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
↳Argument Filtering and Ordering
→DP Problem 2
↳Remaining
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 that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: F(x1, x2) -> F(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Remaining
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
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
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
Innermost Termination of R could not be shown.
Duration:
0:00 minutes