Term Rewriting System R:
[x]
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(s(x))) -> F(f(x))
F(s(s(x))) -> F(x)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

F(s(s(x))) -> F(x)
F(s(s(x))) -> F(f(x))

Rules:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(x))) -> F(f(x))
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(x''))
F(s(s(s(s(x''))))) -> F(s(f(f(x''))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Narrowing Transformation

Dependency Pairs:

F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)

Rules:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
three new Dependency Pairs are created:

F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 3
Polynomial Ordering

Dependency Pairs:

F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))
F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x))) -> F(x)
F(s(s(x''))) -> F(s(x''))

Rules:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

The following dependency pairs can be strictly oriented:

F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))
F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x))) -> F(x)
F(s(s(x''))) -> F(s(x''))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(s(x1)) =  1 + x1 POL(f(x1)) =  1 + x1 POL(F(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes