* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() head(cons(x,xs)) -> x head(nil()) -> error() ifPlus(false(),x,y,z) -> plusIter(x,s(y),s(z)) ifPlus(true(),x,y,z) -> y ifSum(false(),xs,x,y) -> sumIter(tail(xs),y) ifSum(true(),xs,x,y) -> x isempty(cons(x,xs)) -> false() isempty(nil()) -> true() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) plus(x,y) -> plusIter(x,y,0()) plusIter(x,y,z) -> ifPlus(le(x,z),x,y,z) sum(xs) -> sumIter(xs,0()) sumIter(xs,x) -> ifSum(isempty(xs),xs,x,plus(x,head(xs))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {a/0,head/1,ifPlus/4,ifSum/4,isempty/1,le/2,plus/2,plusIter/3,sum/1,sumIter/2,tail/1} / {0/0,b/0,c/0,cons/2 ,error/0,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,head,ifPlus,ifSum,isempty,le,plus,plusIter,sum,sumIter ,tail} and constructors {0,b,c,cons,error,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() head(cons(x,xs)) -> x head(nil()) -> error() ifPlus(false(),x,y,z) -> plusIter(x,s(y),s(z)) ifPlus(true(),x,y,z) -> y ifSum(false(),xs,x,y) -> sumIter(tail(xs),y) ifSum(true(),xs,x,y) -> x isempty(cons(x,xs)) -> false() isempty(nil()) -> true() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) plus(x,y) -> plusIter(x,y,0()) plusIter(x,y,z) -> ifPlus(le(x,z),x,y,z) sum(xs) -> sumIter(xs,0()) sumIter(xs,x) -> ifSum(isempty(xs),xs,x,plus(x,head(xs))) tail(cons(x,xs)) -> xs tail(nil()) -> nil() - Signature: {a/0,head/1,ifPlus/4,ifSum/4,isempty/1,le/2,plus/2,plusIter/3,sum/1,sumIter/2,tail/1} / {0/0,b/0,c/0,cons/2 ,error/0,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,head,ifPlus,ifSum,isempty,le,plus,plusIter,sum,sumIter ,tail} and constructors {0,b,c,cons,error,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] WORST_CASE(Omega(n^1),?)