Termination w.r.t. Q proof of TRS/TRCSREx8_BLR02

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 9 SCCs with 26 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD(X1, mark(X2)) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.

ADD(mark(X1), X2) → ADD(X1, X2)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)

Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = ADD(x2)
mark(x1) = mark(x1)
ok(x1) = x1

Lexicographic Path Order [19].
Precedence:

mark₁ > ADD₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD(mark(X1), X2) → ADD(X1, X2)
ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD(ok(X1), ok(X2)) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.

ADD(mark(X1), X2) → ADD(X1, X2)

Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = ADD(x2)
mark(x1) = mark
ok(x1) = ok(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > [ADD₁, mark]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD(mark(X1), X2) → ADD(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD(mark(X1), X2) → ADD(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = x1
mark(x1) = mark(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.

CONS(mark(X1), X2) → CONS(X1, X2)

Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x2)
mark(x1) = mark
ok(x1) = ok(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > [CONS₁, mark]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = x1
mark(x1) = mark(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.

S(mark(X)) → S(X)

Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
ok(x1) = ok(x1)
mark(x1) = x1

Lexicographic Path Order [19].
Precedence:

[S₁, ok_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
mark(x1) = mark(x1)

Lexicographic Path Order [19].
Precedence:

[S₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FIB1(X1, mark(X2)) → FIB1(X1, X2)
FIB1(mark(X1), X2) → FIB1(X1, X2)
FIB1(ok(X1), ok(X2)) → FIB1(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FIB1(X1, mark(X2)) → FIB1(X1, X2)

The remaining pairs can at least be oriented weakly.

FIB1(mark(X1), X2) → FIB1(X1, X2)
FIB1(ok(X1), ok(X2)) → FIB1(X1, X2)

Used ordering: Combined order from the following AFS and order.
FIB1(x1, x2) = FIB1(x2)
mark(x1) = mark(x1)
ok(x1) = x1

Lexicographic Path Order [19].
Precedence:

mark₁ > FIB1₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FIB1(mark(X1), X2) → FIB1(X1, X2)
FIB1(ok(X1), ok(X2)) → FIB1(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FIB1(ok(X1), ok(X2)) → FIB1(X1, X2)

The remaining pairs can at least be oriented weakly.

FIB1(mark(X1), X2) → FIB1(X1, X2)

Used ordering: Combined order from the following AFS and order.
FIB1(x1, x2) = FIB1(x2)
mark(x1) = mark
ok(x1) = ok(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > [FIB1₁, mark]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FIB1(mark(X1), X2) → FIB1(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FIB1(mark(X1), X2) → FIB1(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
FIB1(x1, x2) = x1
mark(x1) = mark(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.

SEL(ok(X1), ok(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)

Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = SEL(x2)
mark(x1) = mark(x1)
ok(x1) = x1

Lexicographic Path Order [19].
Precedence:

mark₁ > SEL₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL(ok(X1), ok(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

SEL(ok(X1), ok(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.

SEL(mark(X1), X2) → SEL(X1, X2)

Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = SEL(x2)
ok(x1) = ok(x1)
mark(x1) = mark

Lexicographic Path Order [19].
Precedence:

ok₁ > SEL₁
mark > SEL₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = x1
mark(x1) = mark(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FIB(mark(X)) → FIB(X)
FIB(ok(X)) → FIB(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FIB(mark(X)) → FIB(X)

The remaining pairs can at least be oriented weakly.

FIB(ok(X)) → FIB(X)

Used ordering: Combined order from the following AFS and order.
FIB(x1) = FIB(x1)
mark(x1) = mark(x1)
ok(x1) = x1

Lexicographic Path Order [19].
Precedence:

[FIB₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FIB(ok(X)) → FIB(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FIB(ok(X)) → FIB(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
FIB(x1) = FIB(x1)
ok(x1) = ok(x1)

Lexicographic Path Order [19].
Precedence:

[FIB₁, ok_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER(add(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(fib1(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(fib1(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(add(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.

PROPER(s(X)) → PROPER(X)
PROPER(fib(X)) → PROPER(X)

Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
add(x1, x2) = add(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
fib1(x1, x2) = fib1(x1, x2)
fib(x1) = x1
sel(x1, x2) = sel(x1, x2)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(s(X)) → PROPER(X)
PROPER(fib(X)) → PROPER(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.

PROPER(fib(X)) → PROPER(X)

Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
s(x1) = s(x1)
fib(x1) = x1

Lexicographic Path Order [19].
Precedence:

[PROPER₁, s_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(fib(X)) → PROPER(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER(fib(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
fib(x1) = fib(x1)

Lexicographic Path Order [19].
Precedence:

[PROPER₁, fib_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE(fib1(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(fib1(X1, X2)) → ACTIVE(X2)
ACTIVE(add(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(add(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(fib(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
fib1(x1, x2) = fib1(x1, x2)
sel(x1, x2) = sel(x1, x2)
cons(x1, x2) = x1
add(x1, x2) = add(x1, x2)
fib(x1) = x1
s(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(fib(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(fib(X)) → ACTIVE(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
cons(x1, x2) = x1
fib(x1) = x1
s(x1) = s(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(fib(X)) → ACTIVE(X)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE(fib(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
cons(x1, x2) = x1
fib(x1) = fib(x1)

Lexicographic Path Order [19].
Precedence:

fib₁ > ACTIVE₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
cons(x1, x2) = cons(x1)

Lexicographic Path Order [19].
Precedence:

[ACTIVE₁, cons_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ PisEmptyProof

              ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following 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))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
fib1(mark(X1), X2) → mark(fib1(X1, X2))
fib1(X1, mark(X2)) → mark(fib1(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(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))
fib(ok(X)) → ok(fib(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
fib1(ok(X1), ok(X2)) → ok(fib1(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.