YES * Step 1: UnsatPaths YES + Considered Problem: Rules: 0. f2(A,B,C,D,E,F,G,H) -> f8(A,B,0,B,E,F,G,H) [A >= 1 + B] (?,1) 1. f8(A,B,C,D,E,F,G,H) -> f8(A,B,C,D,1 + E,J,I,H) [A >= E && I >= J] (?,1) 2. f8(A,B,C,D,E,F,G,H) -> f8(A,B,J,E,1 + E,I,K,H) [A >= E && I >= 1 + K] (?,1) 3. f19(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,1 + E,F,G,J) [A >= E] (?,1) 4. f27(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,1 + E,F,G,J) [A >= E] (?,1) 5. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [0 >= 1 + C] (?,1) 6. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [C >= 1] (?,1) 7. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [0 >= 1 + I && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 8. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [I >= 1 && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 9. f43(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,1 + E,F,G,H) [A >= E] (?,1) 10. f49(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,1 + E,F,G,H) [A >= E] (?,1) 11. f36(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,0) [A >= D] (?,1) 12. f34(A,B,C,D,E,F,G,H) -> f2(A,1 + B,0,D,E,F,G,H) [C = 0] (?,1) 13. f49(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,H) [E >= 1 + A] (?,1) 14. f43(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 15. f36(A,B,C,D,E,F,G,H) -> f2(A,1 + B,C,D,E,F,G,H) [D >= 1 + A] (?,1) 16. f27(A,B,C,D,E,F,G,H) -> f34(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 17. f19(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 18. f8(A,B,C,D,E,F,G,H) -> f34(A,B,C,B,E,F,G,H) [E >= 1 + A && B = D] (?,1) 19. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [B >= 1 + D && E >= 1 + A] (?,1) 20. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [D >= 1 + B && E >= 1 + A] (?,1) 21. f2(A,B,C,D,E,F,G,H) -> f1(A,B,C,D,E,F,G,H) [B >= A] (?,1) 22. start(A,B,C,D,E,F,G,H) -> f2(A,B,C,D,E,F,G,H) True (1,1) Signature: {(f1,8);(f19,8);(f2,8);(f27,8);(f34,8);(f36,8);(f43,8);(f49,8);(f8,8);(start,8)} Flow Graph: [0->{1,2,18,19,20},1->{1,2,18,19,20},2->{1,2,18,19,20},3->{3,17},4->{4,16},5->{7,8,11,15},6->{7,8,11,15} ,7->{9,14},8->{9,14},9->{9,14},10->{10,13},11->{7,8,11,15},12->{0,21},13->{7,8,11,15},14->{10,13},15->{0,21} ,16->{5,6,12},17->{4,16},18->{5,6,12},19->{3,17},20->{3,17},21->{},22->{0,21}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,19),(0,20),(14,10),(17,4),(19,3),(20,3)] * Step 2: UnreachableRules YES + Considered Problem: Rules: 0. f2(A,B,C,D,E,F,G,H) -> f8(A,B,0,B,E,F,G,H) [A >= 1 + B] (?,1) 1. f8(A,B,C,D,E,F,G,H) -> f8(A,B,C,D,1 + E,J,I,H) [A >= E && I >= J] (?,1) 2. f8(A,B,C,D,E,F,G,H) -> f8(A,B,J,E,1 + E,I,K,H) [A >= E && I >= 1 + K] (?,1) 3. f19(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,1 + E,F,G,J) [A >= E] (?,1) 4. f27(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,1 + E,F,G,J) [A >= E] (?,1) 5. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [0 >= 1 + C] (?,1) 6. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [C >= 1] (?,1) 7. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [0 >= 1 + I && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 8. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [I >= 1 && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 9. f43(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,1 + E,F,G,H) [A >= E] (?,1) 10. f49(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,1 + E,F,G,H) [A >= E] (?,1) 11. f36(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,0) [A >= D] (?,1) 12. f34(A,B,C,D,E,F,G,H) -> f2(A,1 + B,0,D,E,F,G,H) [C = 0] (?,1) 13. f49(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,H) [E >= 1 + A] (?,1) 14. f43(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 15. f36(A,B,C,D,E,F,G,H) -> f2(A,1 + B,C,D,E,F,G,H) [D >= 1 + A] (?,1) 16. f27(A,B,C,D,E,F,G,H) -> f34(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 17. f19(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 18. f8(A,B,C,D,E,F,G,H) -> f34(A,B,C,B,E,F,G,H) [E >= 1 + A && B = D] (?,1) 19. