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