TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60003 ms.

The following DP Processors were used


Problem 1 remains open; application of the following processors failed [DependencyGraph (10591ms), SubtermCriterion (2ms), DependencyGraph (9961ms), PolynomialLinearRange4iUR (10013ms), DependencyGraph (timeout), PolynomialLinearRange8NegiUR (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

app#(app(filter, fun), app(app(cons, x), xs))app#(app(app(app(filter2, app(fun, x)), fun), x), xs)app#(app(minus, app(s, x)), 0)app#(s, x)
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u))app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(f, app(s, x))
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(minus, y), x)app#(app(le, app(s, x)), app(s, y))app#(le, x)
app#(app(map, fun), app(app(cons, x), xs))app#(fun, x)app#(app(map, fun), app(app(cons, x), xs))app#(app(cons, app(fun, x)), app(app(map, fun), xs))
app#(perfectp, app(s, x))app#(s, 0)app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(app(app(f, x), u), z), u)
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(app(f, x), u), z)app#(app(app(app(f, app(s, x)), 0), z), u)app#(minus, z)
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(s, x)app#(app(le, app(s, x)), app(s, y))app#(app(le, x), y)
app#(perfectp, app(s, x))app#(s, x)app#(app(app(app(filter2, true), fun), x), xs)app#(filter, fun)
app#(app(map, fun), app(app(cons, x), xs))app#(app(map, fun), xs)app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(f, x), u)
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(minus, y)app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)
app#(app(app(app(filter2, true), fun), x), xs)app#(cons, x)app#(app(minus, app(s, x)), app(s, y))app#(app(minus, x), y)
app#(app(filter, fun), app(app(cons, x), xs))app#(app(app(filter2, app(fun, x)), fun), x)app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(le, x), y)
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(if, app(app(le, x), y))app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(le, x)
app#(app(filter, fun), app(app(cons, x), xs))app#(fun, x)app#(app(app(app(f, app(s, x)), 0), z), u)app#(s, x)
app#(app(app(app(filter2, false), fun), x), xs)app#(filter, fun)app#(app(app(app(filter2, true), fun), x), xs)app#(app(cons, x), app(app(filter, fun), xs))
app#(app(app(app(filter2, true), fun), x), xs)app#(app(filter, fun), xs)app#(app(app(app(f, app(s, x)), 0), z), u)app#(app(f, x), u)
app#(perfectp, app(s, x))app#(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))app#(app(minus, app(s, x)), app(s, y))app#(minus, x)
app#(perfectp, app(s, x))app#(f, x)app#(app(app(app(filter2, false), fun), x), xs)app#(app(filter, fun), xs)
app#(app(app(app(f, app(s, x)), 0), z), u)app#(app(app(f, x), u), app(app(minus, z), app(s, x)))app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(app(f, app(s, x)), app(app(minus, y), x)), z)
app#(app(app(app(f, app(s, x)), 0), z), u)app#(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app#(app(app(app(f, app(s, x)), 0), z), u)app#(app(minus, z), app(s, x))app#(app(map, fun), app(app(cons, x), xs))app#(map, fun)
app#(app(filter, fun), app(app(cons, x), xs))app#(app(filter2, app(fun, x)), fun)app#(perfectp, app(s, x))app#(app(app(f, x), app(s, 0)), app(s, x))
app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(app(f, app(s, x)), app(app(minus, y), x))app#(app(app(app(f, app(s, x)), app(s, y)), z), u)app#(f, x)
app#(app(app(app(f, app(s, x)), 0), z), u)app#(f, x)app#(app(map, fun), app(app(cons, x), xs))app#(cons, app(fun, x))
app#(app(filter, fun), app(app(cons, x), xs))app#(filter2, app(fun, x))app#(perfectp, app(s, x))app#(app(f, x), app(s, 0))

Rewrite Rules

app(app(minus, 0), y)0app(app(minus, app(s, x)), 0)app(s, x)
app(app(minus, app(s, x)), app(s, y))app(app(minus, x), y)app(app(le, 0), y)true
app(app(le, app(s, x)), 0)falseapp(app(le, app(s, x)), app(s, y))app(app(le, x), y)
app(app(app(if, true), x), y)xapp(app(app(if, false), x), y)y
app(perfectp, 0)falseapp(perfectp, app(s, x))app(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x))
app(app(app(app(f, 0), y), 0), u)trueapp(app(app(app(f, 0), y), app(s, z)), u)false
app(app(app(app(f, app(s, x)), 0), z), u)app(app(app(app(f, x), u), app(app(minus, z), app(s, x))), u)app(app(app(app(f, app(s, x)), app(s, y)), z), u)app(app(app(if, app(app(le, x), y)), app(app(app(app(f, app(s, x)), app(app(minus, y), x)), z), u)), app(app(app(app(f, x), u), z), u))
app(app(map, fun), nil)nilapp(app(map, fun), app(app(cons, x), xs))app(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(filter, fun), nil)nilapp(app(filter, fun), app(app(cons, x), xs))app(app(app(app(filter2, app(fun, x)), fun), x), xs)
app(app(app(app(filter2, true), fun), x), xs)app(app(cons, x), app(app(filter, fun), xs))app(app(app(app(filter2, false), fun), x), xs)app(app(filter, fun), xs)

Original Signature

Termination of terms over the following signature is verified: f, app, minus, true, 0, s, le, if, perfectp, map, false, filter2, filter, nil, cons