Term Rewriting System R:
[x, y, z, u]
:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))

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:

:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(x, y)

Furthermore, R contains one SCC.

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳Argument Filtering and Ordering`

Dependency Pairs:

:'(:(:(:(C, x), y), z), u) -> :'(x, y)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))

Rule:

:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

:'(:(:(:(C, x), y), z), u) -> :'(x, y)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, y), z)
:'(:(:(:(C, x), y), z), u) -> :'(:(:(x, y), z), u)
:'(:(:(:(C, x), y), z), u) -> :'(x, z)
:'(:(:(:(C, x), y), z), u) -> :'(:(x, z), :(:(:(x, y), z), u))

The following usable rule w.r.t. the AFS can be oriented:

:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))

Used ordering: Lexicographic Path Order with Precedence:
:' > :

resulting in one new DP problem.
Used Argument Filtering System:
:'(x1, x2) -> :'(x1, x2)
:(x1, x2) -> :(x1, x2)

`   R`
`     ↳OC`
`       →TRS2`
`         ↳DPs`
`           →DP Problem 1`
`             ↳AFS`
`             ...`
`               →DP Problem 2`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

:(:(:(:(C, x), y), z), u) -> :(:(x, z), :(:(:(x, y), z), u))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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