Term Rewriting System R:
[X, Y, Z, X1, X2]
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)


Rules:


first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
eight new Dependency Pairs are created:

ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfirst(nfrom(X'), X2)) -> FIRST(from(activate(X')), activate(X2))
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, nfrom(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(nfirst(X1, ns(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(nfirst(X1, X2')) -> FIRST(activate(X1), X2')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nfirst(X1, X2')) -> FIRST(activate(X1), X2')
ACTIVATE(nfirst(X1, ns(X'))) -> FIRST(activate(X1), s(activate(X')))
ACTIVATE(nfirst(X1, nfrom(X'))) -> FIRST(activate(X1), from(activate(X')))
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(activate(X')), activate(X2))
ACTIVATE(nfirst(nfrom(X'), X2)) -> FIRST(from(activate(X')), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(ns(X)) -> ACTIVATE(X)


Rules:


first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, nfirst(X, activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X




Termination of R could not be shown.
Duration:
0:01 minutes