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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(a, b) -> +'(b, a)
+'(a, +(b, z)) -> +'(b, +(a, z))
+'(a, +(b, z)) -> +'(a, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
F(+(x, y), z) -> +'(f(x, z), f(y, z))
F(+(x, y), z) -> F(x, z)
F(+(x, y), z) -> F(y, z)

Furthermore, R contains three SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`
`       →DP Problem 3`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

+'(a, +(b, z)) -> +'(a, z)

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`
`       →DP Problem 3`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

+'(a, +(b, z)) -> +'(a, z)

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`
`       →DP Problem 3`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

+'(a, +(b, z)) -> +'(a, z)

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

• Dependency Pairs:

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

Rules:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

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