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