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

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))
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(activate(X1), activate(X2))
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)

Furthermore, R contains three SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo
       →DP Problem 3
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, nadd(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> nadd(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X





The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(ADD(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 4
Dependency Graph
       →DP Problem 2
Polo
       →DP Problem 3
Nar


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polynomial Ordering
       →DP Problem 3
Nar


Dependency Pairs:

ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X1)


Rules:


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





The following dependency pairs can be strictly oriented:

ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nadd(X1, X2)) -> ACTIVATE(X1)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__fib1(x1, x2))=  x1 + x2  
  POL(ACTIVATE(x1))=  x1  
  POL(n__add(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Polynomial Ordering
       →DP Problem 3
Nar


Dependency Pairs:

ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X1)


Rules:


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





The following dependency pairs can be strictly oriented:

ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(nfib1(X1, X2)) -> ACTIVATE(X1)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__fib1(x1, x2))=  1 + x1 + x2  
  POL(ACTIVATE(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 5
Polo
             ...
               →DP Problem 6
Dependency Graph
       →DP Problem 3
Nar


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
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, nadd(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
add(s(X), Y) -> s(add(X, Y))
add(X1, X2) -> nadd(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(nadd(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X





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))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Nar
           →DP Problem 7
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

SEL(s(N), cons(X, nfib1(X1', X2'))) -> SEL(N, fib1(activate(X1'), activate(X2')))
eight new Dependency Pairs are created:

SEL(s(N), cons(X, nfib1(X1'', X2''))) -> SEL(N, cons(activate(X1''), nfib1(activate(X2''), nadd(activate(X1''), activate(X2'')))))
SEL(s(N), cons(X, nfib1(X1'', X2''))) -> SEL(N, nfib1(activate(X1''), activate(X2'')))
SEL(s(N), cons(X, nfib1(nfib1(X1'', X2''), X2'))) -> SEL(N, fib1(fib1(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, nfib1(nadd(X1'', X2''), X2'))) -> SEL(N, fib1(add(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, nfib1(X1'', X2'))) -> SEL(N, fib1(X1'', activate(X2')))
SEL(s(N), cons(X, nfib1(X1', nfib1(X1'', X2'')))) -> SEL(N, fib1(activate(X1'), fib1(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, nfib1(X1', nadd(X1'', X2'')))) -> SEL(N, fib1(activate(X1'), add(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, nfib1(X1', X2''))) -> SEL(N, fib1(activate(X1'), X2''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Nar
           →DP Problem 7
Nar
             ...
               →DP Problem 8
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

SEL(s(N), cons(X, nadd(X1', X2'))) -> SEL(N, add(activate(X1'), activate(X2')))
seven new Dependency Pairs are created:

SEL(s(N), cons(X, nadd(X1'', X2''))) -> SEL(N, nadd(activate(X1''), activate(X2'')))
SEL(s(N), cons(X, nadd(nfib1(X1'', X2''), X2'))) -> SEL(N, add(fib1(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, nadd(nadd(X1'', X2''), X2'))) -> SEL(N, add(add(activate(X1''), activate(X2'')), activate(X2')))
SEL(s(N), cons(X, nadd(X1'', X2'))) -> SEL(N, add(X1'', activate(X2')))
SEL(s(N), cons(X, nadd(X1', nfib1(X1'', X2'')))) -> SEL(N, add(activate(X1'), fib1(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, nadd(X1', nadd(X1'', X2'')))) -> SEL(N, add(activate(X1'), add(activate(X1''), activate(X2''))))
SEL(s(N), cons(X, nadd(X1', X2''))) -> SEL(N, add(activate(X1'), X2''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Nar
           →DP Problem 7
Nar
             ...
               →DP Problem 9
Polynomial Ordering


Dependency Pairs:

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


Rules:


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





The following dependency pairs can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(activate(x1))=  0  
  POL(0)=  0  
  POL(cons(x1, x2))=  0  
  POL(SEL(x1, x2))=  x1  
  POL(n__fib1(x1, x2))=  0  
  POL(s(x1))=  1 + x1  
  POL(fib1(x1, x2))=  0  
  POL(n__add(x1, x2))=  0  
  POL(add(x1, x2))=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Nar
           →DP Problem 7
Nar
             ...
               →DP Problem 10
Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:04 minutes