Term Rewriting System R:

   [X, Y, Z, X1, X2, X3, X4]
   plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
   plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))
   PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
   PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
   PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
   PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X2, X4)

Furthermore, R contains one SCC.


SCC1:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X2, X4)
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))

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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(plus(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X2, X4)
   PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
   PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))

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(plus(x_1, x_2)) = 0
POL(s(x_1)) = 1 + x_1
POL(PLUS(x_1, x_2)) = x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.622 seconds.