUE, FALSE
Equals.knownEqualities.setdefault(self.lhs, set()).add(knownTruth)
Equals.knownEqualities.setdefault(self.rhs, set()).add(knownTruth)
if isIrreducibleValue(self.rhs):
assumptions_sorted = sorted(knownTruth.assumptions,
key=lambda expr: hash(expr))
lhsKey = (self.lhs, tuple(assumptions_sorted))
# n.b.: the values in the known_simplifications
# dictionary consist of single KnownTruths not sets
Equals.known_simplifications[lhsKey] = knownTruth
Equals.known_evaluation_sets.setdefault(self.lhs,
set()).add(knownTruth)
if (self.lhs != self.rhs):
# automatically derive the reversed form which is equivalent
yield self.deriveReversed
if self.rhs == FALSE:
try:
self.lhs.prove(automation=False)
# derive FALSE given lhs=FALSE and lhs.
yield self.deriveContradiction
except ProofFailure:
pass
# Use this form after merging in 'Expression.proven' commite:
#if self.lhs.proven(): # If lhs is proven using default assumptions.
# # derive FALSE given lhs=FALSE and lhs.
# yield self.deriveContradiction
if self.rhs in (TRUE, FALSE):
# automatically derive A from A=TRUE or Not(A) from A=FALSE
yield self.deriveViaBooleanEquality
if hasattr(self.lhs, 'equalitySideEffects'):
for sideEffect in self.lhs.equalitySideEffects(knownTruth):
yield sideEffect
def conclude(self, assumptions):
from proveit.logic import FALSE
if isIrreducibleValue(self.lhs) and isIrreducibleValue(self.rhs):
# prove that two irreducible values are not equal
return self.lhs.notEqual(self.rhs)
if self.lhs == FALSE or self.rhs == FALSE:
try:
# prove something is not false by proving it to be true
return self.concludeViaDoubleNegation(assumptions)
except:
pass
if hasattr(self.lhs, 'notEquals') and isIrreducibleValue(self.rhs):
try:
return self.lhs.notEqual(self.rhs, assumptions)
except:
pass
try:
return self.concludeAsFolded(assumptions)
except:
return Operation.conclude(assumptions) # try the default (reduction)
def evaluation(self, assumptions=USE_DEFAULTS):
'''
If possible, return a KnownTruth of this expression equal to its
irreducible value. Checks for an existing evaluation. If it
doesn't exist, try some default strategies including a reduction.
Attempt the Expression-class-specific "doReducedEvaluation"
when necessary.
'''
from proveit.logic import Equals, defaultSimplification, SimplificationError
from proveit import KnownTruth, ProofFailure
from proveit.logic.irreducible_value import isIrreducibleValue
method_called = None
try:
# First try the default tricks. If a reduction succesfully occurs,
# evaluation will be called on that reduction.
evaluation = defaultSimplification(self.innerExpr(),
mustEvaluate=True,
assumptions=assumptions)
method_called = defaultSimplification
except SimplificationError as e:
# The default failed, let's try the Expression-class specific version.
try:
evaluation = self.doReducedEvaluation(assumptions)
method_called = self.doReducedEvaluation
except NotImplementedError:
# We have nothing but the default evaluation strategy to try, and that failed.
raise e
if not isinstance(evaluation, KnownTruth) or not isinstance(
evaluation.expr, Equals):
msg = ("%s must return an KnownTruth, "
"not %s for %s assuming %s" %
(method_called, evaluation, self, assumptions))
raise ValueError(msg)
if evaluation.lhs != self:
msg = ("%s must return an KnownTruth "
"equality with self on the left side, "
"not %s for %s assuming %s" %
(method_called, evaluation, self, assumptions))
raise ValueError(msg)
if not isIrreducibleValue(evaluation.rhs):
msg = ("%s must return an KnownTruth "
"equality with an irreducible value on the right side, "
"not %s for %s assuming %s" %
(method_name, evaluation, self, assumptions))
raise ValueError(msg)
# Note: No need to store in Equals.evaluations or Equals.simplifications; this
# is done automatically as a side-effect for proven equalities with irreducible
# right sides.
return evaluation
def defaultSimplification(innerExpr,
inPlace=False,
mustEvaluate=False,
operandsOnly=False,
assumptions=USE_DEFAULTS,
automation=True):
'''
Default attempt to simplify the given inner expression under the
given assumptions. If successful, returns a KnownTruth (using a
subset of the given assumptions) that expresses an equality between
the expression (on the left) and one with a simplified form for the
"inner" part (in some canonical sense determined by the Operation).
