Term Rewriting System R:

   [y, x, z]
   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

Termination of R to be shown.


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

   f(nil) -> nil

where the Polynomial interpretation:
POL(nil) = 0
POL(g(x_1)) = x_1
POL(f(x_1)) = 1 + 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: 

   f(.(nil, y)) -> .(nil, f(y))

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(nil) = 1
POL(f(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: 

   g(nil) -> nil

where the Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(nil) = 0
POL(f(x_1)) = 1 + 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: 

   g(.(x, nil)) -> .(g(x), nil)

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(nil) = 1
POL(f(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.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


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

Furthermore, R contains two SCCs.


SCC1:

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

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(F(x_1)) = 1 + x_1
POL(.(x_1, x_2)) = 1 + x_1

 resulting in no subcycles.


SCC2:

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

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(G(x_1)) = x_1
POL(.(x_1, x_2)) = 1 + x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.599 seconds.