Term Rewriting System R:
[f, n, x, xs]
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(hd, app(app(map, f), app(app(cons, 0), nil)))
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(map, f)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(cons, 0)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
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(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)

Rules:

app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Strategy:

innermost

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

APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
three new Dependency Pairs are created:

APP(app(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
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(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)

Rules:

app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Strategy:

innermost

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

APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
one new Dependency Pair is created:

APP(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))

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(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
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(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)

Rules:

app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Strategy:

innermost

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

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
six new Dependency Pairs are created:

APP(app(map, app(f'', 0)), app(app(cons, x'), xs)) -> APP(app(cons, app(app(hd, app(app(map, f''), app(app(cons, 0), nil))), x')), app(app(map, app(f'', 0)), xs))
APP(app(map, app(map, f'')), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, app(map, f'')), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, 0), app(app(cons, x), xs')) -> APP(app(cons, app(0, x)), app(app(hd, app(app(map, map), app(app(cons, 0), nil))), xs'))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 4
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, 0), app(app(cons, x), xs')) -> APP(app(cons, app(0, x)), app(app(hd, app(app(map, map), app(app(cons, 0), nil))), xs'))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, app(f'', 0)), app(app(cons, x'), xs)) -> APP(app(cons, app(app(hd, app(app(map, f''), app(app(cons, 0), nil))), x')), app(app(map, app(f'', 0)), xs))
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))

Rules:

app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:05 minutes