Term Rewriting System R:
[x, y, z]
*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(*(x, y), z) -> *'(x, *(y, z))
*'(*(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pair:

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Non Termination


Dependency Pair:

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))

R = none

s = *'(x'', +(f(z''), f(z''')))
evaluates to t =*'(g(g(x'', z'''), z''), +(f(z''), f(z'')))

Thus, s starts an infinite chain as s matches t.

Innermost Termination of R could not be shown.
Duration:
0:00 minutes