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