* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ack(0(),x) -> plus(x,s(0())) ack(s(x),0()) -> ack(x,s(0())) ack(s(x),s(y)) -> ack(x,ack(s(x),y)) double(x) -> permute(x,x,a()) isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(x)) -> x permute(x,y,a()) -> permute(isZero(x),x,b()) permute(y,x,c()) -> s(s(permute(x,y,a()))) permute(false(),x,b()) -> permute(ack(x,x),p(x),c()) permute(true(),x,b()) -> 0() plus(x,0()) -> x plus(x,s(0())) -> s(x) plus(x,s(s(y))) -> s(plus(s(x),y)) plus(0(),y) -> y plus(s(x),y) -> plus(x,s(y)) - Signature: {ack/2,double/1,isZero/1,p/1,permute/3,plus/2} / {0/0,a/0,b/0,c/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ack,double,isZero,p,permute,plus} and constructors {0,a,b ,c,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(s(x),0()) -> ack(x,s(0())) ack(s(x),s(y)) -> ack(x,ack(s(x),y)) double(x) -> permute(x,x,a()) isZero(0()) -> true() isZero(s(x)) -> false() p(0()) -> 0() p(s(x)) -> x permute(x,y,a()) -> permute(isZero(x),x,b()) permute(y,x,c()) -> s(s(permute(x,y,a()))) permute(false(),x,b()) -> permute(ack(x,x),p(x),c()) permute(true(),x,b()) -> 0() plus(x,0()) -> x plus(x,s(0())) -> s(x) plus(x,s(s(y))) -> s(plus(s(x),y)) plus(0(),y) -> y plus(s(x),y) -> plus(x,s(y)) - Signature: {ack/2,double/1,isZero/1,p/1,permute/3,plus/2} / {0/0,a/0,b/0,c/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ack,double,isZero,p,permute,plus} and constructors {0,a,b ,c,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),?)