app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
↳ QTRS
↳ DependencyPairsProof
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
APP2(asort, z) -> APP2(sort, min)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(sort, f), g), y)
APP2(dsort, z) -> APP2(sort, max)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
APP2(dsort, z) -> APP2(app2(sort, max), min)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(app2(insert, f), g), z)
APP2(app2(app2(app2(insert, f), g), nil), y) -> APP2(cons, y)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(g, x), y)
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(app2(min, x), y)
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(app2(max, x), y)
APP2(asort, z) -> APP2(app2(app2(sort, min), max), z)
APP2(app2(app2(app2(insert, f), g), nil), y) -> APP2(app2(cons, y), nil)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(insert, f), g)
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(max, x)
APP2(asort, z) -> APP2(app2(sort, min), max)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(f, x), y)
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(min, x)
APP2(dsort, z) -> APP2(app2(app2(sort, max), min), z)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(f, x)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y))
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y))
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(g, x)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(insert, f)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(cons, app2(app2(f, x), y))
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP2(asort, z) -> APP2(sort, min)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(sort, f), g), y)
APP2(dsort, z) -> APP2(sort, max)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
APP2(dsort, z) -> APP2(app2(sort, max), min)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(app2(insert, f), g), z)
APP2(app2(app2(app2(insert, f), g), nil), y) -> APP2(cons, y)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(g, x), y)
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(app2(min, x), y)
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(app2(max, x), y)
APP2(asort, z) -> APP2(app2(app2(sort, min), max), z)
APP2(app2(app2(app2(insert, f), g), nil), y) -> APP2(app2(cons, y), nil)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(insert, f), g)
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(max, x)
APP2(asort, z) -> APP2(app2(sort, min), max)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(f, x), y)
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(min, x)
APP2(dsort, z) -> APP2(app2(app2(sort, max), min), z)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(f, x)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y))
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y))
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(g, x)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(insert, f)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(cons, app2(app2(f, x), y))
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(app2(min, x), y)
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP2(app2(min, app2(s, x)), app2(s, y)) -> APP2(app2(min, x), y)
POL(APP2(x1, x2)) = x1
POL(app2(x1, x2)) = 1 + x1 + x2
POL(min) = 0
POL(s) = 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(app2(max, x), y)
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP2(app2(max, app2(s, x)), app2(s, y)) -> APP2(app2(max, x), y)
POL(APP2(x1, x2)) = x1
POL(app2(x1, x2)) = 1 + x1 + x2
POL(max) = 0
POL(s) = 1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(g, x), y)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y))
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(sort, f), g), y)
APP2(asort, z) -> APP2(app2(app2(sort, min), max), z)
APP2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> APP2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
APP2(dsort, z) -> APP2(app2(app2(sort, max), min), z)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(app2(f, x), y)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(g, x)
APP2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> APP2(f, x)
app2(app2(app2(sort, f), g), nil) -> nil
app2(app2(app2(sort, f), g), app2(app2(cons, x), y)) -> app2(app2(app2(app2(insert, f), g), app2(app2(app2(sort, f), g), y)), x)
app2(app2(app2(app2(insert, f), g), nil), y) -> app2(app2(cons, y), nil)
app2(app2(app2(app2(insert, f), g), app2(app2(cons, x), z)), y) -> app2(app2(cons, app2(app2(f, x), y)), app2(app2(app2(app2(insert, f), g), z), app2(app2(g, x), y)))
app2(app2(max, 0), y) -> y
app2(app2(max, x), 0) -> x
app2(app2(max, app2(s, x)), app2(s, y)) -> app2(app2(max, x), y)
app2(app2(min, 0), y) -> 0
app2(app2(min, x), 0) -> 0
app2(app2(min, app2(s, x)), app2(s, y)) -> app2(app2(min, x), y)
app2(asort, z) -> app2(app2(app2(sort, min), max), z)
app2(dsort, z) -> app2(app2(app2(sort, max), min), z)