(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(fib(fib1(X141735_3, X241736_3))) →+ a__fib(cons(mark(mark(X141735_3)), fib1(mark(X241736_3), add(mark(X141735_3), mark(X241736_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X141735_3 / fib(fib1(X141735_3, X241736_3))].
The result substitution is [ ].
The rewrite sequence
mark(fib(fib1(X141735_3, X241736_3))) →+ a__fib(cons(mark(mark(X141735_3)), fib1(mark(X241736_3), add(mark(X141735_3), mark(X241736_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,1,0].
The pumping substitution is [X141735_3 / fib(fib1(X141735_3, X241736_3))].
The result substitution is [ ].
(2) BOUNDS(2^n, INF)