YES(?,O(1)) * Step 1: UnsatPaths WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f5(A,0,C) [A >= 128] (1,1) 1. f0(A,B,C) -> f7(A,0,D) [127 >= A] (1,1) 2. f7(A,B,C) -> f7(A,1 + B,1 + C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && 35 >= B] (?,1) 3. f7(A,B,C) -> f5(A,B,C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && B >= 36] (?,1) Signature: {(f0,3);(f5,3);(f7,3)} Flow Graph: [0->{},1->{2,3},2->{2,3},3->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,3)] * Step 2: TrivialSCCs WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f5(A,0,C) [A >= 128] (1,1) 1. f0(A,B,C) -> f7(A,0,D) [127 >= A] (1,1) 2. f7(A,B,C) -> f7(A,1 + B,1 + C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && 35 >= B] (?,1) 3. f7(A,B,C) -> f5(A,B,C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && B >= 36] (?,1) Signature: {(f0,3);(f5,3);(f7,3)} Flow Graph: [0->{},1->{2},2->{2,3},3->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 3: PolyRank WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f5(A,0,C) [A >= 128] (1,1) 1. f0(A,B,C) -> f7(A,0,D) [127 >= A] (1,1) 2. f7(A,B,C) -> f7(A,1 + B,1 + C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && 35 >= B] (?,1) 3. f7(A,B,C) -> f5(A,B,C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && B >= 36] (1,1) Signature: {(f0,3);(f5,3);(f7,3)} Flow Graph: [0->{},1->{2},2->{2,3},3->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f0) = 36 p(f5) = 36 + -1*x2 p(f7) = 36 + -1*x2 Following rules are strictly oriented: [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && 35 >= B] ==> f7(A,B,C) = 36 + -1*B > 35 + -1*B = f7(A,1 + B,1 + C) Following rules are weakly oriented: [A >= 128] ==> f0(A,B,C) = 36 >= 36 = f5(A,0,C) [127 >= A] ==> f0(A,B,C) = 36 >= 36 = f7(A,0,D) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && B >= 36] ==> f7(A,B,C) = 36 + -1*B >= 36 + -1*B = f5(A,B,C) * Step 4: KnowledgePropagation WORST_CASE(?,O(1)) + Considered Problem: Rules: 0. f0(A,B,C) -> f5(A,0,C) [A >= 128] (1,1) 1. f0(A,B,C) -> f7(A,0,D) [127 >= A] (1,1) 2. f7(A,B,C) -> f7(A,1 + B,1 + C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && 35 >= B] (36,1) 3. f7(A,B,C) -> f5(A,B,C) [B >= 0 && 127 + -1*A + B >= 0 && 127 + -1*A >= 0 && B >= 36] (1,1) Signature: {(f0,3);(f5,3);(f7,3)} Flow Graph: [0->{},1->{2},2->{2,3},3->{}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(1))