Term Rewriting System R:
[x, y, z]
f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(+(x, 0)) -> F(x)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(+(x, 0)) -> F(x)

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

F(+(x, 0)) -> F(x)

The following rules can be oriented:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  1 POL(F(x1)) =  x1 POL(+(x1, x2)) =  x1 + x2 POL(f(x1)) =  x1

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

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

Dependency Pair:

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pairs:

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

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)
f(+(x, 0)) -> f(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(f(x1)) =  x1 POL(+'(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)
f(+(x, 0)) -> f(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(f(x1)) =  x1

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳AFS`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

f(+(x, 0)) -> f(x)
+(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