Term Rewriting System R:
[f, x, xs]
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))
app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
APP(app(treemap, f), app(app(node, x), xs)) -> APP(node, app(f, x))
APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(map, app(treemap, f))

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
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))

Rules:

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))
app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, 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))
five new Dependency Pairs are created:

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, app(treemap, f'')), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(node, app(f'', x'')), app(app(map, app(treemap, f'')), xs''))), app(app(map, app(treemap, f'')), 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
Narrowing Transformation

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, app(treemap, f'')), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(node, app(f'', x'')), app(app(map, app(treemap, f'')), xs''))), app(app(map, app(treemap, f'')), xs))
APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(node, app(f, x)), app(app(map, app(treemap, 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, 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(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)

Rules:

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))
app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

Strategy:

innermost

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

APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))
five new Dependency Pairs are created:

APP(app(treemap, app(map, f'')), app(app(node, nil), xs)) -> APP(app(node, nil), app(app(map, app(treemap, app(map, f''))), xs))
APP(app(treemap, app(map, f'')), app(app(node, app(app(cons, x''), xs'')), xs)) -> APP(app(node, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(treemap, app(map, f''))), xs))
APP(app(treemap, app(treemap, f'')), app(app(node, app(app(node, x''), xs'')), xs)) -> APP(app(node, app(app(node, app(f'', x'')), app(app(map, app(treemap, f'')), xs''))), app(app(map, app(treemap, app(treemap, f''))), xs))
APP(app(treemap, f''), app(app(node, x), nil)) -> APP(app(node, app(f'', x)), nil)
APP(app(treemap, f''), app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(node, app(f'', x)), app(app(cons, app(app(treemap, f''), x'')), app(app(map, app(treemap, f'')), xs'')))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 3
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(treemap, f''), app(app(node, x), app(app(cons, x''), xs''))) -> APP(app(node, app(f'', x)), app(app(cons, app(app(treemap, f''), x'')), app(app(map, app(treemap, f'')), xs'')))
APP(app(treemap, app(treemap, f'')), app(app(node, app(app(node, x''), xs'')), xs)) -> APP(app(node, app(app(node, app(f'', x'')), app(app(map, app(treemap, f'')), xs''))), app(app(map, app(treemap, app(treemap, f''))), xs))
APP(app(treemap, app(map, f'')), app(app(node, app(app(cons, x''), xs'')), xs)) -> APP(app(node, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(treemap, app(map, f''))), xs))
APP(app(map, app(treemap, f'')), app(app(cons, app(app(node, x''), xs'')), xs)) -> APP(app(cons, app(app(node, app(f'', x'')), app(app(map, app(treemap, f'')), xs''))), app(app(map, app(treemap, f'')), xs))
APP(app(treemap, f), app(app(node, x), xs)) -> APP(app(map, app(treemap, f)), xs)
APP(app(treemap, f), app(app(node, x), xs)) -> APP(f, x)
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, 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), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))

Rules:

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))
app(app(treemap, f), app(app(node, x), xs)) -> app(app(node, app(f, x)), app(app(map, app(treemap, f)), xs))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:06 minutes