YES(?,O(n^1)) * Step 1: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{5,6,7,8,9,10,11},2->{},3->{5,6,7,8,9,10,11},4->{},5->{5,6,7,8,9,10,11},6->{},7->{5,6,7,8,9,10 ,11},8->{},9->{5,6,7,8,9,10,11},10->{},11->{},12->{0,1,2,3,4}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,5) ,(1,6) ,(1,11) ,(3,5) ,(3,6) ,(3,10) ,(5,11) ,(7,5) ,(7,6) ,(7,11)] * Step 2: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (?,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{7,8,9,10},2->{},3->{7,8,9,11},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,11},10->{},11->{},12->{0,1,2,3,4}] + 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) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{7,8,9,10},2->{},3->{7,8,9,11},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,11},10->{},11->{},12->{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 + 3*x1 + x7 p(start) = -2 + 3*x1 + x4 p(start0) = 4*x1 p(stop) = -2 + 3*x1 + -1*x4 + 2*x7 Following rules are strictly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = -2 + 3*A + D > -4 + 3*A + D = stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = -2 + 3*A + D > -4 + 3*A + D = stop(A,0,C,D,0,F,-1 + D,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G > -4 + 3*A + -1*D + 2*G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G > -4 + 3*A + -1*D + 2*G = stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G > -2 + 3*A + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G > -4 + 3*A + -1*D + 2*G = stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G > -4 + 3*A + -1*D + 2*G = stop(A,E,C,D,0,F,-1 + G,H) Following rules are weakly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = -2 + 3*A + D >= -2 + 3*A + D = stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = -2 + 3*A + D >= -2 + 3*A + D = cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = -2 + 3*A + D >= -2 + 3*A + D = cut(A,0,C,D,1,F,-1 + D,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G >= -2 + 3*A + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = -1 + 3*A + G >= -2 + 3*A + G = cut(A,B,C,D,0,F,-1 + G,H) True ==> start0(A,B,C,D,E,F,G,H) = 4*A >= -2 + 4*A = start(A,C,C,A,F,F,H,H) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (4*A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{7,8,9,10},2->{},3->{7,8,9,11},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,11},10->{},11->{},12->{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 + x4 + x7 p(start) = 2*x4 p(start0) = 2*x1 p(stop) = x4 + x7 Following rules are strictly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = 2*D > -1 + 2*D = stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = 2*D > -1 + 2*D = stop(A,0,C,D,0,F,-1 + D,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > -1 + D + G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > D + G = cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > -1 + D + G = stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > D + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > -1 + D + G = stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G > -1 + D + G = stop(A,E,C,D,0,F,-1 + G,H) Following rules are weakly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = 2*D >= 2*D = stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = 2*D >= 2*D = cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = 2*D >= 2*D = cut(A,0,C,D,1,F,-1 + D,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + D + G >= D + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) True ==> start0(A,B,C,D,E,F,G,H) = 2*A >= 2*A = start(A,C,C,A,F,F,H,H) * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (?,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (2*A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (4*A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{7,8,9,10},2->{},3->{7,8,9,11},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,11},10->{},11->{},12->{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 + x1 + -1*x4 + x7 p(start) = x1 p(start0) = x1 p(stop) = x1 + -1*x4 + x7 Following rules are strictly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = A > -1 + A = stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] start(A,B,C,D,E,F,G,H) = A > -1 + A = stop(A,0,C,D,0,F,-1 + D,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > A + -1*D + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > -1 + A + -1*D + G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > A + -1*D + G = cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > -1 + A + -1*D + G = stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > A + -1*D + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > -1 + A + -1*D + G = stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 ==> && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] cut(A,B,C,D,E,F,G,H) = 1 + A + -1*D + G > -1 + A + -1*D + G = stop(A,E,C,D,0,F,-1 + G,H) Following rules are weakly oriented: [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = A >= A = stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = A >= A = cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 ==> && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] start(A,B,C,D,E,F,G,H) = A >= A = cut(A,0,C,D,1,F,-1 + D,H) True ==> start0(A,B,C,D,E,F,G,H) = A >= A = start(A,C,C,A,F,F,H,H) * Step 6: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && 0 >= A && B = C && D = A && E = F && G = H] 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && A >= 2 && B = C && D = A && E = F && G = H] 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [G + -1*H >= 0 (1,1) && -1*G + H >= 0 && E + -1*F >= 0 && -1*E + F >= 0 && A + -1*D >= 0 && -1*A + D >= 0 && B + -1*C >= 0 && -1*B + C >= 0 && D = 1 && B = C && A = 1 && E = F && G = H] 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && E >= 2 && G >= 1 && A >= 1 + G && A >= E + G && D = A] 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && E >= 2 && A >= 2 && A >= 1 + E && G = 1 && D = A] 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (2*A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && 1 >= E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && 1 >= E && A >= 2 && A >= 1 + E && G = 1 && D = A] 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (4*A,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && G >= 2 && A >= 2 + E && G >= 1 && A >= 1 + G && A >= E + G && D = A] 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 2 + E && A >= 2 && A >= 1 + E && G = 1 && D = A] 11. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [-1 + D + -1*G >= 0 (1,1) && -1 + A + -1*G >= 0 && -1 + G >= 0 && -1 + E + G >= 0 && -3 + D + G >= 0 && -3 + A + G >= 0 && -1 + D + -1*E >= 0 && -1 + A + -1*E >= 0 && E >= 0 && -2 + D + E >= 0 && -2 + A + E >= 0 && A + -1*D >= 0 && -2 + D >= 0 && -4 + A + D >= 0 && -1*A + D >= 0 && -2 + A >= 0 && A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] 12. start0(A,B,C,D,E,F,G,H) -> start(A,C,C,A,F,F,H,H) True (1,1) Signature: {(cut,8);(start,8);(start0,8);(stop,8)} Flow Graph: [0->{},1->{7,8,9,10},2->{},3->{7,8,9,11},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,11},10->{},11->{},12->{0,1,2,3,4}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))