YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B) -> sumto(A,B) True (1,1) 1. sumto(A,B) -> sumto(1 + A,B) [B >= A] (?,1) 2. sumto(A,B) -> end(A,B) [A >= 1 + B] (?,1) Signature: {(end,2);(start,2);(sumto,2)} Flow Graph: [0->{1,2},1->{1,2},2->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B) -> sumto(A,B) True (1,1) 1. sumto(A,B) -> sumto(1 + A,B) [B >= A] (?,1) 2. sumto(A,B) -> end(A,B) [A >= 1 + B] (1,1) Signature: {(end,2);(start,2);(sumto,2)} Flow Graph: [0->{1,2},1->{1,2},2->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(end) = 1 + -1*x1 + x2 p(start) = 1 + -1*x1 + x2 p(sumto) = 1 + -1*x1 + x2 Following rules are strictly oriented: [B >= A] ==> sumto(A,B) = 1 + -1*A + B > -1*A + B = sumto(1 + A,B) Following rules are weakly oriented: True ==> start(A,B) = 1 + -1*A + B >= 1 + -1*A + B = sumto(A,B) [A >= 1 + B] ==> sumto(A,B) = 1 + -1*A + B >= 1 + -1*A + B = end(A,B) * Step 3: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B) -> sumto(A,B) True (1,1) 1. sumto(A,B) -> sumto(1 + A,B) [B >= A] (1 + A + B,1) 2. sumto(A,B) -> end(A,B) [A >= 1 + B] (1,1) Signature: {(end,2);(start,2);(sumto,2)} Flow Graph: [0->{1,2},1->{1,2},2->{}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))