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) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = A && G = H] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= 1 + C && B = C && D = E && F = A && G = H] 2. start(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && C >= E && 100 >= A && B = C && D = E && F = A && G = H] 3. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (?,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && F >= 101 && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 4. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (?,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && A + C + E >= 1 + 2*B + F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 5. lbl72(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [100 + -1*G >= 0 (?,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && 2*B + F >= A + C + E && 100 >= F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,A,H,H) True (1,1) Signature: {(lbl72,8);(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) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = A && G = H] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= 1 + C && B = C && D = E && F = A && G = H] 2. start(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && C >= E && 100 >= A && B = C && D = E && F = A && G = H] 3. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (1,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && F >= 101 && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 4. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (1,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && A + C + E >= 1 + 2*B + F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 5. lbl72(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [100 + -1*G >= 0 (?,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && 2*B + F >= A + C + E && 100 >= F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,A,H,H) True (1,1) Signature: {(lbl72,8);(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(lbl72) = 202 + -1*x1 + x2 + x3 + -1*x5 + -1*x6 + -1*x7 p(start) = 201 + -2*x1 + 2*x2 + -2*x5 p(start0) = 201 + -2*x1 + 2*x3 + -2*x5 p(stop) = 201 + -2*x1 + 2*x2 + -2*x5 Following rules are strictly oriented: [100 + -1*G >= 0 ==> && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && F >= 101 && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] lbl72(A,B,C,D,E,F,G,H) = 202 + -1*A + B + C + -1*E + -1*F + -1*G > 201 + -2*A + 2*B + -2*E = stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 ==> && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && A + C + E >= 1 + 2*B + F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] lbl72(A,B,C,D,E,F,G,H) = 202 + -1*A + B + C + -1*E + -1*F + -1*G > 201 + -2*A + 2*B + -2*E = stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 ==> && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && 2*B + F >= A + C + E && 100 >= F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] lbl72(A,B,C,D,E,F,G,H) = 202 + -1*A + B + C + -1*E + -1*F + -1*G > 201 + -1*A + B + C + -1*D + -1*E + -1*F = lbl72(A,-1 + B,C,1 + F,E,D,F,H) Following rules are weakly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = A && G = H] start(A,B,C,D,E,F,G,H) = 201 + -2*A + 2*B + -2*E >= 201 + -2*A + 2*B + -2*E = stop(A,B,C,D,E,F,G,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= 1 + C && B = C && D = E && F = A && G = H] start(A,B,C,D,E,F,G,H) = 201 + -2*A + 2*B + -2*E >= 201 + -2*A + 2*B + -2*E = stop(A,B,C,D,E,F,G,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && C >= E && 100 >= A && B = C && D = E && F = A && G = H] start(A,B,C,D,E,F,G,H) = 201 + -2*A + 2*B + -2*E >= 201 + -1*A + B + C + -1*D + -1*E + -1*F = lbl72(A,-1 + B,C,1 + F,E,D,F,H) True ==> start0(A,B,C,D,E,F,G,H) = 201 + -2*A + 2*C + -2*E >= 201 + -2*A + 2*C + -2*E = start(A,C,C,E,E,A,H,H) * 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) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 101 && B = C && D = E && F = A && G = H] 1. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && E >= 1 + C && B = C && D = E && F = A && G = H] 2. start(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && A + -1*F >= 0 && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && C >= E && 100 >= A && B = C && D = E && F = A && G = H] 3. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (1,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && F >= 101 && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 4. lbl72(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,E,F,G,H) [100 + -1*G >= 0 (1,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && A + C + E >= 1 + 2*B + F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 5. lbl72(A,B,C,D,E,F,G,H) -> lbl72(A,-1 + B,C,1 + F,E,D,F,H) [100 + -1*G >= 0 (201 + 2*A + 2*C + 2*E,1) && -1 + D + -1*G >= 0 && 201 + -1*D + -1*G >= 0 && 200 + -1*A + -1*G >= 0 && 1 + -1*D + G >= 0 && C + -1*E >= 0 && 101 + -1*D >= 0 && 201 + -1*A + -1*D >= 0 && -1 + -1*B + C >= 0 && 100 + -1*A >= 0 && 2*B + F >= A + C + E && 100 >= F && 100 >= A && 101 + B + F >= A + C + E && 1 + B >= F && C >= 1 + B && C >= E && 1 + B + F + G = A + C + E && B + D + F = A + C + E] 6. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,E,E,A,H,H) True (1,1) Signature: {(lbl72,8);(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))