* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f() -> g() f() -> h() half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),b,x,y) -> logZeroError() if(true(),false(),x,s(y)) -> y if(true(),true(),x,y) -> logIter(x,y) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) logarithm(x) -> logIter(x,0()) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,half,if,inc,le,logIter,logarithm} and constructors {0 ,false,g,h,logZeroError,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f() -> g() f() -> h() half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(false(),b,x,y) -> logZeroError() if(true(),false(),x,s(y)) -> y if(true(),true(),x,y) -> logIter(x,y) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) logarithm(x) -> logIter(x,0()) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,half,if,inc,le,logIter,logarithm} and constructors {0 ,false,g,h,logZeroError,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),?)