Term Rewriting System R:

   [x, y, z]
   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   if(true, x, y) -> x

where the Polynomial interpretation:
POL(v) = 0
POL(true) = 1
POL(u) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   if(false, x, y) -> y

where the Polynomial interpretation:
POL(v) = 0
POL(u) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   if(x, y, y) -> y

where the Polynomial interpretation:
POL(v) = 0
POL(u) = 0
POL(if(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + x_3
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


R contains the following Dependency Pairs: 


   IF(if(x, y, z), u, v) -> IF(x, if(y, u, v), if(z, u, v))
   IF(if(x, y, z), u, v) -> IF(y, u, v)
   IF(if(x, y, z), u, v) -> IF(z, u, v)

Furthermore, R contains one SCC.


SCC1:

IF(if(x, y, z), u, v) -> IF(z, u, v)
IF(if(x, y, z), u, v) -> IF(y, u, v)

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(v) = 1
POL(u) = 1
POL(if(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
POL(IF(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   IF(if(x, y, z), u, v) -> IF(z, u, v)
   IF(if(x, y, z), u, v) -> IF(y, u, v)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.502 seconds.