Term Rewriting System R:
[x, y]
f(g(x), s(0), y) -> f(y, y, g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(x), s(0), y) -> F(y, y, g(x))
G(s(x)) -> G(x)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳NonTerm
Dependency Pair:
G(s(x)) -> G(x)
Rules:
f(g(x), s(0), y) -> f(y, y, g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
We number the DPs as follows:
- G(s(x)) -> G(x)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Non Termination
Dependency Pair:
F(g(x), s(0), y) -> F(y, y, g(x))
Rules:
f(g(x), s(0), y) -> f(y, y, g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
Found an infinite P-chain over R:
P =
F(g(x), s(0), y) -> F(y, y, g(x))
R =
f(g(x), s(0), y) -> f(y, y, g(x))
g(s(x)) -> s(g(x))
g(0) -> 0
s = F(g(s(0)), g(s(0)), g(s(0)))
evaluates to t =F(g(s(0)), g(s(0)), g(s(0)))
Thus, s starts an infinite chain.
Non-Termination of R could be shown.
Duration:
0:00 minutes