YES(?,O(n^1)) * Step 1: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (?,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (?,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (?,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (?,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (?,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (?,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (?,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + 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) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (1,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (1,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (1,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (1,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (1,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (1,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (1,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (1,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (1,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(cut) = x2 p(lbl101) = x4 p(lbl111) = x2 + x4 p(lbl6) = x4 p(start) = x1 p(start0) = x1 p(stop) = x4 Following rules are strictly oriented: [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] lbl111(A,B,C,D) = B + D > B = cut(A,B,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B + D > D = lbl111(A,B,C,-1*B + D) Following rules are weakly oriented: [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] ==> start(A,B,C,D) = A >= D = stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] ==> start(A,B,C,D) = A >= D = lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] ==> start(A,B,C,D) = A >= B = cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] ==> start(A,B,C,D) = A >= D = lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] ==> start(A,B,C,D) = A >= D = lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 ==> && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] lbl6(A,B,C,D) = D >= D = stop(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] lbl101(A,B,C,D) = D >= B = cut(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = D >= D = lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = D >= D = lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B + D >= D = lbl101(A,B + -1*D,C,D) [C + -1*D >= 0 ==> && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] cut(A,B,C,D) = B >= D = stop(A,B,C,D) True ==> start0(A,B,C,D) = A >= A = start(A,C,C,A) * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (1,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (1,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (1,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (1,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (1,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (1,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (1,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (1,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (?,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (A,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (1,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(cut) = -1 + x2 p(lbl101) = -1 + x4 p(lbl111) = x2 p(lbl6) = x2 p(start) = x2 p(start0) = x3 p(stop) = -1 + x2 Following rules are strictly oriented: [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] lbl111(A,B,C,D) = B > -1 + B = cut(A,B,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B > -1 + D = lbl101(A,B + -1*D,C,D) Following rules are weakly oriented: [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] ==> start(A,B,C,D) = B >= -1 + B = stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] ==> start(A,B,C,D) = B >= B = lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] ==> start(A,B,C,D) = B >= -1 + B = cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] ==> start(A,B,C,D) = B >= -1 + D = lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] ==> start(A,B,C,D) = B >= B = lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 ==> && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] lbl6(A,B,C,D) = B >= -1 + B = stop(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] lbl101(A,B,C,D) = -1 + D >= -1 + B = cut(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = -1 + D >= -1 + D = lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = -1 + D >= B = lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B >= B = lbl111(A,B,C,-1*B + D) [C + -1*D >= 0 ==> && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] cut(A,B,C,D) = -1 + B >= -1 + B = stop(A,B,C,D) True ==> start0(A,B,C,D) = C >= C = start(A,C,C,A) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (1,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (1,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (1,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (1,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (1,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (1,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (1,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (1,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (C,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (A,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (1,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(cut) = x1 + x2 p(lbl101) = x1 + x4 p(lbl111) = -1 + x1 + x2 + x4 p(lbl6) = x1 + x4 p(start) = 2*x1 p(start0) = 2*x1 p(stop) = x1 + x4 Following rules are strictly oriented: [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] ==> start(A,B,C,D) = 2*A > -1 + A + D = lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = A + D > -1 + A + D = lbl111(A,B,C,-1*B + D) Following rules are weakly oriented: [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] ==> start(A,B,C,D) = 2*A >= A + D = stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] ==> start(A,B,C,D) = 2*A >= A + D = lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] ==> start(A,B,C,D) = 2*A >= A + B = cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] ==> start(A,B,C,D) = 2*A >= A + D = lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 ==> && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] lbl6(A,B,C,D) = A + D >= A + D = stop(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] lbl101(A,B,C,D) = A + D >= A + B = cut(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = A + D >= A + D = lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] lbl111(A,B,C,D) = -1 + A + B + D >= A + B = cut(A,B,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = -1 + A + B + D >= A + D = lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = -1 + A + B + D >= -1 + A + D = lbl111(A,B,C,-1*B + D) [C + -1*D >= 0 ==> && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] cut(A,B,C,D) = A + B >= A + D = stop(A,B,C,D) True ==> start0(A,B,C,D) = 2*A >= 2*A = start(A,C,C,A) * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (1,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (1,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (1,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (1,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (1,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (1,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (1,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (?,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (2*A,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (1,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (C,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (A,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (1,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(cut) = x2 p(lbl101) = x2 + x4 p(lbl111) = x2 p(lbl6) = x2 p(start) = x2 p(start0) = x3 p(stop) = x2 Following rules are strictly oriented: [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] lbl101(A,B,C,D) = B + D > B = cut(A,B,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = B + D > B = lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 ==> && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] lbl101(A,B,C,D) = B + D > B = lbl111(A,B,C,-1*B + D) Following rules are weakly oriented: [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] ==> start(A,B,C,D) = B >= B = stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] ==> start(A,B,C,D) = B >= B = lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] ==> start(A,B,C,D) = B >= B = cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] ==> start(A,B,C,D) = B >= B = lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] ==> start(A,B,C,D) = B >= B = lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 ==> && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] lbl6(A,B,C,D) = B >= B = stop(A,B,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] lbl111(A,B,C,D) = B >= B = cut(A,B,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B >= B = lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 ==> && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] lbl111(A,B,C,D) = B >= B = lbl111(A,B,C,-1*B + D) [C + -1*D >= 0 ==> && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] cut(A,B,C,D) = B >= B = stop(A,B,C,D) True ==> start0(A,B,C,D) = C >= C = start(A,C,C,A) * Step 6: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A] (1,1) 1. start(A,B,C,D) -> lbl6(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && 0 >= C && B = C && D = A] (1,1) 2. start(A,B,C,D) -> cut(A,B,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && D = A && B = A && C = A] (1,1) 3. start(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 && C >= 1 + A && B = C && D = A] (1,1) 4. start(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 1 + C && C >= 1 && B = C && D = A] (1,1) 5. lbl6(A,B,C,D) -> stop(A,B,C,D) [A + -1*D >= 0 (1,1) && -1 + D >= 0 && -1 + -1*C + D >= 0 && -1 + -1*B + D >= 0 && -2 + A + D >= 0 && -1*A + D >= 0 && -1*C >= 0 && B + -1*C >= 0 && -1*B + -1*C >= 0 && -1 + A + -1*C >= 0 && -1*B + C >= 0 && -1*B >= 0 && -1 + A + -1*B >= 0 && -1 + A >= 0 && A >= 1 && 0 >= C && D = A && B = C] 6. lbl101(A,B,C,D) -> cut(A,B,C,D) [-1 + C + -1*D >= 0 (1,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= 2*B && D = B] 7. lbl101(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + C + -1*D >= 0 (C,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && B >= 1 + D && A >= D && B >= 1 && D >= 1 && C >= B + D] 8. lbl101(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + C + -1*D >= 0 (2*A,1) && A + -1*D >= 0 && -1 + D >= 0 && -3 + C + D >= 0 && -2 + A + D >= 0 && -2 + C >= 0 && -1 + -1*B + C >= 0 && -3 + A + C >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && D >= 1 + B && A >= D && B >= 1 && D >= 1 && C >= B + D] 9. lbl111(A,B,C,D) -> cut(A,B,C,D) [-1 + A + -1*D >= 0 (1,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && C >= B && B >= 1 && A >= 2*B && D = B] 10. lbl111(A,B,C,D) -> lbl101(A,B + -1*D,C,D) [-1 + A + -1*D >= 0 (C,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && B >= 1 + D && C >= B && B >= 1 && D >= 1 && A >= B + D] 11. lbl111(A,B,C,D) -> lbl111(A,B,C,-1*B + D) [-1 + A + -1*D >= 0 (A,1) && -2 + C + D >= 0 && -2 + B + D >= 0 && -3 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -3 + A + C >= 0 && -1 + A + -1*B >= 0 && -1 + B >= 0 && -3 + A + B >= 0 && -2 + A >= 0 && D >= 1 + B && C >= B && B >= 1 && D >= 1 && A >= B + D] 12. cut(A,B,C,D) -> stop(A,B,C,D) [C + -1*D >= 0 (1,1) && B + -1*D >= 0 && A + -1*D >= 0 && -1 + D >= 0 && -2 + C + D >= 0 && -2 + B + D >= 0 && -1*B + D >= 0 && -2 + A + D >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1*B + C >= 0 && -2 + A + C >= 0 && A + -1*B >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1 + A >= 0 && A >= B && B >= 1 && C >= B && D = B] 13. start0(A,B,C,D) -> start(A,C,C,A) True (1,1) Signature: {(cut,4);(lbl101,4);(lbl111,4);(lbl6,4);(start,4);(start0,4);(stop,4)} Flow Graph: [0->{},1->{5},2->{12},3->{6,7,8},4->{9,10,11},5->{},6->{12},7->{6,7,8},8->{9,10,11},9->{12},10->{6,7,8} ,11->{9,10,11},12->{},13->{0,1,2,3,4}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))