* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +(x,h()) -> x +(+(x,y),z) -> +(x,+(y,z)) +(h(),x) -> x +(s(x),s(y)) -> s(s(+(x,y))) a(l,x,s(y),h()) -> a(l,x,y,s(h())) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,s(x),h(),z) -> a(l,x,z,z) a(h(),h(),h(),x) -> s(x) a(s(l),h(),h(),z) -> a(l,z,h(),z) app(l,nil()) -> l app(cons(x,l),k) -> cons(x,app(l,k)) app(nil(),k) -> k s(h()) -> 1() sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h(),h()),l)) sum(cons(x,nil())) -> cons(x,nil()) - Signature: {+/2,a/4,app/2,s/1,sum/1} / {1/0,cons/2,h/0,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,a,app,s,sum} and constructors {1,cons,h,nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +(x,h()) -> x +(+(x,y),z) -> +(x,+(y,z)) +(h(),x) -> x +(s(x),s(y)) -> s(s(+(x,y))) a(l,x,s(y),h()) -> a(l,x,y,s(h())) a(l,x,s(y),s(z)) -> a(l,x,y,a(l,x,s(y),z)) a(l,s(x),h(),z) -> a(l,x,z,z) a(h(),h(),h(),x) -> s(x) a(s(l),h(),h(),z) -> a(l,z,h(),z) app(l,nil()) -> l app(cons(x,l),k) -> cons(x,app(l,k)) app(nil(),k) -> k s(h()) -> 1() sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h(),h()),l)) sum(cons(x,nil())) -> cons(x,nil()) - Signature: {+/2,a/4,app/2,s/1,sum/1} / {1/0,cons/2,h/0,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,a,app,s,sum} and constructors {1,cons,h,nil} + 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),?)