Term Rewriting System R:

   [y, x, z]
   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   MEM(x, union(y, z)) -> OR(mem(x, y), mem(x, z))
   MEM(x, union(y, z)) -> MEM(x, y)
   MEM(x, union(y, z)) -> MEM(x, z)

Furthermore, R contains one SCC.


SCC1:

MEM(x, union(y, z)) -> MEM(x, z)
MEM(x, union(y, z)) -> MEM(x, y)

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(MEM(x_1, x_2)) = 1 + x_1 + x_2
POL(union(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MEM(x, union(y, z)) -> MEM(x, z)
   MEM(x, union(y, z)) -> MEM(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.512 seconds.