Term Rewriting System R:
[x]
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
SUM(s(x)) -> SQR(s(x))
SUM(s(x)) -> SUM(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pair:
SUM(s(x)) -> SUM(x)
Rules:
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
Strategy:
innermost
The following dependency pair can be strictly oriented:
SUM(s(x)) -> SUM(x)
There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(SUM(x1)) | = x1 |
| POL(s(x1)) | = 1 + x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes