app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
↳ QTRS
↳ DependencyPairsProof
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(f, 0), 1), x) → APP(s, x)
APP(app(app(f, x), y), app(s, z)) → APP(app(app(f, 0), 1), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(app(f, x), y), app(s, z)) → APP(s, app(app(app(f, 0), 1), z))
APP(app(app(f, 0), 1), x) → APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) → APP(app(app(f, app(s, x)), x), x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(app(f, x), y), app(s, z)) → APP(f, 0)
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(f, 0), 1), x) → APP(f, app(s, x))
APP(app(app(f, x), y), app(s, z)) → APP(app(f, 0), 1)
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
APP(app(app(app(filter2, true), 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(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP(app(app(app(filter2, false), fun), x), xs) → APP(filter, fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(filter2, app(fun, x)), fun)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(f, 0), 1), x) → APP(s, x)
APP(app(app(f, x), y), app(s, z)) → APP(app(app(f, 0), 1), z)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(filter2, app(fun, x)), fun), x)
APP(app(app(f, x), y), app(s, z)) → APP(s, app(app(app(f, 0), 1), z))
APP(app(app(f, 0), 1), x) → APP(app(f, app(s, x)), x)
APP(app(app(f, 0), 1), x) → APP(app(app(f, app(s, x)), x), x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(fun, x)), fun), x), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
APP(app(app(f, x), y), app(s, z)) → APP(f, 0)
APP(app(app(app(filter2, true), fun), x), xs) → APP(cons, x)
APP(app(filter, fun), app(app(cons, x), xs)) → APP(filter2, app(fun, x))
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(f, 0), 1), x) → APP(f, app(s, x))
APP(app(app(f, x), y), app(s, z)) → APP(app(f, 0), 1)
APP(app(map, fun), app(app(cons, x), xs)) → APP(cons, app(fun, x))
APP(app(app(app(filter2, true), 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(map, fun), app(app(cons, x), xs)) → APP(app(cons, app(fun, x)), app(app(map, fun), xs))
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
APP(app(app(f, x), y), app(s, z)) → APP(app(app(f, 0), 1), z)
APP(app(app(f, 0), 1), x) → APP(app(app(f, app(s, x)), x), x)
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(app(f, x), y), app(s, z)) → APP(app(app(f, 0), 1), z)
Used ordering: Polynomial interpretation [25,35]:
APP(app(app(f, 0), 1), x) → APP(app(app(f, app(s, x)), x), x)
The value of delta used in the strict ordering is 12.
POL(APP(x1, x2)) = x_2
POL(f) = 0
POL(app(x1, x2)) = 3 + (3)x_1 + (2)x_2
POL(s) = 3
POL(0) = 0
POL(1) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
APP(app(app(f, 0), 1), x) → APP(app(app(f, app(s, x)), x), x)
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(filter, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(app(app(filter2, false), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(app(app(filter2, true), fun), x), xs) → APP(app(filter, fun), xs)
APP(app(map, fun), app(app(cons, x), xs)) → APP(fun, x)
APP(app(map, fun), app(app(cons, x), xs)) → APP(app(map, fun), xs)
The value of delta used in the strict ordering is 336.
POL(APP(x1, x2)) = (4)x_1 + (4)x_2
POL(cons) = 4
POL(true) = 3
POL(map) = 4
POL(false) = 4
POL(app(x1, x2)) = 4 + (4)x_1 + (4)x_2
POL(filter2) = 4
POL(filter) = 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
app(app(app(f, 0), 1), x) → app(app(app(f, app(s, x)), x), x)
app(app(app(f, x), y), app(s, z)) → app(s, app(app(app(f, 0), 1), z))
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)