* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) <=(0(),y) -> true() <=(s(x),0()) -> false() <=(s(x),s(y)) -> <=(x,y) f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,-(z,s(x)),u) f(s(x),s(y),z,u) -> if(<=(x,y),f(s(x),-(y,x),z,u),f(x,u,z,u)) if(false(),x,y) -> y if(true(),x,y) -> x perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {-/2,<=/2,f/4,if/3,perfectp/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,<=,f,if,perfectp} 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: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) <=(0(),y) -> true() <=(s(x),0()) -> false() <=(s(x),s(y)) -> <=(x,y) f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,-(z,s(x)),u) f(s(x),s(y),z,u) -> if(<=(x,y),f(s(x),-(y,x),z,u),f(x,u,z,u)) if(false(),x,y) -> y if(true(),x,y) -> x perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {-/2,<=/2,f/4,if/3,perfectp/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,<=,f,if,perfectp} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: -(x,y){x -> s(x),y -> s(y)} = -(s(x),s(y)) ->^+ -(x,y) = C[-(x,y) = -(x,y){}] WORST_CASE(Omega(n^1),?)