* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f0(x,y,z) -> d() f0(0(),y,x) -> f1(x,y,x) f1(x,y,z) -> c() f1(x,y,z) -> f2(x,y,z) f2(x,1(),z) -> f0(x,z,z) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) ifPlus(false(),x,y) -> plus(p(x),y) ifPlus(true(),x,y) -> p(y) ifTimes(false(),i,x,y,z) -> timesIter(inc(i),x,y,plus(z,y)) ifTimes(true(),i,x,y,z) -> z inc(x) -> s(x) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isZero(0()) -> true() isZero(s(0())) -> false() isZero(s(s(x))) -> isZero(s(x)) p(0()) -> 0() p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) plus(x,y) -> ifPlus(isZero(x),x,inc(y)) times(x,y) -> timesIter(0(),x,y,0()) timesIter(i,x,y,z) -> ifTimes(ge(i,x),i,x,y,z) - Signature: {f0/3,f1/3,f2/3,ge/2,ifPlus/3,ifTimes/5,inc/1,isZero/1,p/1,plus/2,times/2,timesIter/4} / {0/0,1/0,c/0,d/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f0,f1,f2,ge,ifPlus,ifTimes,inc,isZero,p,plus,times ,timesIter} and constructors {0,1,c,d,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f0(x,y,z) -> d() f0(0(),y,x) -> f1(x,y,x) f1(x,y,z) -> c() f1(x,y,z) -> f2(x,y,z) f2(x,1(),z) -> f0(x,z,z) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) ifPlus(false(),x,y) -> plus(p(x),y) ifPlus(true(),x,y) -> p(y) ifTimes(false(),i,x,y,z) -> timesIter(inc(i),x,y,plus(z,y)) ifTimes(true(),i,x,y,z) -> z inc(x) -> s(x) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) isZero(0()) -> true() isZero(s(0())) -> false() isZero(s(s(x))) -> isZero(s(x)) p(0()) -> 0() p(s(x)) -> x p(s(s(x))) -> s(p(s(x))) plus(x,y) -> ifPlus(isZero(x),x,inc(y)) times(x,y) -> timesIter(0(),x,y,0()) timesIter(i,x,y,z) -> ifTimes(ge(i,x),i,x,y,z) - Signature: {f0/3,f1/3,f2/3,ge/2,ifPlus/3,ifTimes/5,inc/1,isZero/1,p/1,plus/2,times/2,timesIter/4} / {0/0,1/0,c/0,d/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f0,f1,f2,ge,ifPlus,ifTimes,inc,isZero,p,plus,times ,timesIter} and constructors {0,1,c,d,false,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),?)