YES(?,O(n^1)) * Step 1: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C) -> eval(A,B,C) True (1,1) 1. eval(A,B,C) -> eval(A,B,1 + C) [A >= 1 + B && B >= C] (?,1) 2. eval(A,B,C) -> eval(A,1 + B,C) [A >= 1 + B && C >= 1 + B] (?,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x1 + -1*x2 p(start) = x1 + -1*x2 Following rules are strictly oriented: [A >= 1 + B && C >= 1 + B] ==> eval(A,B,C) = A + -1*B > -1 + A + -1*B = eval(A,1 + B,C) Following rules are weakly oriented: True ==> start(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,C) [A >= 1 + B && B >= C] ==> eval(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,1 + C) * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C) -> eval(A,B,C) True (1,1) 1. eval(A,B,C) -> eval(A,B,1 + C) [A >= 1 + B && B >= C] (?,1) 2. eval(A,B,C) -> eval(A,1 + B,C) [A >= 1 + B && C >= 1 + B] (A + B,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x1 + -1*x3 p(start) = 1 + x1 + -1*x3 Following rules are strictly oriented: [A >= 1 + B && B >= C] ==> eval(A,B,C) = A + -1*C > -1 + A + -1*C = eval(A,B,1 + C) Following rules are weakly oriented: True ==> start(A,B,C) = 1 + A + -1*C >= A + -1*C = eval(A,B,C) [A >= 1 + B && C >= 1 + B] ==> eval(A,B,C) = A + -1*C >= A + -1*C = eval(A,1 + B,C) * Step 3: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C) -> eval(A,B,C) True (1,1) 1. eval(A,B,C) -> eval(A,B,1 + C) [A >= 1 + B && B >= C] (1 + A + C,1) 2. eval(A,B,C) -> eval(A,1 + B,C) [A >= 1 + B && C >= 1 + B] (A + B,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))