* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),0(),z) -> true() div(0(),s(x),z) -> false() div(s(x),0(),s(z)) -> div(s(x),s(z),s(z)) div(s(x),s(y),z) -> div(x,y,z) divides(x,y) -> div(x,y,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if1(false(),x,y) -> true() if1(true(),x,y) -> if2(divides(x,y),x,y) if2(false(),x,y) -> test(x,s(y)) if2(true(),x,y) -> false() prime(x) -> test(x,s(s(0()))) test(x,y) -> if1(gt(x,y),x,y) - Signature: {div/3,divides/2,gt/2,if1/3,if2/3,prime/1,test/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,divides,gt,if1,if2,prime,test} 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(0(),0(),z) -> true() div(0(),s(x),z) -> false() div(s(x),0(),s(z)) -> div(s(x),s(z),s(z)) div(s(x),s(y),z) -> div(x,y,z) divides(x,y) -> div(x,y,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if1(false(),x,y) -> true() if1(true(),x,y) -> if2(divides(x,y),x,y) if2(false(),x,y) -> test(x,s(y)) if2(true(),x,y) -> false() prime(x) -> test(x,s(s(0()))) test(x,y) -> if1(gt(x,y),x,y) - Signature: {div/3,divides/2,gt/2,if1/3,if2/3,prime/1,test/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,divides,gt,if1,if2,prime,test} and constructors {0 ,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: div(x,y,z){x -> s(x),y -> s(y)} = div(s(x),s(y),z) ->^+ div(x,y,z) = C[div(x,y,z) = div(x,y,z){}] WORST_CASE(Omega(n^1),?)