Term Rewriting System R:
[x, y]
f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))
f(c(x), y) -> f(x, c(y))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, a(b(c(y)))) -> F(b(c(a(b(x)))), y)
F(a(x), y) -> F(x, a(y))
F(b(x), y) -> F(x, b(y))
F(c(x), y) -> F(x, c(y))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Modular Removal of Rules`

Dependency Pairs:

F(c(x), y) -> F(x, c(y))
F(a(x), y) -> F(x, a(y))
F(b(x), y) -> F(x, b(y))
F(x, a(b(c(y)))) -> F(b(c(a(b(x)))), y)

Rules:

f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))
f(c(x), y) -> f(x, c(y))

We have the following set of usable rules: none
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
 POL(c(x1)) =  x1 POL(b(x1)) =  x1 POL(a(x1)) =  x1 POL(F(x1, x2)) =  1 + x1 + x2

We have the following set D of usable symbols: {c, b, a, F}
No Dependency Pairs can be deleted.
4 non usable rules have been deleted.

The result of this processor delivers one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳MRR`
`           →DP Problem 2`
`             ↳Non-Overlappingness Check`

Dependency Pairs:

F(c(x), y) -> F(x, c(y))
F(a(x), y) -> F(x, a(y))
F(b(x), y) -> F(x, b(y))
F(x, a(b(c(y)))) -> F(b(c(a(b(x)))), y)

Rule:

none

R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳MRR`
`           →DP Problem 2`
`             ↳NOC`
`             ...`
`               →DP Problem 3`
`                 ↳Scc To SRS`

Dependency Pairs:

F(c(x), y) -> F(x, c(y))
F(a(x), y) -> F(x, a(y))
F(b(x), y) -> F(x, b(y))
F(x, a(b(c(y)))) -> F(b(c(a(b(x)))), y)

Rule:

none

Strategy:

innermost

It has been determined that showing finiteness of this DP problem is equivalent to showing termination of a string rewrite system.
(Re)applying the dependency pair method (incl. the dependency graph) for the following SRS:

a(b(c(x))) -> b(a(c(b(x))))
The graph does not contain any SCC and, thus, we obtain no new DP problems.

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