* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(oddNs()) -> mark(incr(pairNs())) active(pair(X1,X2)) -> pair(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(repItems(X)) -> repItems(active(X)) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(repItems(nil())) -> mark(nil()) active(s(X)) -> s(active(X)) active(tail(X)) -> tail(active(X)) active(tail(cons(X,XS))) -> mark(XS) active(take(X1,X2)) -> take(X1,active(X2)) active(take(X1,X2)) -> take(active(X1),X2) active(take(0(),XS)) -> mark(nil()) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(X,nil())) -> mark(nil()) active(zip(X1,X2)) -> zip(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(zip(nil(),XS)) -> mark(nil()) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) incr(ok(X)) -> ok(incr(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(oddNs()) -> ok(oddNs()) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(pairNs()) -> ok(pairNs()) proper(repItems(X)) -> repItems(proper(X)) proper(s(X)) -> s(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) repItems(mark(X)) -> mark(repItems(X)) repItems(ok(X)) -> ok(repItems(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) tail(mark(X)) -> mark(tail(X)) tail(ok(X)) -> ok(tail(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) - Signature: {active/1,cons/2,incr/1,pair/2,proper/1,repItems/1,s/1,tail/1,take/2,top/1,zip/2} / {0/0,mark/1,nil/0 ,oddNs/0,ok/1,pairNs/0} - Obligation: innermost runtime complexity wrt. defined symbols {active,cons,incr,pair,proper,repItems,s,tail,take,top ,zip} and constructors {0,mark,nil,oddNs,ok,pairNs} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(incr(X)) -> incr(active(X)) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(oddNs()) -> mark(incr(pairNs())) active(pair(X1,X2)) -> pair(X1,active(X2)) active(pair(X1,X2)) -> pair(active(X1),X2) active(pairNs()) -> mark(cons(0(),incr(oddNs()))) active(repItems(X)) -> repItems(active(X)) active(repItems(cons(X,XS))) -> mark(cons(X,cons(X,repItems(XS)))) active(repItems(nil())) -> mark(nil()) active(s(X)) -> s(active(X)) active(tail(X)) -> tail(active(X)) active(tail(cons(X,XS))) -> mark(XS) active(take(X1,X2)) -> take(X1,active(X2)) active(take(X1,X2)) -> take(active(X1),X2) active(take(0(),XS)) -> mark(nil()) active(take(s(N),cons(X,XS))) -> mark(cons(X,take(N,XS))) active(zip(X,nil())) -> mark(nil()) active(zip(X1,X2)) -> zip(X1,active(X2)) active(zip(X1,X2)) -> zip(active(X1),X2) active(zip(cons(X,XS),cons(Y,YS))) -> mark(cons(pair(X,Y),zip(XS,YS))) active(zip(nil(),XS)) -> mark(nil()) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) incr(mark(X)) -> mark(incr(X)) incr(ok(X)) -> ok(incr(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(incr(X)) -> incr(proper(X)) proper(nil()) -> ok(nil()) proper(oddNs()) -> ok(oddNs()) proper(pair(X1,X2)) -> pair(proper(X1),proper(X2)) proper(pairNs()) -> ok(pairNs()) proper(repItems(X)) -> repItems(proper(X)) proper(s(X)) -> s(proper(X)) proper(tail(X)) -> tail(proper(X)) proper(take(X1,X2)) -> take(proper(X1),proper(X2)) proper(zip(X1,X2)) -> zip(proper(X1),proper(X2)) repItems(mark(X)) -> mark(repItems(X)) repItems(ok(X)) -> ok(repItems(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) tail(mark(X)) -> mark(tail(X)) tail(ok(X)) -> ok(tail(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) zip(X1,mark(X2)) -> mark(zip(X1,X2)) zip(mark(X1),X2) -> mark(zip(X1,X2)) zip(ok(X1),ok(X2)) -> ok(zip(X1,X2)) - Signature: {active/1,cons/2,incr/1,pair/2,proper/1,repItems/1,s/1,tail/1,take/2,top/1,zip/2} / {0/0,mark/1,nil/0 ,oddNs/0,ok/1,pairNs/0} - Obligation: innermost runtime complexity wrt. defined symbols {active,cons,incr,pair,proper,repItems,s,tail,take,top ,zip} and constructors {0,mark,nil,oddNs,ok,pairNs} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: cons(x,y){x -> mark(x)} = cons(mark(x),y) ->^+ mark(cons(x,y)) = C[cons(x,y) = cons(x,y){}] WORST_CASE(Omega(n^1),?)