Term Rewriting System R:

   [x]
   a(f, 0) -> a(s, 0)
   a(d, 0) -> 0
   a(d, a(s, x)) -> a(s, a(s, a(d, a(p, a(s, x)))))
   a(f, a(s, x)) -> a(d, a(f, a(p, a(s, x))))
   a(p, a(s, x)) -> x

Termination of R to be shown.


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

   a(f, 0) -> a(s, 0)

where the Polynomial interpretation:
POL(s) = 0
POL(d) = 0
POL(a(x_1, x_2)) = x_1 + x_2
POL(f) = 1
POL(0) = 0
POL(p) = 0
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: 


   A(f, a(s, x)) -> A(d, a(f, a(p, a(s, x))))
   A(f, a(s, x)) -> A(f, a(p, a(s, x)))
   A(f, a(s, x)) -> A(p, a(s, x))
   A(d, a(s, x)) -> A(s, a(s, a(d, a(p, a(s, x)))))
   A(d, a(s, x)) -> A(s, a(d, a(p, a(s, x))))
   A(d, a(s, x)) -> A(d, a(p, a(s, x)))
   A(d, a(s, x)) -> A(p, a(s, x))

Furthermore, R contains two SCCs.


SCC1:

A(d, a(s, x)) -> A(d, a(p, a(s, x)))

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

   A(d, a(s, x)) -> A(d, a(p, a(s, x)))
one new Dependency Pair
is created:


   A(d, a(s, x'')) -> A(d, x'')

The transformation is resulting in one subcycle:


SCC1.Nar1:

A(d, a(s, x'')) -> A(d, x'')

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(s) = 1
POL(d) = 1
POL(a(x_1, x_2)) = 1 + x_1 + x_2
POL(A(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   A(d, a(s, x'')) -> A(d, x'')



 This transformation is resulting in no new subcycles.


SCC2:

A(f, a(s, x)) -> A(f, a(p, a(s, x)))

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

   A(f, a(s, x)) -> A(f, a(p, a(s, x)))
one new Dependency Pair
is created:


   A(f, a(s, x'')) -> A(f, x'')

The transformation is resulting in one subcycle:


SCC2.Nar1:

A(f, a(s, x'')) -> A(f, x'')

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(s) = 1
POL(a(x_1, x_2)) = 1 + x_1 + x_2
POL(A(x_1, x_2)) = 1 + x_1 + x_2
POL(f) = 1

The following Dependency Pairs can be deleted:

   A(f, a(s, x'')) -> A(f, x'')



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 20.789 seconds.