Term Rewriting System R:
[X, Y, Z, X1]
2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

2ND(cons(X, X1)) -> 2ND(cons1(X, activate(X1)))
2ND(cons(X, X1)) -> ACTIVATE(X1)
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
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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)) -> ACTIVATE(X)
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(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(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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(ns(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(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:

ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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''))) -> ACTIVATE(nfrom(X''))
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(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:

ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(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:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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(ns(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(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(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
two new Dependency Pairs are created:

ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(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:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Polynomial Ordering


Dependency Pairs:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(n__s(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Dependency Graph


Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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 10
Polynomial Ordering


Dependency Pair:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__from(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 11
Dependency Graph


Dependency Pair:


Rules:


2nd(cons1(X, cons(Y, Z))) -> Y
2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(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