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [B >= 1 + D && E >= 1 + A] (?,1) 20. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [D >= 1 + B && E >= 1 + A] (?,1) 21. f2(A,B,C,D,E,F,G,H) -> f1(A,B,C,D,E,F,G,H) [B >= A] (?,1) 22. start(A,B,C,D,E,F,G,H) -> f2(A,B,C,D,E,F,G,H) True (1,1) Signature: {(f1,8);(f19,8);(f2,8);(f27,8);(f34,8);(f36,8);(f43,8);(f49,8);(f8,8);(start,8)} Flow Graph: [0->{1,2,18},1->{1,2,18,19,20},2->{1,2,18,19,20},3->{3,17},4->{4,16},5->{7,8,11,15},6->{7,8,11,15},7->{9 ,14},8->{9,14},9->{9,14},10->{10,13},11->{7,8,11,15},12->{0,21},13->{7,8,11,15},14->{13},15->{0,21},16->{5,6 ,12},17->{16},18->{5,6,12},19->{17},20->{17},21->{},22->{0,21}] + Applied Processor: UnreachableRules + Details: Following transitions are not reachable from the starting states and are revomed: [3,4,10] * Step 3: FromIts YES + Considered Problem: Rules: 0. f2(A,B,C,D,E,F,G,H) -> f8(A,B,0,B,E,F,G,H) [A >= 1 + B] (?,1) 1. f8(A,B,C,D,E,F,G,H) -> f8(A,B,C,D,1 + E,J,I,H) [A >= E && I >= J] (?,1) 2. f8(A,B,C,D,E,F,G,H) -> f8(A,B,J,E,1 + E,I,K,H) [A >= E && I >= 1 + K] (?,1) 5. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [0 >= 1 + C] (?,1) 6. f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [C >= 1] (?,1) 7. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [0 >= 1 + I && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 8. f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [I >= 1 && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] (?,1) 9. f43(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,1 + E,F,G,H) [A >= E] (?,1) 11. f36(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,0) [A >= D] (?,1) 12. f34(A,B,C,D,E,F,G,H) -> f2(A,1 + B,0,D,E,F,G,H) [C = 0] (?,1) 13. f49(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,H) [E >= 1 + A] (?,1) 14. f43(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 15. f36(A,B,C,D,E,F,G,H) -> f2(A,1 + B,C,D,E,F,G,H) [D >= 1 + A] (?,1) 16. f27(A,B,C,D,E,F,G,H) -> f34(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 17. f19(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,E,F,G,H) [E >= 1 + A] (?,1) 18. f8(A,B,C,D,E,F,G,H) -> f34(A,B,C,B,E,F,G,H) [E >= 1 + A && B = D] (?,1) 19. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [B >= 1 + D && E >= 1 + A] (?,1) 20. f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [D >= 1 + B && E >= 1 + A] (?,1) 21. f2(A,B,C,D,E,F,G,H) -> f1(A,B,C,D,E,F,G,H) [B >= A] (?,1) 22. start(A,B,C,D,E,F,G,H) -> f2(A,B,C,D,E,F,G,H) True (1,1) Signature: {(f1,8);(f19,8);(f2,8);(f27,8);(f34,8);(f36,8);(f43,8);(f49,8);(f8,8);(start,8)} Flow Graph: [0->{1,2,18},1->{1,2,18,19,20},2->{1,2,18,19,20},5->{7,8,11,15},6->{7,8,11,15},7->{9,14},8->{9,14},9->{9 ,14},11->{7,8,11,15},12->{0,21},13->{7,8,11,15},14->{13},15->{0,21},16->{5,6,12},17->{16},18->{5,6,12} ,19->{17},20->{17},21->{},22->{0,21}] + Applied Processor: FromIts + Details: () * Step 4: Decompose YES + Considered Problem: Rules: f2(A,B,C,D,E,F,G,H) -> f8(A,B,0,B,E,F,G,H) [A >= 1 + B] f8(A,B,C,D,E,F,G,H) -> f8(A,B,C,D,1 + E,J,I,H) [A >= E && I >= J] f8(A,B,C,D,E,F,G,H) -> f8(A,B,J,E,1 + E,I,K,H) [A >= E && I >= 1 + K] f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [0 >= 1 + C] f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [C >= 1] f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [0 >= 1 + I && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [I >= 1 && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] f43(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,1 + E,F,G,H) [A >= E] f36(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,0) [A >= D] f34(A,B,C,D,E,F,G,H) -> f2(A,1 + B,0,D,E,F,G,H) [C = 0] f49(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,H) [E >= 1 + A] f43(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,E,F,G,H) [E >= 1 + A] f36(A,B,C,D,E,F,G,H) -> f2(A,1 + B,C,D,E,F,G,H) [D >= 1 + A] f27(A,B,C,D,E,F,G,H) -> f34(A,B,C,D,E,F,G,H) [E >= 1 + A] f19(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,E,F,G,H) [E >= 1 + A] f8(A,B,C,D,E,F,G,H) -> f34(A,B,C,B,E,F,G,H) [E >= 1 + A && B = D] f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [B >= 1 + D && E >= 1 + A] f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [D >= 1 + B && E >= 1 + A] f2(A,B,C,D,E,F,G,H) -> f1(A,B,C,D,E,F,G,H) [B >= A] start(A,B,C,D,E,F,G,H) -> f2(A,B,C,D,E,F,G,H) True Signature: {(f1,8);(f19,8);(f2,8);(f27,8);(f34,8);(f36,8);(f43,8);(f49,8);(f8,8);(start,8)} Rule Graph: [0->{1,2,18},1->{1,2,18,19,20},2->{1,2,18,19,20},5->{7,8,11,15},6->{7,8,11,15},7->{9,14},8->{9,14},9->{9 ,14},11->{7,8,11,15},12->{0,21},13->{7,8,11,15},14->{13},15->{0,21},16->{5,6,12},17->{16},18->{5,6,12} ,19->{17},20->{17},21->{},22->{0,21}] + Applied Processor: Decompose NoGreedy + Details: We construct a looptree: P: [0,1,2,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22] | `- p:[0,12,16,17,19,1,2,20,18,15,5,6,11,13,14,7,8,9] c: [1] | `- p:[0,12,16,17,19,2,20,18,15,5,6,11,13,14,7,8,9] c: [2,9,16,17,19,20] | `- p:[0,12,18,15,5,6,11,13,14,7,8] c: [0,5,6,12,15,18] | `- p:[7,11,13,14,8] c: [7,8,11,13,14] * Step 5: CloseWith YES + Considered Problem: (Rules: f2(A,B,C,D,E,F,G,H) -> f8(A,B,0,B,E,F,G,H) [A >= 1 + B] f8(A,B,C,D,E,F,G,H) -> f8(A,B,C,D,1 + E,J,I,H) [A >= E && I >= J] f8(A,B,C,D,E,F,G,H) -> f8(A,B,J,E,1 + E,I,K,H) [A >= E && I >= 1 + K] f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [0 >= 1 + C] f34(A,B,C,D,E,F,G,H) -> f36(A,B,C,D,E,F,G,H) [C >= 1] f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [0 >= 1 + I && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] f36(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,E,F,G,J) [I >= 1 && A >= D && I >= C*K && C*K + K >= 1 + I && K >= J && I >= C*L && C*L + L >= 1 + I && J >= L] f43(A,B,C,D,E,F,G,H) -> f43(A,B,C,D,1 + E,F,G,H) [A >= E] f36(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,0) [A >= D] f34(A,B,C,D,E,F,G,H) -> f2(A,1 + B,0,D,E,F,G,H) [C = 0] f49(A,B,C,D,E,F,G,H) -> f36(A,B,C,1 + D,E,F,G,H) [E >= 1 + A] f43(A,B,C,D,E,F,G,H) -> f49(A,B,C,D,E,F,G,H) [E >= 1 + A] f36(A,B,C,D,E,F,G,H) -> f2(A,1 + B,C,D,E,F,G,H) [D >= 1 + A] f27(A,B,C,D,E,F,G,H) -> f34(A,B,C,D,E,F,G,H) [E >= 1 + A] f19(A,B,C,D,E,F,G,H) -> f27(A,B,C,D,E,F,G,H) [E >= 1 + A] f8(A,B,C,D,E,F,G,H) -> f34(A,B,C,B,E,F,G,H) [E >= 1 + A && B = D] f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [B >= 1 + D && E >= 1 + A] f8(A,B,C,D,E,F,G,H) -> f19(A,B,C,D,E,F,G,H) [D >= 1 + B && E >= 1 + A] f2(A,B,C,D,E,F,G,H) -> f1(A,B,C,D,E,F,G,H) [B >= A] start(A,B,C,D,E,F,G,H) -> f2(A,B,C,D,E,F,G,H) True Signature: {(f1,8);(f19,8);(f2,8);(f27,8);(f34,8);(f36,8);(f43,8);(f49,8);(f8,8);(start,8)} Rule Graph: [0->{1,2,18},1->{1,2,18,19,20},2->{1,2,18,19,20},5->{7,8,11,15},6->{7,8,11,15},7->{9,14},8->{9,14},9->{9 ,14},11->{7,8,11,15},12->{0,21},13->{7,8,11,15},14->{13},15->{0,21},16->{5,6,12},17->{16},18->{5,6,12} ,19->{17},20->{17},21->{},22->{0,21}] ,We construct a looptree: P: [0,1,2,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22] | `- p:[0,12,16,17,19,1,2,20,18,15,5,6,11,13,14,7,8,9] c: [1] | `- p:[0,12,16,17,19,2,20,18,15,5,6,11,13,14,7,8,9] c: [2,9,16,17,19,20] | `- p:[0,12,18,15,5,6,11,13,14,7,8] c: [0,5,6,12,15,18] | `- p:[7,11,13,14,8] c: [7,8,11,13,14]) + Applied Processor: CloseWith True + Details: () YES