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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))
DFIB(s(s(x)), y) -> DFIB(x, y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

DFIB(s(s(x)), y) -> DFIB(x, y)
DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))

Rule:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

DFIB(s(s(x)), y) -> DFIB(x, y)
DFIB(s(s(x)), y) -> DFIB(s(x), dfib(x, y))

The following usable rule for innermost w.r.t. to the AFS can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(DFIB(x1, x2)) =  1 + x1 + x2 POL(s(x1)) =  1 + x1

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

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

Dependency Pair:

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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