(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FIB(N) → SEL(N, fib1(s(0), s(0)))
FIB(N) → FIB1(s(0), s(0))
ADD(s(X), Y) → ADD(X, Y)
SEL(s(N), cons(X, XS)) → SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__fib1(X1, X2)) → FIB1(activate(X1), activate(X2))
ACTIVATE(n__fib1(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__fib1(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
The TRS R consists of the following rules:
fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ADD(s(X), Y) → ADD(X, Y)
The TRS R consists of the following rules:
fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__fib1(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__fib1(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
The TRS R consists of the following rules:
fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SEL(s(N), cons(X, XS)) → SEL(N, activate(XS))
The TRS R consists of the following rules:
fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.