YES(?,O(1)) * Step 1: TrivialSCCs WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A) -> f5(0) True (1,1) 1. f5(A) -> f5(1 + A) [A >= 0 && 1 >= A] (?,1) 2. f5(A) -> f13(A) [A >= 0 && A >= 2 && 0 >= 1 + B] (?,1) 3. f5(A) -> f13(A) [A >= 0 && A >= 2] (?,1) Signature: {(f0,1);(f13,1);(f5,1)} Flow Graph: [0->{1,2,3},1->{1,2,3},2->{},3->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A) -> f5(0) True (1,1) 1. f5(A) -> f5(1 + A) [A >= 0 && 1 >= A] (?,1) 2. f5(A) -> f13(A) [A >= 0 && A >= 2 && 0 >= 1 + B] (1,1) 3. f5(A) -> f13(A) [A >= 0 && A >= 2] (1,1) Signature: {(f0,1);(f13,1);(f5,1)} Flow Graph: [0->{1,2,3},1->{1,2,3},2->{},3->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,2),(0,3)] * Step 3: PolyRank WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A) -> f5(0) True (1,1) 1. f5(A) -> f5(1 + A) [A >= 0 && 1 >= A] (?,1) 2. f5(A) -> f13(A) [A >= 0 && A >= 2 && 0 >= 1 + B] (1,1) 3. f5(A) -> f13(A) [A >= 0 && A >= 2] (1,1) Signature: {(f0,1);(f13,1);(f5,1)} Flow Graph: [0->{1},1->{1,2,3},2->{},3->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f0) = 2 p(f13) = 2 + -1*x1 p(f5) = 2 + -1*x1 Following rules are strictly oriented: [A >= 0 && 1 >= A] ==> f5(A) = 2 + -1*A > 1 + -1*A = f5(1 + A) Following rules are weakly oriented: True ==> f0(A) = 2 >= 2 = f5(0) [A >= 0 && A >= 2 && 0 >= 1 + B] ==> f5(A) = 2 + -1*A >= 2 + -1*A = f13(A) [A >= 0 && A >= 2] ==> f5(A) = 2 + -1*A >= 2 + -1*A = f13(A) * Step 4: KnowledgePropagation WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A) -> f5(0) True (1,1) 1. f5(A) -> f5(1 + A) [A >= 0 && 1 >= A] (2,1) 2. f5(A) -> f13(A) [A >= 0 && A >= 2 && 0 >= 1 + B] (1,1) 3. f5(A) -> f13(A) [A >= 0 && A >= 2] (1,1) Signature: {(f0,1);(f13,1);(f5,1)} Flow Graph: [0->{1},1->{1,2,3},2->{},3->{}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(1))