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
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


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




As we are in the innermost case, we can delete all 9 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. ADD(s(X), Y) -> ADD(X, Y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


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




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 4
             ↳Size-Change Principle


Dependency Pair:

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


Rules:


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


Strategy:

innermost




We number the DPs as follows:
  1. SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)

We obtain no new DP problems.

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