Term Rewriting System R:
[X]
a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
A -> G(c)
F(g(X), b) -> F(a, X)
F(g(X), b) -> A
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pair:
F(g(X), b) -> F(a, X)
Rules:
a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)
Strategy:
innermost
The following dependency pair can be strictly oriented:
F(g(X), b) -> F(a, X)
The following usable rules for innermost w.r.t. to the AFS can be oriented:
a -> g(c)
g(a) -> b
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(c) | = 0 |
| POL(g(x1)) | = 1 + x1 |
| POL(b) | = 1 |
| POL(a) | = 1 |
| POL(F(x1, x2)) | = x1 + x2 |
resulting in one new DP problem.
Used Argument Filtering System: F(x1, x2) -> F(x1, x2)
g(x1) -> g(x1)
a -> a
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes