* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) head(cons(x,xs)) -> x head(nil()) -> error() ifProd(false(),xs,x) -> prodIter(tail(xs),times(x,head(xs))) ifProd(true(),xs,x) -> x ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u)) ifTimes(true(),x,y,z,u) -> z isempty(cons(x,xs)) -> false() isempty(nil()) -> true() plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) prod(xs) -> prodIter(xs,s(0())) prodIter(xs,x) -> ifProd(isempty(xs),xs,x) tail(cons(x,xs)) -> xs tail(nil()) -> nil() times(x,y) -> timesIter(x,y,0(),0()) timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u) - Signature: {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4} / {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,ge,head,ifProd,ifTimes,isempty,plus,prod,prodIter,tail ,times,timesIter} 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() ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) head(cons(x,xs)) -> x head(nil()) -> error() ifProd(false(),xs,x) -> prodIter(tail(xs),times(x,head(xs))) ifProd(true(),xs,x) -> x ifTimes(false(),x,y,z,u) -> timesIter(x,y,plus(y,z),s(u)) ifTimes(true(),x,y,z,u) -> z isempty(cons(x,xs)) -> false() isempty(nil()) -> true() plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) prod(xs) -> prodIter(xs,s(0())) prodIter(xs,x) -> ifProd(isempty(xs),xs,x) tail(cons(x,xs)) -> xs tail(nil()) -> nil() times(x,y) -> timesIter(x,y,0(),0()) timesIter(x,y,z,u) -> ifTimes(ge(u,x),x,y,z,u) - Signature: {a/0,ge/2,head/1,ifProd/3,ifTimes/5,isempty/1,plus/2,prod/1,prodIter/2,tail/1,times/2,timesIter/4} / {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,ge,head,ifProd,ifTimes,isempty,plus,prod,prodIter,tail ,times,timesIter} 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: ge(x,y){x -> s(x),y -> s(y)} = ge(s(x),s(y)) ->^+ ge(x,y) = C[ge(x,y) = ge(x,y){}] WORST_CASE(Omega(n^1),?)