class as_function:
    def __init__(self, C, g):
        self.curve = C
        F = C.base_ring
        n = C.height
        variable_names = 'x, y'
        for i in range(n):
            variable_names += ', z' + str(i)
        Rxyz = PolynomialRing(F, n+2, variable_names)
        x, y = Rxyz.gens()[:2]
        z = Rxyz.gens()[2:]
        RxyzQ = FractionField(Rxyz)
        self.function = RxyzQ(g)
        #self.function = as_reduction(AS, RxyzQ(g))
        
    def __repr__(self):
        return str(self.function)

    def __add__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        return as_function(C, g1 + g2)

    def __sub__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        return as_function(C, g1 - g2)

    def __rmul__(self, constant):
        C = self.curve
        g = self.function
        return as_function(C, constant*g)
    
    def __mul__(self, other):
        if isinstance(other, as_function):
            C = self.curve
            g1 = self.function
            g2 = other.function
            return as_function(C, g1*g2)
        if isinstance(other, as_form):
            C = self.curve
            g1 = self.function
            g2 = other.form
            return as_form(C, g1*g2)
    
    def __truediv__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        return as_function(C, g1/g2)
    
    def __pow__(self, exponent):
        C = self.curve
        g1 = self.function
        return as_function(C, g1^(exponent))
    
    def expansion_at_infty(self, i = 0):
        C = self.curve
        delta = C.nb_of_pts_at_infty
        F = C.base_ring
        x_series = C.x_series[i]
        y_series = C.y_series[i]
        z_series = C.z_series[i]
        n = C.height
        variable_names = 'x, y'
        for j in range(n):
            variable_names += ', z' + str(j)
        Rxyz = PolynomialRing(F, n+2, variable_names)
        x, y = Rxyz.gens()[:2]
        z = Rxyz.gens()[2:]
        RxyzQ = FractionField(Rxyz)
        prec = C.prec
        Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
        g = self.function
        g = RxyzQ(g)
        sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
        return g.substitute(sub_list)

    def group_action(self, ZN_tuple):
        C = self.curve
        n = C.height
        F = C.base_ring
        variable_names = 'x, y'
        for j in range(n):
            variable_names += ', z' + str(j)
        Rxyz = PolynomialRing(F, n+2, variable_names)
        x, y = Rxyz.gens()[:2]
        z = Rxyz.gens()[2:]
        RxyzQ = FractionField(Rxyz)
        sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
        g = self.function
        return as_function(C, g.substitute(sub_list))

    def trace(self):
        C = self.curve
        C_super = C.quotient
        n = C.height
        F = C.base_ring
        variable_names = 'x, y'
        for j in range(n):
            variable_names += ', z' + str(j)
        Rxyz = PolynomialRing(F, n+2, variable_names)
        x, y = Rxyz.gens()[:2]
        z = Rxyz.gens()[2:]
        RxyzQ = FractionField(Rxyz)
        result = as_function(C, 0)
        G = C.group
        for a in G:
            result += self.group_action(a)
        result = result.function
        Rxy.<x, y> = PolynomialRing(F, 2)
        Qxy = FractionField(Rxy)
        result = as_reduction(AS, result)
        return superelliptic_function(C_super, Qxy(result))
    
    
    def diffn(self):
        C = self.curve
        C_super = C.quotient
        n = C.height
        RxyzQ, Rxyz, x, y, z = C.fct_field
        fcts = C.functions
        f = self.function
        y_super = superelliptic_function(C_super, y)
        dy_super = y_super.diffn().form
        dz = []
        for i in range(n):
            dfct = fcts[i].diffn().form
            dz += [-dfct]
        result = f.derivative(x)
        result += f.derivative(y)*dy_super
        for i in range(n):
            result += f.derivative(z[i])*dz[i]
        return as_form(C, result)
        
    def valuation(self, i = 0):
        '''Return valuation at i-th place at infinity.'''
        C = self.curve
        F = C.base_ring
        Rt.<t> = LaurentSeriesRing(F)
        return Rt(self.expansion_at_infty(i)).valuation()