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))

Innermost 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
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pairs:

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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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:
SEL(x1, x2) -> SEL(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 10
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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:
FIB1(x1, x2) -> FIB1(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 11
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pairs:

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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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:
CONS(x1, x2) -> CONS(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 12
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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:
S(x1) -> S(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 13
Dependency Graph
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Argument Filtering and Ordering
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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:
FIB(x1) -> FIB(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
           →DP Problem 14
Dependency Graph
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
Argument Filtering and Ordering
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 15
Dependency Graph
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
Argument Filtering and Ordering
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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)


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:
ACTIVE(x1) -> ACTIVE(x1)
add(x1, x2) -> add(x1, x2)
s(x1) -> s(x1)
sel(x1, x2) -> sel(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
cons(x1, x2) -> cons(x1, x2)
fib(x1) -> fib(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
           →DP Problem 16
Dependency Graph
       →DP Problem 8
AFS
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Argument Filtering and Ordering
       →DP Problem 9
Remaining


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))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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)


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:
PROPER(x1) -> PROPER(x1)
add(x1, x2) -> add(x1, x2)
sel(x1, x2) -> sel(x1, x2)
cons(x1, x2) -> cons(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
fib(x1) -> fib(x1)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
           →DP Problem 17
Dependency Graph
       →DP Problem 9
Remaining


Dependency Pair:


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))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
       →DP Problem 9
Remaining Obligation(s)




The following remains to be proven:
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))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:04 minutes