0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
↳7 QDP
↳8 QDPOrderProof (⇔)
↳9 QDP
↳10 PisEmptyProof (⇔)
↳11 TRUE
↳12 QDP
↳13 QDPOrderProof (⇔)
↳14 QDP
↳15 PisEmptyProof (⇔)
↳16 TRUE
↳17 QDP
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
APP(app(max, app(s, x)), app(s, y)) → APP(max, x)
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
APP(app(min, app(s, x)), app(s, y)) → APP(min, x)
APP(app(app(app(insert, f), g), nil), x) → APP(app(cons, x), nil)
APP(app(app(app(insert, f), g), nil), x) → APP(cons, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(cons, app(app(f, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(f, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(app(app(insert, f), g), t), app(app(g, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(app(insert, f), g), t)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(g, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(g, x)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(app(insert, f), g), app(app(app(sort, f), g), t))
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(insert, f), g)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(insert, f)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(app(sort, f), g), t)
APP(ascending_sort, l) → APP(app(app(sort, min), max), l)
APP(ascending_sort, l) → APP(app(sort, min), max)
APP(ascending_sort, l) → APP(sort, min)
APP(descending_sort, l) → APP(app(app(sort, max), min), l)
APP(descending_sort, l) → APP(app(sort, max), min)
APP(descending_sort, l) → APP(sort, max)
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
min1(s(x), s(y)) → min1(x, y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(min, app(s, x)), app(s, y)) → APP(app(min, x), y)
[min11, s1]
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
max1(s(x), s(y)) → max1(x, y)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(max, app(s, x)), app(s, y)) → APP(app(max, x), y)
[max11, s1]
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(f, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(app(app(insert, f), g), t), app(app(g, x), h))
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(app(g, x), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) → APP(g, x)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(sort, f), g), app(app(cons, h), t)) → APP(app(app(sort, f), g), t)
APP(ascending_sort, l) → APP(app(app(sort, min), max), l)
APP(descending_sort, l) → APP(app(app(sort, max), min), l)
app(app(max, 0), x) → x
app(app(max, x), 0) → x
app(app(max, app(s, x)), app(s, y)) → app(app(max, x), y)
app(app(min, 0), x) → 0
app(app(min, x), 0) → 0
app(app(min, app(s, x)), app(s, y)) → app(app(min, x), y)
app(app(app(app(insert, f), g), nil), x) → app(app(cons, x), nil)
app(app(app(app(insert, f), g), app(app(cons, h), t)), x) → app(app(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
app(app(app(sort, f), g), nil) → nil
app(app(app(sort, f), g), app(app(cons, h), t)) → app(app(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
app(ascending_sort, l) → app(app(app(sort, min), max), l)
app(descending_sort, l) → app(app(app(sort, max), min), l)
app(app(max, 0), x0)
app(app(max, x0), 0)
app(app(max, app(s, x0)), app(s, x1))
app(app(min, 0), x0)
app(app(min, x0), 0)
app(app(min, app(s, x0)), app(s, x1))
app(app(app(app(insert, x0), x1), nil), x2)
app(app(app(app(insert, x0), x1), app(app(cons, x2), x3)), x4)
app(app(app(sort, x0), x1), nil)
app(app(app(sort, x0), x1), app(app(cons, x2), x3))
app(ascending_sort, x0)
app(descending_sort, x0)