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
↳Argument Filtering and 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 that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial
resulting in one new DP problem.
Used Argument Filtering System: SUM(x1) -> SUM(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→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