YES
* Step 1: UnsatPaths YES
    + Considered Problem:
        Rules:
          0.  f2(A,B) -> f2(1 + A,B)  [A + B >= 0 && B >= 1 + A && 0 >= 2 + A && B >= 1]     (?,1)
          1.  f2(A,B) -> f2(1 + A,B)  [A + B >= 0 && B >= 1 + A && A >= 0 && B >= 1]         (?,1)
          2.  f2(A,B) -> f2(A,1 + B)  [B >= A && 0 >= 2 + B && 0 >= 1 + A && 0 >= 1 + A + B] (?,1)
          3.  f2(A,B) -> f2(A,1 + B)  [B >= A && B >= 0 && 0 >= 1 + A && 0 >= 1 + A + B]     (?,1)
          4.  f2(A,B) -> f2(-1 + A,B) [1 >= A + B && A >= 1 + B && 0 >= A && 0 >= 1 + B]     (?,1)
          5.  f2(A,B) -> f2(-1 + A,B) [1 >= A + B && A >= 1 + B && A >= 2 && 0 >= 1 + B]     (?,1)
          6.  f2(A,B) -> f2(A,-1 + B) [A >= B && 0 >= B && A >= 1 && A + B >= 2]             (?,1)
          7.  f2(A,B) -> f2(A,-1 + B) [A >= B && B >= 2 && A >= 1 && A + B >= 2]             (?,1)
          8.  f0(A,B) -> f2(A,B)      [0 >= 1 + A && 0 >= 1 + B]                             (1,1)
          9.  f0(A,B) -> f2(A,B)      [0 >= 1 + A && B >= 1]                                 (1,1)
          10. f0(A,B) -> f2(A,B)      [A >= 1 && 0 >= 1 + B]                                 (1,1)
          11. f0(A,B) -> f2(A,B)      [A >= 1 && B >= 1]                                     (1,1)
        Signature:
          {(f0,2);(f2,2)}
        Flow Graph:
          [0->{0,1,2,3,4,5,6,7},1->{0,1,2,3,4,5,6,7},2->{0,1,2,3,4,5,6,7},3->{0,1,2,3,4,5,6,7},4->{0,1,2,3,4,5,6,7}
          ,5->{0,1,2,3,4,5,6,7},6->{0,1,2,3,4,5,6,7},7->{0,1,2,3,4,5,6,7},8->{0,1,2,3,4,5,6,7},9->{0,1,2,3,4,5,6,7}
          ,10->{0,1,2,3,4,5,6,7},11->{0,1,2,3,4,5,6,7}]
        
    + Applied Processor:
        UnsatPaths
    + Details:
        We remove following edges from the transition graph: [(0,1)
                                                             ,(0,2)
                                                             ,(0,3)
                                                             ,(0,4)
                                                             ,(0,5)
                                                             ,(0,6)
                                                             ,(0,7)
                                                             ,(1,0)
                                                             ,(1,2)
                                                             ,(1,3)
                                                             ,(1,4)
                                                             ,(1,5)
                                                             ,(1,6)
                                                             ,(2,0)
                                                             ,(2,1)
                                                             ,(2,3)
                                                             ,(2,4)
                                                             ,(2,5)
                                                             ,(2,6)
                                                             ,(2,7)
                                                             ,(3,1)
                                                             ,(3,2)
                                                             ,(3,4)
                                                             ,(3,5)
                                                             ,(3,6)
                                                             ,(3,7)
                                                             ,(4,0)
                                                             ,(4,1)
                                                             ,(4,3)
                                                             ,(4,5)
                                                             ,(4,6)
                                                             ,(4,7)
                                                             ,(5,0)
                                                             ,(5,1)
                                                             ,(5,2)
                                                             ,(5,3)
                                                             ,(5,4)
                                                             ,(5,6)
                                                             ,(5,7)
                                                             ,(6,0)
                                                             ,(6,1)
