* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: app(cons(N,L),Y) -> cons(N,app(L,Y)) app(nil(),Y) -> Y high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L)) high(N,nil()) -> nil() ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L)) ifhigh(true(),N,cons(M,L)) -> high(N,L) iflow(false(),N,cons(M,L)) -> low(N,L) iflow(true(),N,cons(M,L)) -> cons(M,low(N,L)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L)) low(N,nil()) -> nil() quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L)))) quicksort(nil()) -> nil() - Signature: {app/2,high/2,ifhigh/3,iflow/3,le/2,low/2,quicksort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,high,ifhigh,iflow,le,low ,quicksort} 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: app(cons(N,L),Y) -> cons(N,app(L,Y)) app(nil(),Y) -> Y high(N,cons(M,L)) -> ifhigh(le(M,N),N,cons(M,L)) high(N,nil()) -> nil() ifhigh(false(),N,cons(M,L)) -> cons(M,high(N,L)) ifhigh(true(),N,cons(M,L)) -> high(N,L) iflow(false(),N,cons(M,L)) -> low(N,L) iflow(true(),N,cons(M,L)) -> cons(M,low(N,L)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) low(N,cons(M,L)) -> iflow(le(M,N),N,cons(M,L)) low(N,nil()) -> nil() quicksort(cons(N,L)) -> app(quicksort(low(N,L)),cons(N,quicksort(high(N,L)))) quicksort(nil()) -> nil() - Signature: {app/2,high/2,ifhigh/3,iflow/3,le/2,low/2,quicksort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,high,ifhigh,iflow,le,low ,quicksort} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: app(y,z){y -> cons(x,y)} = app(cons(x,y),z) ->^+ cons(x,app(y,z)) = C[app(y,z) = app(y,z){}] WORST_CASE(Omega(n^1),?)