def join(self, secondSummation, assumptions=frozenset()):
'''
Join the "second summation" with "this" summation, deducing and returning
the equivalence of these summations added with the joined summation.
Both summation must be over Intervals.
The relation between the first summation upper bound, UB1, and the second
summation lower bound, LB2 must be explicitly either UB1 = LB2-1 or LB2=UB1+1.
'''
from theorems import sumSplitAfter, sumSplitBefore
from proveit.number.common import one
from proveit.number import Sub, Add
if not isinstance(self.domain, Interval) or not isinstance(secondSummation.domain, Interval):
raise Exception('Sum joining only implemented for Interval domains')
if self.summand != secondSummation.summand:
raise Exception('Sum joining only allowed when the summands are the same')
if self.domain.upperBound == Sub(secondSummation.domain.lowerBound, one):
sumSplit = sumSplitBefore
splitIndex = secondSummation.domain.lowerBound
elif secondSummation.domain.lowerBound == Add(self.domain.upperBound, one):
sumSplit = sumSplitAfter
splitIndex = self.domain.upperBound
else:
raise Exception('Sum joining only implemented when there is an explicit increment of one from the upper bound and the second summations lower bound')
lowerBound, upperBound = self.domain.lowerBound, secondSummation.domain.upperBound
deduceInIntegers(lowerBound, assumptions)
deduceInIntegers(upperBound, assumptions)
deduceInIntegers(splitIndex, assumptions)
return sumSplit.specialize({Operation(f, self.instanceVars):self.summand}).specialize({a:lowerBound, b:splitIndex, c:upperBound, x:self.indices[0]}).deriveReversed()
greaterThanEqualsInBools = Forall([a, b],
InSet(GreaterThanEquals(a, b), Booleans),
domain=Reals)
greaterThanEqualsInBools
notEqualsIsLessThanOrGreaterThan = Forall((a, x),
Or(LessThan(x, a), GreaterThan(x,
a)),
domain=Reals,
conditions=[NotEquals(x, a)])
notEqualsIsLessThanOrGreaterThan
shiftLessThanToLessThanEquals = Forall((a, b),
LessThanEquals(a, b),
domain=Integers,
conditions=[LessThan(Sub(a, one), b)])
shiftLessThanToLessThanEquals
lessThanEqualsAddRight = Forall([a, b, c],
LessThanEquals(Add(a, c), Add(b, c)),
domain=Reals,
conditions=[LessThanEquals(a, b)])
lessThanEqualsAddRight
lessThanEqualsAddLeft = Forall([a, b, c],
LessThanEquals(Add(c, a), Add(c, b)),
domain=Reals,
conditions=[LessThanEquals(a, b)])
lessThanEqualsAddLeft
lessThanEqualsSubtract = Forall([a, b, c],
from proveit import Etcetera
from proveit.logic import Forall, InSet, Equals, NotEquals, Iff, And, SetOfAll
from proveit.number import Integers, Interval, Reals, RealsPos, Complexes
from proveit.number import Abs, Mod, ModAbs, GreaterThanEquals, LessThanEquals, Add, Sub, Neg, Mult, Fraction, IntervalCO
from proveit.common import a, b, c, x, y, N, xEtc, xMulti
from proveit.number.common import zero, one
from proveit import beginTheorems, endTheorems
beginTheorems(locals())
modIntClosure = Forall((a, b), InSet(Mod(a, b), Integers), domain=Integers)
modIntClosure
modInInterval = Forall((a, b),
InSet(Mod(a, b), Interval(zero, Sub(b, one))),
domain=Integers)
modInInterval
modRealClosure = Forall((a, b), InSet(Mod(a, b), Reals), domain=Reals)
modRealClosure
modAbsRealClosure = Forall((a, b), InSet(ModAbs(a, b), Reals), domain=Reals)
modAbsRealClosure
absComplexClosure = Forall([a], InSet(Abs(a), Reals), domain=Complexes)
absComplexClosure
absNonzeroClosure = Forall([a],
InSet(Abs(a), RealsPos),
domain=Complexes,
conditions=[NotEquals(a, zero)])
# m: Random variable for the measurement of Psi as an integer from the register's binary representation.
m_ = Literal(pkg, 'm')
# phase_m: Random variable for the phase result of the quantum phase estimation.
# phase_m = m / 2^t
phase_m_ = Literal(pkg, 'phase_m', {LATEX: r'\varphi_m'})
# b: The "best" outcome of m such that phase_m is as close as possible to phase.
b_ = Literal(pkg, 'b')
# 2^t
two_pow_t = Exp(two, t_)
# 2^{t-1}
two_pow_t_minus_one = Exp(two, Sub(t_, one))
# amplitude of output register as indexted
alpha_ = Literal(pkg, 'alpha', {STRING: 'alpha', LATEX: r'\alpha'})
alpha_l = SubIndexed(alpha_, l)
abs_alpha_l = Abs(alpha_l)
alpha_l_sqrd = Exp(Abs(alpha_l), two)
# delta: difference between the phase and the best phase_m
delta_ = Literal(pkg, 'delta', {LATEX: r'\delta'})
fullDomain = Interval(Add(Neg(Exp(two, Sub(t_, one))), one),
Exp(two, Sub(t_, one)))
negDomain = Interval(Add(Neg(two_pow_t_minus_one), one), Neg(Add(eps, one)))
posDomain = Interval(Add(eps, one), two_pow_t_minus_one)
epsDomain = Interval(one, Sub(two_pow_t_minus_one, two))
greaterThanEqualsInBools = Forall([a, b],
InSet(GreaterEq(a, b), Booleans),
domain=Reals)
greaterThanEqualsInBools
notEqualsIsLessThanOrGreaterThan = Forall((a, x),
Or(Less(x, a), Greater(x, a)),
domain=Reals,
conditions=[NotEquals(x, a)])
notEqualsIsLessThanOrGreaterThan
shiftLessThanToLessThanEquals = Forall((a, b),
LessEq(a, b),
domain=Integers,
conditions=[Less(Sub(a, one), b)])
shiftLessThanToLessThanEquals
lessThanEqualsAddRight = Forall([a, b, c],
LessEq(Add(a, c), Add(b, c)),
domain=Reals,
conditions=[LessEq(a, b)])
lessThanEqualsAddRight
lessThanEqualsAddLeft = Forall([a, b, c],
LessEq(Add(c, a), Add(c, b)),
domain=Reals,
conditions=[LessEq(a, b)])
lessThanEqualsAddLeft
lessThanEqualsSubtract = Forall([a, b, c],
addOneRightInExpRev
addOneLeftInExp = Forall([a, b],
Equals(Exp(a, Add(one, b)), Mult(a, Exp(a, b))),
domain=Complexes,
conditions=[NotEquals(a, zero)])
addOneLeftInExp
addOneLeftInExpRev = Forall([a, b],
Equals(Mult(a, Exp(a, b)), Exp(a, Add(one, b))),
domain=Complexes,
conditions=[NotEquals(a, zero)])
addOneLeftInExpRev
diffInExp = Forall([a, b, c],
Equals(Exp(a, Sub(b, c)), Mult(Exp(a, b), Exp(a, Neg(c)))),
domain=Complexes,
conditions=[NotEquals(a, zero)])
diffInExp
diffInExpRev = Forall([a, b, c],
Equals(Mult(Exp(a, b), Exp(a, Neg(c))),
Exp(a, Sub(b, c))),
domain=Complexes,
conditions=[NotEquals(a, zero)])
diffInExpRev
diffFracInExp = Forall([a, b, c, d],
Equals(Exp(a, Sub(b, frac(c, d))),
Mult(Exp(a, b), Exp(a, frac(Neg(c), d)))),
domain=Complexes,
domain=Complexes,
conditions=[NotEquals(y, zero)])
distributeFractionThroughSum
distributeFractionThroughSumRev = Forall([xEtc, y],
Equals(
Add(Etcetera(Fraction(xMulti,
y))),
Fraction(Add(xEtc), y)),
domain=Complexes,
conditions=[NotEquals(y, zero)])
distributeFractionThroughSumRev
distributeFractionThroughSubtract = Forall([x, y, z],
Equals(
Fraction(Sub(x, y), z),
Sub(Fraction(x, z),
Fraction(y, z))),
domain=Complexes,
conditions=[NotEquals(z, zero)])
distributeFractionThroughSubtract
distributeFractionThroughSubtractRev = Forall([x, y, z],
Equals(
Sub(Fraction(x, z),
Fraction(y, z)),
Fraction(Sub(x, y), z)),
domain=Complexes,
conditions=[NotEquals(z, zero)])
distributeFractionThroughSubtractRev
Forall([a, b, c],
Equals(
Sum(x, fx, Interval(a, c)),
Add(Sum(x, fx, Interval(a, b)),
Sum(x, fx, Interval(Add(b, one), c)))),
domain=Integers,
conditions=[LessThanEquals(a, b),
LessThan(b, c)]))
sumSplitAfter
sumSplitBefore = Forall(
f,
Forall([a, b, c],
Equals(
Sum(x, fx, Interval(a, c)),
Add(Sum(x, fx, Interval(a, Sub(b, one))),
Sum(x, fx, Interval(b, c)))),
domain=Integers,
