Term Rewriting System R:
[y, x, f, xs]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(sumwith, f), nil) -> nil
app(app(sumwith, f), app(app(cons, x), xs)) -> app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(plus, app(f, x)), app(app(sumwith, f), xs))
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(plus, app(f, x))
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(sumwith, f), xs)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Forward Instantiation Transformation

Dependency Pairs:

APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(sumwith, f), xs)
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(plus, app(f, x)), app(app(sumwith, f), xs))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(sumwith, f), nil) -> nil
app(app(sumwith, f), app(app(cons, x), xs)) -> app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Strategy:

innermost

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

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

APP(app(sumwith, app(sumwith, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(sumwith, f''), app(app(cons, x''), xs''))
APP(app(sumwith, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(plus, app(s, x'')), x0)

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
FwdInst
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

APP(app(sumwith, app(plus, app(s, x''))), app(app(cons, x0), xs)) -> APP(app(plus, app(s, x'')), x0)
APP(app(sumwith, app(sumwith, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(sumwith, f''), app(app(cons, x''), xs''))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(plus, app(f, x)), app(app(sumwith, f), xs))
APP(app(sumwith, f), app(app(cons, x), xs)) -> APP(app(sumwith, f), xs)

Rules:

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(sumwith, f), nil) -> nil
app(app(sumwith, f), app(app(cons, x), xs)) -> app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes