Term Rewriting System R:
[x]
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(x)) -> G(f(x))
F(s(x)) -> F(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Forward Instantiation Transformation
Dependency Pair:
F(s(x)) -> F(x)
Rules:
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))
On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule
F(s(x)) -> F(x)
one new Dependency Pair
is created:
F(s(s(x''))) -> F(s(x''))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
F(s(s(x''))) -> F(s(x''))
Rules:
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))
Termination of R could not be shown.
Duration:
0:00 minutes