* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),x,y) -> log2(half(x),y) if(true(),x,s(y)) -> y inc(0()) -> 0() inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(x) -> log2(x,0()) log2(x,y) -> if(le(x,s(0())),x,inc(y)) - Signature: {half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {half,if,inc,le,log,log2} 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: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),x,y) -> log2(half(x),y) if(true(),x,s(y)) -> y inc(0()) -> 0() inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(x) -> log2(x,0()) log2(x,y) -> if(le(x,s(0())),x,inc(y)) - Signature: {half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {half,if,inc,le,log,log2} 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),?)