* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) id_inc(x) -> x id_inc(x) -> s(x) if(false(),b,x,y,z) -> div_by_zero() if(true(),false(),x,y,z) -> z if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z)) minus(x,0()) -> x minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(x,y) quot(x,y) -> div(x,y,0()) - Signature: {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2} / {0/0,div_by_zero/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,id_inc,if,minus,quot} and constructors {0 ,div_by_zero,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,z) -> if(ge(y,s(0())),ge(x,y),x,y,z) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) id_inc(x) -> x id_inc(x) -> s(x) if(false(),b,x,y,z) -> div_by_zero() if(true(),false(),x,y,z) -> z if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z)) minus(x,0()) -> x minus(0(),y) -> 0() minus(s(x),s(y)) -> minus(x,y) quot(x,y) -> div(x,y,0()) - Signature: {div/3,ge/2,id_inc/1,if/5,minus/2,quot/2} / {0/0,div_by_zero/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,ge,id_inc,if,minus,quot} and constructors {0 ,div_by_zero,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ge(x,y){x -> s(x),y -> s(y)} = ge(s(x),s(y)) ->^+ ge(x,y) = C[ge(x,y) = ge(x,y){}] WORST_CASE(Omega(n^1),?)