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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

+'(-(x, y), z) -> +'(x, z)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

-(+(x, y), y) -> x

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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