Term Rewriting System R:
[x]
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(a, x) -> F(g(x), x)
F(a, x) -> G(x)
H(g(x)) -> H(a)
G(h(x)) -> G(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
G(h(x)) -> G(x)
Rules:
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
The following dependency pair can be strictly oriented:
G(h(x)) -> G(x)
There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(G(x1)) | = x1 |
| POL(h(x1)) | = 1 + x1 |
resulting in one new DP problem.
Used Argument Filtering System: G(x1) -> G(x1)
h(x1) -> h(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
F(a, x) -> F(g(x), x)
Rules:
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
The following dependency pair can be strictly oriented:
F(a, x) -> F(g(x), x)
The following usable rule w.r.t. to the AFS can be oriented:
g(h(x)) -> g(x)
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g) | = 0 |
| 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
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes