TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60018 ms.

The following DP Processors were used


Problem 1 remains open; application of the following processors failed [DependencyGraph (837ms), SubtermCriterion (1ms), DependencyGraph (786ms), PolynomialLinearRange4iUR (8850ms), DependencyGraph (725ms), PolynomialLinearRange8NegiUR (25385ms), DependencyGraph (697ms), ReductionPairSAT (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

a#(a(append, a(a(cons, x), xs)), ys)a#(a(append, xs), ys)a#(qs, a(a(cons, x), xs))a#(qs, a(a(filter, a(not, a(le, x))), xs))
a#(qs, a(a(cons, x), xs))a#(a(filter, a(not, a(le, x))), xs)a#(a(append, a(a(cons, x), xs)), ys)a#(append, xs)
a#(a(append, a(a(cons, x), xs)), ys)a#(a(cons, x), a(a(append, xs), ys))a#(a(not, f), b)a#(not2, a(f, b))
a#(a(not, f), b)a#(f, b)a#(a(a(if, true), x), xs)a#(cons, x)
a#(a(append, a(a(cons, x), xs)), ys)a#(cons, x)a#(qs, a(a(cons, x), xs))a#(filter, a(not, a(le, x)))
a#(a(filter, f), a(a(cons, x), xs))a#(f, x)a#(qs, a(a(cons, x), xs))a#(qs, a(a(filter, a(le, x)), xs))
a#(qs, a(a(cons, x), xs))a#(a(cons, x), a(qs, a(a(filter, a(not, a(le, x))), xs)))a#(qs, a(a(cons, x), xs))a#(not, a(le, x))
a#(qs, a(a(cons, x), xs))a#(a(filter, a(le, x)), xs)a#(a(le, a(s, x)), a(s, y))a#(le, x)
a#(a(filter, f), a(a(cons, x), xs))a#(a(if, a(f, x)), x)a#(qs, a(a(cons, x), xs))a#(a(append, a(qs, a(a(filter, a(le, x)), xs))), a(a(cons, x), a(qs, a(a(filter, a(not, a(le, x))), xs))))
a#(a(a(if, true), x), xs)a#(a(cons, x), xs)a#(a(le, a(s, x)), a(s, y))a#(a(le, x), y)
a#(qs, a(a(cons, x), xs))a#(append, a(qs, a(a(filter, a(le, x)), xs)))a#(qs, a(a(cons, x), xs))a#(filter, a(le, x))
a#(qs, a(a(cons, x), xs))a#(cons, x)a#(a(filter, f), a(a(cons, x), xs))a#(a(a(if, a(f, x)), x), a(a(filter, f), xs))
a#(a(filter, f), a(a(cons, x), xs))a#(filter, f)a#(a(filter, f), a(a(cons, x), xs))a#(if, a(f, x))
a#(a(filter, f), a(a(cons, x), xs))a#(a(filter, f), xs)a#(qs, a(a(cons, x), xs))a#(le, x)

Rewrite Rules

a(a(append, nil), ys)ysa(a(append, a(a(cons, x), xs)), ys)a(a(cons, x), a(a(append, xs), ys))
a(a(filter, f), nil)nila(a(filter, f), a(a(cons, x), xs))a(a(a(if, a(f, x)), x), a(a(filter, f), xs))
a(a(le, 0), y)truea(a(le, a(s, x)), 0)false
a(a(le, a(s, x)), a(s, y))a(a(le, x), y)a(a(a(if, true), x), xs)a(a(cons, x), xs)
a(a(a(if, false), x), xs)xsa(a(not, f), b)a(not2, a(f, b))
a(not2, true)falsea(not2, false)true
a(qs, nil)nila(qs, a(a(cons, x), xs))a(a(append, a(qs, a(a(filter, a(le, x)), xs))), a(a(cons, x), a(qs, a(a(filter, a(not, a(le, x))), xs))))

Original Signature

Termination of terms over the following signature is verified: append, a, true, not, 0, le, s, if, not2, false, qs, filter, nil, cons