(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__nats → cons(0, incr(nats))
a__pairs → cons(0, incr(odds))
a__odds → a__incr(a__pairs)
a__incr(cons(X, XS)) → cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) → mark(X)
a__tail(cons(X, XS)) → mark(XS)
mark(nats) → a__nats
mark(incr(X)) → a__incr(mark(X))
mark(pairs) → a__pairs
mark(odds) → a__odds
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__nats → nats
a__incr(X) → incr(X)
a__pairs → pairs
a__odds → odds
a__head(X) → head(X)
a__tail(X) → tail(X)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
a__tail(cons(X, tail(cons(X18065_3, X28066_3)))) →+ a__tail(cons(mark(X18065_3), X28066_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [X28066_3 / tail(cons(X18065_3, X28066_3))].
The result substitution is [X / mark(X18065_3)].
(2) BOUNDS(n^1, INF)