Term Rewriting System R:
[x, y]
average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))
AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)
AVERAGE(s(x), y) -> AVERAGE(x, s(y))

Rules:

average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(AVERAGE(x1, x2)) =  1 + x1 + x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
AVERAGE(x1, x2) -> AVERAGE(x1, x2)
s(x1) -> s(x1)

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

Dependency Pair:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))

Rules:

average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Strategy:

innermost

The following dependency pair can be strictly oriented:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
AVERAGE(x1, x2) -> x1
s(x1) -> s(x1)

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

Dependency Pair:

Rules:

average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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