class as_polyform:
    def __init__(self, form, mult):
        self.form = form
        self.curve = form.curve
        self.mult =  mult
        
    def __add__(self, other):
        return as_polyform(self.form + other.form, self.mult)
    
    def __repr__(self):
        return '(' + str(self.form) + ') dx⊗' + str(self.mult)
    
    def expansion_at_infty(self):
        return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
    
    def coordinates(self, basis = 0):
        """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
        AS = self.curve
        if basis == 0:
            basis = AS.holomorphic_differentials_basis()
        RxyzQ, Rxyz, x, y, z = AS.fct_field
        # We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
        # and sometimes basis elements have denominators. Thus we multiply by them.
        denom = LCM([denominator(omega.form.function) for omega in basis])
        basis = [denom*omega.form.function for omega in basis]
        self_with_no_denominator = denom*self.form.function
        return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
    
    
def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
    v = AS.dx.valuation()
    result = AS.at_most_poles(mult*v, threshold=threshold)
    result = [as_polyform(omega, mult) for omega in result]
    if mult == 1 and len(result) < AS.genus():
        raise ValueError('Increase threshold, not all forms found.')
    if mult > 1 and len(result) < (2*mult - 1)*(AS.genus() - 1):
        raise ValueError('Increase threshold, not all forms found.')
    return result

as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis

def as_symmetric_power_basis(AS, n, threshold = 8):
    g = AS.genus()
    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
    from itertools import product
    indices = [list(range(g)) for i in range(n)]
    indices_nonrepeating = []
    for i in product(*indices):
        if non_decreasing(i):
            indices_nonrepeating += [i]
    result = []
    for i in indices_nonrepeating:
        tensor_form = [as_function(AS, B0[i[j]].form) for j in range(n)]
        tensor_form2 = [B0[i[j]] for j in range(n)]
        print(hash(tuple(tensor_form)))
        print([tensor_form], tuple(tensor_form))
        aux_dict = {}
        aux_dict[tuple(list(tensor_form))] = 1
        print(aux_dict)
        result += [as_symmetric_product_forms([tensor_form], aux_dict)]
    print(binomial(g + n - 1, n), len(result))
    return result

def as_canonical_ideal(AS, n, threshold=8):
    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
    F = AS.base_ring
    g = AS.genus()
    RxyzQ, Rxyz, x, y, z = AS.fct_field
    from itertools import product
    #B1 = as_symmetric_power_basis(AS, n, threshold = 8)
    B1 = [[B0[i[j]] for j in range(n)] for i in product(*indices)]
    B2 = AS.holo_polydifferentials_basis(n, threshold = threshold)
    g = AS.genus()
    r = len(B2)
    M = matrix(F, len(B1), r)
    for i in range(0, len(B1)):
        atuple = B1[i]
        c = Rxyz(1)
        for a in atuple:
            c = c*a.form
        c = as_reduction(AS, c)
        c = as_function(AS, c)
        c = as_polyform(c, n)
        #return M, c.coordinates(basis=B2)
        M[i, :] = vector(c.coordinates(basis=B2))
    K = M.kernel().basis()
    result = []
    for v in K:
        coeffs = {tuple(b) : 0 for b in B1}
        for i in range(r):
            if v[i] != 0:
                coeffs[tuple(B1[i])] += v[i]
        result += [as_symmetric_product_forms(B1, coeffs)]
    return result

as_cover.canonical_ideal = as_canonical_ideal

def as_canonical_ideal_polynomials(AS, n, threshold=8):
    return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]

as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials

class as_tensor_product_forms:
    def __init__(self, pairs_of_forms, coeffs):
        self.pairs = pairs_of_forms
        self.coeffs = coeffs #dictionary
        self.curve = pairs_of_forms[0][0].curve
        
    def coordinates(self, basis = 0):
        AS = self.curve
        g = AS.genus()
        F = AS.base_ring
        if basis == 0:
            basis = AS.holomorphic_differentials_basis()
        result = matrix(F, g, g)
        for (omega1, omega2) in self.pairs:
            c = omega1.coordinates(basis = basis)
            d = omega2.coordinates(basis = basis)
            for i in range(g):
                for j in range(g):
                    result[i, j] += self.coeffs[(omega1, omega2)]*c[i]*d[j]
        return result
    
    def __repr__(self):
        result = ''
        for (omega1, omega2) in self.pairs:
            if self.coeffs[(omega1, omega2)] !=0:
                result += str(self.coeffs[(omega1, omega2)]) + ' * ' + str(omega1) + '⊗' + str(omega2) + ' + '
        return result
    
    def polynomial(self):
        AS = self.curve
        F = AS.base_ring
        g = AS.genus()
        M = self.coordinates()
        Rg = PolynomialRing(F, 'X', g)
        X = Rg.gens()
        return sum(M[i, j] * X[i]*X[j] for i in range(g) for j in range(g))
    
    def group_action(self, elt):
        p = self.base_ring.characteristic()
        n = self.height
        elt1 = [p^n - a for a in elt]
        pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
        coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
        return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
    
class as_symmetric_product_forms:
    def __init__(self, tuples_of_forms, coeffs):
        self.n = len(tuples_of_forms[0])
        self.tuples = tuples_of_forms
        self.coeffs = coeffs #dictionary
        self.curve = tuples_of_forms[0][0].curve
        
    def coordinates(self, basis = 0):
        AS = self.curve
        g = AS.genus()
        F = AS.base_ring
        if basis == 0:
            basis = AS.holomorphic_differentials_basis()
        from itertools import product
        indices = [list(range(g)) for i in range(self.n)]
        result = {i : 0 for i in product(*indices)}
        for atuple in self.tuples:
            coors = [omega.coordinates(basis = basis) for omega in atuple]
            for i in product(*indices):
                aux_product = 1
                for j in range(n):
                    aux_product *= coors[j][i[j]]
                result[i] += self.coeffs[tuple(atuple)]*aux_product
        return result
    
    def __repr__(self):
        result = ''
        for atuple in self.tuples:
            if self.coeffs[tuple(atuple)] !=0:
                result += str(self.coeffs[tuple(atuple)]) + ' * '
                for j in range(self.n):
                    result += "(" + str(atuple[j]) + ")"
                    if j != self.n-1:
                        result + '⊗'
            result += ' + '
        return result
    
    def multiply(self):
        n = self.n
        AS = self.curve
        RxyzQ, Rxyz, x, y, z = AS.fct_field
        result = as_polyform(0*AS.x, n)
        for atuple in self.tuples:
            aux_product = Rxyz(1)
            for fct in atuple:
                aux_product = aux_product * fct.form
            aux_product = as_function(AS, aux_product)
            result += as_polyform(self.coeffs[tuple(atuple)]*aux_product, n)
        return result

    def polynomial(self):
        AS = self.curve
        F = AS.base_ring
        g = AS.genus()
        M = self.coordinates()
        n = self.n
        Rg = PolynomialRing(F, 'X', g)
        X = Rg.gens()
        from itertools import product
        indices = [list(range(g)) for i in range(self.n)]
        result = Rg(0)
        for i in product(*indices):
            aux_product = Rg(1)
            for j in range(n):
                aux_product *= X[i[j]]
            result += M[i] * aux_product
        return result
    
    def group_action(self, elt):
        p = self.base_ring.characteristic()
        n = self.height
        elt1 = [p^n - a for a in elt]
        pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
        coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
        return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
    
def non_decreasing(L):
    return all(x<=y for x, y in zip(L, L[1:]))