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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

.'(.(x, y), z) -> .'(x, .(y, z))
.'(.(x, y), z) -> .'(y, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

.'(.(x, y), z) -> .'(y, z)
.'(.(x, y), z) -> .'(x, .(y, z))

Rule:

.(.(x, y), z) -> .(x, .(y, z))

The following dependency pair can be strictly oriented:

.'(.(x, y), z) -> .'(y, z)

The following rule can be oriented:

.(.(x, y), z) -> .(x, .(y, z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(.(x1, x2)) =  1 + x1 + x2 POL(.'(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

.'(.(x, y), z) -> .'(x, .(y, z))

Rule:

.(.(x, y), z) -> .(x, .(y, z))

The following dependency pair can be strictly oriented:

.'(.(x, y), z) -> .'(x, .(y, z))

The following rule can be oriented:

.(.(x, y), z) -> .(x, .(y, z))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(.(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

.(.(x, y), z) -> .(x, .(y, z))

Using the Dependency Graph resulted in no new DP problems.

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