Term Rewriting System R:
[x, y, z]
:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))

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)
:'(+(x, y), z) -> :'(x, z)
:'(+(x, y), z) -> :'(y, z)
:'(z, +(x, f(y))) -> :'(g(z, y), +(x, a))

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

Rules:

:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(:(x1, x2)) =  1 + x1 + x2 POL(+(x1, x2)) =  x1 + x2

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

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

Dependency Pairs:

:'(+(x, y), z) -> :'(y, z)
:'(+(x, y), z) -> :'(x, z)

Rules:

:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))

The following dependency pairs can be strictly oriented:

:'(+(x, y), z) -> :'(y, z)
:'(+(x, y), z) -> :'(x, z)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(:'(x1, x2)) =  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`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

:(:(x, y), z) -> :(x, :(y, z))
:(+(x, y), z) -> +(:(x, z), :(y, z))
:(z, +(x, f(y))) -> :(g(z, y), +(x, a))

Using the Dependency Graph resulted in no new DP problems.

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