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