Term Rewriting System R:

   [x, y, z, u]
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, *(y, z)), *(y, u)) -> +(x, *(y, +(z, u)))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   *'(x, +(y, z)) -> +'(*(x, y), *(x, z))
   *'(x, +(y, z)) -> *'(x, y)
   *'(x, +(y, z)) -> *'(x, z)
   +'(x, +(y, z)) -> +'(+(x, y), z)
   +'(x, +(y, z)) -> +'(x, y)
   +'(+(x, *(y, z)), *(y, u)) -> +'(x, *(y, +(z, u)))
   +'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
   +'(+(x, *(y, z)), *(y, u)) -> +'(z, u)

Furthermore, R contains one SCC.


SCC1:

+'(+(x, *(y, z)), *(y, u)) -> +'(z, u)
*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
+'(+(x, *(y, z)), *(y, u)) -> *'(y, +(z, u))
+'(+(x, *(y, z)), *(y, u)) -> +'(x, *(y, +(z, u)))
+'(x, +(y, z)) -> +'(x, y)
+'(x, +(y, z)) -> +'(+(x, y), z)
*'(x, +(y, z)) -> +'(*(x, y), *(x, z))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(+(x, *(y, z)), *(y, u)) -> +(x, *(y, +(z, u)))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(*(x_1, x_2)) = x_2
POL(*'(x_1, x_2)) = x_2
POL(+(x_1, x_2)) = 1 + x_1 + x_2
POL(+'(x_1, x_2)) = x_1 + x_2

 resulting in one subcycle.


SCC1.Polo1:

+'(x, +(y, z)) -> +'(+(x, y), z)
+'(+(x, *(y, z)), *(y, u)) -> +'(x, *(y, +(z, u)))

Found an infinite P-chain over R:
P = +'(x, +(y, z)) -> +'(+(x, y), z)
+'(+(x, *(y, z)), *(y, u)) -> +'(x, *(y, +(z, u)))
R = [*(x, +(y, z)) -> +(*(x, y), *(x, z)), +(x, +(y, z)) -> +(+(x, y), z), +(+(x, *(y, z)), *(y, u)) -> +(x, *(y, +(z, u)))]
s = +'(x''', +(*(y'''', z''''), *(y'''', u'''))) evaluates to t = +'(x''', +(*(y'''', z''''), *(y'''', u''')))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.876 seconds.