Term Rewriting System R:

   [x, u, v, z, y]
   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, 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: 


   ADMIT(x, .(u, .(v, .(w, z)))) -> COND(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)

Furthermore, R contains one SCC.


SCC1:

ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), 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(carry(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(w) = 0
POL(ADMIT(x_1, x_2)) = 1 + x_1 + x_2
POL(.(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.556 seconds.