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