TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used


Problem 1 remains open; application of the following processors failed [DependencyGraph (878ms), SubtermCriterion (0ms), DependencyGraph (791ms), PolynomialLinearRange4iUR (10000ms), PolynomialLinearRange8NegiUR (timeout), PolynomialLinearRange8NegiUR (-4ms), ReductionPairSAT (timeout)].

The following open problems remain:

Open Dependency Pair Problem 1

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs)) → app#(f, x)app#(app(map, f), app(app(cons, x), xs)) → app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(app(app(filter2, true), f), x), xs) → app#(app(cons, x), app(app(filter, f), xs))app#(app(filter, f), app(app(cons, x), xs)) → app#(f, x)
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(:, x)app#(app(app(app(filter2, true), f), x), xs) → app#(filter, f)
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(:, app(app(:, x), z))app#(app(app(app(filter2, true), f), x), xs) → app#(app(filter, f), xs)
app#(app(filter, f), app(app(cons, x), xs)) → app#(app(app(app(filter2, app(f, x)), f), x), xs)app#(app(filter, f), app(app(cons, x), xs)) → app#(filter2, app(f, x))
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(app(:, x), z)app#(app(map, f), app(app(cons, x), xs)) → app#(map, f)
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(app(:, x), y)app#(app(app(app(filter2, false), f), x), xs) → app#(filter, f)
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(app(:, app(app(:, app(app(:, x), y)), z)), u)
app#(app(map, f), app(app(cons, x), xs)) → app#(app(map, f), xs)app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(app(:, app(app(:, x), y)), z)
app#(app(filter, f), app(app(cons, x), xs)) → app#(app(app(filter2, app(f, x)), f), x)app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(:, app(app(:, app(app(:, x), y)), z))
app#(app(filter, f), app(app(cons, x), xs)) → app#(app(filter2, app(f, x)), f)app#(app(app(app(filter2, true), f), x), xs) → app#(cons, x)
app#(app(app(app(filter2, false), f), x), xs) → app#(app(filter, f), xs)app#(app(map, f), app(app(cons, x), xs)) → app#(cons, app(f, x))
app#(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app#(:, app(app(:, x), y))

Rewrite Rules

app(app(:, app(app(:, app(app(:, app(app(:, C), x)), y)), z)), u) → app(app(:, app(app(:, x), z)), app(app(:, app(app(:, app(app(:, x), y)), z)), u))app(app(map, f), nil) → nil
app(app(map, f), app(app(cons, x), xs)) → app(app(cons, app(f, x)), app(app(map, f), xs))app(app(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)

Original Signature

Termination of terms over the following signature is verified: app, C, map, false, true, :, filter2, filter, cons, nil