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)])
absNonzeroClosure
modInIntervalCO = Forall((a, b),
InSet(Mod(a, b), IntervalCO(zero, b)),
domain=Reals)
modInIntervalCO
absIsNonNeg = Forall(a, GreaterThanEquals(Abs(a), zero), domain=Complexes)
absIsNonNeg
sqrtRealPosClosure = Forall([a], InSet(sqrt(a), RealsPos), domain=RealsPos)
sqrtRealPosClosure
sqrtComplexClosure = Forall([a], InSet(sqrt(a), Complexes), domain=Complexes)
sqrtComplexClosure
# Should generalize to even power closure, but need to define and implement evens set to do this.
sqrdPosClosure = Forall(a,
InSet(Exp(a, two), RealsPos),
domain=Reals,
conditions=[NotEquals(a, zero)])
sqrdPosClosure
squarePosIneq = Forall([a, b],
LessThanEquals(Exp(Abs(a), two), Exp(b, two)),
domain=Reals,
conditions=(LessThanEquals(Abs(a), b), ))
squarePosIneq
squarePosEq = Forall(a, Equals(Exp(Abs(a), two), Exp(a, two)), domain=Reals)
squarePosEq
expNotEqZero = Forall([a, b],
NotEquals(Exp(a, b), zero),
domain=Complexes,
conditions=[NotEquals(a, zero)])
expNotEqZero
expZeroEqOne = Forall([a],
Equals(Exp(a, zero), one),
# 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))
conditions=[GreaterThanEquals(a, zero)])
sqrtRealClosure
sqrtRealPosClosure = Forall([a], InSet(sqrt(a), RealsPos), domain=RealsPos)
sqrtRealPosClosure
sqrtComplexClosure = Forall([a], InSet(sqrt(a), Complexes), domain=Complexes)
sqrtComplexClosure
# Should generalize to even power closure, but need to define and implement evens set to do this.
sqrdPosClosure = Forall(a, InSet(Exp(a, two), RealsPos),
domain=Reals, conditions=[NotEquals(a, zero)])
sqrdPosClosure
squarePosIneq = Forall([a,b],
LessThanEquals(Exp(Abs(a),two),Exp(b,two)),
domain = Reals,
conditions = (LessThanEquals(Abs(a),b),))
squarePosIneq
squarePosEq = Forall(a,
Equals(Exp(Abs(a),two),Exp(a,two)),
domain = Reals)
squarePosEq
expNotEqZero = Forall([a, b], NotEquals(Exp(a,b), zero), domain=Complexes, conditions=[NotEquals(a, zero)])
expNotEqZero
expZeroEqOne = Forall([a], Equals(Exp(a, zero), one), domain=Complexes, conditions=[NotEquals(a, zero)])
expZeroEqOne