* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: active(cons(X1,X2)) -> cons(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(first(X1,X2)) -> first(active(X1),X2) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(s(X)) -> s(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,cons/2,first/2,from/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,cons,first,from,proper,s,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: active(cons(X1,X2)) -> cons(active(X1),X2) active(first(X1,X2)) -> first(X1,active(X2)) active(first(X1,X2)) -> first(active(X1),X2) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> from(active(X)) active(from(X)) -> mark(cons(X,from(s(X)))) active(s(X)) -> s(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) first(mark(X1),X2) -> mark(first(X1,X2)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) from(mark(X)) -> mark(from(X)) from(ok(X)) -> ok(from(X)) proper(0()) -> ok(0()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(from(X)) -> from(proper(X)) proper(nil()) -> ok(nil()) proper(s(X)) -> s(proper(X)) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,cons/2,first/2,from/1,proper/1,s/1,top/1} / {0/0,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,cons,first,from,proper,s,top} and constructors {0 ,mark,nil,ok} + 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),?)