Term Rewriting System R:

   [n, x, m, y]
   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0, x), y) -> sum(x, y)
   sum(nil, y) -> y
   weight(cons(n, cons(m, x))) -> weight(sum(cons(n, cons(m, x)), cons(0, x)))
   weight(cons(n, nil)) -> n

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: 


   WEIGHT(cons(n, cons(m, x))) -> WEIGHT(sum(cons(n, cons(m, x)), cons(0, x)))
   WEIGHT(cons(n, cons(m, x))) -> SUM(cons(n, cons(m, x)), cons(0, x))
   SUM(cons(s(n), x), cons(m, y)) -> SUM(cons(n, x), cons(s(m), y))
   SUM(cons(0, x), y) -> SUM(x, y)

Furthermore, R contains two SCCs.


SCC1:

SUM(cons(0, x), y) -> SUM(x, y)
SUM(cons(s(n), x), cons(m, y)) -> SUM(cons(n, x), cons(s(m), 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(SUM(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   SUM(cons(0, x), y) -> SUM(x, y)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

SUM(cons(s(n), x), cons(m, y)) -> SUM(cons(n, x), cons(s(m), 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(SUM(x_1, x_2)) = 1 + x_1
POL(cons(x_1, x_2)) = x_1 + x_2

 resulting in no subcycles.


SCC2:

WEIGHT(cons(n, cons(m, x))) -> WEIGHT(sum(cons(n, cons(m, x)), cons(0, x)))

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

Additionally, the following rules can be oriented: 

   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0, x), y) -> sum(x, y)
   sum(nil, y) -> y


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(s(x_1)) = 0
POL(sum(x_1, x_2)) = x_2
POL(WEIGHT(x_1)) = 1 + x_1
POL(0) = 0
POL(cons(x_1, x_2)) = 1 + x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.767 seconds.