def test_conversion_of_fractions_in_z3(self):
self.assertValid(Equals(Real(Fraction(1,9)),
Div(Real(1), Real(9))),
solver_name="msat")
self.assertValid(Equals(Real(Fraction(1,9)),
Div(Real(1), Real(9))),
solver_name="z3")
def test_div_by_0(self):
varA = Symbol('A', REAL)
varB = Symbol('B', REAL)
f = And(Equals(varA, varB),
Not(Equals(Div(varA, Real(0)), Div(varB, Real(0)))))
self.assertUnsat(f)
def cosine_law_crit_angle(self):
"""Use cosine law to find cos^2(theta) between three points
node1---node2---node3 to assert that it is less than cos^2(thetaC)
where thetaC is the critical crossing angle
:param node1: Outside node
:param node2: Middle connecting node
:param node3: Outside node
:returns: cos^2 as calculated using cosine law (a_dot_b^2/a^2*b^2)
"""
node1 = self.get_input_nodes().values()[0]
node2 = self.get_input_nodes().values()[1]
node3 = self.get_output_node()
# Lengths of channels
aX = Minus(node1.get_x(), node2.get_x())
aY = Minus(node1.get_y(), node2.get_y())
bX = Minus(node3.get_x(), node2.get_x())
bY = Minus(node3.get_y(), node2.get_y())
# Dot products between each channel
a_dot_b_squared = Pow(Plus(Times(aX, bX), Times(aY, bY)), Real(2))
a_squared_b_squared = Times(
Plus(Times(aX, aX), Times(aY, aY)),
Plus(Times(bX, bX), Times(bY, bY)),
)
return Div(a_dot_b_squared, a_squared_b_squared)
def test_div_pow(self):
x = FreshSymbol(REAL)
f = Equals(Times(Real(4), Pow(x, Real(-1))), Real(2))
self.assertTrue(is_sat(f))
f = Equals(Div(Real(4), x), Real(2))
self.assertTrue(is_sat(f, solver_name="z3"))
f = Equals(Times(x, x), Real(16))
self.assertTrue(is_sat(f))
def test_integer(self):
x = FreshSymbol(INT)
f = Equals(Times(x, x), Int(2))
with Solver(name="z3") as s:
self.assertFalse(s.is_sat(f))
# f = Equals(Times(Int(4), Pow(x, Int(-1))), Int(2))
# self.assertTrue(is_sat(f, solver_name="z3"))
f = Equals(Div(Int(4), x), Int(2))
self.assertTrue(is_sat(f, solver_name="z3"))
f = Equals(Times(x, x), Int(16))
self.assertTrue(is_sat(f))
def calculate_resistance(self):
"""Calculate the droplet resistance in a channel using:
R = (12 * mu * L) / (w * h^3 * (1 - 0.630 (h/w)) )
This formula assumes that channel height < width, so
the first term returned is the assertion for that
Unit for resistance is kg/(m^4*s)
:param str channel_name: Name of the channel
:returns: list -- two SMT expressions, first asserts
that channel height is less than width, second
is the above expression in SMT form
"""
w = self.get_width()
h = self.get_height()
mu = self.get_viscosity()
chL = self.get_length()
return (LT(h, w),
Div(
Times(Real(12), Times(mu, chL)),
Times(
w,
Times(Pow(h, Real(3)),
Minus(Real(1), Times(Real(0.63), Div(h, w)))))))
def test_misc(self):
bool_list = [
And(self.x, self.y),
Or(self.x, self.y),
Not(self.x),
self.x,
Equals(self.p, self.q),
GE(self.p, self.q),
LE(self.p, self.q),
GT(self.p, self.q),
LT(self.p, self.q),
Bool(True),
Ite(self.x, self.y, self.x)
]
# TODO: FORALL EXISTS
real_list = [
self.r,
Real(4),
Plus(self.r, self.s),
Plus(self.r, Real(2)),
Minus(self.s, self.r),
Times(self.r, Real(1)),
Div(self.r, Real(1)),
Ite(self.x, self.r, self.s),
]
int_list = [
self.p,
Int(4),
Plus(self.p, self.q),
Plus(self.p, Int(2)),
Minus(self.p, self.q),
Times(self.p, Int(1)),
Ite(self.x, self.p, self.q),
]
for f in bool_list:
t = self.tc.walk(f)
self.assertEqual(t, BOOL, f)
for f in real_list:
t = self.tc.walk(f)
self.assertEqual(t, REAL, f)
for f in int_list:
t = self.tc.walk(f)
self.assertEqual(t, INT, f)
def test_div_pow(self):
x = FreshSymbol(REAL)
f = Equals(Times(Real(4), Pow(x, Real(-1))), Real(2))
try:
self.assertTrue(is_sat(f))
except SolverReturnedUnknownResultError:
pass
f = Equals(Div(Real(4), x), Real(2))
try:
self.assertTrue(is_sat(f, solver_name="z3"))
except SolverReturnedUnknownResultError:
pass
f = Equals(Times(x, x), Real(16))
try:
self.assertTrue(is_sat(f))
except SolverReturnedUnknownResultError:
pass
def calculate_flow_rate(self):
"""Calculate the flow rate into a port based on the cross sectional
area of the channel it flows into, the pressure and the density
eqn from https://en.wikipedia.org/wiki/Hagen-Poiseuille_equation
flow_rate = area * sqrt(2*pressure/density)
Unit for flow rate is m^3/s
:param str name: Name of the port
:returns: Flow rate determined from port pressure and area of
connected channels
"""
areas = []
for outgoing_channel in self.outgoing_channels:
areas.append(outgoing_channel.calculate_channel_area())
total_area = Plus(areas)
return Times(
total_area,
Pow(Div(Times(Real(2), self.get_pressure()), self.get_density()),
Real(0.5)))
def channels_in_straight_line(self):
"""Create expressions to assert that 2 channels are in a straight
line with each other by asserting that a triangle between the 2
end nodes and the middle node has zero area
:returns: Expression asserting area of triangle formed between all
three nodes to be 0
"""
# TODO: will this be an issue with classes!
# Check that these nodes connect
# try:
# self.dg[node1_name][node2_name]
# self.dg[node2_name][node3_name]
# except TypeError as e:
# raise TypeError("Tried asserting that 2 channels are in a straight\
# line but they aren't connected")
node1 = self.get_input_nodes().values()[0]
node2 = self.get_input_nodes().values()[1]
node3 = self.get_output_node()
# Constrain that continuous and output ports are in a straight line by
# setting the area of the triangle formed between those two points and
# the center of the t-junct to be 0
# Formula for area of a triangle given 3 points
# x_i (y_p − y_j ) + x_p (y_j − y_i ) + x_j (y_i − y_p ) / 2
return Equals(
Real(0),
Div(
Plus(
Times(node1.get_x(), Minus(node3.get_y(), node2.get_y())),
Plus(
Times(node3.get_x(), Minus(node2.get_y(),
node1.get_y())),
Times(node2.get_x(), Minus(node1.get_y(),
node3.get_y())))), Real(2)))
def get_full_example_formulae(environment=None):
"""Return a list of Examples using the given environment."""
if environment is None:
environment = get_env()
with environment:
x = Symbol("x", BOOL)
y = Symbol("y", BOOL)
p = Symbol("p", INT)
q = Symbol("q", INT)
r = Symbol("r", REAL)
s = Symbol("s", REAL)
aii = Symbol("aii", ARRAY_INT_INT)
ari = Symbol("ari", ArrayType(REAL, INT))
arb = Symbol("arb", ArrayType(REAL, BV8))
abb = Symbol("abb", ArrayType(BV8, BV8))
nested_a = Symbol("a_arb_aii",
ArrayType(ArrayType(REAL, BV8), ARRAY_INT_INT))
rf = Symbol("rf", FunctionType(REAL, [REAL, REAL]))
rg = Symbol("rg", FunctionType(REAL, [REAL]))
ih = Symbol("ih", FunctionType(INT, [REAL, INT]))
ig = Symbol("ig", FunctionType(INT, [INT]))
bf = Symbol("bf", FunctionType(BOOL, [BOOL]))
bg = Symbol("bg", FunctionType(BOOL, [BOOL]))
bv8 = Symbol("bv1", BV8)
bv16 = Symbol("bv2", BV16)
result = [
# Formula, is_valid, is_sat, is_qf
Example(hr="(x & y)",
expr=And(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="(x <-> y)",
expr=Iff(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="((x | y) & (! (x | y)))",
expr=And(Or(x, y), Not(Or(x, y))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BOOL),
Example(hr="(x & (! y))",
expr=And(x, Not(y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
Example(hr="(False -> True)",
expr=Implies(FALSE(), TRUE()),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
#
# LIA
#
Example(hr="((q < p) & (x -> y))",
expr=And(GT(p, q), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_IDL),
Example(hr="(((p + q) = 5) & (q < p))",
expr=And(Equals(Plus(p, q), Int(5)), GT(p, q)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="((q <= p) | (p <= q))",
expr=Or(GE(p, q), LE(p, q)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_IDL),
Example(hr="(! (p < (q * 2)))",
expr=Not(LT(p, Times(q, Int(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(p < (p - (5 - 2)))",
expr=GT(Minus(p, Minus(Int(5), Int(2))), p),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_IDL),
Example(hr="((x ? 7 : ((p + -1) * 3)) = q)",
expr=Equals(
Ite(x, Int(7), Times(Plus(p, Int(-1)), Int(3))), q),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(p < (q + 1))",
expr=LT(p, Plus(q, Int(1))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
#
# LRA
#
Example(hr="((s < r) & (x -> y))",
expr=And(GT(r, s), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="(((r + s) = 28/5) & (s < r))",
expr=And(Equals(Plus(r, s), Real(Fraction("5.6"))),
GT(r, s)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="((s <= r) | (r <= s))",
expr=Or(GE(r, s), LE(r, s)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="(! ((r * 2.0) < (s * 2.0)))",
expr=Not(LT(Div(r, Real((1, 2))), Times(s, Real(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="(! (r < (r - (5.0 - 2.0))))",
expr=Not(GT(Minus(r, Minus(Real(5), Real(2))), r)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
Example(hr="((x ? 7.0 : ((s + -1.0) * 3.0)) = r)",
expr=Equals(
Ite(x, Real(7), Times(Plus(s, Real(-1)), Real(3))), r),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
#
# EUF
#
Example(hr="(bf(x) <-> bg(x))",
expr=Iff(Function(bf, (x, )), Function(bg, (x, ))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UF),
Example(hr="(rf(5.0, rg(r)) = 0.0)",
expr=Equals(Function(rf, (Real(5), Function(rg, (r, )))),
Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
Example(hr="((rg(r) = (5.0 + 2.0)) <-> (rg(r) = 7.0))",
expr=Iff(Equals(Function(rg, [r]), Plus(Real(5), Real(2))),
Equals(Function(rg, [r]), Real(7))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
Example(
hr="((r = (s + 1.0)) & (rg(s) = 5.0) & (rg((r - 1.0)) = 7.0))",
expr=And([
Equals(r, Plus(s, Real(1))),
Equals(Function(rg, [s]), Real(5)),
Equals(Function(rg, [Minus(r, Real(1))]), Real(7))
]),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLRA),
#
# BV
#
Example(hr="((1_32 & 0_32) = 0_32)",
expr=Equals(BVAnd(BVOne(32), BVZero(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((! 2_3) = 5_3)",
expr=Equals(BVNot(BV("010")), BV("101")),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((7_3 xor 0_3) = 0_3)",
expr=Equals(BVXor(BV("111"), BV("000")), BV("000")),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv1::bv1) u< 0_16)",
expr=BVULT(BVConcat(bv8, bv8), BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(1_32[0:7] = 1_8)",
expr=Equals(BVExtract(BVOne(32), end=7), BVOne(8)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_8 u< (((bv1 + 1_8) * 5_8) u/ 5_8))",
expr=BVUGT(
BVUDiv(BVMul(BVAdd(bv8, BVOne(8)), BV(5, width=8)),
BV(5, width=8)), BVZero(8)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_16 u<= bv2)",
expr=BVUGE(bv16, BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(0_16 s<= bv2)",
expr=BVSGE(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(
hr="((0_32 u< (5_32 u% 2_32)) & ((5_32 u% 2_32) u<= 1_32))",
expr=And(
BVUGT(BVURem(BV(5, width=32), BV(2, width=32)),
BVZero(32)),
BVULE(BVURem(BV(5, width=32), BV(2, width=32)),
BVOne(32))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((((1_32 + (- 1_32)) << 1_32) >> 1_32) = 1_32)",
expr=Equals(
BVLShr(BVLShl(BVAdd(BVOne(32), BVNeg(BVOne(32))), 1),
1), BVOne(32)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((1_32 - 1_32) = 0_32)",
expr=Equals(BVSub(BVOne(32), BVOne(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Rotations
Example(hr="(((1_32 ROL 1) ROR 1) = 1_32)",
expr=Equals(BVRor(BVRol(BVOne(32), 1), 1), BVOne(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Extensions
Example(hr="((0_5 ZEXT 11) = (0_1 SEXT 15))",
expr=Equals(BVZExt(BVZero(5), 11), BVSExt(BVZero(1), 15)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 - bv2) = 0_16)",
expr=Equals(BVSub(bv16, bv16), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 - bv2)[0:7] = bv1)",
expr=Equals(BVExtract(BVSub(bv16, bv16), 0, 7), bv8),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2[0:7] bvcomp bv1) = 1_1)",
expr=Equals(BVComp(BVExtract(bv16, 0, 7), bv8), BVOne(1)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 bvcomp bv2) = 0_1)",
expr=Equals(BVComp(bv16, bv16), BVZero(1)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 s< bv2)",
expr=BVSLT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 s< 0_16)",
expr=BVSLT(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s< 0_16) | (0_16 s<= bv2))",
expr=Or(BVSGT(BVZero(16), bv16), BVSGE(bv16, BVZero(16))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 u< bv2)",
expr=BVULT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="(bv2 u< 0_16)",
expr=BVULT(bv16, BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 | 0_16) = bv2)",
expr=Equals(BVOr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 & 0_16) = 0_16)",
expr=Equals(BVAnd(bv16, BVZero(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s< bv2) & ((bv2 s/ 65535_16) s< 0_16))",
expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSDiv(bv16, SBV(-1, 16)), BVZero(16))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s< bv2) & ((bv2 s% 1_16) s< 0_16))",
expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSRem(bv16, BVOne(16)), BVZero(16))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 u% 1_16) = 0_16)",
expr=Equals(BVURem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s% 1_16) = 0_16)",
expr=Equals(BVSRem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 s% (- 1_16)) = 0_16)",
expr=Equals(BVSRem(bv16, BVNeg(BVOne(16))), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((bv2 a>> 0_16) = bv2)",
expr=Equals(BVAShr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
Example(hr="((0_16 s<= bv2) & ((bv2 a>> 1_16) = (bv2 >> 1_16)))",
expr=And(
BVSLE(BVZero(16), bv16),
Equals(BVAShr(bv16, BVOne(16)),
BVLShr(bv16, BVOne(16)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
#
# Quantification
#
Example(hr="(forall y . (x -> y))",
expr=ForAll([y], Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.BOOL),
Example(hr="(forall p, q . ((p + q) = 0))",
expr=ForAll([p, q], Equals(Plus(p, q), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LIA),
Example(
hr="(forall r, s . (((0.0 < r) & (0.0 < s)) -> ((r - s) < r)))",
expr=ForAll([r, s],
Implies(And(GT(r, Real(0)), GT(s, Real(0))),
(LT(Minus(r, s), r)))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
Example(hr="(exists x, y . (x -> y))",
expr=Exists([x, y], Implies(x, y)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.BOOL),
Example(hr="(exists p, q . ((p + q) = 0))",
expr=Exists([p, q], Equals(Plus(p, q), Int(0))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LIA),
Example(hr="(exists r . (forall s . (r < (r - s))))",
expr=Exists([r], ForAll([s], GT(Minus(r, s), r))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
Example(hr="(forall r . (exists s . (r < (r - s))))",
expr=ForAll([r], Exists([s], GT(Minus(r, s), r))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
Example(hr="(x & (forall r . ((r + s) = 5.0)))",
expr=And(x, ForAll([r], Equals(Plus(r, s), Real(5)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
Example(hr="(exists x . ((x <-> (5.0 < s)) & (s < 3.0)))",
expr=Exists([x],
(And(Iff(x, GT(s, Real(5))), LT(s, Real(3))))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.LRA),
#
# UFLIRA
#
Example(hr="((p < ih(r, q)) & (x -> y))",
expr=And(GT(Function(ih, (r, q)), p), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(((p - 3) = q) -> ((p < ih(r, (q + 3))) | (ih(r, p) <= p)))",
expr=Implies(
Equals(Minus(p, Int(3)), q),
Or(GT(Function(ih, (r, Plus(q, Int(3)))), p),
LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(((ToReal((p - 3)) = r) & (ToReal(q) = r)) -> ((p < ih(ToReal((p - 3)), (q + 3))) | (ih(r, p) <= p)))",
expr=Implies(
And(Equals(ToReal(Minus(p, Int(3))), r),
Equals(ToReal(q), r)),
Or(
GT(
Function(
ih,
(ToReal(Minus(p, Int(3))), Plus(q, Int(3)))),
p), LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"(! (((ToReal((p - 3)) = r) & (ToReal(q) = r)) -> ((p < ih(ToReal((p - 3)), (q + 3))) | (ih(r, p) <= p))))",
expr=Not(
Implies(
And(Equals(ToReal(Minus(p, Int(3))), r),
Equals(ToReal(q), r)),
Or(
GT(
Function(ih, (ToReal(Minus(
p, Int(3))), Plus(q, Int(3)))), p),
LE(Function(ih, (r, p)), p)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLIRA),
Example(
hr=
"""("Did you know that any string works? #yolo" & "10" & "|#somesolverskeepthe||" & " ")""",
expr=And(Symbol("Did you know that any string works? #yolo"),
Symbol("10"), Symbol("|#somesolverskeepthe||"),
Symbol(" ")),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
#
# Arrays
#
Example(hr="((q = 0) -> (aii[0 := 0] = aii[0 := q]))",
expr=Implies(
Equals(q, Int(0)),
Equals(Store(aii, Int(0), Int(0)),
Store(aii, Int(0), q))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ALIA),
Example(hr="(aii[0 := 0][0] = 0)",
expr=Equals(Select(Store(aii, Int(0), Int(0)), Int(0)),
Int(0)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ALIA),
Example(hr="((Array{Int, Int}(0)[1 := 1] = aii) & (aii[1] = 0))",
expr=And(Equals(Array(INT, Int(0), {Int(1): Int(1)}), aii),
Equals(Select(aii, Int(1)), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_ALIA*")),
Example(hr="((Array{Int, Int}(0)[1 := 3] = aii) & (aii[1] = 3))",
expr=And(Equals(Array(INT, Int(0), {Int(1): Int(3)}), aii),
Equals(Select(aii, Int(1)), Int(3))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.get_logic_by_name("QF_ALIA*")),
Example(hr="((Array{Real, Int}(10) = ari) & (ari[6/5] = 0))",
expr=And(Equals(Array(REAL, Int(10)), ari),
Equals(Select(ari, Real((6, 5))), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((Array{Real, Int}(0)[1.0 := 10][2.0 := 20][3.0 := 30][4.0 := 40] = ari) & (! ((ari[0.0] = 0) & (ari[1.0] = 10) & (ari[2.0] = 20) & (ari[3.0] = 30) & (ari[4.0] = 40))))",
expr=And(
Equals(
Array(
REAL, Int(0), {
Real(1): Int(10),
Real(2): Int(20),
Real(3): Int(30),
Real(4): Int(40)
}), ari),
Not(
And(Equals(Select(ari, Real(0)), Int(0)),
Equals(Select(ari, Real(1)), Int(10)),
Equals(Select(ari, Real(2)), Int(20)),
Equals(Select(ari, Real(3)), Int(30)),
Equals(Select(ari, Real(4)), Int(40))))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((Array{Real, Int}(0)[1.0 := 10][2.0 := 20][3.0 := 30][4.0 := 40][5.0 := 50] = ari) & (! ((ari[0.0] = 0) & (ari[1.0] = 10) & (ari[2.0] = 20) & (ari[3.0] = 30) & (ari[4.0] = 40) & (ari[5.0] = 50))))",
expr=And(
Equals(
Array(
REAL, Int(0), {
Real(1): Int(10),
Real(2): Int(20),
Real(3): Int(30),
Real(4): Int(40),
Real(5): Int(50)
}), ari),
Not(
And(Equals(Select(ari, Real(0)), Int(0)),
Equals(Select(ari, Real(1)), Int(10)),
Equals(Select(ari, Real(2)), Int(20)),
Equals(Select(ari, Real(3)), Int(30)),
Equals(Select(ari, Real(4)), Int(40)),
Equals(Select(ari, Real(5)), Int(50))))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(
hr=
"((a_arb_aii = Array{Array{Real, BV{8}}, Array{Int, Int}}(Array{Int, Int}(7))) -> (a_arb_aii[arb][42] = 7))",
expr=Implies(
Equals(nested_a,
Array(ArrayType(REAL, BV8), Array(INT, Int(7)))),
Equals(Select(Select(nested_a, arb), Int(42)), Int(7))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.get_logic_by_name("QF_AUFBVLIRA*")),
Example(hr="(abb[bv1 := y_][bv1 := z_] = abb[bv1 := z_])",
expr=Equals(
Store(Store(abb, bv8, Symbol("y_", BV8)), bv8,
Symbol("z_", BV8)),
Store(abb, bv8, Symbol("z_", BV8))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_ABV),
Example(hr="((r / s) = (r * s))",
expr=Equals(Div(r, s), Times(r, s)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="(2.0 = (r * r))",
expr=Equals(Real(2), Times(r, r)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((p ^ 2) = 0)",
expr=Equals(Pow(p, Int(2)), Int(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NIA),
Example(hr="((r ^ 2.0) = 0.0)",
expr=Equals(Pow(r, Real(2)), Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((r * r * r) = 25.0)",
expr=Equals(Times(r, r, r), Real(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(hr="((5.0 * r * 5.0) = 25.0)",
expr=Equals(Times(Real(5), r, Real(5)), Real(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="((p * p * p) = 25)",
expr=Equals(Times(p, p, p), Int(25)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_NIA),
Example(hr="((5 * p * 5) = 25)",
expr=Equals(Times(Int(5), p, Int(5)), Int(25)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(hr="(((1 - 1) * p * 1) = 0)",
expr=Equals(Times(Minus(Int(1), Int(1)), p, Int(1)),
Int(0)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_LIA),
# Huge Fractions:
Example(
hr=
"((r * 1606938044258990275541962092341162602522202993782792835301376/7) = -20480000000000000000000000.0)",
expr=Equals(Times(r, Real(Fraction(2**200, 7))),
Real(-200**11)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
Example(hr="(((r + 5.0 + s) * (s + 2.0 + r)) = 0.0)",
expr=Equals(
Times(Plus(r, Real(5), s), Plus(s, Real(2), r)),
Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_NRA),
Example(
hr=
"(((p + 5 + q) * (p - (q - 5))) = ((p * p) + (10 * p) + 25 + (-1 * q * q)))",
expr=Equals(
Times(Plus(p, Int(5), q), Minus(p, Minus(q, Int(5)))),
Plus(Times(p, p), Times(Int(10), p), Int(25),
Times(Int(-1), q, q))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_NIA),
]
return result
def test_div(self):
x = FreshSymbol(REAL)
f = Equals(Div(x, x), x)
with Solver(name="z3") as s:
self.assertTrue(s.is_sat(f))
def get_example_formulae(environment=None):
if environment is None:
environment = get_env()
with environment:
x = Symbol("x", BOOL)
y = Symbol("y", BOOL)
p = Symbol("p", INT)
q = Symbol("q", INT)
r = Symbol("r", REAL)
s = Symbol("s", REAL)
rf = Symbol("rf", FunctionType(REAL, [REAL, REAL]))
rg = Symbol("rg", FunctionType(REAL, [REAL]))
ih = Symbol("ih", FunctionType(INT, [REAL, INT]))
ig = Symbol("ig", FunctionType(INT, [INT]))
bv8 = Symbol("bv1", BV8)
bv16 = Symbol("bv2", BV16)
result = [
# Formula, is_valid, is_sat, is_qf
# x /\ y
Example(expr=And(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
# x <-> y
Example(expr=Iff(x, y),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
# (x \/ y ) /\ ! ( x \/ y )
Example(expr=And(Or(x, y), Not(Or(x, y))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BOOL),
# (x /\ !y)
Example(expr=And(x, Not(y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
# False -> True
Example(expr=Implies(FALSE(), TRUE()),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
#
# LIA
#
# (p > q) /\ x -> y
Example(expr=And(GT(p, q), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_IDL),
# (p + q) = 5 /\ (p > q)
Example(expr=And(Equals(Plus(p, q), Int(5)), GT(p, q)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
# (p >= q) \/ ( p <= q)
Example(expr=Or(GE(p, q), LE(p, q)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_IDL),
# !( p < q * 2 )
Example(expr=Not(LT(p, Times(q, Int(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
# p - (5 - 2) > p
Example(expr=GT(Minus(p, Minus(Int(5), Int(2))), p),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_IDL),
# x ? 7: (p + -1) * 3 = q
Example(expr=Equals(
Ite(x, Int(7), Times(Plus(p, Int(-1)), Int(3))), q),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
Example(expr=LT(p, Plus(q, Int(1))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LIA),
#
# LRA
#
# (r > s) /\ x -> y
Example(expr=And(GT(r, s), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_RDL),
# (r + s) = 5.6 /\ (r > s)
Example(expr=And(Equals(Plus(r, s), Real(Fraction("5.6"))),
GT(r, s)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
# (r >= s) \/ ( r <= s)
Example(expr=Or(GE(r, s), LE(r, s)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
# !( (r / (1/2)) < s * 2 )
Example(expr=Not(LT(Div(r, Real((1, 2))), Times(s, Real(2)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
# ! ( r - (5 - 2) > r )
Example(expr=Not(GT(Minus(r, Minus(Real(5), Real(2))), r)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_RDL),
# x ? 7: (s + -1) * 3 = r
Example(expr=Equals(
Ite(x, Real(7), Times(Plus(s, Real(-1)), Real(3))), r),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_LRA),
#
# EUF
#
# rf(5, rg(2)) = 0
Example(expr=Equals(Function(rf, (Real(5), Function(rg, (r, )))),
Real(0)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
# (rg(r) = 5 + 2) <-> (rg(r) = 7)
Example(expr=Iff(Equals(Function(rg, [r]), Plus(Real(5), Real(2))),
Equals(Function(rg, [r]), Real(7))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLRA),
# (r = s + 1) & (rg(s) = 5) & (rg(r - 1) = 7)
Example(expr=And([
Equals(r, Plus(s, Real(1))),
Equals(Function(rg, [s]), Real(5)),
Equals(Function(rg, [Minus(r, Real(1))]), Real(7))
]),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLRA),
#
# BV
#
# bv_one & bv_zero == bv_zero
Example(expr=Equals(BVAnd(BVOne(32), BVZero(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# ~(010) == 101
Example(expr=Equals(BVNot(BV("010")), BV("101")),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# "111" xor "000" == "000"
Example(expr=Equals(BVXor(BV("111"), BV("000")), BV("000")),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# bv8 :: bv8 < bv_zero
Example(expr=BVULT(BVConcat(bv8, bv8), BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# bv_one[:7] == bv_one
Example(expr=Equals(BVExtract(BVOne(32), end=7), BVOne(8)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (((bv8 + bv_one) * bv(5)) / bv(5)) > bv(0)
Example(expr=BVUGT(
BVUDiv(BVMul(BVAdd(bv8, BVOne(8)), BV(5, width=8)),
BV(5, width=8)), BVZero(8)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 >=u bv(0)
Example(expr=BVUGE(bv16, BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 >=s bv(0)
Example(expr=BVSGE(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (BV(5) rem BV(2) > bv_zero) /\ (BV(5) rem BV(2) < bv_one)
Example(expr=And(
BVUGT(BVURem(BV(5, width=32), BV(2, width=32)), BVZero(32)),
BVULE(BVURem(BV(5, width=32), BV(2, width=32)), BVOne(32))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# ((bv_one + (- bv_one)) << 1) >> 1 == bv_one
Example(expr=Equals(
BVLShr(BVLShl(BVAdd(BVOne(32), BVNeg(BVOne(32))), 1), 1),
BVOne(32)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# bv_one - bv_one == bv_zero
Example(expr=Equals(BVSub(BVOne(32), BVOne(32)), BVZero(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Rotations
Example(expr=Equals(BVRor(BVRol(BVOne(32), 1), 1), BVOne(32)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# Extensions
Example(expr=Equals(BVZExt(BVZero(5), 11), BVSExt(BVZero(1), 15)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 - bv16 = 0_16
Example(expr=Equals(BVSub(bv16, bv16), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (bv16 - bv16)[0:7] = bv8
Example(expr=Equals(BVExtract(BVSub(bv16, bv16), 0, 7), bv8),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (bv16[0,7] comp bv8) = bv1
Example(expr=Equals(BVComp(BVExtract(bv16, 0, 7), bv8), BVOne(1)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (bv16 comp bv16) = bv0
Example(expr=Equals(BVComp(bv16, bv16), BVZero(1)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# (bv16 s< bv16)
Example(expr=BVSLT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# (bv16 s< 0_16)
Example(expr=BVSLT(bv16, BVZero(16)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (bv16 u< bv16)
Example(expr=BVULT(bv16, bv16),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# (bv16 s< 0_16)
Example(expr=BVULT(bv16, BVZero(16)),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# (bv16 | 0_16) = bv16
Example(expr=Equals(BVOr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# (bv16 & 0_16) = 0_16
Example(expr=Equals(BVAnd(bv16, BVZero(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# 0_16 s< bv16 & ((bv16 s/ -1) s< 0)
Example(expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSDiv(bv16, SBV(-1, 16)), BVZero(16))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
# 0_16 s< bv16 & ((bv16 s% -1) s< 0)
Example(expr=And(BVSLT(BVZero(16), bv16),
BVSLT(BVSRem(bv16, BVOne(16)), BVZero(16))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_BV),
# bv16 u% 1 = 0_16
Example(expr=Equals(BVURem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 s% 1 = 0_16
Example(expr=Equals(BVSRem(bv16, BVOne(16)), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 s% -1 = 0_16
Example(expr=Equals(BVSRem(bv16, BVNeg(BVOne(16))), BVZero(16)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# bv16 a>> 0 = bv16
Example(expr=Equals(BVAShr(bv16, BVZero(16)), bv16),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_BV),
# 0 s<= bv16 & bv16 a>> 1 = bv16 >> 1
Example(expr=And(
BVSLE(BVZero(16), bv16),
Equals(BVAShr(bv16, BVOne(16)), BVLShr(bv16, BVOne(16)))),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BV),
#
# Quantification
#
# forall y . x -> y
Example(expr=ForAll([y], Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.BOOL),
# forall p,q . p + q = 0
Example(expr=ForAll([p, q], Equals(Plus(p, q), Int(0))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LIA),
# forall r,s . ((r > 0) & (s > 0)) -> (r - s < r)
Example(expr=ForAll([r, s],
Implies(And(GT(r, Real(0)), GT(s, Real(0))),
(LT(Minus(r, s), r)))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
# exists x,y . x -> y
Example(expr=Exists([x, y], Implies(x, y)),
is_valid=True,
is_sat=True,
logic=pysmt.logics.BOOL),
# exists p,q . p + q = 0
Example(expr=Exists([p, q], Equals(Plus(p, q), Int(0))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LIA),
# exists r . forall s . (r - s > r)
Example(expr=Exists([r], ForAll([s], GT(Minus(r, s), r))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
# forall r . exists s . (r - s > r)
Example(expr=ForAll([r], Exists([s], GT(Minus(r, s), r))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.LRA),
# x /\ forall r. (r + s = 5)
Example(expr=And(x, ForAll([r], Equals(Plus(r, s), Real(5)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.LRA),
#
# UFLIRA
#
# ih(r,q) > p /\ (x -> y)
Example(expr=And(GT(Function(ih, (r, q)), p), Implies(x, y)),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
# ( (p - 3) = q ) -> ( ih(r, q + 3) > p \/ ih(r, p) <= p )
Example(expr=Implies(
Equals(Minus(p, Int(3)), q),
Or(GT(Function(ih, (r, Plus(q, Int(3)))), p),
LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
# ( (ToReal(p - 3) = r) /\ (ToReal(q) = r) ) ->
# ( ( ih(ToReal(p - 3), q + 3) > p ) \/ (ih(r, p) <= p) )
Example(expr=Implies(
And(Equals(ToReal(Minus(p, Int(3))), r), Equals(ToReal(q), r)),
Or(
GT(
Function(ih,
(ToReal(Minus(p, Int(3))), Plus(q, Int(3)))),
p), LE(Function(ih, (r, p)), p))),
is_valid=True,
is_sat=True,
logic=pysmt.logics.QF_UFLIRA),
# ! ( (ToReal(p - 3) = r /\ ToReal(q) = r) ->
# ( ih(ToReal(p - 3), q + 3) > p \/
# ih(r,p) <= p ) )
Example(expr=Not(
Implies(
And(Equals(ToReal(Minus(p, Int(3))), r),
Equals(ToReal(q), r)),
Or(
GT(
Function(
ih,
(ToReal(Minus(p, Int(3))), Plus(q, Int(3)))),
p), LE(Function(ih, (r, p)), p)))),
is_valid=False,
is_sat=False,
logic=pysmt.logics.QF_UFLIRA),
# Test complex names
Example(expr=And(
Symbol("Did you know that any string works? #yolo"),
Symbol("10"), Symbol("|#somesolverskeepthe||"), Symbol(" ")),
is_valid=False,
is_sat=True,
logic=pysmt.logics.QF_BOOL),
]
return result
def calculate_droplet_volume(self, h, w, wIn, epsilon, qD, qC):
"""From paper DOI:10.1039/c002625e.
Calculating the droplet volume created in a T-junction
Unit is volume in m^3
:param Symbol h: Height of channel
:param Symbol w: Width of continuous/output channel
:param Symbol wIn: Width of dispersed_channel
:param Symbol epsilon: Equals 0.414*radius of rounded edge where
channels join
:param Symbol qD: Flow rate in dispersed_channel
:param Symbol qC: Flow rate in continuous_channel
"""
q_gutter = Real(0.1)
# normalizedVFill = 3pi/8 - (pi/2)(1 - pi/4)(h/w)
v_fill_simple = Minus(
Times(Real((3, 8)), Real(math.pi)),
Times(
Times(Div(Real(math.pi), Real(2)),
Minus(Real(1), Div(Real(math.pi), Real(4)))), Div(h, w)))
hw_parallel = Div(Times(h, w), Plus(h, w))
# r_pinch = w+((wIn-(hw_parallel - eps))+sqrt(2*((wIn-hw_parallel)*(w-hw_parallel))))
r_pinch = Plus(
w,
Plus(
Minus(wIn, Minus(hw_parallel, epsilon)),
Pow(
Times(
Real(2),
Times(Minus(wIn, hw_parallel), Minus(w, hw_parallel))),
Real(0.5))))
r_fill = w
alpha = Times(
Minus(Real(1), Div(Real(math.pi), Real(4))),
Times(
Pow(Minus(Real(1), q_gutter), Real(-1)),
Plus(
Minus(Pow(Div(r_pinch, w), Real(2)),
Pow(Div(r_fill, w), Real(2))),
Times(
Div(Real(math.pi), Real(4)),
Times(Minus(Div(r_pinch, w), Div(r_fill, w)),
Div(h, w))))))
return Times(Times(h, Times(w, w)),
Plus(v_fill_simple, Times(alpha, Div(qD, qC))))