* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: bitIter(x,y) -> if(zero(x),x,inc(y)) bits(x) -> bitIter(x,0()) half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),x,y) -> bitIter(half(x),y) if(true(),x,y) -> p(y) inc(0()) -> 0() inc(s(x)) -> s(inc(x)) p(0()) -> 0() p(s(x)) -> x zero(0()) -> true() zero(s(x)) -> false() - Signature: {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {bitIter,bits,half,if,inc,p,zero} and constructors {0 ,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: bitIter(x,y) -> if(zero(x),x,inc(y)) bits(x) -> bitIter(x,0()) half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),x,y) -> bitIter(half(x),y) if(true(),x,y) -> p(y) inc(0()) -> 0() inc(s(x)) -> s(inc(x)) p(0()) -> 0() p(s(x)) -> x zero(0()) -> true() zero(s(x)) -> false() - Signature: {bitIter/2,bits/1,half/1,if/3,inc/1,p/1,zero/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {bitIter,bits,half,if,inc,p,zero} and constructors {0 ,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: half(x){x -> s(s(x))} = half(s(s(x))) ->^+ s(half(x)) = C[half(x) = half(x){}] WORST_CASE(Omega(n^1),?)