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

Termination of R to be shown.

`   R`
`     ↳Overlay and local confluence Check`

The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(x, x) -> F(a, b)
F(x, x) -> B

Furthermore, R contains one SCC.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Usable Rules (Innermost)`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

As we are in the innermost case, we can delete all 1 non-usable-rules.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳UsableRules`
`             ...`
`               →DP Problem 2`
`                 ↳Rewriting Transformation`

Dependency Pair:

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

Rule:

b -> c

Strategy:

innermost

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

F(x, x) -> F(a, b)
one new Dependency Pair is created:

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

The transformation is resulting in no new DP problems.

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