If inPlace is True, the top-level expression must be a KnownTruth
and the simplified KnownTruth is derived instead of an equivalence
relation.
If mustEvaluate=True, the simplified form must be an irreducible
value (see isIrreducibleValue). Specifically, this method checks to
see if an appropriate simplification/evaluation has already been
proven. If not, but if it is an Operation, call the
simplification/evaluation method on all operands, make these
substitions, then call simplification/evaluation on the expression
with operands substituted for simplified forms. It also treats,
as a special case, evaluating the expression to be true if it is in
the set of assumptions [also see KnownTruth.evaluation and
evaluateTruth]. If operandsOnly = True, only simplify the operands
of the inner expression.
'''
# among other things, convert any assumptions=None
# to assumptions=() to avoid len(None) errors
assumptions = defaults.checkedAssumptions(assumptions)
from proveit.logic import TRUE, FALSE
from proveit.logic.boolean._axioms_ import trueAxiom
topLevel = innerExpr.exprHierarchy[0]
inner = innerExpr.exprHierarchy[-1]
if operandsOnly:
# Just do the reduction of the operands at the level below the
# "inner expression"
reducedInnerExpr = reduceOperands(innerExpr, inPlace, mustEvaluate,
assumptions)
if inPlace:
try:
return reducedInnerExpr.exprHierarchy[0].prove(
assumptions, automation=False)
except:
assert False
try:
eq = Equals(topLevel, reducedInnerExpr.exprHierarchy[0])
return eq.prove(assumptions, automation=False)
except:
assert False
def innerSimplification(innerEquivalence):
if inPlace:
return innerEquivalence.subRightSideInto(innerExpr,
assumptions=assumptions)
return innerEquivalence.substitution(innerExpr,
assumptions=assumptions)
if isIrreducibleValue(inner):
return Equals(inner, inner).prove()
assumptionsSet = set(defaults.checkedAssumptions(assumptions))
# See if the expression is already known to be true as a special
# case.
try:
inner.prove(assumptionsSet, automation=False)
trueEval = evaluateTruth(inner, assumptionsSet) # A=TRUE given A
if inner == topLevel:
if inPlace: return trueAxiom
else: return trueEval
return innerSimplification(trueEval)
except:
pass
# See if the negation of the expression is already known to be true
# as a special case.
try:
inner.disprove(assumptionsSet, automation=False)
falseEval = evaluateFalsehood(inner,
assumptionsSet) # A=FALSE given Not(A)
return innerSimplification(falseEval)
except:
pass
# ================================================================ #
# See if the expression already has a proven simplification #
# ================================================================ #
# construct the key for the known_simplifications dictionary
assumptions_sorted = sorted(assumptions, key=lambda expr: hash(expr))
known_simplifications_key = (inner, tuple(assumptions_sorted))
if (mustEvaluate and inner in Equals.known_evaluation_sets):
evaluations = Equals.known_evaluation_sets[inner]
candidates = []
for knownTruth in evaluations:
if knownTruth.isSufficient(assumptionsSet):
# Found existing evaluation suitable for the assumptions
candidates.append(knownTruth)
if len(candidates) >= 1:
# Return the "best" candidate with respect to fewest number
# of steps.
min_key = lambda knownTruth: knownTruth.proof().numSteps()
simplification = min(candidates, key=min_key)
return innerSimplification(simplification)
elif (not mustEvaluate
and known_simplifications_key in Equals.known_simplifications):
simplification = Equals.known_simplifications[
known_simplifications_key]
return innerSimplification(simplification)
# ================================================================ #
if not automation:
msg = 'Unknown evaluation (without automation): ' + str(inner)
raise SimplificationError(msg)
# See if the expression is equal to something that has an evaluation
# or is already known to be true.
if inner in Equals.knownEqualities:
for knownEq in Equals.knownEqualities[inner]:
try:
if knownEq.isSufficient(assumptionsSet):
if inPlace:
