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)
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
↳Size-Change Principle
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)
We number the DPs as follows:
- SUM(s(x)) -> SUM(x)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes