Term Rewriting System R:
[x, y]
f(x, x) -> a
f(g(x), y) -> f(x, y)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(x), y) -> F(x, y)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Forward Instantiation Transformation
Dependency Pair:
F(g(x), y) -> F(x, y)
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
Strategy:
innermost
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
F(g(x), y) -> F(x, y)
one new Dependency Pair
is created:
F(g(g(x'')), y'') -> F(g(x''), y'')
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Forward Instantiation Transformation
Dependency Pair:
F(g(g(x'')), y'') -> F(g(x''), y'')
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
Strategy:
innermost
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
F(g(g(x'')), y'') -> F(g(x''), y'')
one new Dependency Pair
is created:
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 3
↳Polynomial Ordering
Dependency Pair:
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
Strategy:
innermost
The following dependency pair can be strictly oriented:
F(g(g(g(x''''))), y'''') -> F(g(g(x'''')), y'''')
There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g(x1)) | = 1 + x1 |
| POL(F(x1, x2)) | = 1 + x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(x, x) -> a
f(g(x), y) -> f(x, y)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes