Term Rewriting System R:
[x]
f(h(x)) -> f(i(x))
f(i(x)) -> a
i(x) -> h(x)

Termination of R to be shown. 



   TRS
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

f(i(x)) -> a

where the Polynomial interpretation:
  POL(i(x1))=  x1  
  POL(h(x1))=  x1  
  POL(a)=  0  
  POL(f(x1))=  1 + x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   TRS
     ↳RRRPolo
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(h(x)) -> F(i(x))
F(h(x)) -> I(x)

Furthermore, R contains one SCC. 


   TRS
     ↳RRRPolo
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Non Termination


Dependency Pair:

F(h(x)) -> F(i(x))


Rules:


i(x) -> h(x)
f(h(x)) -> f(i(x))





Found an infinite P-chain over R:
P =

F(h(x)) -> F(i(x))

R =

i(x) -> h(x)
f(h(x)) -> f(i(x))

s = F(i(x'''))
evaluates to t =F(i(x'''))

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:01 minutes