The TRS could not be proven terminating. The proof attempt took 60046 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (6988ms), SubtermCriterion (1ms), DependencyGraph (6656ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (6711ms), PolynomialLinearRange8NegiUR (timeout)].
| 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(filter, fun), app(app(cons, x), xs)) | → | app#(app(app(app(filter2, app(fun, x)), fun), x), xs) | 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(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(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(filter, fun), app(app(cons, x), xs)) | → | app#(fun, x) | app#(app(map, 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(map, fun), app(app(cons, x), xs)) | → | app#(app(cons, app(fun, x)), app(app(map, fun), xs)) | 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#(s, 0) | app#(perfectp, app(s, x)) | → | app#(app(app(app(f, x), app(s, 0)), app(s, x)), app(s, x)) | |
| app#(app(app(app(f, app(s, x)), app(s, y)), z), u) | → | app#(app(app(app(f, x), u), z), u) | 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(f, x), u), z) | app#(app(app(app(f, app(s, x)), 0), z), u) | → | app#(minus, z) | |
| app#(app(map, fun), app(app(cons, x), xs)) | → | app#(map, fun) | app#(app(app(app(f, app(s, x)), 0), z), u) | → | app#(app(minus, z), app(s, x)) | |
| 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)), app(s, y)), z), u) | → | app#(s, x) | |
| app#(perfectp, app(s, x)) | → | app#(s, x) | 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#(f, 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(filter2, true), fun), x), xs) | → | app#(filter, fun) | |
| app#(app(app(app(f, app(s, x)), 0), z), u) | → | app#(f, x) | app#(app(filter, fun), app(app(cons, x), xs)) | → | app#(filter2, app(fun, x)) | |
| app#(app(map, fun), app(app(cons, x), xs)) | → | app#(app(map, fun), xs) | app#(app(map, fun), app(app(cons, x), xs)) | → | app#(cons, app(fun, x)) | |
| app#(perfectp, app(s, x)) | → | app#(app(f, x), app(s, 0)) |
| app(perfectp, 0) | → | false | app(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) | → | true | app(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) | → | nil | app(app(map, fun), app(app(cons, x), xs)) | → | app(app(cons, app(fun, x)), app(app(map, fun), xs)) | |
| app(app(filter, fun), nil) | → | nil | app(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) |
Termination of terms over the following signature is verified: f, app, minus, true, 0, s, le, if, perfectp, map, false, filter2, filter, nil, cons