Term Rewriting System R:
[x, y]
f(0) -> s(0)
f(s(0)) -> s(s(0))
f(s(0)) -> *(s(s(0)), f(0))
f(+(x, s(0))) -> +(s(s(0)), f(x))
f(+(x, y)) -> *(f(x), f(y))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(0)) -> F(0)
F(+(x, s(0))) -> F(x)
F(+(x, y)) -> F(x)
F(+(x, y)) -> F(y)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pairs:
F(+(x, y)) -> F(y)
F(+(x, y)) -> F(x)
F(+(x, s(0))) -> F(x)
Rules:
f(0) -> s(0)
f(s(0)) -> s(s(0))
f(s(0)) -> *(s(s(0)), f(0))
f(+(x, s(0))) -> +(s(s(0)), f(x))
f(+(x, y)) -> *(f(x), f(y))
Strategy:
innermost
The following dependency pairs can be strictly oriented:
F(+(x, y)) -> F(y)
F(+(x, y)) -> F(x)
F(+(x, s(0))) -> F(x)
There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
+(x1, x2) -> +(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(0) -> s(0)
f(s(0)) -> s(s(0))
f(s(0)) -> *(s(s(0)), f(0))
f(+(x, s(0))) -> +(s(s(0)), f(x))
f(+(x, y)) -> *(f(x), f(y))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes