* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ack(0(),x) -> plus(x,s(0())) ack(0(),x) -> s(x) ack(s(x),0()) -> ack(x,s(0())) ack(s(x),s(y)) -> ack(x,ack(s(x),y)) div(x,y) -> quot(x,y,0()) minus(x,0()) -> x minus(0(),y) -> 0() minus(minus(x,y),z) -> minus(x,plus(y,z)) minus(s(x),s(y)) -> minus(x,y) p(0()) -> 0() p(s(x)) -> x plus(0(),y) -> y plus(s(x),y) -> plus(x,s(y)) plus(s(x),y) -> s(plus(y,x)) quot(0(),s(y),z) -> z quot(s(x),s(y),z) -> quot(minus(p(ack(0(),x)),y),s(y),s(z)) zero(0()) -> true() zero(s(x)) -> false() - Signature: {ack/2,div/2,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 {ack,div,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: ack(0(),x) -> plus(x,s(0())) ack(0(),x) -> s(x) ack(s(x),0()) -> ack(x,s(0())) ack(s(x),s(y)) -> ack(x,ack(s(x),y)) div(x,y) -> quot(x,y,0()) minus(x,0()) -> x minus(0(),y) -> 0() minus(minus(x,y),z) -> minus(x,plus(y,z)) minus(s(x),s(y)) -> minus(x,y) p(0()) -> 0() p(s(x)) -> x plus(0(),y) -> y plus(s(x),y) -> plus(x,s(y)) plus(s(x),y) -> s(plus(y,x)) quot(0(),s(y),z) -> z quot(s(x),s(y),z) -> quot(minus(p(ack(0(),x)),y),s(y),s(z)) zero(0()) -> true() zero(s(x)) -> false() - Signature: {ack/2,div/2,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 {ack,div,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: ack(s(x),y){y -> s(y)} = ack(s(x),s(y)) ->^+ ack(x,ack(s(x),y)) = C[ack(s(x),y) = ack(s(x),y){}] WORST_CASE(Omega(n^1),?)