YES * Step 1: TrivialSCCs YES + Considered Problem: Rules: 0. f0(A,B,C,D,E,F,G) -> f12(2,H,I,0,E,F,G) True (1,1) 1. f12(A,B,C,D,E,F,G) -> f15(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 2. f15(A,B,C,D,E,F,G) -> f15(A,B,C,D,1 + E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 3. f23(A,B,C,D,E,F,G) -> f26(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 4. f26(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,0,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 5. f30(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,1 + F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + F] 6. f30(A,B,C,D,E,F,G) -> f26(A,B,C,D,1 + E,F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && F >= A] 7. f26(A,B,C,D,E,F,G) -> f23(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 8. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,0) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (?,1) 9. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 49 >= H] (?,1) 10. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (?,1) 11. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 42 >= H] (?,1) 12. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 21 >= H] (?,1) 13. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 18 >= H] (?,1) 14. f15(A,B,C,D,E,F,G) -> f12(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 15. f12(A,B,C,D,E,F,G) -> f23(A,B,C,0,E,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (?,1) Signature: {(f0,7);(f12,7);(f15,7);(f23,7);(f26,7);(f30,7);(f52,7)} Flow Graph: [0->{1,15},1->{2,14},2->{2,14},3->{4,7},4->{5,6},5->{5,6},6->{4,7},7->{3,8,9,10,11,12,13},8->{},9->{} ,10->{},11->{},12->{},13->{},14->{1,15},15->{3,8,9,10,11,12,13}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 2: UnsatPaths YES + Considered Problem: Rules: 0. f0(A,B,C,D,E,F,G) -> f12(2,H,I,0,E,F,G) True (1,1) 1. f12(A,B,C,D,E,F,G) -> f15(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 2. f15(A,B,C,D,E,F,G) -> f15(A,B,C,D,1 + E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 3. f23(A,B,C,D,E,F,G) -> f26(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 4. f26(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,0,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 5. f30(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,1 + F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + F] 6. f30(A,B,C,D,E,F,G) -> f26(A,B,C,D,1 + E,F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && F >= A] 7. f26(A,B,C,D,E,F,G) -> f23(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 8. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,0) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) 9. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 49 >= H] (1,1) 10. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) 11. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 42 >= H] (1,1) 12. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 21 >= H] (1,1) 13. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 18 >= H] (1,1) 14. f15(A,B,C,D,E,F,G) -> f12(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 15. f12(A,B,C,D,E,F,G) -> f23(A,B,C,0,E,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) Signature: {(f0,7);(f12,7);(f15,7);(f23,7);(f26,7);(f30,7);(f52,7)} Flow Graph: [0->{1,15},1->{2,14},2->{2,14},3->{4,7},4->{5,6},5->{5,6},6->{4,7},7->{3,8,9,10,11,12,13},8->{},9->{} ,10->{},11->{},12->{},13->{},14->{1,15},15->{3,8,9,10,11,12,13}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,15) ,(1,14) ,(3,7) ,(4,6) ,(15,8) ,(15,9) ,(15,10) ,(15,11) ,(15,12) ,(15,13)] * Step 3: Looptree YES + Considered Problem: Rules: 0. f0(A,B,C,D,E,F,G) -> f12(2,H,I,0,E,F,G) True (1,1) 1. f12(A,B,C,D,E,F,G) -> f15(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 2. f15(A,B,C,D,E,F,G) -> f15(A,B,C,D,1 + E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 3. f23(A,B,C,D,E,F,G) -> f26(A,B,C,D,0,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + D] (?,1) 4. f26(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,0,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + E] 5. f30(A,B,C,D,E,F,G) -> f30(A,B,C,D,E,1 + F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && A >= 1 + F] 6. f30(A,B,C,D,E,F,G) -> f26(A,B,C,D,1 + E,F,G) [F >= 0 (?,1) && E + F >= 0 && D + F >= 0 && -2 + A + F >= 0 && 2 + -1*A + F >= 0 && E >= 0 && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && F >= A] 7. f26(A,B,C,D,E,F,G) -> f23(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 8. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,0) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) 9. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 49 >= H] (1,1) 10. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) 11. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 42 >= H] (1,1) 12. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 21 >= H] (1,1) 13. f23(A,B,C,D,E,F,G) -> f52(A,B,C,D,E,F,1) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A && 18 >= H] (1,1) 14. f15(A,B,C,D,E,F,G) -> f12(A,B,C,1 + D,E,F,G) [E >= 0 (?,1) && D + E >= 0 && -2 + A + E >= 0 && 2 + -1*A + E >= 0 && D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && E >= A] 15. f12(A,B,C,D,E,F,G) -> f23(A,B,C,0,E,F,G) [D >= 0 && -2 + A + D >= 0 && 2 + -1*A + D >= 0 && 2 + -1*A >= 0 && -2 + A >= 0 && D >= A] (1,1) Signature: {(f0,7);(f12,7);(f15,7);(f23,7);(f26,7);(f30,7);(f52,7)} Flow Graph: [0->{1},1->{2},2->{2,14},3->{4},4->{5},5->{5,6},6->{4,7},7->{3,8,9,10,11,12,13},8->{},9->{},10->{},11->{} ,12->{},13->{},14->{1,15},15->{3}] + Applied Processor: Looptree + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] | +- p:[1,14,2] c: [1] | | | `- p:[2] c: [2] | `- p:[3,7,6,5,4] c: [3] | `- p:[4,6,5] c: [4] | `- p:[5] c: [5] YES