Term Rewriting System R:
[x, y, z]
ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(x, y), 0)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
two new Dependency Pairs are created:

AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))

The transformation is resulting in one new DP problem:

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

The following remains to be proven:
Dependency Pairs:

AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(s, z)) -> AP(ap(ap(g, ap(ap(g, x''), y'')), ap(s, z'')), ap(ap(ap(ap(g, x''), y''), ap(ap(x'', y''), 0)), 0))
AP(ap(ap(g, f), y'), ap(s, z)) -> AP(ap(ap(g, f), y'), ap(y', 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Rules:

ap(f, x) -> x
ap(ap(ap(g, x), y), ap(s, z)) -> ap(ap(ap(g, x), y), ap(ap(x, y), 0))

Strategy:

innermost

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