YES(?,O(n^1)) * Step 1: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f1(A,B) -> f2(A,B) [A >= 1 && B >= 1] (1,1) 1. f2(A,B) -> f2(-1 + A,B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 2 && B >= 1] (?,1) 2. f2(A,B) -> f2(A,-1 + B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 1 && B >= 2] (?,1) Signature: {(f1,2);(f2,2)} 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(f1) = x2 p(f2) = x2 Following rules are strictly oriented: [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 1 && B >= 2] ==> f2(A,B) = B > -1 + B = f2(A,-1 + B) Following rules are weakly oriented: [A >= 1 && B >= 1] ==> f1(A,B) = B >= B = f2(A,B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 2 && B >= 1] ==> f2(A,B) = B >= B = f2(-1 + A,B) * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f1(A,B) -> f2(A,B) [A >= 1 && B >= 1] (1,1) 1. f2(A,B) -> f2(-1 + A,B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 2 && B >= 1] (?,1) 2. f2(A,B) -> f2(A,-1 + B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 1 && B >= 2] (B,1) Signature: {(f1,2);(f2,2)} 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(f1) = x1 p(f2) = x1 Following rules are strictly oriented: [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 2 && B >= 1] ==> f2(A,B) = A > -1 + A = f2(-1 + A,B) Following rules are weakly oriented: [A >= 1 && B >= 1] ==> f1(A,B) = A >= A = f2(A,B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 1 && B >= 2] ==> f2(A,B) = A >= A = f2(A,-1 + B) * Step 3: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f1(A,B) -> f2(A,B) [A >= 1 && B >= 1] (1,1) 1. f2(A,B) -> f2(-1 + A,B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 2 && B >= 1] (A,1) 2. f2(A,B) -> f2(A,-1 + B) [-1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= 1 && B >= 2] (B,1) Signature: {(f1,2);(f2,2)} Flow Graph: [0->{1,2},1->{1,2},2->{1,2}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))