YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (?,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = G && H = A] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (?,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && G >= 1 + E && B = C && D = E && F = G && H = A] 2. start(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [A + -1*H >= 0 (?,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= G && 100 >= A && B = C && D = E && F = G && H = A] 3. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (?,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && A + E + G >= 102 + B + D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 4. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (?,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && B >= D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 5. lbl71(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [E + -1*H >= 0 (?,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && D >= 1 + B && 101 + B + D >= A + E + G && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,G,G,A) True (1,1) Signature: {(lbl71,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{},2->{3,4,5},3->{},4->{},5->{3,4,5},6->{0,1,2}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = G && H = A] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && G >= 1 + E && B = C && D = E && F = G && H = A] 2. start(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= G && 100 >= A && B = C && D = E && F = G && H = A] 3. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (1,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && A + E + G >= 102 + B + D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 4. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (1,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && B >= D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 5. lbl71(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [E + -1*H >= 0 (?,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && D >= 1 + B && 101 + B + D >= A + E + G && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,G,G,A) True (1,1) Signature: {(lbl71,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{},2->{3,4,5},3->{},4->{},5->{3,4,5},6->{0,1,2}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl71) = 1 + -1*x1 + x4 + x5 + -1*x7 p(start) = -1*x1 + x4 + x5 + -1*x7 p(start0) = -1*x1 + 2*x5 + -1*x7 p(stop) = -1*x1 + x4 + x5 + -1*x7 Following rules are strictly oriented: [E + -1*H >= 0 ==> && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && A + E + G >= 102 + B + D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] lbl71(A,B,C,D,E,F,G,H) = 1 + -1*A + D + E + -1*G > -1*A + D + E + -1*G = stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 ==> && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && D >= 1 + B && 101 + B + D >= A + E + G && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] lbl71(A,B,C,D,E,F,G,H) = 1 + -1*A + D + E + -1*G > -1*A + D + E + -1*G = lbl71(A,H,C,-1 + D,E,1 + H,G,F) Following rules are weakly oriented: [A + -1*H >= 0 ==> && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = G && H = A] start(A,B,C,D,E,F,G,H) = -1*A + D + E + -1*G >= -1*A + D + E + -1*G = stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 ==> && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && G >= 1 + E && B = C && D = E && F = G && H = A] start(A,B,C,D,E,F,G,H) = -1*A + D + E + -1*G >= -1*A + D + E + -1*G = stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 ==> && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= G && 100 >= A && B = C && D = E && F = G && H = A] start(A,B,C,D,E,F,G,H) = -1*A + D + E + -1*G >= -1*A + D + E + -1*G = lbl71(A,H,C,-1 + D,E,1 + H,G,F) [E + -1*H >= 0 ==> && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && B >= D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] lbl71(A,B,C,D,E,F,G,H) = 1 + -1*A + D + E + -1*G >= -1*A + D + E + -1*G = stop(A,B,C,D,E,F,G,H) True ==> start0(A,B,C,D,E,F,G,H) = -1*A + 2*E + -1*G >= -1*A + 2*E + -1*G = start(A,C,C,E,E,G,G,A) * Step 3: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = G && H = A] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && G >= 1 + E && B = C && D = E && F = G && H = A] 2. start(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [A + -1*H >= 0 (1,1) && -1*A + H >= 0 && F + -1*G >= 0 && -1*F + G >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= G && 100 >= A && B = C && D = E && F = G && H = A] 3. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (1,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && A + E + G >= 102 + B + D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 4. lbl71(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [E + -1*H >= 0 (1,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && B >= D && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 5. lbl71(A,B,C,D,E,F,G,H) -> lbl71(A,H,C,-1 + D,E,1 + H,G,F) [E + -1*H >= 0 (A + 2*E + G,1) && 1 + D + -1*H >= 0 && E + -1*G >= 0 && 1 + B + -1*F >= 0 && -1 + -1*B + F >= 0 && -1 + -1*D + E >= 0 && 100 + -1*A >= 0 && D >= 1 + B && 101 + B + D >= A + E + G && E >= 1 + D && 100 >= A && E >= G && 2 + B + 2*D >= A + E + G && 100 >= B && F = 1 + B && 1 + B + D + H = A + E + G] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,G,G,A) True (1,1) Signature: {(lbl71,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{},2->{3,4,5},3->{},4->{},5->{3,4,5},6->{0,1,2}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))