* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: head(cons(x,xs)) -> x if(false(),false(),y,xs,ys,x) -> sumList(ys,x) if(false(),true(),y,xs,ys,x) -> sumList(xs,y) if(true(),b,y,xs,ys,x) -> y inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sumList(xs,0()) sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList ,tail} 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: head(cons(x,xs)) -> x if(false(),false(),y,xs,ys,x) -> sumList(ys,x) if(false(),true(),y,xs,ys,x) -> sumList(xs,y) if(true(),b,y,xs,ys,x) -> y inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isEmpty(cons(x,xs)) -> false() isEmpty(nil()) -> true() isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sumList(xs,0()) sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y)) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList ,tail} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: inc(x){x -> s(x)} = inc(s(x)) ->^+ s(inc(x)) = C[inc(x) = inc(x){}] WORST_CASE(Omega(n^1),?)