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