# this file will be run by the remote sage server, so should not import local files.
from functools import reduce
from typing import Optional, Iterator
from attr import dataclass
from sage.rings.polynomial.polydict import ETuple

def mk_app(*args: str) -> str:
    return "(" + " ".join(args) + ")"

def const_to_string(coeff: QQ) -> str:
    return mk_app("poly.const", str(coeff.numerator()) + "/" + str(coeff.denominator()))

def power_to_string(var: int, pow: int) -> str:
    assert pow != 0
    var_s = mk_app("poly.var", str(var))
    if pow == 1:
        return var_s
    return mk_app("poly.pow", var_s, str(pow))

@dataclass
class MonomForm:
    """
    `MonomForm` stores the string representation of a monomial.

    To cleanly format sums of monomials, we need to be careful with negation:
    if the first monomial in a sum is negative, we print the negation symbol;
    if a subsequent monomial is negative, we subtract the non-negated version.
    `MonomForm` always stores the *positive* representation of a monomial in the `pos_form` field.
    If the monomial is in fact negative, it also stores the full (negative) representation
    in the `neg_form` field. For instance, putting `-2*x` into a `MonomForm` would store
    the representation of `2*x` in `pos_form` and `-2*x` in `neg_form`.
    """
    pos_form: str
    neg_form: Optional[str] = None

def sum_to_string(terms: Iterator[MonomForm]) -> str:
    try:
        first = next(terms)
    except StopIteration:
        return const_to_string(QQ(0))
    else:
        first_form = first.neg_form if first.neg_form is not None else first.pos_form
        return reduce(
            lambda old, nxt: mk_app("poly.sub" if nxt.neg_form is not None else "poly.add", old, nxt.pos_form),
            terms, first_form)

def monomial_to_string(etuple: ETuple, coeff: QQ) -> MonomForm:
    etuple = list(etuple.sparse_iter())
    if abs(coeff) == 1 and len(etuple) > 0:
        powforms = [power_to_string(t[0], t[1]) for t in etuple]
        pos_form = reduce(
            lambda s, t: mk_app("poly.mul", s, t), powforms)
        return MonomForm(pos_form, mk_app("poly.neg", pos_form) if coeff < 0 else None)
    else:
        pos_form = reduce(
            lambda s, t: mk_app("poly.mul", s, power_to_string(t[0], t[1])),
            etuple,
            const_to_string(abs(coeff)))
        neg_form = reduce(
            lambda s, t: mk_app("poly.mul", s, power_to_string(t[0], t[1])),
            etuple,
            const_to_string(coeff)) if coeff < 0 else None
    return MonomForm(pos_form, neg_form)


def polynomial_to_string(p) -> str:
    return sum_to_string(monomial_to_string(pows, coeff) for pows, coeff in p.dict().items())