Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
MINUS(h(x)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)
MINUS(f(x, y)) -> MINUS(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pairs:
MINUS(f(x, y)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)
MINUS(h(x)) -> MINUS(x)
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
Strategy:
innermost
The following dependency pairs can be strictly oriented:
MINUS(f(x, y)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(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(MINUS(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polynomial Ordering
Dependency Pair:
MINUS(h(x)) -> MINUS(x)
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
Strategy:
innermost
The following dependency pair can be strictly oriented:
MINUS(h(x)) -> MINUS(x)
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(MINUS(x1)) | = x1 |
| POL(h(x1)) | = 1 + x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Polo
...
→DP Problem 3
↳Dependency Graph
Dependency Pair:
Rules:
minus(minus(x)) -> x
minus(h(x)) -> h(minus(x))
minus(f(x, y)) -> f(minus(y), minus(x))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes