from util import *


@apply(simplify=None)
def apply(gt):
    x, a = gt.of(Abs > Expr)
    return Or(x < -a, x > a)


@prove
def prove(Eq):
    from axiom import algebra

    x, a = Symbol(real=True, given=True)
    Eq << apply(abs(x) > a)

    Eq << ~Eq[0]

    Eq << Eq[-1].this.apply(algebra.abs_le.imply.et)

    Eq <<= Eq[1] & Eq[-1]

    


if __name__ == '__main__':
    run()
# created on 2018-07-31
# updated on 2022-01-07