Term Rewriting System R:
[X]
f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(X), X) -> F(X, a(X))
F(X, c(X)) -> F(s(X), X)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Inst`

Dependency Pair:

F(s(X), X) -> F(X, a(X))

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

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

F(s(X), X) -> F(X, a(X))
one new Dependency Pair is created:

F(s(a(X''')), a(X''')) -> F(a(X'''), a(a(X''')))

The transformation is resulting in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`       →DP Problem 2`
`         ↳Instantiation Transformation`

Dependency Pair:

F(X, c(X)) -> F(s(X), X)

Rules:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)

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

F(X, c(X)) -> F(s(X), X)
one new Dependency Pair is created:

F(s(X'''), c(s(X'''))) -> F(s(s(X''')), s(X'''))

The transformation is resulting in no new DP problems.

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