YES(?,O(n^1)) * Step 1: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f6(A,B,C) -> f4(A,B,1) [A >= 1] (1,1) 1. f4(A,B,C) -> f3(1 + -1*A,B,0) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] (?,1) 2. f3(A,B,C) -> f4(-1 + -1*A,B,1) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] (?,1) 3. f3(A,B,C) -> f7(0,D,C) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] (?,1) 4. f4(A,B,C) -> f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] (?,1) Signature: {(f3,3);(f4,3);(f6,3);(f7,3)} Flow Graph: [0->{1,4},1->{2,3},2->{1,4},3->{},4->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,4)] * Step 2: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f6(A,B,C) -> f4(A,B,1) [A >= 1] (1,1) 1. f4(A,B,C) -> f3(1 + -1*A,B,0) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] (?,1) 2. f3(A,B,C) -> f4(-1 + -1*A,B,1) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] (?,1) 3. f3(A,B,C) -> f7(0,D,C) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] (?,1) 4. f4(A,B,C) -> f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] (?,1) Signature: {(f3,3);(f4,3);(f6,3);(f7,3)} Flow Graph: [0->{1},1->{2,3},2->{1,4},3->{},4->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f6(A,B,C) -> f4(A,B,1) [A >= 1] (1,1) 1. f4(A,B,C) -> f3(1 + -1*A,B,0) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] (?,1) 2. f3(A,B,C) -> f4(-1 + -1*A,B,1) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] (?,1) 3. f3(A,B,C) -> f7(0,D,C) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] (1,1) 4. f4(A,B,C) -> f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] (1,1) Signature: {(f3,3);(f4,3);(f6,3);(f7,3)} Flow Graph: [0->{1},1->{2,3},2->{1,4},3->{},4->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f3) = 1 + -1*x1 p(f4) = x1 p(f6) = x1 p(f7) = 0 Following rules are strictly oriented: [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] ==> f3(A,B,C) = 1 + -1*A > -1 + -1*A = f4(-1 + -1*A,B,1) Following rules are weakly oriented: [A >= 1] ==> f6(A,B,C) = A >= A = f4(A,B,1) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] ==> f4(A,B,C) = A >= A = f3(1 + -1*A,B,0) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] ==> f3(A,B,C) = 1 + -1*A >= 0 = f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] ==> f4(A,B,C) = A >= 0 = f7(0,D,C) * Step 4: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f6(A,B,C) -> f4(A,B,1) [A >= 1] (1,1) 1. f4(A,B,C) -> f3(1 + -1*A,B,0) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] (?,1) 2. f3(A,B,C) -> f4(-1 + -1*A,B,1) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] (A,1) 3. f3(A,B,C) -> f7(0,D,C) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] (1,1) 4. f4(A,B,C) -> f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] (1,1) Signature: {(f3,3);(f4,3);(f6,3);(f7,3)} Flow Graph: [0->{1},1->{2,3},2->{1,4},3->{},4->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f6(A,B,C) -> f4(A,B,1) [A >= 1] (1,1) 1. f4(A,B,C) -> f3(1 + -1*A,B,0) [1 + -1*C >= 0 && -1 + C >= 0 && A >= 1 && C = 1] (1 + A,1) 2. f3(A,B,C) -> f4(-1 + -1*A,B,1) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && 0 >= 1 + A && 0 >= C] (A,1) 3. f3(A,B,C) -> f7(0,D,C) [-1*C >= 0 && -1*A + -1*C >= 0 && C >= 0 && -1*A + C >= 0 && -1*A >= 0 && A = 0] (1,1) 4. f4(A,B,C) -> f7(0,D,C) [1 + -1*C >= 0 && -1 + C >= 0 && A = 0] (1,1) Signature: {(f3,3);(f4,3);(f6,3);(f7,3)} Flow Graph: [0->{1},1->{2,3},2->{1,4},3->{},4->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))