class Derivative(Expr):
@staticmethod
def add_order():
return Mul.add_order()-1
@staticmethod
def mul_order():
return AppliedFunction.mul_order()+1
rules = RuleList(
Rule(
'd(-...) -> -d(...)',
lambda obj: isinstance(obj.arg, Neg),
lambda obj, v: Neg( Derivative(obj.arg.arg, obj.wrt) )
),
Rule(
'd(sum(...)) -> sum(d(...))',
lambda obj: isinstance(obj.arg, Summation),
lambda obj, v: Summation( Derivative(obj.arg.arg, obj.wrt), *obj.arg.lims )
),
Rule(
'd(add(...)) -> add(d(...))',
lambda obj: isinstance(obj.arg, Add),
lambda obj, v: Add( *[Derivative(arg, obj.wrt) for arg in obj.arg.args] )
),
Rule(
'derivative rule',
lambda obj: True,
lambda obj, v: rules_derivative(obj, v)
)
)
def __new__(cls, *args, **kwargs):
arg, wrt, = args
if isinstance(wrt, Number):
raise Exception('trying to derivate with respect to a number')
if isinstance(arg, Number):
return Number(0)
if arg == wrt:
return Number(1)
obj = super().__new__(cls, *args)
if not hasattr(obj, 'wrt'):
obj.arg = arg
obj.wrt = wrt
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
wrt = self.wrt
if not isinstance(wrt, Symbol):
wrt = delim(wrt)
if isinstance(self.arg, Expr):
return fr'\frac{{ {{\rm d}} }}{{ {{\rm d}}{self.wrt} }}{delim(self.arg)}'
return fr'\frac{{ {{\rm d}}{self.arg} }}{{ {{\rm d}}{self.wrt} }}'
class Neg(Expr):
def add_order(self):
return self.arg.add_order() + .1
def mul_order(self):
return self.arg.mul_order()
rules = RuleList(
# Rule(
# r'negative number',
# lambda obj: isinstance(obj.arg, Number),
# lambda obj, v: NegNumber(obj.arg.value)
# ),
Rule(r'double negative', lambda obj: isinstance(obj.arg, Neg),
lambda obj, v: obj.arg.arg))
def __new__(cls, arg):
obj = super().__new__(cls, arg)
if not hasattr(obj, 'arg'):
obj.arg = arg
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
arg = self.arg
s = str(arg)
if not isinstance(arg, Symbol):
if not hasattr(arg, 'base'):
s = delim(s)
return fr'-{s}'
class Pow(Expr):
@staticmethod
def add_order():
return Mul.add_order() - 1
@staticmethod
def mul_order():
return Add.mul_order() - 1
rules = RuleList(
Rule('(x!=0)^0 -> 1',
lambda obj: obj.base == Number(0) and not obj.power == Number(0),
lambda obj, v: Number(1)),
Rule('0^{x!=0} -> 0',
lambda obj: obj.power == Number(0) and not obj.base == Number(0),
lambda obj, v: Number(0)),
Rule('x^1 -> x', lambda obj: obj.power == Number(1),
lambda obj, v: obj.base),
Rule('1^x -> 1', lambda obj: obj.base == Number(1),
lambda obj, v: Number(1)),
Rule('(x^a)^b -> x^{a*b}', lambda obj: isinstance(obj.base, Pow),
lambda obj, v: obj.base.base**(obj.base.power * obj.power)),
)
def __new__(cls, *args, **kwargs):
obj = super().__new__(cls, *args)
if not hasattr(obj, 'base'):
obj.base = args[0]
obj.power = args[1]
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
base = self.base
if not isinstance(base, (Symbol, Indexed)):
base = delim(base)
power = self.power
if isinstance(power, Neg):
if power == -Number(1):
return fr'\frac{{1}}{{{base}}}'
return fr'\frac{{1}}{{{base}^{{{power.arg}}} }}'
return fr'{base}^{{ {power} }}'
class Add(Expr):
@staticmethod
def add_order():
return Symbol.add_order() - 1
@staticmethod
def mul_order():
return Symbol.mul_order() + 2
rules = RuleList(
Rule(r'flatten args', lambda obj: any(areinstance(obj.args, Add)),
lambda obj, v: Add(*flatten_args(obj))),
Rule(
