Term Rewriting System R:
[N, X, Y, X1, X2, XS]
fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

FIB(N) -> SEL(N, fib1(s(0), s(0)))
FIB(N) -> FIB1(s(0), s(0))
FIB1(X, Y) -> ADD(X, Y)
ADD(s(X), Y) -> ADD(X, Y)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(nfib1(X1, X2)) -> FIB1(X1, X2)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Nar


Dependency Pair:

ADD(s(X), Y) -> ADD(X, Y)


Rules:


fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X


Strategy:

innermost




The following dependency pair can be strictly oriented:

ADD(s(X), Y) -> ADD(X, Y)


There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ADD(x1, x2) -> ADD(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Nar


Dependency Pair:


Rules:


fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Narrowing Transformation


Dependency Pair:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))


Rules:


fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X


Strategy:

innermost




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

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
two new Dependency Pairs are created:

SEL(s(N), cons(X, nfib1(X1', X2'))) -> SEL(N, fib1(X1', X2'))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳Forward Instantiation Transformation


Dependency Pairs:

SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, nfib1(X1', X2'))) -> SEL(N, fib1(X1', X2'))


Rules:


fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X


Strategy:

innermost




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

SEL(s(N), cons(X, XS')) -> SEL(N, XS')
two new Dependency Pairs are created:

SEL(s(s(N'')), cons(X, cons(X'', nfib1(X1''', X2''')))) -> SEL(s(N''), cons(X'', nfib1(X1''', X2''')))
SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Nar
           →DP Problem 4
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))
SEL(s(s(N'')), cons(X, cons(X'', nfib1(X1''', X2''')))) -> SEL(s(N''), cons(X'', nfib1(X1''', X2''')))
SEL(s(N), cons(X, nfib1(X1', X2'))) -> SEL(N, fib1(X1', X2'))


Rules:


fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes