YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (?,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (?,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: 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) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl101) = 1 + -1*x2 + x5 + x6 p(lbl91) = -1*x2 + x5 p(start) = -1*x2 + x4 p(start0) = -1*x3 + x5 p(stop) = -1*x2 + x5 Following rules are strictly oriented: [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 1 + -1*B + E + F > -1*B + E = lbl101(A,1 + B + F,C,D,E,F) Following rules are weakly oriented: [A + -1*F >= 0 ==> && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + E = stop(A,B,C,D,E,F) [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 >= 0 && C >= 1 + E && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + E = stop(A,B,C,D,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + E = lbl91(A,B,C,-1 + D + -1*F,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + E = lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*B + E >= -1*B + E = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*B + E >= -1*B + E = lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*B + E >= -1*B + E = lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 1 + -1*B + E + F >= -1*B + E = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 1 + -1*B + E + F >= -1*B + E = lbl91(A,B,C,-1 + D + -1*F,E,F) True ==> start0(A,B,C,D,E,F) = -1*C + E >= -1*C + E = start(A,C,C,E,E,A) * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl101) = -1*x3 + x4 p(lbl91) = -1*x3 + x4 + x6 p(start) = -1*x2 + x4 p(start0) = -1*x3 + x5 p(stop) = -1*x2 + x4 Following rules are strictly oriented: [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = -1*C + D > -1 + -1*C + D = lbl91(A,B,C,-1 + D + -1*F,E,F) Following rules are weakly oriented: [A + -1*F >= 0 ==> && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + D = stop(A,B,C,D,E,F) [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 >= 0 && C >= 1 + E && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*B + D = stop(A,B,C,D,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1 + -1*C + D = lbl91(A,B,C,-1 + D + -1*F,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = -1*B + D >= -1*C + D = lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*C + D + F >= -1*B + D = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*C + D + F >= -1 + -1*C + D = lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = -1*C + D + F >= -1*C + D = lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = -1*C + D >= -1*B + D = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = -1*C + D >= -1*C + D = lbl101(A,1 + B + F,C,D,E,F) True ==> start0(A,B,C,D,E,F) = -1*C + E >= -1*C + E = start(A,C,C,E,E,A) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(lbl101) = 2 + -1*x2 + x4 + x6 p(lbl91) = 1 + -1*x2 + x4 + x6 p(start) = 1 + -1*x2 + x4 p(start0) = 1 + -1*x3 + x5 p(stop) = 1 + -1*x2 + x4 Following rules are strictly oriented: [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = 1 + -1*B + D + F > -1*B + D = lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 2 + -1*B + D + F > 1 + -1*B + D = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 2 + -1*B + D + F > -1*B + D = lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] lbl101(A,B,C,D,E,F) = 2 + -1*B + D + F > 1 + -1*B + D = lbl101(A,1 + B + F,C,D,E,F) Following rules are weakly oriented: [A + -1*F >= 0 ==> && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] start(A,B,C,D,E,F) = 1 + -1*B + D >= 1 + -1*B + D = stop(A,B,C,D,E,F) [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 >= 0 && C >= 1 + E && B = C && D = E && F = A] start(A,B,C,D,E,F) = 1 + -1*B + D >= 1 + -1*B + D = stop(A,B,C,D,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = 1 + -1*B + D >= -1*B + D = lbl91(A,B,C,-1 + D + -1*F,E,F) [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 >= 0 && E >= C && B = C && D = E && F = A] start(A,B,C,D,E,F) = 1 + -1*B + D >= 1 + -1*B + D = lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = 1 + -1*B + D + F >= 1 + -1*B + D = stop(A,B,C,D,E,F) [A + -1*F >= 0 ==> && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] lbl91(A,B,C,D,E,F) = 1 + -1*B + D + F >= 1 + -1*B + D = lbl101(A,1 + B + F,C,D,E,F) True ==> start0(A,B,C,D,E,F) = 1 + -1*C + E >= 1 + -1*C + E = start(A,C,C,E,E,A) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1 + C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (?,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= 1 + A && B = C && D = E && F = A] 1. start(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && C >= 1 + E && B = C && D = E && F = A] 2. start(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 3. start(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (1,1) && -1*A + F >= 0 && D + -1*E >= 0 && -1*D + E >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 0 && E >= C && B = C && D = E && F = A] 4. lbl91(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && B >= 1 + D && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 5. lbl91(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (1 + C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 6. lbl91(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (2 + 2*C + 2*E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1 + -1*D + E >= 0 && -1*C + E >= 0 && -1*B + E >= 0 && B + -1*C >= 0 && A >= 0 && D >= B && B >= C && 1 + A + D >= B && E >= 1 + A + D && F = A] 7. lbl101(A,B,C,D,E,F) -> stop(A,B,C,D,E,F) [A + -1*F >= 0 (1,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && B >= 1 + D && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 8. lbl101(A,B,C,D,E,F) -> lbl91(A,B,C,-1 + D + -1*F,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 9. lbl101(A,B,C,D,E,F) -> lbl101(A,1 + B + F,C,D,E,F) [A + -1*F >= 0 (C + E,1) && F >= 0 && A + F >= 0 && -1*A + F >= 0 && -1*D + E >= 0 && -1*C + E >= 0 && -1*C + D >= 0 && -1 + B + -1*C >= 0 && A >= 0 && D >= B && E >= D && B >= 1 + A + C && 1 + A + D >= B && F = A] 10. start0(A,B,C,D,E,F) -> start(A,C,C,E,E,A) True (1,1) Signature: {(lbl101,6);(lbl91,6);(start,6);(start0,6);(stop,6)} Flow Graph: [0->{},1->{},2->{4,5,6},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: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))