r'add numbers', lambda obj: any(areinstance(obj.args, Number)),
lambda obj, v: Add(Number(sum(getnumbervalues(
obj))), *[arg for arg in obj.args if not isnumber(arg)])),
Rule(r'add rules', lambda obj: True, lambda obj, v: rules_add(obj, v)),
)
def __new__(cls, *args, **kwargs):
if len(args) == 0:
return Zero
if len(args) == 1:
return args[0]
if Zero in args:
return Add(*[arg for arg in args if not arg == Zero])
args = sorted(args, key=lambda x: x.add_order())
obj = super().__new__(cls, *args)
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
s = '+'.join([
str(arg) if not isinstance(arg, Add) else delim(str(arg))
for arg in self.args
])
return s.replace('+-', '-')
def __lshift__(self, arg):
def _(obj, v):
print(arg.label)
print(self.args[0].label)
print(obj.args[0].label)
string = arg.label.replace(self.args[0].label, obj.args[0].label)
print('string', string)
return eval(
arg.label.replace(self.args[0].label, obj.args[0].label))
rule = Rule(fr'{self} := {arg}',
lambda obj: obj.operator == self.operator,
lambda obj, v: _(obj, v))
self.__class__.rules.append(rule)
return
class DiracDelta(AppliedFunction):
rules = RuleList(
Rule(
'delta dirac zero',
lambda obj: obj.arg == Number(0),
lambda obj, v: Number(1)
)
)
def __new__(cls, arg, **kwargs):
if arg == Number(0):
return Number(1)
symbol = Symbol(r'{\delta}')
obj = super().__new__(cls, symbol, arg)
if not hasattr(obj, 'arg'):
obj.arg = arg
return obj
class KroneckerDelta(Indexed):
@staticmethod
def add_order():
return Mul.add_order() - 1
@staticmethod
def mul_order():
return AppliedFunction.mul_order() + 2
rules = RuleList(
Rule('KroneckerDelta rule',
lambda obj: obj.indices[0] == obj.indices[1],
lambda obj, v: Number(1)))
def __new__(cls, ind0, ind1, **kwargs):
if ind0 == ind1:
return Number(1)
symbol = Symbol(r'{\delta}')
obj = super().__new__(cls, symbol, [ind0, ind1])
return obj
class Mul(Expr):
@staticmethod
def add_order():
return Symbol.add_order()+2
@staticmethod
def mul_order():
return Symbol.mul_order()+1
rules = RuleList(
Rule(
r'mul rules',
lambda obj: True,
lambda obj, v: rules_mul(obj, v)
),
)
def __new__(cls, *args, **kwargs):
if len(args) == 0:
return Zero
if len(args) == 1:
return args[0]
if Zero in args:
return Zero
if One in args:
return Mul(*[arg for arg in args if not arg == One])
args = sorted( args, key=lambda x: x.mul_order() )
obj = super().__new__(cls, *args )
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
s = ''
for arg in self.args:
if isinstance(arg, (Add, Neg, Mul)):
s += delim(str(arg))
else:
s += str(arg)
return s
class Summation(Expr):
@staticmethod
def add_order():
return Number.add_order() + 1
@staticmethod
def mul_order():
return AppliedFunction.mul_order() + 1
rules = RuleList(
Rule('summation rule', lambda obj: True,
lambda obj, v: rules_summation(obj, v)))
def __new__(cls, arg, *lims, **kwargs):
lims = list(map(translate, lims))
if len(lims) < 2:
lims = (lims[0], -Infinity, Infinity)
if len(lims) < 3:
lims = (lims[0], lims[1], Infinity)
if arg == Number(1):
return lims[2] - lims[1]
obj = super().__new__(cls, arg, *lims)
if not hasattr(obj, 'name'):
obj.arg = arg
obj.lims = lims
return obj
def __init__(self, *args, **kwargs):
pass
def __str__(self):
lims = self.lims
arg = self.arg
if isinstance(arg, Add):
arg = delim(arg)
return fr'\sum_{{{lims[0]}={lims[1]}}}^{{{lims[2]}}}{arg}'