# Should first substitute in the known
# equivalence then simplify that.
if inner == knownEq.lhs:
knownEq.subRightSideInto(innerExpr, assumptions)
elif inner == knownEq.rhs:
knownEq.subLeftSideInto(innerExpr, assumptions)
# Use mustEvaluate=True. Simply being equal to
# something simplified isn't necessarily the
# appropriate simplification for "inner" itself.
alt_inner = knownEq.otherSide(inner).innerExpr()
equivSimp = \
defaultSimplification(alt_inner, inPlace=inPlace,
mustEvaluate=True,
assumptions=assumptions,
automation=False)
if inPlace:
# Returns KnownTruth with simplification:
return equivSimp
innerEquiv = knownEq.applyTransitivity(
equivSimp, assumptions)
if inner == topLevel: return innerEquiv
return innerEquiv.substitution(innerExpr,
assumptions=assumptions)
except SimplificationError:
pass
# try to simplify via reduction
if not isinstance(inner, Operation):
if mustEvaluate:
raise EvaluationError('Unknown evaluation: ' + str(inner),
assumptions)
else:
# don't know how to simplify, so keep it the same
return innerSimplification(Equals(inner, inner).prove())
reducedInnerExpr = reduceOperands(innerExpr, inPlace, mustEvaluate,
assumptions)
if reducedInnerExpr == innerExpr:
if mustEvaluate:
# Since it wasn't irreducible to begin with, it must change
# in order to evaluate.
raise SimplificationError('Unable to evaluate: ' + str(inner))
else:
raise SimplificationError('Unable to simplify: ' + str(inner))
# evaluate/simplify the reduced inner expression
inner = reducedInnerExpr.exprHierarchy[-1]
if mustEvaluate:
innerEquiv = inner.evaluation(assumptions)
else:
innerEquiv = inner.simplification(assumptions)
value = innerEquiv.rhs
if value == TRUE:
# Attempt to evaluate via proving the expression;
# This should result in a shorter proof if allowed
# (e.g., if theorems are usable).
try:
evaluateTruth(inner, assumptions)
except:
pass
if value == FALSE:
# Attempt to evaluate via disproving the expression;
# This should result in a shorter proof if allowed
# (e.g., if theorems are usable).
try:
evaluateFalsehood(inner, assumptions)
except:
pass
reducedSimplification = innerSimplification(innerEquiv)
if inPlace:
simplification = reducedSimplification
else:
# Via transitivity, go from the original expression to the
# reduced expression (simplified inner operands) and then the
# final simplification (simplified inner expression).
reduced_top_level = reducedInnerExpr.exprHierarchy[0]
eq1 = Equals(topLevel, reduced_top_level)
eq1.prove(assumptions, automation=False)
eq2 = Equals(reduced_top_level, reducedSimplification.rhs)
eq2.prove(assumptions, automation=False)
simplification = eq1.applyTransitivity(eq2, assumptions)
if not inPlace and topLevel == inner:
# Store direct simplifications in the known_simplifications
# dictionary for next time.
Equals.known_simplifications[
known_simplifications_key] = simplification
if isIrreducibleValue(value):
# also store it in the known_evaluation_sets dictionary for
# next time, since it evaluated to an irreducible value.
Equals.known_evaluation_sets.setdefault(topLevel,
set()).add(simplification)
return simplification
def reduceOperands(innerExpr,
inPlace=True,
mustEvaluate=False,
assumptions=USE_DEFAULTS):
'''
Attempt to return an InnerExpr object that is provably equivalent to
the given innerExpr but with simplified operands at the
inner-expression level.
If inPlace is True, the top-level expression must be a KnownTruth
and the simplified KnownTruth is derived instead of an equivalence
relation.
If mustEvaluate is True, the simplified
operands must be irreducible values (see isIrreducibleValue).
'''
# Any of the operands that can be simplified must be replaced with
# their simplification.
from proveit import InnerExpr, ExprRange
assert isinstance(innerExpr, InnerExpr), \
"Expecting 'innerExpr' to be of type 'InnerExpr'"
inner = innerExpr.exprHierarchy[-1]
substitutions = []
while True:
allReduced = True
for operand in inner.operands:
if (not isIrreducibleValue(operand)
and not isinstance(operand, ExprRange)):
