Term Rewriting System R:
[x]
a(b(x)) -> b(a(x))
a(c(x)) -> x
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
A(b(x)) -> A(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pair:
A(b(x)) -> A(x)
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x
Strategy:
innermost
The following dependency pair can be strictly oriented:
A(b(x)) -> A(x)
There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(b(x1)) | = 1 + x1 |
| POL(A(x1)) | = x1 |
resulting in one new DP problem.
Used Argument Filtering System: A(x1) -> A(x1)
b(x1) -> b(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes