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
         ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


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




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Inst
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:

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