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
↳Forward Instantiation Transformation
Dependency Pair:
A(b(x)) -> A(x)
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x
Strategy:
innermost
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
A(b(x)) -> A(x)
one new Dependency Pair
is created:
A(b(b(x''))) -> A(b(x''))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Forward Instantiation Transformation
Dependency Pair:
A(b(b(x''))) -> A(b(x''))
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x
Strategy:
innermost
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
A(b(b(x''))) -> A(b(x''))
one new Dependency Pair
is created:
A(b(b(b(x'''')))) -> A(b(b(x'''')))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 3
↳Argument Filtering and Ordering
Dependency Pair:
A(b(b(b(x'''')))) -> A(b(b(x'''')))
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x
Strategy:
innermost
The following dependency pair can be strictly oriented:
A(b(b(b(x'''')))) -> A(b(b(x'''')))
There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(b(x1)) | = 1 + x1 |
| POL(A(x1)) | = 1 + x1 |
resulting in one new DP problem.
Used Argument Filtering System: A(x1) -> A(x1)
b(x1) -> b(x1)
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 4
↳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