Term Rewriting System R:
[N, X, Y, XS, X1, X2]
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(fib(N)) -> SEL(N, fib1(s(0), s(0)))
ACTIVE(fib(N)) -> FIB1(s(0), s(0))
ACTIVE(fib(N)) -> S(0)
ACTIVE(fib1(X, Y)) -> CONS(X, fib1(Y, add(X, Y)))
ACTIVE(fib1(X, Y)) -> FIB1(Y, add(X, Y))
ACTIVE(fib1(X, Y)) -> ADD(X, Y)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(sel(s(N), cons(X, XS))) -> SEL(N, XS)
ACTIVE(fib(X)) -> FIB(active(X))
ACTIVE(fib(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(fib1(X1, X2)) -> FIB1(active(X1), X2)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
ACTIVE(fib1(X1, X2)) -> FIB1(X1, active(X2))
ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
FIB(mark(X)) -> FIB(X)
FIB(ok(X)) -> FIB(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
FIB1(mark(X1), X2) -> FIB1(X1, X2)
FIB1(X1, mark(X2)) -> FIB1(X1, X2)
FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
PROPER(fib(X)) -> FIB(proper(X))
PROPER(fib(X)) -> PROPER(X)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(fib1(X1, X2)) -> FIB1(proper(X1), proper(X2))
PROPER(fib1(X1, X2)) -> PROPER(X1)
PROPER(fib1(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains nine SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
  2. SEL(X1, mark(X2)) -> SEL(X1, X2)
  3. SEL(mark(X1), X2) -> SEL(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
  2. CONS(mark(X1), X2) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
FIB1(X1, mark(X2)) -> FIB1(X1, X2)
FIB1(mark(X1), X2) -> FIB1(X1, X2)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
  2. FIB1(X1, mark(X2)) -> FIB1(X1, X2)
  3. FIB1(mark(X1), X2) -> FIB1(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
  2. ADD(X1, mark(X2)) -> ADD(X1, X2)
  3. ADD(mark(X1), X2) -> ADD(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

FIB(ok(X)) -> FIB(X)
FIB(mark(X)) -> FIB(X)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(fib(X)) -> ACTIVE(X)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(add(X1, X2)) -> ACTIVE(X2)
  2. ACTIVE(add(X1, X2)) -> ACTIVE(X1)
  3. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(s(X)) -> ACTIVE(X)
  5. ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
  6. ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
  7. ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
  8. ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
  9. ACTIVE(fib(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 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)
fib(x1) -> fib(x1)
s(x1) -> s(x1)
sel(x1, x2) -> sel(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳Nar


Dependency Pairs:

PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(fib1(X1, X2)) -> PROPER(X2)
PROPER(fib1(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(fib(X)) -> PROPER(X)


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(add(X1, X2)) -> PROPER(X2)
  2. PROPER(add(X1, X2)) -> PROPER(X1)
  3. PROPER(cons(X1, X2)) -> PROPER(X2)
  4. PROPER(cons(X1, X2)) -> PROPER(X1)
  5. PROPER(s(X)) -> PROPER(X)
  6. PROPER(fib1(X1, X2)) -> PROPER(X2)
  7. PROPER(fib1(X1, X2)) -> PROPER(X1)
  8. PROPER(sel(X1, X2)) -> PROPER(X2)
  9. PROPER(sel(X1, X2)) -> PROPER(X1)
  10. PROPER(fib(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {10, 9, 8, 7, 6, 5, 4, 3, 2, 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)
fib(x1) -> fib(x1)
sel(x1, x2) -> sel(x1, x2)
s(x1) -> s(x1)
fib1(x1, x2) -> fib1(x1, x2)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

TOP(mark(X)) -> TOP(proper(X))
seven new Dependency Pairs are created:

TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar
           →DP Problem 10
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

TOP(ok(X)) -> TOP(active(X))
15 new Dependency Pairs are created:

TOP(ok(fib(N'))) -> TOP(mark(sel(N', fib1(s(0), s(0)))))
TOP(ok(fib1(X'', Y'))) -> TOP(mark(cons(X'', fib1(Y', add(X'', Y')))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(sel(0, cons(X'', XS')))) -> TOP(mark(X''))
TOP(ok(sel(s(N'), cons(X'', XS')))) -> TOP(mark(sel(N', XS')))
TOP(ok(fib(X''))) -> TOP(fib(active(X'')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(active(X1'), X2'))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(X1', active(X2')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 11
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(X1', active(X2')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(fib(X''))) -> TOP(fib(active(X'')))
TOP(ok(sel(s(N'), cons(X'', XS')))) -> TOP(mark(sel(N', XS')))
TOP(ok(sel(0, cons(X'', XS')))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(fib1(X'', Y'))) -> TOP(mark(cons(X'', fib1(Y', add(X'', Y')))))
TOP(ok(fib(N'))) -> TOP(mark(sel(N', fib1(s(0), s(0)))))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))


Rules:


active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes