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)

Innermost Termination of R to be shown.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Usable Rules (Innermost)`

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)

Strategy:

innermost

As we are in the innermost case, we can delete all 3 non-usable-rules.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳UsableRules`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pair:

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

Rule:

f(a) -> g(h(a))

Strategy:

innermost

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 no new DP problems.

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