YES(?,O(n^1)) * Step 1: ArgumentFilter WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [A >= 2] (1,1) 1. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f9(A,B,0,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [A >= 1 + B] (?,1) 2. f9(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f9(A,B,C,1 + D,C,S,S,H,I,J,K,L,M,N,O,P,Q,R) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f9(A,B,S,1 + D,C,S,S,H,I,J,K,L,M,N,O,P,Q,R) [S >= C && A >= D] (?,1) 4. f26(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f26(A,B,C,1 + D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [A >= D] (?,1) 5. f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f32(A,B,C,1 + D,E,F,G,0,0,J,K,L,M,N,O,P,Q,R) [A >= D] (?,1) 6. f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f32(A,B,C,1 + D,E,F,G,S,T,J + T,K,L,M,N,O,P,Q,R) [0 >= 1 + S && A >= D] (?,1) 7. f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f32(A,B,C,1 + D,E,F,G,S,T,J + T,K,L,M,N,O,P,Q,R) [S >= 1 && A >= D] (?,1) 8. f52(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f55(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [A >= K] (?,1) 9. f55(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f55(A,B,C,1 + D,E,F,G,H,I,S,K,L,M,N,O,P,Q,R) [A >= D] (?,1) 10. f62(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f62(A,B,C,1 + D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [A >= D] (?,1) 11. f62(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f52(A,B,C,D,E,F,G,H,I,J,1 + K,L,M,N,O,P,Q,R) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f62(A,B,C,D,E,F,G,H,I,J,K,S,M,N,O,P,Q,R) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f5(A,1 + B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f52(A,B,C,D,E,F,G,H,I,J,K,L,S,T,T,P,Q,R) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f52(A,B,C,D,E,F,G,H,I,J,K,L,M,N,-1*S,T,S,R) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f32(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f5(A,1 + B,0,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,0) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f26(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f26(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [B >= A && 0 >= 1 + S] (?,1) 21. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [B >= A && S >= 1] (?,1) 22. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) -> f1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R) [B >= A] (?,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17,18,19},2->{2,3,17,18,19},3->{2,3,17,18,19},4->{4,16},5->{5,6,7,14,15},6->{5,6 ,7,14,15},7->{5,6,7,14,15},8->{9,12},9->{9,12},10->{10,11},11->{8,13},12->{10,11},13->{1,20,21,22},14->{8 ,13},15->{8,13},16->{5,6,7,14,15},17->{1,20,21,22},18->{4,16},19->{4,16},20->{},21->{},22->{}] + Applied Processor: ArgumentFilter [4,5,6,7,8,9,11,12,13,14,15,16,17] + Details: We remove following argument positions: [4,5,6,7,8,9,11,12,13,14,15,16,17]. * Step 2: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (?,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 4. f26(A,B,C,D,K) -> f26(A,B,C,1 + D,K) [A >= D] (?,1) 5. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [A >= D] (?,1) 6. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [0 >= 1 + S && A >= D] (?,1) 7. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [S >= 1 && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (?,1) 10. f62(A,B,C,D,K) -> f62(A,B,C,1 + D,K) [A >= D] (?,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (?,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (?,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (?,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17,18,19},2->{2,3,17,18,19},3->{2,3,17,18,19},4->{4,16},5->{5,6,7,14,15},6->{5,6 ,7,14,15},7->{5,6,7,14,15},8->{9,12},9->{9,12},10->{10,11},11->{8,13},12->{10,11},13->{1,20,21,22},14->{8 ,13},15->{8,13},16->{5,6,7,14,15},17->{1,20,21,22},18->{4,16},19->{4,16},20->{},21->{},22->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(1,18) ,(1,19) ,(12,10) ,(16,5) ,(16,6) ,(16,7) ,(18,4) ,(19,4)] * Step 3: UnreachableRules WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (?,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 4. f26(A,B,C,D,K) -> f26(A,B,C,1 + D,K) [A >= D] (?,1) 5. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [A >= D] (?,1) 6. