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

Innermost 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(X1, X2)
ACTIVATE(nfrom(X)) -> FROM(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
one new Dependency Pair is created:

FIRST(s(X), cons(Y, nfirst(X1'', X2''))) -> ACTIVATE(nfirst(X1'', X2''))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

FIRST(s(X), cons(Y, nfirst(X1'', X2''))) -> ACTIVATE(nfirst(X1'', X2''))
ACTIVATE(nfirst(X1, X2)) -> FIRST(X1, X2)


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(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(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(nfirst(X1, X2)) -> FIRST(X1, X2)
one new Dependency Pair is created:

ACTIVATE(nfirst(s(X''), cons(Y'', nfirst(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', nfirst(X1'''', X2'''')))

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(nfirst(s(X''), cons(Y'', nfirst(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', nfirst(X1'''', X2'''')))
FIRST(s(X), cons(Y, nfirst(X1'', X2''))) -> ACTIVATE(nfirst(X1'', X2''))


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

FIRST(s(X), cons(Y, nfirst(X1'', X2''))) -> ACTIVATE(nfirst(X1'', X2''))
one new Dependency Pair is created:

FIRST(s(X), cons(Y, nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))) -> ACTIVATE(nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))

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:

FIRST(s(X), cons(Y, nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))) -> ACTIVATE(nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))
ACTIVATE(nfirst(s(X''), cons(Y'', nfirst(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', nfirst(X1'''', X2'''')))


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(s(X)))
from(X) -> nfrom(X)
activate(nfirst(X1, X2)) -> first(X1, X2)
activate(nfrom(X)) -> from(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(nfirst(s(X''), cons(Y'', nfirst(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', nfirst(X1'''', X2'''')))
one new Dependency Pair is created:

ACTIVATE(nfirst(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pairs:

ACTIVATE(nfirst(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))
FIRST(s(X), cons(Y, nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))) -> ACTIVATE(nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVATE(nfirst(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', nfirst(s(X''''''), cons(Y'''''', nfirst(X1'''''''', X2'''''''')))))


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(cons(x1, x2))=  x2  
  POL(FIRST(x1, x2))=  x2  
  POL(s(x1))=  1  
  POL(ACTIVATE(x1))=  x1  
  POL(n__first(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:

FIRST(s(X), cons(Y, nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))) -> ACTIVATE(nfirst(s(X''''), cons(Y'''', nfirst(X1'''''', X2''''''))))


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes