Term Rewriting System R:
[x, y, f, g, h, t, 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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(ascendingsort, l) -> APP(app(sort, min), max)
APP(ascendingsort, l) -> APP(sort, min)
APP(descendingsort, l) -> APP(app(app(sort, max), min), l)
APP(descendingsort, l) -> APP(app(sort, max), min)
APP(descendingsort, l) -> APP(sort, max)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

APP(descendingsort, l) -> APP(app(sort, max), min)
APP(descendingsort, l) -> APP(app(app(sort, max), min), l)
APP(ascendingsort, l) -> APP(app(sort, min), max)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> 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(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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(app(app(insert, f), g), t)
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(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(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), nil), x) -> APP(app(cons, x), nil)
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(app(app(insert, f), g), nil), x) -> APP(app(cons, x), nil)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Narrowing Transformation

Dependency Pairs:

APP(descendingsort, l) -> APP(app(app(sort, max), min), l)
APP(ascendingsort, l) -> APP(app(sort, min), max)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> 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(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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(app(app(insert, f), g), t)
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(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
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(descendingsort, l) -> APP(app(sort, max), min)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(app(insert, f), g), t)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 3
Narrowing Transformation

Dependency Pairs:

APP(descendingsort, l) -> APP(app(sort, max), min)
APP(ascendingsort, l) -> APP(app(sort, min), max)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> 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(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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(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(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(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
APP(descendingsort, l) -> APP(app(app(sort, max), min), l)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(insert, f), g)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 4
Narrowing Transformation

Dependency Pairs:

APP(descendingsort, l) -> APP(app(app(sort, max), min), l)
APP(ascendingsort, l) -> APP(app(sort, min), max)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(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(f, x)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
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(descendingsort, l) -> APP(app(sort, max), min)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(ascendingsort, l) -> APP(app(sort, min), max)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 5
Narrowing Transformation

Dependency Pairs:

APP(descendingsort, l) -> APP(app(sort, max), min)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(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(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(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
APP(descendingsort, l) -> APP(app(app(sort, max), min), l)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

APP(descendingsort, l) -> APP(app(sort, max), min)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 6
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(descendingsort, l) -> APP(app(app(sort, max), min), l)
APP(app(app(sort, f), g), app(app(cons, h), t)) -> APP(app(app(sort, f), g), t)
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(app(app(insert, f), g), app(app(app(sort, f), g), t)), h)
APP(app(app(app(insert, f), g), app(app(cons, h), t)), x) -> APP(g, x)
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(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(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(cons, app(app(f, x), h)), app(app(app(app(insert, f), g), t), app(app(g, x), h)))
APP(app(min, app(s, x)), app(s, y)) -> APP(app(min, x), y)
APP(app(max, app(s, x)), app(s, y)) -> APP(app(max, x), y)
APP(ascendingsort, l) -> APP(app(app(sort, min), max), l)

Rules:

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(ascendingsort, l) -> app(app(app(sort, min), max), l)
app(descendingsort, l) -> app(app(app(sort, max), min), l)

Termination of R could not be shown.
Duration:
0:01 minutes