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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

AND(true, X) -> ACTIVATE(X)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ADD(0, X) -> ACTIVATE(X)
ADD(s(X), Y) -> S(nadd(activate(X), activate(Y)))
ADD(s(X), Y) -> ACTIVATE(X)
ADD(s(X), Y) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FROM(X) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
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(X)
ACTIVATE(ns(X)) -> S(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

ADD(s(X), Y) -> ACTIVATE(Y)
ADD(s(X), Y) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)


Rules:


and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(nadd(activate(X), activate(Y)))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nadd(X1, X2)) -> add(activate(X1), X2)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X





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

ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), X2)
five new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nadd(X1', X2)) -> ADD(X1', X2)
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), X2)
ACTIVATE(nadd(nfrom(X'), X2)) -> ADD(from(X'), X2)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), X2)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), X2''), X2)
FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
ACTIVATE(nfirst(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(Y)


Rules:


and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(nadd(activate(X), activate(Y)))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nadd(X1, X2)) -> add(activate(X1), X2)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(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))
10 new Dependency Pairs are created:

ACTIVATE(nfirst(nadd(X1'', X2''), X2)) -> FIRST(add(activate(X1''), X2''), activate(X2))
ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(nfirst(nfrom(X'), X2)) -> FIRST(from(X'), activate(X2))
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(X'), activate(X2))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(X1, nadd(X1'', X2''))) -> FIRST(activate(X1), add(activate(X1''), X2''))
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, nfrom(X'))) -> FIRST(activate(X1), from(X'))
ACTIVATE(nfirst(X1, ns(X'))) -> FIRST(activate(X1), s(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
Nar
             ...
               →DP Problem 3
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(X'))
ACTIVATE(nfirst(X1, nfrom(X'))) -> FIRST(activate(X1), from(X'))
ACTIVATE(nfirst(X1, nfirst(X1'', X2''))) -> FIRST(activate(X1), first(activate(X1''), activate(X2'')))
ACTIVATE(nfirst(X1, nadd(X1'', X2''))) -> FIRST(activate(X1), add(activate(X1''), X2''))
ACTIVATE(nfirst(X1', X2)) -> FIRST(X1', activate(X2))
ACTIVATE(nfirst(ns(X'), X2)) -> FIRST(s(X'), activate(X2))
ACTIVATE(nfirst(nfrom(X'), X2)) -> FIRST(from(X'), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(nfirst(nfirst(X1'', X2''), X2)) -> FIRST(first(activate(X1''), activate(X2'')), activate(X2))
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
ACTIVATE(nfirst(nadd(X1'', X2''), X2)) -> FIRST(add(activate(X1''), X2''), activate(X2))
ACTIVATE(nadd(X1', X2)) -> ADD(X1', X2)
ACTIVATE(nadd(ns(X'), X2)) -> ADD(s(X'), X2)
ADD(s(X), Y) -> ACTIVATE(Y)
ACTIVATE(nadd(nfrom(X'), X2)) -> ADD(from(X'), X2)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(nadd(nfirst(X1'', X2''), X2)) -> ADD(first(activate(X1''), activate(X2'')), X2)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(nadd(nadd(X1'', X2''), X2)) -> ADD(add(activate(X1''), X2''), X2)
FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfirst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)


Rules:


and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(nadd(activate(X), activate(Y)))
add(X1, X2) -> nadd(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), nfirst(activate(X), activate(Z)))
first(X1, X2) -> nfirst(X1, X2)
from(X) -> cons(activate(X), nfrom(ns(activate(X))))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nadd(X1, X2)) -> add(activate(X1), X2)
activate(nfirst(X1, X2)) -> first(activate(X1), activate(X2))
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes