Term Rewriting System R:
[x]
s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(x)

Furthermore, R contains one SCC.

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

Dependency Pairs:

S1(s1(s0(s0(x)))) -> S1(x)
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))

Rule:

s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))

Strategy:

innermost

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

S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))
two new Dependency Pairs are created:

S1(s1(s0(s0(s0(s0(x'')))))) -> S1(s0(s0(s0(s1(s1(s1(x'')))))))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(x)

Rule:

s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))

Strategy:

innermost

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

S1(s1(s0(s0(x)))) -> S1(x)
two new Dependency Pairs are created:

S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(x''))))
S1(s1(s0(s0(s1(s0(s0(s1(s0(s0(x'''')))))))))) -> S1(s1(s0(s0(s1(s0(s0(x'''')))))))

The transformation is resulting in one new DP problem:

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

The following remains to be proven:
Dependency Pairs:

S1(s1(s0(s0(s1(s0(s0(s1(s0(s0(x'''')))))))))) -> S1(s1(s0(s0(s1(s0(s0(x'''')))))))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(x''))))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))

Rule:

s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))

Strategy:

innermost

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