human
Please write code following the instructions in jupyter notebook style.
 ... Write me a code to and the other steps required?
assistant
#Here is code to and the other steps required.

delta, antidifference, I, E, f = symbols(r'\Delta \Delta^{-1} I E f', cls=Function) # combinators

I_eval_rule = define(I(x), x) # identity
E_eval_rule = define(E(x),x+1) # forward shift

I_eval_rule, E_eval_rule

delta_eval_rule = define(delta(f(x)), f(x+1)-f(x),) # forward difference def
delta_pow_rule = define(delta(f(x))**k, delta(delta(f(x))**(k-1)),) # repeated applications of differentiation

delta_eval_rule, delta_pow_rule

delta_EI_conv = define(delta(x), E(x)-I(x)) # conversion via combinators

delta_EI_conv

def rewrite(eq, rule, **kwds):
    return eq.replace(query=rule.lhs, value=rule.rhs, **kwds)

s = Wild('s')
rewrite(delta_eval_rule, define(f, Lambda([s], ff(s,n))))

f_wild = WildFunction('f')
D=define(delta(f_wild**k), delta(delta(f_wild**(k-1))))
rewrite(delta_pow_rule, D)

class ForwardDifference(Function):
    
    def _latex(self, *_):
        if len(self.args) < 2:
            func, args = self.args[0].func, self.args[0].args
            v = args[0]
        else:
            func, args = self.args[0].func, self.args[0].args
            v = self.args[1]
            
        D = Function(r'\Delta_{{{}}}'.format(latex(v)))
        expr = D(func(*args))
        return latex(expr)
    
    def doit(self):
        
        if len(self.args) < 2:
            func, args = self.args[0].func, self.args[0].args
            v = args[0]
        else:
            func, args = self.args[0].func, self.args[0].args
            v = self.args[1]
            
        return func(*map(lambda a: a.subs({v:v+1}, simulataneous=True),args)) - func(*args)
                
    def _e2val_power(self, k):
        if k.is_Number:
            return ForwardDifference(Pow(ForwardDifference(self.args), k-1, evaluate=False))
        else:
            return super()._eval_power(k)