Term Rewriting System R:
[x]
half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

HALF(s(s(x))) -> HALF(x)
LOG(s(s(x))) -> LOG(s(half(x)))
LOG(s(s(x))) -> HALF(x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

HALF(s(s(x))) -> HALF(x)

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

The following dependency pair can be strictly oriented:

HALF(s(s(x))) -> HALF(x)

Additionally, the following rules can be oriented:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(HALF(x1)) =  1 + x1 POL(log(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(half(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

LOG(s(s(x))) -> LOG(s(half(x)))

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

The following dependency pair can be strictly oriented:

LOG(s(s(x))) -> LOG(s(half(x)))

Additionally, the following rules can be oriented:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(log(x1)) =  x1 POL(s(x1)) =  1 + x1 POL(half(x1)) =  x1 POL(LOG(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Using the Dependency Graph resulted in no new DP problems.

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