"""Block stacking (overhang) problem, closed form of the constraints; checked in exact rational arithmetic.

Upstream (block_stacking_problem.ipynb) used a float tolerance of 100*eps on every constraint, which lets a
submission exceed the true optimum H_n/2 by accumulated slack; here all constraints are exact (non-strict).
"""
from fractions import Fraction

from run import Invalid, float_list


def check(inst, ans):
    n = inst["n"]
    x = [Fraction(v) for v in float_list(ans.get("x"), n, "x")]
    xs = [Fraction(0)] + x  # x_0 = 0
    for i in range(n):
        if xs[i + 1] < xs[i]:
            raise Invalid(f"need x_{i} <= x_{i + 1}")
    tail = Fraction(0)
    for i in range(n - 1, -1, -1):  # tail = x_{i+1} + ... + x_n
        tail += xs[i + 1]
        if tail > (n - i) * (xs[i] + Fraction(1, 2)):
            raise Invalid(f"stability constraint violated at i = {i}")
    return float(xs[n]), {}