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'')

The following rules can be oriented:

p(s(x)) -> x
fac(0) -> s(0)
fac(s(x)) -> times(s(x), fac(p(s(x))))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  0 POL(FAC(x1)) =  x1 POL(fac(x1)) =  1 + x1 POL(s(x1)) =  1 + x1 POL(p(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
FAC(x1) -> FAC(x1)
s(x1) -> s(x1)
p(x1) -> p(x1)
fac(x1) -> fac(x1)
times(x1, x2) -> 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