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

Innermost 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(X1, X2)
ACTIVATE(nfirst(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(ns(X)) -> S(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

FROM(X) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> FROM(X)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ADD(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(X1, X2)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X


Strategy:

innermost




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

ADD(0, X) -> ACTIVATE(X)
two new Dependency Pairs are created:

ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(X)) -> FROM(X)
ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))
ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nadd(X1, X2)) -> ADD(X1, X2)
FROM(X) -> ACTIVATE(X)


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


Strategy:

innermost




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

FROM(X) -> ACTIVATE(X)
two new Dependency Pairs are created:

FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))
ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))
ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nadd(X1, X2)) -> ADD(X1, X2)
FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nfrom(X)) -> FROM(X)


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


Strategy:

innermost




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

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

ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 4
Forward Instantiation Transformation


Dependency Pairs:

ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nfrom(X)) -> FROM(X)
FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))


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


Strategy:

innermost




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

ACTIVATE(nfrom(X)) -> FROM(X)
two new Dependency Pairs are created:

ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))


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


Strategy:

innermost




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

ADD(0, nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
two new Dependency Pairs are created:

ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 6
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))


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


Strategy:

innermost




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

ADD(0, nfrom(X'')) -> ACTIVATE(nfrom(X''))
two new Dependency Pairs are created:

ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 7
Forward Instantiation Transformation


Dependency Pairs:

ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))


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


Strategy:

innermost




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

FROM(nadd(X1'', X2'')) -> ACTIVATE(nadd(X1'', X2''))
two new Dependency Pairs are created:

FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Forward Instantiation Transformation


Dependency Pairs:

FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))


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


Strategy:

innermost




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

FROM(nfrom(X'')) -> ACTIVATE(nfrom(X''))
two new Dependency Pairs are created:

FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Forward Instantiation Transformation


Dependency Pairs:

FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))


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


Strategy:

innermost




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

ACTIVATE(nadd(0, nadd(X1'''', X2''''))) -> ADD(0, nadd(X1'''', X2''''))
two new Dependency Pairs are created:

ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 10
Forward Instantiation Transformation


Dependency Pairs:

FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))


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


Strategy:

innermost




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

ACTIVATE(nadd(0, nfrom(X''''))) -> ADD(0, nfrom(X''''))
two new Dependency Pairs are created:

ACTIVATE(nadd(0, nfrom(nadd(X1'''''''', X2'''''''')))) -> ADD(0, nfrom(nadd(X1'''''''', X2'''''''')))
ACTIVATE(nadd(0, nfrom(nfrom(X'''''''')))) -> ADD(0, nfrom(nfrom(X'''''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 11
Forward Instantiation Transformation


Dependency Pairs:

FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ACTIVATE(nadd(0, nfrom(nfrom(X'''''''')))) -> ADD(0, nfrom(nfrom(X'''''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(nadd(X1'''''''', X2'''''''')))) -> ADD(0, nfrom(nadd(X1'''''''', X2'''''''')))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))


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


Strategy:

innermost




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

ACTIVATE(nfrom(nadd(X1'''', X2''''))) -> FROM(nadd(X1'''', X2''''))
two new Dependency Pairs are created:

ACTIVATE(nfrom(nadd(0, nadd(X1'''''''', X2'''''''')))) -> FROM(nadd(0, nadd(X1'''''''', X2'''''''')))
ACTIVATE(nfrom(nadd(0, nfrom(X'''''''')))) -> FROM(nadd(0, nfrom(X'''''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 12
Forward Instantiation Transformation


Dependency Pairs:

ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ACTIVATE(nadd(0, nfrom(nfrom(X'''''''')))) -> ADD(0, nfrom(nfrom(X'''''''')))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nfrom(nadd(0, nfrom(X'''''''')))) -> FROM(nadd(0, nfrom(X'''''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(nadd(X1'''''''', X2'''''''')))) -> ADD(0, nfrom(nadd(X1'''''''', X2'''''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(0, nadd(X1'''''''', X2'''''''')))) -> FROM(nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))


