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