* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: eq(x,x) -> true() eq(0(),0()) -> true() eq(0(),s(x)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) f() -> g() f() -> h() ifPlus(false(),x,y,z) -> plus(x,z) ifPlus(true(),x,y,z) -> y ifTimes(false(),x,y,z,u) -> timesIter(x,y,u) ifTimes(true(),x,y,z,u) -> z inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) minus(x,x) -> 0() minus(x,0()) -> x minus(0(),x) -> 0() minus(s(x),s(y)) -> minus(x,y) plus(x,y) -> ifPlus(eq(x,0()),minus(x,s(0())),x,inc(x)) times(x,y) -> timesIter(x,y,0()) timesIter(x,y,z) -> ifTimes(eq(x,0()),minus(x,s(0())),y,z,plus(y,z)) - Signature: {eq/2,f/0,ifPlus/4,ifTimes/5,inc/1,minus/2,plus/2,times/2,timesIter/3} / {0/0,false/0,g/0,h/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,f,ifPlus,ifTimes,inc,minus,plus,times ,timesIter} and constructors {0,false,g,h,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: eq(x,x) -> true() eq(0(),0()) -> true() eq(0(),s(x)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) f() -> g() f() -> h() ifPlus(false(),x,y,z) -> plus(x,z) ifPlus(true(),x,y,z) -> y ifTimes(false(),x,y,z,u) -> timesIter(x,y,u) ifTimes(true(),x,y,z,u) -> z inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) minus(x,x) -> 0() minus(x,0()) -> x minus(0(),x) -> 0() minus(s(x),s(y)) -> minus(x,y) plus(x,y) -> ifPlus(eq(x,0()),minus(x,s(0())),x,inc(x)) times(x,y) -> timesIter(x,y,0()) timesIter(x,y,z) -> ifTimes(eq(x,0()),minus(x,s(0())),y,z,plus(y,z)) - Signature: {eq/2,f/0,ifPlus/4,ifTimes/5,inc/1,minus/2,plus/2,times/2,timesIter/3} / {0/0,false/0,g/0,h/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,f,ifPlus,ifTimes,inc,minus,plus,times ,timesIter} and constructors {0,false,g,h,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,y){x -> s(x),y -> s(y)} = eq(s(x),s(y)) ->^+ eq(x,y) = C[eq(x,y) = eq(x,y){}] WORST_CASE(Omega(n^1),?)