Term Rewriting System R:

   [x, y, z]
   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z
   .(.(x, y), z) -> .(x, .(y, z))
   i(1) -> 1
   i(i(x)) -> x
   i(.(x, y)) -> .(i(y), i(x))

Termination of R to be shown.


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

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z

where the Polynomial interpretation:
POL(1) = 0
POL(i(x_1)) = x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2
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: 

   i(1) -> 1

where the Polynomial interpretation:
POL(1) = 1
POL(i(x_1)) = 2*x_1
POL(.(x_1, x_2)) = x_1 + x_2
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: 

   i(i(x)) -> x

where the Polynomial interpretation:
POL(i(x_1)) = 1 + 2*x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2
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: 

   i(.(x, y)) -> .(i(y), i(x))

where the Polynomial interpretation:
POL(i(x_1)) = 2*x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2
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: 

   .(.(x, y), z) -> .(x, .(y, z))

where the Polynomial interpretation:
POL(.(x_1, x_2)) = 1 + 2*x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.495 seconds.