YES(?,O(n^1)) * Step 1: UnsatRules 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) [0 >= A && B = C && D = A && E = F && G = H] (?,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 11. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,0,F,-1 + G,H) [G >= 2 && 1 + E >= A && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (?,1) 13. 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->{5,6,7,8,9,10,11,12},2->{},3->{5,6,7,8,9,10,11,12},4->{},5->{5,6,7,8,9,10,11,12},6->{},7->{5,6,7 ,8,9,10,11,12},8->{},9->{5,6,7,8,9,10,11,12},10->{},11->{5,6,7,8,9,10,11,12},12->{},13->{0,1,2,3,4}] + Applied Processor: UnsatRules + Details: Following transitions have unsatisfiable constraints and are removed: [11] * Step 2: 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) [0 >= A && B = C && D = A && E = F && G = H] (?,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (?,1) 13. 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->{5,6,7,8,9,10,12},2->{},3->{5,6,7,8,9,10,12},4->{},5->{5,6,7,8,9,10,12},6->{},7->{5,6,7,8,9,10 ,12},8->{},9->{5,6,7,8,9,10,12},10->{},12->{},13->{0,1,2,3,4}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,5) ,(1,6) ,(1,12) ,(3,5) ,(3,6) ,(3,10) ,(5,12) ,(7,5) ,(7,6) ,(7,12)] * Step 3: 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) [0 >= A && B = C && D = A && E = F && G = H] (?,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (?,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (?,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (?,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (?,1) 13. 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->{7,8,9,12},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,12},10->{},12->{},13->{0,1,2,3,4}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * 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) [0 >= A && B = C && D = A && E = F && G = H] (1,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (1,1) 13. 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->{7,8,9,12},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,12},10->{},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) = x7 p(start) = x1 p(start0) = x1 p(stop) = x7 Following rules are strictly oriented: [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = cut(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = stop(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = cut(A,0,C,D,1,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = stop(A,0,C,D,0,F,-1 + D,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,B,C,D,0,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,E,C,D,0,F,-1 + G,H) Following rules are weakly oriented: [0 >= A && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A >= D = stop(A,B,C,D,0,F,D,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G >= -1 + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G >= -1 + G = cut(A,B,C,D,0,F,-1 + G,H) True ==> start0(A,B,C,D,E,F,G,H) = A >= 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) [0 >= A && B = C && D = A && E = F && G = H] (1,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (A,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (1,1) 13. 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->{7,8,9,12},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,12},10->{},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) = x7 p(start) = x1 p(start0) = x1 p(stop) = x7 Following rules are strictly oriented: [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = cut(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = stop(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = cut(A,0,C,D,1,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A > -1 + D = stop(A,0,C,D,0,F,-1 + D,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = cut(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,B,C,D,0,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] ==> cut(A,B,C,D,E,F,G,H) = G > -1 + G = stop(A,E,C,D,0,F,-1 + G,H) Following rules are weakly oriented: [0 >= A && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = A >= D = stop(A,B,C,D,0,F,D,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = G >= -1 + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) True ==> start0(A,B,C,D,E,F,G,H) = A >= A = start(A,C,C,A,F,F,H,H) * Step 6: 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) [0 >= A && B = C && D = A && E = F && G = H] (1,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (?,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (A,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (A,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (1,1) 13. 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->{7,8,9,12},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,12},10->{},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) = -5 + 2*x1 + x7 p(start) = -4 + 3*x1 p(start0) = -4 + 3*x1 p(stop) = -4 + 2*x1 + x7 Following rules are strictly oriented: [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G > -6 + 2*A + G = cut(A,B,C,D,-1 + E,F,-1 + G,H) Following rules are weakly oriented: [0 >= A && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = -4 + 3*A >= -4 + 2*A + D = stop(A,B,C,D,0,F,D,H) [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = -4 + 3*A >= -6 + 2*A + D = cut(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = -4 + 3*A >= -5 + 2*A + D = stop(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = -4 + 3*A >= -6 + 2*A + D = cut(A,0,C,D,1,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] ==> start(A,B,C,D,E,F,G,H) = -4 + 3*A >= -5 + 2*A + D = stop(A,0,C,D,0,F,-1 + D,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -5 + 2*A + G = stop(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -6 + 2*A + G = cut(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -5 + 2*A + G = stop(A,B,C,D,0,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -6 + 2*A + G = cut(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -5 + 2*A + G = stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] ==> cut(A,B,C,D,E,F,G,H) = -5 + 2*A + G >= -5 + 2*A + G = stop(A,E,C,D,0,F,-1 + G,H) True ==> start0(A,B,C,D,E,F,G,H) = -4 + 3*A >= -4 + 3*A = start(A,C,C,A,F,F,H,H) * Step 7: 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) [0 >= A && B = C && D = A && E = F && G = H] (1,1) 1. start(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 2. start(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 3. start(A,B,C,D,E,F,G,H) -> cut(A,0,C,D,1,F,-1 + D,H) [A >= 2 && B = C && D = A && E = F && G = H] (1,1) 4. start(A,B,C,D,E,F,G,H) -> stop(A,0,C,D,0,F,-1 + D,H) [D = 1 && B = C && A = 1 && E = F && G = H] (1,1) 5. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,-1 + E,F,-1 + G,H) [G >= 2 && E >= 2 && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (4 + 3*A,1) 6. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,-1 + E,F,-1 + G,H) [E >= 2 && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 7. cut(A,B,C,D,E,F,G,H) -> cut(A,B,C,D,0,F,-1 + G,H) [G >= 2 && 1 >= E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (A,1) 8. cut(A,B,C,D,E,F,G,H) -> stop(A,B,C,D,0,F,-1 + G,H) [1 >= E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 9. cut(A,B,C,D,E,F,G,H) -> cut(A,E,C,D,1 + E,F,-1 + G,H) [G >= 2 && A >= 2 + E && G >= 1 && E >= 0 && A >= 1 + G && A >= E + G && D = A] (A,1) 10. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,1 + E,F,-1 + G,H) [A >= 2 + E && E >= 0 && A >= 2 && A >= 1 + E && G = 1 && D = A] (1,1) 12. cut(A,B,C,D,E,F,G,H) -> stop(A,E,C,D,0,F,-1 + G,H) [A >= 1 && A >= 2 && G = 1 && 1 + E = A && D = A] (1,1) 13. 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->{7,8,9,12},4->{},5->{5,6,7,8,9,10},6->{},7->{7,8,9,10},8->{},9->{5,6,7,8,9 ,10,12},10->{},12->{},13->{0,1,2,3,4}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))