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


Strategy:

innermost




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

ACTIVATE(nfrom(nfrom(X''''))) -> FROM(nfrom(X''''))
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(nadd(X1'''''''', X2'''''''')))) -> FROM(nfrom(nadd(X1'''''''', X2'''''''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''''''')))) -> FROM(nfrom(nfrom(X'''''''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 13
Polynomial Ordering


Dependency Pairs:

FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''''''')))) -> FROM(nfrom(nfrom(X'''''''')))
ACTIVATE(nadd(0, nfrom(nfrom(X'''''''')))) -> ADD(0, nfrom(nfrom(X'''''''')))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nfrom(nadd(0, nfrom(X'''''''')))) -> FROM(nadd(0, nfrom(X'''''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(nadd(X1'''''''', X2'''''''')))) -> ADD(0, nfrom(nadd(X1'''''''', X2'''''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nadd(0, nadd(X1'''''''', X2'''''''')))) -> FROM(nadd(0, nadd(X1'''''''', X2'''''''')))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nfrom(nfrom(nadd(X1'''''''', X2'''''''')))) -> FROM(nfrom(nadd(X1'''''''', X2'''''''')))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

FROM(nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))
FROM(nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
FROM(nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
FROM(nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FROM(x1))=  1 + x1  
  POL(n__from(x1))=  1 + x1  
  POL(0)=  0  
  POL(ACTIVATE(x1))=  x1  
  POL(n__add(x1, x2))=  x2  
  POL(ADD(x1, x2))=  x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 14
Dependency Graph


Dependency Pairs:

ACTIVATE(nfrom(nfrom(nfrom(X'''''''')))) -> FROM(nfrom(nfrom(X'''''''')))
ACTIVATE(nadd(0, nfrom(nfrom(X'''''''')))) -> ADD(0, nfrom(nfrom(X'''''''')))
ACTIVATE(nfrom(nadd(0, nfrom(X'''''''')))) -> FROM(nadd(0, nfrom(X'''''''')))
ADD(0, nfrom(nadd(X1'''''', X2''''''))) -> ACTIVATE(nfrom(nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nfrom(nadd(X1'''''''', X2'''''''')))) -> ADD(0, nfrom(nadd(X1'''''''', X2'''''''')))
ADD(0, nadd(0, nfrom(X''''''))) -> ACTIVATE(nadd(0, nfrom(X'''''')))
ACTIVATE(nadd(0, nadd(0, nfrom(X'''''''')))) -> ADD(0, nadd(0, nfrom(X'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(X1'''''', X2'''''')))
ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
ACTIVATE(nfrom(nadd(0, nadd(X1'''''''', X2'''''''')))) -> FROM(nadd(0, nadd(X1'''''''', X2'''''''')))
ACTIVATE(nfrom(nfrom(nadd(X1'''''''', X2'''''''')))) -> FROM(nfrom(nadd(X1'''''''', X2'''''''')))
ADD(0, nfrom(nfrom(X''''''))) -> ACTIVATE(nfrom(nfrom(X'''''')))


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


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 15
Polynomial Ordering


Dependency Pairs:

ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))
ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(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(X1, X2)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVATE(nadd(0, nadd(0, nadd(X1'''''''', X2'''''''')))) -> ADD(0, nadd(0, nadd(X1'''''''', X2'''''''')))


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  1  
  POL(ACTIVATE(x1))=  x1  
  POL(n__add(x1, x2))=  x1 + x2  
  POL(ADD(x1, x2))=  x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 16
Dependency Graph


Dependency Pair:

ADD(0, nadd(0, nadd(X1'''''', X2''''''))) -> ACTIVATE(nadd(0, nadd(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(X1, X2)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(X)
activate(ns(X)) -> s(X)
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes