YES(?,O(n^1)) * Step 1: UnsatRules WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] (?,1) 1. start(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,1 + D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] (?,1) 2. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,-1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] (?,1) 3. start(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] (?,1) 4. lbl71(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] (?,1) 5. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] (?,1) 6. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && A >= 1 && E = 1 && J = A && H = I && G = C] (?,1) 7. lbl81(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] (?,1) 8. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (?,1) 9. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (?,1) 10. start0(A,B,C,D,E,F,G,H,I,J) -> start(A,B,C,B,F,F,C,I,I,A) True (1,1) Signature: {(lbl71,10);(lbl81,10);(start,10);(start0,10);(stop,10)} Flow Graph: [0->{},1->{4,5,6},2->{},3->{7,8,9},4->{},5->{4,5,6},6->{7,8,9},7->{},8->{4,5,6},9->{7,8,9},10->{0,1,2,3}] + Applied Processor: UnsatRules + Details: Following transitions have unsatisfiable constraints and are removed: [6,8] * Step 2: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] (?,1) 1. start(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,1 + D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] (?,1) 2. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,-1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] (?,1) 3. start(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] (?,1) 4. lbl71(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] (?,1) 5. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] (?,1) 7. lbl81(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] (?,1) 9. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (?,1) 10. start0(A,B,C,D,E,F,G,H,I,J) -> start(A,B,C,B,F,F,C,I,I,A) True (1,1) Signature: {(lbl71,10);(lbl81,10);(start,10);(start0,10);(stop,10)} Flow Graph: [0->{},1->{4,5},2->{},3->{7,9},4->{},5->{4,5},7->{},9->{7,9},10->{0,1,2,3}] + 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. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] (1,1) 1. start(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,1 + D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] (1,1) 2. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,-1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 3. start(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 4. lbl71(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] (1,1) 5. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] (?,1) 7. lbl81(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] (1,1) 9. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (?,1) 10. start0(A,B,C,D,E,F,G,H,I,J) -> start(A,B,C,B,F,F,C,I,I,A) True (1,1) Signature: {(lbl71,10);(lbl81,10);(start,10);(start0,10);(stop,10)} Flow Graph: [0->{},1->{4,5},2->{},3->{7,9},4->{},5->{4,5},7->{},9->{7,9},10->{0,1,2,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl71) = 0 p(lbl81) = 1 + x3 + -1*x4 p(start) = -1*x2 + x3 p(start0) = -1*x2 + x3 p(stop) = x3 + -1*x4 Following rules are strictly oriented: [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] ==> lbl81(A,B,C,D,E,F,G,H,I,J) = 1 + C + -1*D > 1 + C + -1*D + E = lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) Following rules are weakly oriented: [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= 0 = lbl71(A,B,C,1 + D,1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = stop(A,B,C,D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] ==> lbl71(A,B,C,D,E,F,G,H,I,J) = 0 >= C + -1*D = stop(A,B,C,D,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] ==> lbl71(A,B,C,D,E,F,G,H,I,J) = 0 >= 0 = lbl71(A,B,C,D + E,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] ==> lbl81(A,B,C,D,E,F,G,H,I,J) = 1 + C + -1*D >= C + -1*D = stop(A,B,C,D,E,F,G,H,I,J) True ==> start0(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= -1*B + C = start(A,B,C,B,F,F,C,I,I,A) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] (1,1) 1. start(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,1 + D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] (1,1) 2. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,-1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 3. start(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 4. lbl71(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] (1,1) 5. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] (?,1) 7. lbl81(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] (1,1) 9. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (B + C,1) 10. start0(A,B,C,D,E,F,G,H,I,J) -> start(A,B,C,B,F,F,C,I,I,A) True (1,1) Signature: {(lbl71,10);(lbl81,10);(start,10);(start0,10);(stop,10)} Flow Graph: [0->{},1->{4,5},2->{},3->{7,9},4->{},5->{4,5},7->{},9->{7,9},10->{0,1,2,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl71) = 1 + x3 + -1*x4 p(lbl81) = 0 p(start) = -1*x2 + x3 p(start0) = -1*x2 + x3 p(stop) = x3 + -1*x4 Following rules are strictly oriented: [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] ==> lbl71(A,B,C,D,E,F,G,H,I,J) = 1 + C + -1*D > 1 + C + -1*D + -1*E = lbl71(A,B,C,D + E,E,F,G,H,I,J) Following rules are weakly oriented: [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = lbl71(A,B,C,1 + D,1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= C + -1*D = stop(A,B,C,D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] ==> start(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= 0 = lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] ==> lbl71(A,B,C,D,E,F,G,H,I,J) = 1 + C + -1*D >= C + -1*D = stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] ==> lbl81(A,B,C,D,E,F,G,H,I,J) = 0 >= C + -1*D = stop(A,B,C,D,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] ==> lbl81(A,B,C,D,E,F,G,H,I,J) = 0 >= 0 = lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) True ==> start0(A,B,C,D,E,F,G,H,I,J) = -1*B + C >= -1*B + C = start(A,B,C,B,F,F,C,I,I,A) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,1,F,G,H,I,J) [A >= 1 && B >= 1 + C && D = B && E = F && G = C && H = I && J = A] (1,1) 1. start(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,1 + D,1,F,G,H,I,J) [A >= 1 && C >= B && D = B && E = F && G = C && H = I && J = A] (1,1) 2. start(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,-1,F,G,H,I,J) [B >= 1 + C && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 3. start(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,1 + D,-1,F,G,H,I,J) [C >= B && 0 >= A && D = B && E = F && G = C && H = I && J = A] (1,1) 4. lbl71(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [C >= B && A >= 1 && D = 1 + C && E = 1 && J = A && H = I && G = C] (1,1) 5. lbl71(A,B,C,D,E,F,G,H,I,J) -> lbl71(A,B,C,D + E,E,F,G,H,I,J) [A >= 1 && C >= D && D >= 1 + B && 1 + C >= D && E = 1 && J = A && H = I && G = C] (B + C,1) 7. lbl81(A,B,C,D,E,F,G,H,I,J) -> stop(A,B,C,D,E,F,G,H,I,J) [0 >= A && C >= B && D = 1 + C && 1 + E = 0 && J = A && H = I && G = C] (1,1) 9. lbl81(A,B,C,D,E,F,G,H,I,J) -> lbl81(A,B,C,D + -1*E,E,F,G,H,I,J) [C >= D && 0 >= A && D >= 1 + B && 1 + C >= D && 1 + E = 0 && J = A && H = I && G = C] (B + C,1) 10. start0(A,B,C,D,E,F,G,H,I,J) -> start(A,B,C,B,F,F,C,I,I,A) True (1,1) Signature: {(lbl71,10);(lbl81,10);(start,10);(start0,10);(stop,10)} Flow Graph: [0->{},1->{4,5},2->{},3->{7,9},4->{},5->{4,5},7->{},9->{7,9},10->{0,1,2,3}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))