conditions=[LessThan(a, b), LessThanEquals(b, c)]))
sumSplitBefore
sumSplitFirst = Forall(
f,
Forall([a, b],
Equals(Sum(x, fx, Interval(a, b)),
Add(fa, Sum(x, fx, Interval(Add(a, one), b)))),
domain=Integers,
conditions=[LessThan(a, b)]))
sumSplitFirst
sumZeroAndOne = Forall(
distributeThroughSum = Forall([xEtc, yEtc, zEtc],
Equals(Mult(xEtc, Add(yEtc), zEtc),
Add(Etcetera(Mult(xEtc, yMulti, zEtc)))),
domain=Complexes)
distributeThroughSum
distributeThroughSumRev = Forall([xEtc, yEtc, zEtc],
Equals(
Add(Etcetera(Mult(xEtc, yMulti, zEtc))),
Mult(xEtc, Add(yEtc), zEtc)),
domain=Complexes)
distributeThroughSumRev
distributeThroughSubtract = Forall([wEtc, x, y, zEtc],
Equals(
Mult(wEtc, Sub(x, y), zEtc),
Sub(Mult(wEtc, x, zEtc),
Mult(wEtc, y, zEtc))),
domain=Complexes)
distributeThroughSubtract
distributeThroughSubtractRev = Forall([wEtc, x, y, zEtc],
Equals(
Sub(Mult(wEtc, x, zEtc),
Mult(wEtc, y, zEtc)),
Mult(wEtc, Sub(x, y), zEtc)),
domain=Complexes)
distributeThroughSubtractRev
distributeThroughSummation = Forall(
[P, S],
addOneLeftInExp = Forall([a,b],
Equals(Exp(a,Add(one, b)),
Mult(a, Exp(a,b))),
domain = Complexes, conditions=[NotEquals(a, zero)])
addOneLeftInExp
addOneLeftInExpRev = Forall([a,b],
Equals(Mult(a, Exp(a,b)),
Exp(a,Add(one, b))),
domain = Complexes, conditions=[NotEquals(a, zero)])
addOneLeftInExpRev
diffInExp = Forall([a,b,c],
Equals(Exp(a,Sub(b,c)),
Mult(Exp(a,b),Exp(a,Neg(c)))),
domain = Complexes, conditions=[NotEquals(a, zero)])
diffInExp
diffInExpRev = Forall([a,b,c],
Equals(Mult(Exp(a,b),Exp(a,Neg(c))),
Exp(a,Sub(b,c))),
domain = Complexes, conditions=[NotEquals(a, zero)])
diffInExpRev
diffFracInExp = Forall([a,b,c,d],
Equals(Exp(a,Sub(b,Fraction(c, d))),
Mult(Exp(a,b),Exp(a,Fraction(Neg(c), d)))),
domain = Complexes, conditions=[NotEquals(a, zero), NotEquals(d, zero)])
from proveit.logic import Forall, Equals, And, InSet
from proveit.number import Floor, Sub, IntervalCO, Integers, Reals
from proveit.common import x
from proveit.number.common import zero, one
from proveit import beginAxioms, endAxioms
beginAxioms(locals())
floorDef = Forall(x,
And(InSet(Floor(x), Integers),
InSet(Sub(x, Floor(x)), IntervalCO(zero, one))),
domain=Reals)
floorDef
endAxioms(locals(), __package__)
negNotEqZero
distributeNegThroughSum = Forall([xEtc],
Equals(Neg(Add(xEtc)),
Add(Etcetera(Neg(xMulti)))),
domain=Complexes)
distributeNegThroughSum
distributeNegThroughSumRev = Forall([xEtc],
Equals(Add(Etcetera(Neg(xMulti))),
Neg(Add(xEtc))),
domain=Complexes)
distributeNegThroughSumRev
distributeNegThroughSubtract = Forall([x, y],
Equals(Neg(Sub(x, y)), Add(Neg(x), y)),
domain=Complexes)
distributeNegThroughSubtract
negTimesPos = Forall([x, y],
Equals(Mult(Neg(x), y), Neg(Mult(x, y))),
domain=Complexes)
negTimesPos
negTimesPosRev = Forall([x, y],
Equals(Neg(Mult(x, y)), Mult(Neg(x), y)),
domain=Complexes)
negTimesPosRev
posTimesNeg = Forall([x, y],
Equals(Mult(x, Neg(y)), Neg(Mult(x, y))),
from proveit import Etcetera from proveit.logic import Forall, InSet, NotEquals, Equals from proveit.number import Sub, Naturals, NaturalsPos, Integers, Reals, Complexes, Add, Neg, GreaterThan, GreaterThanEquals from proveit.common import a, b, w, x, y, z, xEtc, yEtc, vEtc, wEtc, zEtc, yMulti from proveit.number.common import zero, one from proveit import beginTheorems, endTheorems beginTheorems(locals()) subtractIntClosure = Forall([a, b], InSet(Sub(a, b), Integers), domain=Integers) subtractIntClosure subtractClosureNats = Forall([a, b], InSet(Sub(a, b), Naturals), domain=Integers, conditions=[GreaterThanEquals(a, b)]) subtractClosureNats subtractClosureNatsPos = Forall([a, b], InSet(Sub(a, b), NaturalsPos), domain=Integers, conditions=[GreaterThan(a, b)]) subtractClosureNatsPos subtractComplexClosure = Forall([a, b], InSet(Sub(a, b), Complexes), domain=Complexes) subtractComplexClosure subtractRealClosure = Forall([a, b], InSet(Sub(a, b), Reals), domain=Reals) subtractRealClosure subtractOneInNats = Forall(a, InSet(Sub(a, one), Naturals), domain=NaturalsPos) subtractOneInNats diffNotEqZero = Forall((a, b), NotEquals(Sub(a, b), zero), domain=Complexes, conditions=[NotEquals(a, b)]) diffNotEqZero subtractAsAddNeg = Forall([x, y], Equals(Sub(x, y),
from proveit.logic import Forall, Equals
from proveit.number import Sum, Integers, Interval, LessThan, Add, Sub
from proveit.common import a, b, f, x, fa, fb, fx
from proveit.number.common import one
from proveit import beginAxioms, endAxioms
beginAxioms(locals())
sumSingle = Forall(f, Forall(a,
Equals(Sum(x,fx,Interval(a,a)),
fa),
domain=Integers))
sumSingle
sumSplitLast = Forall(f,
Forall([a,b],
Equals(Sum(x,fx,Interval(a,b)),
Add(Sum(x,fx,Interval(a,Sub(b, one))),
fb)),
domain=Integers, conditions=[LessThan(a, b)]))
sumSplitLast
endAxioms(locals(), __package__)
lessThanInBools = Forall([a, b], InSet(LessThan(a, b), Booleans), domain=Reals) lessThanInBools lessThanEqualsInBools = Forall([a, b], InSet(LessThanEquals(a, b), Booleans), domain=Reals) lessThanEqualsInBools greaterThanInBools = Forall([a, b], InSet(GreaterThan(a, b), Booleans), domain=Reals) greaterThanInBools greaterThanEqualsInBools = Forall([a, b], InSet(GreaterThanEquals(a, b), Booleans), domain=Reals) greaterThanEqualsInBools notEqualsIsLessThanOrGreaterThan = Forall((a, x), Or(LessThan(x, a), GreaterThan(x, a)), domain=Reals, conditions=[NotEquals(x, a)]) notEqualsIsLessThanOrGreaterThan shiftLessThanToLessThanEquals = Forall((a, b), LessThanEquals(a, b), domain=Integers, conditions=[LessThan(Sub(a, one), b)]) shiftLessThanToLessThanEquals lessThanEqualsAddRight = Forall([a, b, c], LessThanEquals(Add(a, c), Add(b, c)), domain=Reals, conditions=[LessThanEquals(a, b)]) lessThanEqualsAddRight lessThanEqualsAddLeft = Forall([a, b, c], LessThanEquals(Add(c, a), Add(c, b)), domain=Reals, conditions=[LessThanEquals(a, b)]) lessThanEqualsAddLeft lessThanEqualsSubtract = Forall([a, b, c], LessThanEquals(Sub(a, c), Sub(b, c)), domain=Reals, conditions=[LessThanEquals(a, b)]) lessThanEqualsSubtract lessThanAddRight = Forall([a, b, c], LessThan(Add(a, c), Add(b, c)), domain=Reals, conditions=[LessThan(a, b)]) lessThanAddRight lessThanAddLeft = Forall([a, b, c], LessThan(Add(c, a), Add(c, b)), domain=Reals, conditions=[LessThan(a, b)])
distributefracThroughSum = Forall([xEtc, y],
Equals(frac(Add(xEtc), y),
Add(Etcetera(frac(xMulti, y)))),
domain=Complexes,
conditions=[NotEquals(y, zero)])
distributefracThroughSum
distributefracThroughSumRev = Forall([xEtc, y],
Equals(Add(Etcetera(frac(xMulti, y))),
frac(Add(xEtc), y)),
domain=Complexes,
conditions=[NotEquals(y, zero)])
distributefracThroughSumRev
distributefracThroughSubtract = Forall([x, y, z],
Equals(frac(Sub(x, y), z),
Sub(frac(x, z), frac(y, z))),
domain=Complexes,
conditions=[NotEquals(z, zero)])
distributefracThroughSubtract
distributefracThroughSubtractRev = Forall([x, y, z],
Equals(Sub(frac(x, z), frac(y, z)),
frac(Sub(x, y), z)),
domain=Complexes,
conditions=[NotEquals(z, zero)])
distributefracThroughSubtractRev
distributefracThroughSummation = Forall(
[P, S],
Implies(
