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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(x), y) -> F(x, s(x))
F(x, s(y)) -> F(y, x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pairs:

F(x, s(y)) -> F(y, x)
F(s(x), y) -> F(x, s(x))

Rules:

f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)

Strategy:

innermost

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

F(x, s(y)) -> F(y, x)
two new Dependency Pairs are created:

F(s(y''), s(y0)) -> F(y0, s(y''))
F(x0, s(s(x''))) -> F(s(x''), x0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Forward Instantiation Transformation`

Dependency Pairs:

F(x0, s(s(x''))) -> F(s(x''), x0)
F(s(y''), s(y0)) -> F(y0, s(y''))
F(s(x), y) -> F(x, s(x))

Rules:

f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)

Strategy:

innermost

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

F(s(x), y) -> F(x, s(x))
three new Dependency Pairs are created:

F(s(s(x'')), y) -> F(s(x''), s(s(x'')))
F(s(s(y'''')), y) -> F(s(y''''), s(s(y'''')))
F(s(s(x'''')), y) -> F(s(x''''), s(s(x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

F(s(s(x'''')), y) -> F(s(x''''), s(s(x'''')))
F(s(s(y'''')), y) -> F(s(y''''), s(s(y'''')))
F(s(s(x'')), y) -> F(s(x''), s(s(x'')))
F(s(y''), s(y0)) -> F(y0, s(y''))
F(x0, s(s(x''))) -> F(s(x''), x0)

Rules:

f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)

Strategy:

innermost

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