Term Rewriting System R:
[x]
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
FAC(s(x)) -> FAC(p(s(x)))
FAC(s(x)) -> P(s(x))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
Dependency Pair:
FAC(s(x)) -> FAC(p(s(x)))
Rules:
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule
FAC(s(x)) -> FAC(p(s(x)))
one new Dependency Pair
is created:
FAC(s(x'')) -> FAC(x'')
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
FAC(s(x'')) -> FAC(x'')
Rules:
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
The following dependency pair can be strictly oriented:
FAC(s(x'')) -> FAC(x'')
There are no usable rules w.r.t. to the AFS 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: FAC(x1) -> FAC(x1)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes