Termination Proof using AProVE

Term Rewriting System R:
[x, y]
f(a) -> g(h(a))
h(g(x)) -> g(h(f(x)))
k(x, h(x), a) -> h(x)
k(f(x), y, x) -> f(x)

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(a) -> H(a)
H(g(x)) -> H(f(x))
H(g(x)) -> F(x)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pair:

H(g(x)) -> H(f(x))

Rules:

f(a) -> g(h(a))
h(g(x)) -> g(h(f(x)))
k(x, h(x), a) -> h(x)
k(f(x), y, x) -> f(x)

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

H(g(x)) -> H(f(x))

one new Dependency Pair is created:

H(g(a)) -> H(g(h(a)))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pair:

H(g(a)) -> H(g(h(a)))

Rules:

f(a) -> g(h(a))
h(g(x)) -> g(h(f(x)))
k(x, h(x), a) -> h(x)
k(f(x), y, x) -> f(x)

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

H(g(a)) -> H(g(h(a)))

no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes