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https://github.com/Z3Prover/z3
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127 lines
3.5 KiB
Python
127 lines
3.5 KiB
Python
############################################
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# Copyright (c) 2012 Microsoft Corporation
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#
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# Complex numbers in Z3
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# See http://research.microsoft.com/en-us/um/people/leonardo/blog/2013/01/26/complex.html
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#
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# Author: Leonardo de Moura (leonardo)
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############################################
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from __future__ import print_function
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import sys
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if sys.version_info.major >= 3:
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from functools import reduce
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from z3 import *
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def _to_complex(a):
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if isinstance(a, ComplexExpr):
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return a
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else:
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return ComplexExpr(a, RealVal(0))
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def _is_zero(a):
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return (isinstance(a, int) and a == 0) or (is_rational_value(a) and a.numerator_as_long() == 0)
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class ComplexExpr:
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def __init__(self, r, i):
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self.r = r
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self.i = i
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def __add__(self, other):
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other = _to_complex(other)
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return ComplexExpr(self.r + other.r, self.i + other.i)
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def __radd__(self, other):
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other = _to_complex(other)
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return ComplexExpr(other.r + self.r, other.i + self.i)
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def __sub__(self, other):
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other = _to_complex(other)
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return ComplexExpr(self.r - other.r, self.i - other.i)
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def __rsub__(self, other):
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other = _to_complex(other)
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return ComplexExpr(other.r - self.r, other.i - self.i)
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def __mul__(self, other):
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other = _to_complex(other)
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return ComplexExpr(self.r*other.r - self.i*other.i, self.r*other.i + self.i*other.r)
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def __mul__(self, other):
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other = _to_complex(other)
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return ComplexExpr(other.r*self.r - other.i*self.i, other.i*self.r + other.r*self.i)
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def __pow__(self, k):
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if k == 0:
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return ComplexExpr(1, 0)
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if k == 1:
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return self
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if k < 0:
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return (self ** (-k)).inv()
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return reduce(lambda x, y: x * y, [self for _ in range(k)], ComplexExpr(1, 0))
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def inv(self):
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den = self.r*self.r + self.i*self.i
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return ComplexExpr(self.r/den, -self.i/den)
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def __div__(self, other):
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inv_other = _to_complex(other).inv()
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return self.__mul__(inv_other)
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if sys.version_info.major >= 3:
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# In python 3 the meaning of the '/' operator
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# was changed.
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def __truediv__(self, other):
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return self.__div__(other)
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def __rdiv__(self, other):
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other = _to_complex(other)
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return self.inv().__mul__(other)
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def __eq__(self, other):
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other = _to_complex(other)
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return And(self.r == other.r, self.i == other.i)
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def __neq__(self, other):
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return Not(self.__eq__(other))
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def simplify(self):
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return ComplexExpr(simplify(self.r), simplify(self.i))
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def repr_i(self):
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if is_rational_value(self.i):
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return "%s*I" % self.i
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else:
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return "(%s)*I" % str(self.i)
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def __repr__(self):
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if _is_zero(self.i):
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return str(self.r)
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elif _is_zero(self.r):
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return self.repr_i()
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else:
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return "%s + %s" % (self.r, self.repr_i())
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def Complex(a):
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return ComplexExpr(Real('%s.r' % a), Real('%s.i' % a))
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I = ComplexExpr(RealVal(0), RealVal(1))
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def evaluate_cexpr(m, e):
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return ComplexExpr(m[e.r], m[e.i])
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x = Complex("x")
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s = Tactic('qfnra-nlsat').solver()
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s.add(x*x == -2)
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print(s)
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print(s.check())
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m = s.model()
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print('x = %s' % evaluate_cexpr(m, x))
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print((evaluate_cexpr(m,x)*evaluate_cexpr(m,x)).simplify())
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s.add(x.i != -1)
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print(s)
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print(s.check())
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print(s.model())
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s.add(x.i != 1)
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print(s.check())
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# print(s.model())
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print(((3 + I) ** 2)/(5 - I))
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print(((3 + I) ** -3)/(5 - I))
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