class as_cech:
    def __init__(self, C, omega, f):
        self.curve = C
        n = C.height
        F = C.base_ring
        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.omega0 = omega
        self.f = f
        self.omega8 = self.omega0 - self.f.diffn()
        if self.omega0.form not in Rxyz or self.omega8.valuation() < 0:
            raise ValueError('cech cocycle not regular')

    def __repr__(self):
        return "( " + str(self.omega0)+", " + str(self.f) + " )"

    def __add__(self, other):
        C = self.curve
        omega = self.omega0
        f = self.f
        omega1 = other.omega0
        f1 = other.f
        return as_cech(C, omega + omega1, f+f1)

    def __sub__(self, other):
        C = self.curve
        omega = self.omega0
        f = self.f
        omega1 = other.omega0
        f1 = other.f
        return as_cech(C, omega - omega1, f - f1)
    
    def __rmul__(self, constant):
        C = self.curve
        omega = self.omega0
        f = self.f
        return as_cech(C, constant*omega, constant*f)
    
    def coordinates(self, threshold=10, basis = 0):
        '''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
        AS = self.curve
        RxyzQ, Rxyz, x, y, z = AS.fct_field
        if basis == 0:
            basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
        holo_diffs = basis[0]
        coh_basis =  basis[1]
        dR = basis[2]
        F = AS.base_ring
        f_products = []
        for f in coh_basis:
            f_products += [[omega.serre_duality_pairing(f) for omega in holo_diffs]]
        print(f_products)
        product_of_fct_and_omegas = []
        fct = self.f
        product_of_fct_and_omegas = [omega.serre_duality_pairing(fct) for omega in holo_diffs]

        V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
        coh_coordinates = V.coordinates(product_of_fct_and_omegas) #coeficients of self in the basis elts coming from cohomology of OX
        for i in range(AS.genus()):
            self -= coh_coordinates[i]*dR[i+AS.genus()]
        #We remove now from f the summands which are obviously regular at infty
        print(self, [])
        f_num = numerator(self.f.function)
        f_den = denominator(self.f.function)
        v_f_den = as_function(AS, f_den).valuation()
        for a in f_num.monomials():
            if as_function(AS, a).valuation() >= v_f_den:
                self.f.function -= f_num.monomial_coefficient(a)*a/f_den
        f_num = numerator(self.f.function)
        f_den = denominator(self.f.function)
        quo, rem = f_num.quo_rem(f_den)
        if as_function(AS, rem/f_den).valuation() >= 0:
            self.f = as_function(AS, quo)
            hol_form = self.omega0 - self.f.diffn() #now this should be a holomorphic form
            hol_form = as_form(AS, as_reduction(AS, hol_form.form))
            print('hol_form', hol_form)
            return hol_form.coordinates() + coh_coordinates
        print(self, [omega.serre_duality_pairing(self.f) for omega in holo_diffs])
        raise ValueError('I arrived at a form (omega, 0), in which omega is not regular on U0. I hoped this wouldn t happen.')
        
    
    def group_action(self, g):
        AS = self.curve
        omega = self.omega0
        f = self.f
        return as_cech(self.curve, omega.group_action(g), f.group_action(g))