* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: del(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs) del(x,nil()) -> nil() eq(0(),0()) -> true() eq(0(),s(y)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) ge(0(),0()) -> true() ge(0(),s(x)) -> false() ge(s(x),0()) -> true() ge(s(x),s(y)) -> ge(x,y) if1(false(),x,y,xs) -> max(cons(y,xs)) if1(true(),x,y,xs) -> max(cons(x,xs)) if2(false(),x,y,xs) -> cons(y,del(x,xs)) if2(true(),x,y,xs) -> xs max(cons(x,cons(y,xs))) -> if1(ge(x,y),x,y,xs) max(cons(x,nil())) -> x max(nil()) -> 0() sort(cons(x,xs)) -> cons(max(cons(x,xs)),sort(del(max(cons(x,xs)),cons(x,xs)))) sort(nil()) -> nil() - Signature: {del/2,eq/2,ge/2,if1/4,if2/4,max/1,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {del,eq,ge,if1,if2,max,sort} and constructors {0,cons ,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: del(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs) del(x,nil()) -> nil() eq(0(),0()) -> true() eq(0(),s(y)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) ge(0(),0()) -> true() ge(0(),s(x)) -> false() ge(s(x),0()) -> true() ge(s(x),s(y)) -> ge(x,y) if1(false(),x,y,xs) -> max(cons(y,xs)) if1(true(),x,y,xs) -> max(cons(x,xs)) if2(false(),x,y,xs) -> cons(y,del(x,xs)) if2(true(),x,y,xs) -> xs max(cons(x,cons(y,xs))) -> if1(ge(x,y),x,y,xs) max(cons(x,nil())) -> x max(nil()) -> 0() sort(cons(x,xs)) -> cons(max(cons(x,xs)),sort(del(max(cons(x,xs)),cons(x,xs)))) sort(nil()) -> nil() - Signature: {del/2,eq/2,ge/2,if1/4,if2/4,max/1,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {del,eq,ge,if1,if2,max,sort} and constructors {0,cons ,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,y){x -> s(x),y -> s(y)} = eq(s(x),s(y)) ->^+ eq(x,y) = C[eq(x,y) = eq(x,y){}] WORST_CASE(Omega(n^1),?)