# The operand isn't already irreducible, so try to
# simplify it.
if mustEvaluate:
operandEval = operand.evaluation(assumptions=assumptions)
else:
operandEval = operand.simplification(
assumptions=assumptions)
if mustEvaluate and not isIrreducibleValue(operandEval.rhs):
msg = 'Evaluations expected to be irreducible values'
raise EvaluationError(msg, assumptions)
if operandEval.lhs != operandEval.rhs:
# Compose map to replace all instances of the
# operand within the inner expression.
global_repl = Lambda.globalRepl(inner, operand)
lambdaMap = innerExpr.repl_lambda().compose(global_repl)
# substitute in the evaluated value
if inPlace:
subbed = operandEval.subRightSideInto(lambdaMap)
innerExpr = InnerExpr(subbed, innerExpr.innerExprPath)
else:
sub = operandEval.substitution(lambdaMap)
innerExpr = InnerExpr(sub.rhs, innerExpr.innerExprPath)
substitutions.append(sub)
allReduced = False
# Start over since there may have been multiple
# substitutions:
break
if allReduced: break # done!
inner = innerExpr.exprHierarchy[-1]
if not inPlace and len(substitutions) > 1:
# When there have been multiple substitutions, apply
# transtivity over the chain of substitutions to equate the
# end-points.
Equals.applyTransitivities(substitutions, assumptions)
return innerExpr
def conclude(self, assumptions):
'''
Attempt to conclude the equality various ways:
simple reflexivity (x=x), via an evaluation (if one side is an
irreducible), or via transitivity.
'''
from proveit.logic import TRUE, FALSE, Implies, Iff, inBool
if self.lhs == self.rhs:
# Trivial x=x
return self.concludeViaReflexivity()
if self.lhs or self.rhs in (TRUE, FALSE):
try:
# Try to conclude as TRUE or FALSE.
return self.concludeBooleanEquality(assumptions)
except ProofFailure:
pass
if isIrreducibleValue(self.rhs):
try:
evaluation = self.lhs.evaluation(assumptions)
if evaluation.rhs != self.rhs:
raise ProofFailure(
self, assumptions,
"Does not match with evaluation: %s" % str(evaluation))
return evaluation
except EvaluationError as e:
raise ProofFailure(self, assumptions,
"Evaluation error: %s" % e.message)
elif isIrreducibleValue(self.lhs):
try:
evaluation = self.rhs.evaluation(assumptions)
if evaluation.rhs != self.lhs:
raise ProofFailure(
self, assumptions,
"Does not match with evaluation: %s" % str(evaluation))
return evaluation.deriveReversed()
except EvaluationError as e:
raise ProofFailure(self, assumptions,
"Evaluation error: %s" % e.message)
if (Implies(self.lhs, self.rhs).proven(assumptions)
and Implies(self.rhs, self.lhs).proven(assumptions)
and inBool(self.lhs).proven(assumptions)
and inBool(self.rhs).proven(assumptions)):
# There is mutual implication both sides are known to be
# boolean. Conclude equality via mutual implication.
return Iff(self.lhs, self.rhs).deriveEquality(assumptions)
if hasattr(self.lhs, 'deduceEquality'):
# If there is a 'deduceEquality' method, use that.
# The responsibility then shifts to that method for
# determining what strategies should be attempted
# (with the recommendation that it should not attempt
# multiple non-trivial automation strategies).
eq = self.lhs.deduceEquality(self, assumptions)
if eq.expr != self:
raise ValueError("'deduceEquality' not implemented "
"correctly; must deduce the 'equality' "
"that it is given if it can: "
"'%s' != '%s'" % (eq.expr, self))
return eq
else:
'''
If there is no 'deduceEquality' method, we'll try
simplifying each side to see if they are equal.
'''
# Try to prove equality via simplifying both sides.
lhs_simplification = self.lhs.simplification(assumptions)
rhs_simplification = self.rhs.simplification(assumptions)
simplified_lhs = lhs_simplification.rhs
simplified_rhs = rhs_simplification.rhs
try:
if simplified_lhs != self.lhs or simplified_rhs != self.rhs:
simplified_eq = Equals(simplified_lhs,
simplified_rhs).prove(assumptions)
return Equals.applyTransitivities([
lhs_simplification, simplified_eq, rhs_simplification
], assumptions)
except ProofFailure:
pass
raise ProofFailure(
self, assumptions, "Unable to automatically conclude by "
"standard means. To try to prove this via "
"transitive implication relations, try "
"'concludeViaTransitivity'.")
