Term Rewriting System R:
[x]
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

W(r(x)) -> W(x)
B(r(x)) -> B(x)
B(w(x)) -> W(b(x))
B(w(x)) -> B(x)

Furthermore, R contains two SCCs.

R
DPs
→DP Problem 1
Argument Filtering and Ordering
→DP Problem 2
AFS

Dependency Pair:

W(r(x)) -> W(x)

Rules:

w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

The following dependency pair can be strictly oriented:

W(r(x)) -> W(x)

There are no usable rules using the Ce-refinement 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:
W(x1) -> W(x1)
r(x1) -> r(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 3
Dependency Graph
→DP Problem 2
AFS

Dependency Pair:

Rules:

w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Using the Dependency Graph resulted in no new DP problems.

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Argument Filtering and Ordering

Dependency Pairs:

B(w(x)) -> B(x)
B(r(x)) -> B(x)

Rules:

w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

The following dependency pairs can be strictly oriented:

B(w(x)) -> B(x)
B(r(x)) -> B(x)

There are no usable rules using the Ce-refinement 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:
B(x1) -> B(x1)
w(x1) -> w(x1)
r(x1) -> r(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Using the Dependency Graph resulted in no new DP problems.

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