Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.


QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

AND2(true, X) -> ACTIVATE1(X)
ADD2(0, X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> S1(X)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
ACTIVATE1(n__from1(X)) -> FROM1(X)
IF3(true, X, Y) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> S1(n__add2(activate1(X), activate1(Y)))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Y)
ADD2(s1(X), Y) -> ACTIVATE1(X)
IF3(false, X, Y) -> ACTIVATE1(Y)
ADD2(s1(X), Y) -> ACTIVATE1(Y)
ACTIVATE1(n__add2(X1, X2)) -> ADD2(X1, X2)
FROM1(X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

AND2(true, X) -> ACTIVATE1(X)
ADD2(0, X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> S1(X)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
ACTIVATE1(n__from1(X)) -> FROM1(X)
IF3(true, X, Y) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> S1(n__add2(activate1(X), activate1(Y)))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Y)
ADD2(s1(X), Y) -> ACTIVATE1(X)
IF3(false, X, Y) -> ACTIVATE1(Y)
ADD2(s1(X), Y) -> ACTIVATE1(Y)
ACTIVATE1(n__add2(X1, X2)) -> ADD2(X1, X2)
FROM1(X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)

The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

ADD2(0, X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(Y)
ACTIVATE1(n__add2(X1, X2)) -> ADD2(X1, X2)
FROM1(X) -> ACTIVATE1(X)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__from1(X)) -> FROM1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Y)

The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ADD2(0, X) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__add2(X1, X2)) -> ADD2(X1, X2)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(X1, X2)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__from1(X)) -> FROM1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Y)
The remaining pairs can at least be oriented weakly.

ADD2(s1(X), Y) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(Y)
FROM1(X) -> ACTIVATE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( n__add2(x1, x2) ) = x1 + 2x2 + 2


POL( ACTIVATE1(x1) ) = 2x1 + 1


POL( n__from1(x1) ) = 2x1 + 1


POL( FIRST2(x1, x2) ) = x1 + 3x2 + 2


POL( 0 ) = 3


POL( s1(x1) ) = 2x1 + 2


POL( ADD2(x1, x2) ) = max{0, 2x1 + 3x2 - 3}


POL( FROM1(x1) ) = 3x1 + 1


POL( cons2(x1, x2) ) = 2x1 + x2


POL( n__first2(x1, x2) ) = 2x1 + 2x2 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
QDP
              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

ADD2(s1(X), Y) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(Y)
FROM1(X) -> ACTIVATE1(X)

The TRS R consists of the following rules:

and2(true, X) -> activate1(X)
and2(false, Y) -> false
if3(true, X, Y) -> activate1(X)
if3(false, X, Y) -> activate1(Y)
add2(0, X) -> activate1(X)
add2(s1(X), Y) -> s1(n__add2(activate1(X), activate1(Y)))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(activate1(Y), n__first2(activate1(X), activate1(Z)))
from1(X) -> cons2(activate1(X), n__from1(n__s1(activate1(X))))
add2(X1, X2) -> n__add2(X1, X2)
first2(X1, X2) -> n__first2(X1, X2)
from1(X) -> n__from1(X)
s1(X) -> n__s1(X)
activate1(n__add2(X1, X2)) -> add2(X1, X2)
activate1(n__first2(X1, X2)) -> first2(X1, X2)
activate1(n__from1(X)) -> from1(X)
activate1(n__s1(X)) -> s1(X)
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 3 less nodes.