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