Term Rewriting System R:
[X]
c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
C(b(a(X))) -> A(a(b(b(c(c(X))))))
C(b(a(X))) -> A(b(b(c(c(X)))))
C(b(a(X))) -> B(b(c(c(X))))
C(b(a(X))) -> B(c(c(X)))
C(b(a(X))) -> C(c(X))
C(b(a(X))) -> C(X)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Forward Instantiation Transformation
Dependency Pairs:
C(b(a(X))) -> C(X)
C(b(a(X))) -> C(c(X))
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
C(b(a(X))) -> C(c(X))
no new Dependency Pairs
are created.
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
C(b(a(X))) -> C(X)
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e
The following dependency pair can be strictly oriented:
C(b(a(X))) -> C(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: C(x1) -> C(x1)
b(x1) -> b(x1)
a(x1) -> a(x1)
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳AFS
...
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes