def is_equivalent(self, other):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> x, y = symbols('x y')
>>> AbstractSet(x, x-y>1).is_equivalent(AbstractSet(x, x-y>1))
True
>>> AbstractSet(x, x<1).is_equivalent(AbstractSet(x, x-1<0))
A. x st (x - 1 < 0) <=> (x < 1)
>>> AbstractSet(x, abs(x)>1).is_equivalent(Interval(1,2))
A. x st (x =< 2) /\ (x >= 1) <=> (|x| > 1)
"""
if isinstance(other, AbstractSet):
vars1 = self.variables
vars2 = other.variables
if not type_match(vars1, vars2):
return false
vars12 = rename_variables_in(vars1, free_symbols(self) | free_symbols(other))
return Forall(
vars12,
Equivalent(self.expr.xreplace(dict(zip(vars1, vars12))), other.expr.xreplace(dict(zip(vars2, vars12)))),
)
else:
other_ = as_abstract(other)
if isinstance(other_, AbstractSet):
return self.is_equivalent(other_)
else:
raise ValueError("Unknown argument '%s'" % other)
def _complement(self, other):
"""complement two set builder
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> x, y = symbols('x y')
>>> AbstractSet(y, abs(y)<=3)._complement(AbstractSet(x, abs(x)>1))
{y | (|y| > 1) /\ (|y| > 3)}
>>> AbstractSet((x,y), abs(x-y)>1)._complement(AbstractSet(x, abs(x)>1))
{x | |x| > 1}
>>> AbstractSet(y, y<=x)._complement(AbstractSet(x, abs(x)>1))
{y | (y > x) /\ (|y| > 1)}
>>> x2 = symbols('x2', positive=True)
>>> AbstractSet(x, abs(x)>1)._complement(AbstractSet(x2, x2<=y))
{x | (x =< y) /\ (|x| =< 1)}
>>> m, n = MatrixSymbol('m', 2, 2), MatrixSymbol('n', 2, 2)
>>> AbstractSet(n, Eq(det(n),0))._complement(AbstractSet(m, Eq(m[0,0]+m[1,1],0)))
{n | (0 =/= ||n||) /\ (0 == n[0, 0] + n[1, 1])}
>>> AbstractSet(x, abs(x)>1)._complement(AbstractSet(m, Eq(m[0,0]+m[1,1],0)))
{m | 0 == m[0, 0] + m[1, 1]}
>>> AbstractSet(x, abs(x)>1)._complement(Interval(1,2))
"""
if isinstance(other, AbstractSet):
vars1 = self.variables
vars2 = other.variables
if not type_match(vars1, vars2):
return other
vars12 = rename_variables_in(vars1, free_symbols(self) | free_symbols(other))
expr12 = And(
other.expr.xreplace(dict(zip(vars2, vars12))), Not(self.expr.xreplace(dict(zip(vars1, vars12))))
)
return AbstractSet(vars12, expr12)
def _union(self, other):
"""union two set builder
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> x, y = symbols('x y')
>>> AbstractSet(x, abs(x)>1)._union(AbstractSet(y, abs(y)<3))
{x | (|x| < 3) \/ (|x| > 1)}
>>> AbstractSet(x, abs(x)>1)._union(AbstractSet((x,y), abs(x-y)>1))
>>> AbstractSet(x, abs(x)>1)._union(AbstractSet(y, y<x))
{x_ | (x_ < x) \/ (|x_| > 1)}
>>> x2 = symbols('x2', positive=True)
>>> AbstractSet(x, abs(x)>1)._union(AbstractSet(x2, x2<y))
{x | (x < y) \/ (|x| > 1)}
>>> m, n = MatrixSymbol('m', 2, 2), MatrixSymbol('n', 2, 2)
>>> AbstractSet(m, Eq(m[0,0]+m[1,1],0))._union(AbstractSet(n, Eq(det(n),0)))
{m | (0 == m[0, 0] + m[1, 1]) \/ (0 == ||m||)}
>>> AbstractSet(m, Eq(m[0,0]+m[1,1],0))._union(AbstractSet(x, abs(x)>1))
>>> AbstractSet(x, abs(x)>1)._union(Interval(1,2))
"""
if isinstance(other, AbstractSet):
vars1 = self.variables
vars2 = other.variables
if not type_match(vars1, vars2):
return None
vars12 = rename_variables_in(vars1, free_symbols(self) | free_symbols(other))
expr12 = Or(self.expr.xreplace(dict(zip(vars1, vars12))), other.expr.xreplace(dict(zip(vars2, vars12))))
return AbstractSet(vars12, expr12)
def _contains(self, other):
"""
>>> from sympy import *
>>> from symplus.strplus import init_mprinting
>>> init_mprinting()
>>> x, y, z = symbols('x y z')
>>> AbstractSet(x, abs(x)>1)._contains(2)
True
>>> AbstractSet((x,y), abs(x-y)>1)._contains((3,1))
True
>>> AbstractSet((x,y), abs(x-y)>1)._contains((x+3,x+1))
True
>>> AbstractSet(x, x>y)._contains(z)
z > y
>>> AbstractSet(x, x>y)._contains(Matrix(2,2,[1,2,3,4]))
False
"""
var = self.variables
val = pack_if_not(other)
if not type_match(var, val):
return false
return self.expr.xreplace(dict(zip(var, val)))