Term Rewriting System R:

   [X, XS, N, Y, YS]
   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   MINUS(s(X), s(Y)) -> MINUS(X, Y)
   FROM(X) -> FROM(s(X))
   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MINUS(X, Y)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> ZWQUOT(XS, YS)
   SEL(s(N), cons(X, XS)) -> SEL(N, XS)

Furthermore, R contains five SCCs.


SCC1:

MINUS(s(X), s(Y)) -> MINUS(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(s(x_1)) = 1 + x_1
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MINUS(s(X), s(Y)) -> MINUS(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

FROM(X) -> FROM(s(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(x_1)) = x_1
POL(FROM(x_1)) = 1 + x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   from(X) -> cons(X, from(s(X)))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))
   zWquot(XS, nil) -> nil
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

FROM(X) -> FROM(s(X))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC2.MRR1.NOC1:

FROM(X) -> FROM(s(X))

Found an infinite P-chain over R:
P = FROM(X) -> FROM(s(X))
R = []
s = FROM(X) evaluates to t = FROM(s(X))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.701 seconds.