Term Rewriting System R:
[Z, X, Y, X1, X2]
fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
s(X) -> ns(X)
activate(nfst(X1, X2)) -> fst(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nlen(X)) -> len(activate(X))
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ADD(s(X), Y) -> S(nadd(activate(X), Y))
ADD(s(X), Y) -> ACTIVATE(X)
LEN(cons(X, Z)) -> S(nlen(activate(Z)))
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nfst(X1, X2)) -> FST(activate(X1), activate(X2))
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(X)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nlen(X)) -> LEN(activate(X))
ACTIVATE(nlen(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

ACTIVATE(nlen(X)) -> ACTIVATE(X)
ACTIVATE(nlen(X)) -> LEN(activate(X))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X1)
LEN(cons(X, Z)) -> ACTIVATE(Z)


Rules:


fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
s(X) -> ns(X)
activate(nfst(X1, X2)) -> fst(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nlen(X)) -> len(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nlen(X)) -> LEN(activate(X))
six new Dependency Pairs are created:

ACTIVATE(nlen(nfst(X1', X2'))) -> LEN(fst(activate(X1'), activate(X2')))
ACTIVATE(nlen(nfrom(X''))) -> LEN(from(activate(X'')))
ACTIVATE(nlen(ns(X''))) -> LEN(s(X''))
ACTIVATE(nlen(nadd(X1', X2'))) -> LEN(add(activate(X1'), activate(X2')))
ACTIVATE(nlen(nlen(X''))) -> LEN(len(activate(X'')))
ACTIVATE(nlen(X'')) -> LEN(X'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

ACTIVATE(nlen(X'')) -> LEN(X'')
ACTIVATE(nlen(nlen(X''))) -> LEN(len(activate(X'')))
ACTIVATE(nlen(nadd(X1', X2'))) -> LEN(add(activate(X1'), activate(X2')))
ACTIVATE(nlen(ns(X''))) -> LEN(s(X''))
ACTIVATE(nlen(nfrom(X''))) -> LEN(from(activate(X'')))
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nlen(nfst(X1', X2'))) -> LEN(fst(activate(X1'), activate(X2')))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nlen(X)) -> ACTIVATE(X)


Rules:


fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
s(X) -> ns(X)
activate(nfst(X1, X2)) -> fst(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nlen(X)) -> len(activate(X))
activate(X) -> X


Strategy:

innermost




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

ACTIVATE(nlen(ns(X''))) -> LEN(s(X''))
one new Dependency Pair is created:

ACTIVATE(nlen(ns(X''))) -> LEN(ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

ACTIVATE(nlen(nlen(X''))) -> LEN(len(activate(X'')))
ACTIVATE(nlen(nadd(X1', X2'))) -> LEN(add(activate(X1'), activate(X2')))
ACTIVATE(nlen(nfrom(X''))) -> LEN(from(activate(X'')))
ACTIVATE(nlen(nfst(X1', X2'))) -> LEN(fst(activate(X1'), activate(X2')))
ACTIVATE(nlen(X)) -> ACTIVATE(X)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfst(X1, X2)) -> ACTIVATE(X1)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(nlen(X'')) -> LEN(X'')


Rules:


fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, nfst(activate(X), activate(Z)))
fst(X1, X2) -> nfst(X1, X2)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
add(0, X) -> X
add(s(X), Y) -> s(nadd(activate(X), Y))
add(X1, X2) -> nadd(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(nlen(activate(Z)))
len(X) -> nlen(X)
s(X) -> ns(X)
activate(nfst(X1, X2)) -> fst(activate(X1), activate(X2))
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(X)
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(nlen(X)) -> len(activate(X))
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:09 minutes