Term Rewriting System R:

   [x, y]
   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

Termination of R to be shown.


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

   f(x) -> x

where the Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(d(x_1)) = x_1
POL(c(x_1, x_2)) = 1 + x_1 + x_2
POL(f(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


R contains the following Dependency Pairs: 


   G(c(x, s(y))) -> G(c(s(x), y))
   F(c(s(x), y)) -> F(c(x, s(y)))
   F(f(x)) -> F(d(f(x)))

Furthermore, R contains two SCCs.


SCC1:

G(c(x, s(y))) -> G(c(s(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(G(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(c(x_1, x_2)) = 1 + x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

G(c(x, s(y))) -> G(c(s(x), y))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

G(c(x, s(y))) -> G(c(s(x), y))

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(s(x_1)) = 1 + x_1
POL(c(x_1, x_2)) = 1 + x_2

 resulting in no subcycles.


SCC2:

F(c(s(x), y)) -> F(c(x, s(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(s(x_1)) = x_1
POL(F(x_1)) = 1 + x_1
POL(c(x_1, x_2)) = 1 + x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

F(c(s(x), y)) -> F(c(x, s(y)))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC2.MRR1.NOC1:

F(c(s(x), y)) -> F(c(x, s(y)))

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(s(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1
POL(c(x_1, x_2)) = x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.632 seconds.