* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: *(#(),x) -> #() *(*(x,y),z) -> *(x,*(y,z)) *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) +(x,#()) -> x +(#(),x) -> x +(+(x,y),z) -> +(x,+(y,z)) +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() prod(cons(x,l)) -> *(x,prod(l)) prod(nil()) -> 1(#()) sum(cons(x,l)) -> +(x,sum(l)) sum(nil()) -> 0(#()) - Signature: {*/2,+/2,0/1,prod/1,sum/1} / {#/0,1/1,cons/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,+,0,prod,sum} and constructors {#,1,cons,nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: *(#(),x) -> #() *(*(x,y),z) -> *(x,*(y,z)) *(0(x),y) -> 0(*(x,y)) *(1(x),y) -> +(0(*(x,y)),y) +(x,#()) -> x +(#(),x) -> x +(+(x,y),z) -> +(x,+(y,z)) +(0(x),0(y)) -> 0(+(x,y)) +(0(x),1(y)) -> 1(+(x,y)) +(1(x),0(y)) -> 1(+(x,y)) +(1(x),1(y)) -> 0(+(+(x,y),1(#()))) 0(#()) -> #() prod(cons(x,l)) -> *(x,prod(l)) prod(nil()) -> 1(#()) sum(cons(x,l)) -> +(x,sum(l)) sum(nil()) -> 0(#()) - Signature: {*/2,+/2,0/1,prod/1,sum/1} / {#/0,1/1,cons/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,+,0,prod,sum} and constructors {#,1,cons,nil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: *(x,y){x -> 1(x)} = *(1(x),y) ->^+ +(0(*(x,y)),y) = C[*(x,y) = *(x,y){}] WORST_CASE(Omega(n^1),?)