Term Rewriting System R:

   [x, y]
   f(0, 1, x) -> f(h(x), h(x), x)
   h(0) -> 0
   h(g(x, y)) -> y

Termination of R to be shown.


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


R contains the following Dependency Pairs: 


   F(0, 1, x) -> F(h(x), h(x), x)
   F(0, 1, x) -> H(x)

Furthermore, R contains one SCC.


SCC1:

F(0, 1, x) -> F(h(x), h(x), x)

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   F(0, 1, x) -> F(h(x), h(x), x)
four new Dependency Pairs
are created:


   F(0, 1, 0) -> F(0, h(0), 0)
   F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
   F(0, 1, 0) -> F(h(0), 0, 0)
   F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))

The transformation is resulting in two subcycles:


SCC1.Nar1:

F(0, 1, 0) -> F(0, h(0), 0)

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   F(0, 1, 0) -> F(0, h(0), 0)
one new Dependency Pair
is created:


   F(0, 1, 0) -> F(0, 0, 0)

The transformation is resulting in no subcycles.


SCC1.Nar2:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   F(0, 1, g(x'', y')) -> F(y', h(g(x'', y')), g(x'', y'))
one new Dependency Pair
is created:


   F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in one subcycle:


SCC1.Nar2.Rewr1:

F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   F(0, 1, g(x'', y')) -> F(h(g(x'', y')), y', g(x'', y'))
one new Dependency Pair
is created:


   F(0, 1, g(x'', y')) -> F(y', y', g(x'', y'))

The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 10.570 seconds.