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`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

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

Rules:

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

• 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)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

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

Rules:

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

• 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)

Termination of R could not be shown.
Duration:
0:00 minutes