* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(s(X)) -> s(mark(X)) mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__incr(X) -> incr(X) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__nats() -> cons(0(),incr(nats())) a__nats() -> nats() a__odds() -> a__incr(a__pairs()) a__odds() -> odds() a__pairs() -> cons(0(),incr(odds())) a__pairs() -> pairs() a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(head(X)) -> a__head(mark(X)) mark(incr(X)) -> a__incr(mark(X)) mark(nats()) -> a__nats() mark(odds()) -> a__odds() mark(pairs()) -> a__pairs() mark(s(X)) -> s(mark(X)) mark(tail(X)) -> a__tail(mark(X)) - Signature: {a__head/1,a__incr/1,a__nats/0,a__odds/0,a__pairs/0,a__tail/1,mark/1} / {0/0,cons/2,head/1,incr/1,nats/0 ,odds/0,pairs/0,s/1,tail/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__head,a__incr,a__nats,a__odds,a__pairs,a__tail ,mark} and constructors {0,cons,head,incr,nats,odds,pairs,s,tail} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> cons(x,y)} = mark(cons(x,y)) ->^+ cons(mark(x),y) = C[mark(x) = mark(x){}] WORST_CASE(Omega(n^1),?)