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [0 >= 1 + S && A >= D] (?,1) 7. f32(A,B,C,D,K) -> f32(A,B,C,1 + D,K) [S >= 1 && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (?,1) 10. f62(A,B,C,D,K) -> f62(A,B,C,1 + D,K) [A >= D] (?,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (?,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (?,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (?,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},4->{4,16},5->{5,6,7,14,15},6->{5,6,7,14 ,15},7->{5,6,7,14,15},8->{9,12},9->{9,12},10->{10,11},11->{8,13},12->{11},13->{1,20,21,22},14->{8,13},15->{8 ,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: UnreachableRules + Details: Following transitions are not reachable from the starting states and are revomed: [4,5,6,7,10] * Step 4: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (?,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (?,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (?,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (?,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (?,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (?,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (?,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x2 + x18 p(f2) = x1 + -1*x2 + x18 p(f26) = x1 + -1*x2 p(f32) = x1 + -1*x2 p(f5) = x1 + -1*x2 + x18 p(f52) = x1 + -1*x2 p(f55) = x1 + -1*x2 p(f62) = x1 + -1*x2 p(f9) = x1 + -1*x2 Following rules are strictly oriented: [A >= 1 + B] ==> f5(A,B,C,D,K) = 1 + A + -1*B > A + -1*B = f9(A,B,0,D,K) Following rules are weakly oriented: [A >= 2] ==> f2(A,B,C,D,K) = 1 + A + -1*B >= 1 + A + -1*B = f5(A,B,C,D,K) [C >= 1 + S && A >= D] ==> f9(A,B,C,D,K) = A + -1*B >= A + -1*B = f9(A,B,C,1 + D,K) [S >= C && A >= D] ==> f9(A,B,C,D,K) = A + -1*B >= A + -1*B = f9(A,B,S,1 + D,K) [A >= K] ==> f52(A,B,C,D,K) = A + -1*B >= A + -1*B = f55(A,B,C,D,K) [A >= D] ==> f55(A,B,C,D,K) = A + -1*B >= A + -1*B = f55(A,B,C,1 + D,K) [D >= 1 + A] ==> f62(A,B,C,D,K) = A + -1*B >= A + -1*B = f52(A,B,C,D,1 + K) [D >= 1 + A] ==> f55(A,B,C,D,K) = A + -1*B >= A + -1*B = f62(A,B,C,D,K) [K >= 1 + A] ==> f52(A,B,C,D,K) = A + -1*B >= A + -1*B = f5(A,1 + B,C,D,K) [U >= 0 && D >= 1 + A] ==> f32(A,B,C,D,K) = A + -1*B >= A + -1*B = f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] ==> f32(A,B,C,D,K) = A + -1*B >= A + -1*B = f52(A,B,C,D,K) [D >= 1 + A] ==> f26(A,B,C,D,K) = A + -1*B >= A + -1*B = f32(A,B,C,D,K) [D >= 1 + A && C = 0] ==> f9(A,B,C,D,K) = A + -1*B >= A + -1*B = f5(A,1 + B,0,D,K) [0 >= 1 + C && D >= 1 + A] ==> f9(A,B,C,D,K) = A + -1*B >= A + -1*B = f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] ==> f9(A,B,C,D,K) = A + -1*B >= A + -1*B = f26(A,B,C,D,K) [B >= A && 0 >= 1 + S] ==> f5(A,B,C,D,K) = 1 + A + -1*B >= 1 + A + -1*B = f1(A,B,C,D,K) [B >= A && S >= 1] ==> f5(A,B,C,D,K) = 1 + A + -1*B >= 1 + A + -1*B = f1(A,B,C,D,K) [B >= A] ==> f5(A,B,C,D,K) = 1 + A + -1*B >= 1 + A + -1*B = f1(A,B,C,D,K) * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (?,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x4 + x18 p(f2) = x1 + -1*x4 + x18 p(f26) = x1 + -1*x4 + x18 p(f32) = x1 + -1*x4 + x18 p(f5) = x1 + -1*x4 + x18 p(f52) = x1 + -1*x4 + x18 p(f55) = x1 + -1*x4 + x18 p(f62) = x1 + -1*x4 + x18 p(f9) = x1 + -1*x4 + x18 Following rules are strictly oriented: [A >= D] ==> f55(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f55(A,B,C,1 + D,K) Following rules are weakly oriented: [A >= 2] ==> f2(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,B,C,D,K) [A >= 1 + B] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f9(A,B,0,D,K) [C >= 1 + S && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= A + -1*D = f9(A,B,C,1 + D,K) [S >= C && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= A + -1*D = f9(A,B,S,1 + D,K) [A >= K] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f55(A,B,C,D,K) [D >= 1 + A] ==> f62(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,1 + K) [D >= 1 + A] ==> f55(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f62(A,B,C,D,K) [K >= 1 + A] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,C,D,K) [U >= 0 && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [D >= 1 + A] ==> f26(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f32(A,B,C,D,K) [D >= 1 + A && C = 0] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,0,D,K) [0 >= 1 + C && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [B >= A && 0 >= 1 + S] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A && S >= 1] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) * Step 7: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (?,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x5 + x18 p(f2) = x1 + -1*x5 + x18 p(f26) = x1 + -1*x5 + x18 p(f32) = x1 + -1*x5 + x18 p(f5) = x1 + -1*x5 + x18 p(f52) = x1 + -1*x5 + x18 p(f55) = x1 + -1*x5 p(f62) = x1 + -1*x5 p(f9) = x1 + -1*x5 + x18 Following rules are strictly oriented: [A >= K] ==> f52(A,B,C,D,K) = 1 + A + -1*K > A + -1*K = f55(A,B,C,D,K) Following rules are weakly oriented: [A >= 2] ==> f2(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f5(A,B,C,D,K) [A >= 1 + B] ==> f5(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f9(A,B,0,D,K) [C >= 1 + S && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f9(A,B,C,1 + D,K) [S >= C && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f9(A,B,S,1 + D,K) [A >= D] ==> f55(A,B,C,D,K) = A + -1*K >= A + -1*K = f55(A,B,C,1 + D,K) [D >= 1 + A] ==> f62(A,B,C,D,K) = A + -1*K >= A + -1*K = f52(A,B,C,D,1 + K) [D >= 1 + A] ==> f55(A,B,C,D,K) = A + -1*K >= A + -1*K = f62(A,B,C,D,K) [K >= 1 + A] ==> f52(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f5(A,1 + B,C,D,K) [U >= 0 && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f52(A,B,C,D,K) [D >= 1 + A] ==> f26(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f32(A,B,C,D,K) [D >= 1 + A && C = 0] ==> f9(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f5(A,1 + B,0,D,K) [0 >= 1 + C && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f26(A,B,C,D,K) [B >= A && 0 >= 1 + S] ==> f5(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f1(A,B,C,D,K) [B >= A && S >= 1] ==> f5(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f1(A,B,C,D,K) [B >= A] ==> f5(A,B,C,D,K) = 1 + A + -1*K >= 1 + A + -1*K = f1(A,B,C,D,K) * Step 8: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (1 + A + K,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (?,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (?,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 9: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (?,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (1 + A + K,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (2 + 2*A + D + K,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (2 + 2*A + D + K,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x4 + x18 p(f2) = x1 + -1*x4 + x18 p(f26) = x1 + -1*x4 + x18 p(f32) = x1 + -1*x4 + x18 p(f5) = x1 + -1*x4 + x18 p(f52) = x1 + -1*x4 + x18 p(f55) = x1 + -1*x4 + x18 p(f62) = x1 + -1*x4 + x18 p(f9) = x1 + -1*x4 + x18 Following rules are strictly oriented: [S >= C && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f9(A,B,S,1 + D,K) [A >= D] ==> f55(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f55(A,B,C,1 + D,K) Following rules are weakly oriented: [A >= 2] ==> f2(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,B,C,D,K) [A >= 1 + B] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f9(A,B,0,D,K) [C >= 1 + S && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= A + -1*D = f9(A,B,C,1 + D,K) [A >= K] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f55(A,B,C,D,K) [D >= 1 + A] ==> f62(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,1 + K) [D >= 1 + A] ==> f55(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f62(A,B,C,D,K) [K >= 1 + A] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,C,D,K) [U >= 0 && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [D >= 1 + A] ==> f26(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f32(A,B,C,D,K) [D >= 1 + A && C = 0] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,0,D,K) [0 >= 1 + C && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [B >= A && 0 >= 1 + S] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A && S >= 1] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) * Step 10: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (?,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (1 + A + D,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (1 + A + K,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (2 + 2*A + D + K,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (2 + 2*A + D + K,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x4 + x18 p(f2) = x1 + -1*x4 + x18 p(f26) = x1 + -1*x4 + x18 p(f32) = x1 + -1*x4 + x18 p(f5) = x1 + -1*x4 + x18 p(f52) = x1 + -1*x4 + x18 p(f55) = x1 + -1*x4 + x18 p(f62) = x1 + -1*x4 + x18 p(f9) = x1 + -1*x4 + x18 Following rules are strictly oriented: [C >= 1 + S && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f9(A,B,C,1 + D,K) [S >= C && A >= D] ==> f9(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f9(A,B,S,1 + D,K) [A >= D] ==> f55(A,B,C,D,K) = 1 + A + -1*D > A + -1*D = f55(A,B,C,1 + D,K) Following rules are weakly oriented: [A >= 2] ==> f2(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,B,C,D,K) [A >= 1 + B] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f9(A,B,0,D,K) [A >= K] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f55(A,B,C,D,K) [D >= 1 + A] ==> f62(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,1 + K) [D >= 1 + A] ==> f55(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f62(A,B,C,D,K) [K >= 1 + A] ==> f52(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,C,D,K) [U >= 0 && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] ==> f32(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f52(A,B,C,D,K) [D >= 1 + A] ==> f26(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f32(A,B,C,D,K) [D >= 1 + A && C = 0] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f5(A,1 + B,0,D,K) [0 >= 1 + C && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] ==> f9(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f26(A,B,C,D,K) [B >= A && 0 >= 1 + S] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A && S >= 1] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) [B >= A] ==> f5(A,B,C,D,K) = 1 + A + -1*D >= 1 + A + -1*D = f1(A,B,C,D,K) * Step 11: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (1 + A + D,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (1 + A + D,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (1 + A + K,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (2 + 2*A + D + K,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (2 + 2*A + D + K,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (?,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (?,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (?,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (?,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (?,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (?,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (?,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 12: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,K) -> f5(A,B,C,D,K) [A >= 2] (1,1) 1. f5(A,B,C,D,K) -> f9(A,B,0,D,K) [A >= 1 + B] (1 + A + B,1) 2. f9(A,B,C,D,K) -> f9(A,B,C,1 + D,K) [C >= 1 + S && A >= D] (1 + A + D,1) 3. f9(A,B,C,D,K) -> f9(A,B,S,1 + D,K) [S >= C && A >= D] (1 + A + D,1) 8. f52(A,B,C,D,K) -> f55(A,B,C,D,K) [A >= K] (1 + A + K,1) 9. f55(A,B,C,D,K) -> f55(A,B,C,1 + D,K) [A >= D] (1 + A + D,1) 11. f62(A,B,C,D,K) -> f52(A,B,C,D,1 + K) [D >= 1 + A] (2 + 2*A + D + K,1) 12. f55(A,B,C,D,K) -> f62(A,B,C,D,K) [D >= 1 + A] (2 + 2*A + D + K,1) 13. f52(A,B,C,D,K) -> f5(A,1 + B,C,D,K) [K >= 1 + A] (10 + 10*A + 9*D + K,1) 14. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [U >= 0 && D >= 1 + A] (4 + 4*A + 4*D,1) 15. f32(A,B,C,D,K) -> f52(A,B,C,D,K) [0 >= 1 + U && D >= 1 + A] (4 + 4*A + 4*D,1) 16. f26(A,B,C,D,K) -> f32(A,B,C,D,K) [D >= 1 + A] (4 + 4*A + 4*D,1) 17. f9(A,B,C,D,K) -> f5(A,1 + B,0,D,K) [D >= 1 + A && C = 0] (3 + 3*A + B + 2*D,1) 18. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [0 >= 1 + C && D >= 1 + A] (2 + 2*A + 2*D,1) 19. f9(A,B,C,D,K) -> f26(A,B,C,D,K) [C >= 1 && D >= 1 + A] (2 + 2*A + 2*D,1) 20. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && 0 >= 1 + S] (1,1) 21. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A && S >= 1] (1,1) 22. f5(A,B,C,D,K) -> f1(A,B,C,D,K) [B >= A] (1,1) Signature: {(f1,18);(f2,18);(f26,18);(f32,18);(f5,18);(f52,18);(f55,18);(f62,18);(f9,18)} Flow Graph: [0->{1,20,21,22},1->{2,3,17},2->{2,3,17,18,19},3->{2,3,17,18,19},8->{9,12},9->{9,12},11->{8,13},12->{11} ,13->{1,20,21,22},14->{8,13},15->{8,13},16->{14,15},17->{1,20,21,22},18->{16},19->{16},20->{},21->{},22->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))