Term Rewriting System R:
[x]
f(g(x), s(0)) -> f(g(x), 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)) -> F(g(x), 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)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0





We number the DPs as follows:
  1. G(s(x)) -> G(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Non Termination


Dependency Pair:

F(g(x), s(0)) -> F(g(x), g(x))


Rules:


f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0





Found an infinite P-chain over R:
P =

F(g(x), s(0)) -> F(g(x), g(x))

R =

f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0

s = F(g(s(0)), g(s(0)))
evaluates to t =F(g(s(0)), g(s(0)))

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:00 minutes