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class Unique(object):
def __init__(self,name):
self.name = name
def __str__(self):
return self.name
def __repr__(self):
return "Unique(\"%s\")" % self.name
class Idealized(object):
UNION = ["UNION"]
def __init__(self, R, idealMap = 0, vars = {}):
self.varnames = vars
if not isinstance(idealMap,dict):
idealMap = {()*R:idealMap}
self.idealMap = idealMap
self.R = R
self._sqrt = None
self._isqrt = None
@staticmethod
def uvar(x):
return Idealized.var(Unique(x))
@staticmethod
def var(x):
name = str(x)
R = PolynomialRing(QQ,[name])
rx = R.gens()[0]
return Idealized(R,rx,{x:(name,rx)})
@staticmethod
def vars(xs):
return tuple((Idealized.var(x) for x in xs))
@staticmethod
def uvars(xs):
return tuple((Idealized.uvar(x) for x in xs))
def __str__(self):
def rep(I,x):
x = str(x)
gs = I.gens()
gs = [g for g in gs if g != 0]
if len(gs) == 0: return x
else:
g = ", ".join(["(%s)" % str(gen) for gen in gs])
return g + ": " + x
return "\n".join([rep(I,self.idealMap[I]) for I in self.idealMap])
def __repr__(self):
# HACK!
if len(self.idealMap) == 0:
return "undef"
if len(self.idealMap) > 1:
return str(self)
for _,v in self.idealMap.iteritems():
return str(v)
def prune(self):
self.idealMap = {I:v for I,v in self.idealMap.iteritems() if not (I*self.R).is_one()}
return self
def __add__(self,other):
def f(x,y): return x+y
return self.op(other,f)
def __radd__(self,other):
def f(x,y): return y+x
return self.op(other,f)
def __rsub__(self,other):
def f(x,y): return y-x
return self.op(other,f)
def __neg__(self):
def f(x,y): return y-x
return self.op(0,f)
def __sub__(self,other):
def f(x,y): return x-y
return self.op(other,f)
def is_square(self):
for _,v in self.idealMap.iteritems():
if not is_square(v): return False
return True
def sqrt(self):
if self._sqrt is None:
s = Idealized.uvar("s")
self._sqrt = s.assuming(s^2 - self)
return self._sqrt
def isqrt(self):
if self._isqrt is None:
s = Idealized.uvar("s")
z = Idealized(0).assuming(Self)
self._isqrt = s.assuming(s^2*self-1).union(z)
return self._isqrt
def __mul__(self,other):
def f(x,y): return x*y
return self.op(other,f)
def __rmul__(self,other):
def f(x,y): return y*x
return self.op(other,f)
def __pow__(self,n):
if n < 0: return 1/self^(-n)
if n == 0: return 1
if n == 1: return self
if is_even(n): return (self*self)^(n//2)
if is_odd(n): return (self*self)^(n//2) * self
def __div__(self,other):
def f(x,y): return x/y
return self.op(other,f)
def __rdiv__(self,other):
def f(x,y): return y/x
return self.op(other,f)
def union(self,other):
return self.op(other,Idealized.UNION)
def __eq__(self,other):
return (self - other).is_zero()
def __ne__(self,other):
return not (self==other)
def __hash__(self):
return 0
def assume_zero(self):
out = {}
for I,J in self.idealMap.iteritems():
IJ = I+J.numerator()
if IJ.is_one(): continue
out[IJ] = self.R(0)
if len(out) == 0:
raise Exception("Inconsistent assumption")
return Idealized(self.R,out,self.varnames)
def assuming(self,other):
return self + other.assume_zero()
def is_zero(self):
for I,v in self.idealMap.iteritems():
if v.denominator() in I: return False
if v.numerator() not in I: return False
return True
def op(self,other,f):
if not isinstance(other,Idealized):
other = Idealized(self.R,other,self.varnames)
bad = False
for v in self.varnames:
if v not in other.varnames or self.varnames[v] != other.varnames[v]:
bad = True
break
for v in other.varnames:
if v not in self.varnames or self.varnames[v] != other.varnames[v]:
bad = True
break
if bad:
def incrVar(v):
if v[-1] not in "0123456789": return v + "1"
elif v[-1] == 9: return incrVar(v[:-1]) + "0"
else: return v[:-1] + str(int(v[-1])+1)
vars = {}
names = set()
for v,(name,_) in self.varnames.iteritems():
assert(name not in names)
names.add(name)
vars[v] = name
subMe = {n:n for n in names}
subThem = {}
for v,(name,_) in other.varnames.iteritems():
if v in self.varnames:
subThem[name] = self.varnames[v][0]
else:
oname = name
while name in names:
name = incrVar(name)
names.add(name)
subThem[oname] = name
vars[v] = name
R = PolynomialRing(QQ,sorted(list(names)),order="degrevlex")
gd = R.gens_dict()
subMe = {m:gd[n] for m,n in subMe.iteritems()}
subThem = {m:gd[n] for m,n in subThem.iteritems()}
vars = {v:(n,gd[n]) for v,n in vars.iteritems()}
def subIdeal(I,sub):
return [g(**sub) for g in I.gens()]*R
idealMe = {subIdeal(I,subMe):v(**subMe) for I,v in self.idealMap.iteritems()}
idealThem = {subIdeal(I,subThem):v(**subThem) for I,v in other.idealMap.iteritems()}
else:
R = self.R
idealMe = self.idealMap
idealThem = other.idealMap
vars = self.varnames
def consist(I,x,y):
if (x-y).numerator() not in I:
raise Exception("Inconsistent: %s != %s in ideal %s" %
(str(x),str(y),str(I)))
out = {}
if f is Idealized.UNION:
for I,v in idealMe.iteritems():
if I in idealThem:
consist(I,v,idealThem[I])
out[I] = v
for I,v in idealThem.iteritems():
if I in idealMe:
consist(I,v,idealMe[I])
out[I] = v
else:
for I,v in idealMe.iteritems():
if I in idealThem:
x = f(v,idealThem[I])
if I in out:
consist(I,x,out[I])
else: out[I] = x
else:
for J,w in idealThem.iteritems():
IJ = I+J
if not IJ.is_one():
x = f(v,w)
if IJ in out:
consist(IJ,x,out[IJ])
else:
out[IJ] = x
def gb(I):
II = [0]*R
for g in I.gens():
if g not in II: II = II+[g]*R
return II
def red(I,v):
if I.is_zero(): return v
return I.reduce(R(v.numerator())) / I.reduce(R(v.denominator()))
out = {gb(I):v for I,v in out.iteritems()}
out = {I:red(I,v) for I,v in out.iteritems()}
return Idealized(R,out,vars)
def reduce(self):
def red(I,v):
if I.is_zero(): return v
return I.reduce(R(v.numerator())) / I.reduce(R(v.denominator()))
out = {I:red(I,v) for I,v in self.idealMap.iteritems()}
return Idealized(self.R,out,self.vars)
Idealized.INF = Idealized.uvar("inf")
Idealized.ZOZ = Idealized.uvar("zoz")
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