Term Rewriting System R:
[x, y]
f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(x), s(y)) -> F(x, y)
G(0, x) -> G(f(x, x), x)
G(0, x) -> F(x, x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Polo
Dependency Pair:
F(s(x), s(y)) -> F(x, y)
Rules:
f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)
Strategy:
innermost
The following dependency pair can be strictly oriented:
F(s(x), s(y)) -> F(x, y)
There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(s(x1)) | = 1 + x1 |
| POL(F(x1, x2)) | = 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(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)
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(0, x) -> G(f(x, x), x)
Rules:
f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)
Strategy:
innermost
The following dependency pair can be strictly oriented:
G(0, x) -> G(f(x, x), x)
Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:
f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(0) | = 1 |
| POL(G(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = 0 |
| POL(f(x1, x2)) | = 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(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))
g(0, x) -> g(f(x, x), x)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes