Term Rewriting System R:

   [x, y, z]
   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   INF(x) -> CONS(x, inf(s(x)))
   INF(x) -> INF(s(x))

Furthermore, R contains one SCC.


SCC1:

INF(x) -> INF(s(x))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = x_1
POL(INF(x_1)) = 1 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   cons(x, cons(y, z)) -> big
   inf(x) -> cons(x, inf(s(x)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

INF(x) -> INF(s(x))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

INF(x) -> INF(s(x))

Found an infinite P-chain over R:
P = INF(x) -> INF(s(x))
R = []
s = INF(x) evaluates to t = INF(s(x))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.482 seconds.