* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: app(l,nil()) -> l app(cons(x,l),k) -> cons(x,app(l,k)) app(nil(),k) -> k minus(x,0()) -> x minus(minus(x,y),z) -> minus(x,plus(y,z)) minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) sum(cons(x,nil())) -> cons(x,nil()) - Signature: {app/2,minus/2,plus/2,quot/2,sum/1} / {0/0,cons/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {app,minus,plus,quot,sum} and constructors {0,cons,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: app(l,nil()) -> l app(cons(x,l),k) -> cons(x,app(l,k)) app(nil(),k) -> k minus(x,0()) -> x minus(minus(x,y),z) -> minus(x,plus(y,z)) minus(s(x),s(y)) -> minus(x,y) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) sum(cons(x,nil())) -> cons(x,nil()) - Signature: {app/2,minus/2,plus/2,quot/2,sum/1} / {0/0,cons/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {app,minus,plus,quot,sum} and constructors {0,cons,nil,s} + 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),?)