                                                             ,(6,2)
                                                             ,(6,3)
                                                             ,(6,4)
                                                             ,(6,7)
                                                             ,(7,0)
                                                             ,(7,1)
                                                             ,(7,2)
                                                             ,(7,3)
                                                             ,(7,4)
                                                             ,(7,5)
                                                             ,(7,6)
                                                             ,(8,0)
                                                             ,(8,1)
                                                             ,(8,3)
                                                             ,(8,5)
                                                             ,(8,6)
                                                             ,(8,7)
                                                             ,(9,1)
                                                             ,(9,2)
                                                             ,(9,4)
                                                             ,(9,5)
                                                             ,(9,6)
                                                             ,(9,7)
                                                             ,(10,0)
                                                             ,(10,1)
                                                             ,(10,2)
                                                             ,(10,3)
                                                             ,(10,4)
                                                             ,(10,7)
                                                             ,(11,0)
                                                             ,(11,2)
                                                             ,(11,3)
                                                             ,(11,4)
                                                             ,(11,5)
                                                             ,(11,6)]
* Step 2: FromIts YES
    + Considered Problem:
        Rules:
          0.  f2(A,B) -> f2(1 + A,B)  [A + B >= 0 && B >= 1 + A && 0 >= 2 + A && B >= 1]     (?,1)
          1.  f2(A,B) -> f2(1 + A,B)  [A + B >= 0 && B >= 1 + A && A >= 0 && B >= 1]         (?,1)
          2.  f2(A,B) -> f2(A,1 + B)  [B >= A && 0 >= 2 + B && 0 >= 1 + A && 0 >= 1 + A + B] (?,1)
          3.  f2(A,B) -> f2(A,1 + B)  [B >= A && B >= 0 && 0 >= 1 + A && 0 >= 1 + A + B]     (?,1)
          4.  f2(A,B) -> f2(-1 + A,B) [1 >= A + B && A >= 1 + B && 0 >= A && 0 >= 1 + B]     (?,1)
          5.  f2(A,B) -> f2(-1 + A,B) [1 >= A + B && A >= 1 + B && A >= 2 && 0 >= 1 + B]     (?,1)
          6.  f2(A,B) -> f2(A,-1 + B) [A >= B && 0 >= B && A >= 1 && A + B >= 2]             (?,1)
          7.  f2(A,B) -> f2(A,-1 + B) [A >= B && B >= 2 && A >= 1 && A + B >= 2]             (?,1)
          8.  f0(A,B) -> f2(A,B)      [0 >= 1 + A && 0 >= 1 + B]                             (1,1)
          9.  f0(A,B) -> f2(A,B)      [0 >= 1 + A && B >= 1]                                 (1,1)
          10. f0(A,B) -> f2(A,B)      [A >= 1 && 0 >= 1 + B]                                 (1,1)
          11. f0(A,B) -> f2(A,B)      [A >= 1 && B >= 1]                                     (1,1)
        Signature:
          {(f0,2);(f2,2)}
        Flow Graph:
          [0->{0},1->{1,7},2->{2},3->{0,3},4->{2,4},5->{5},6->{5,6},7->{7},8->{2,4},9->{0,3},10->{5,6},11->{1,7}]
        
    + Applied Processor:
        FromIts
    + Details:
        ()
* Step 3: Decompose YES
    + Considered Problem:
        Rules:
          f2(A,B) -> f2(1 + A,B)   [A + B >= 0 && B >= 1 + A && 0 >= 2 + A && B >= 1]
          f2(A,B) -> f2(1 + A,B)   [A + B >= 0 && B >= 1 + A && A >= 0 && B >= 1]
          f2(A,B) -> f2(A,1 + B)   [B >= A && 0 >= 2 + B && 0 >= 1 + A && 0 >= 1 + A + B]
          f2(A,B) -> f2(A,1 + B)   [B >= A && B >= 0 && 0 >= 1 + A && 0 >= 1 + A + B]
          f2(A,B) -> f2(-1 + A,B)  [1 >= A + B && A >= 1 + B && 0 >= A && 0 >= 1 + B]
          f2(A,B) -> f2(-1 + A,B)  [1 >= A + B && A >= 1 + B && A >= 2 && 0 >= 1 + B]
          f2(A,B) -> f2(A,-1 + B)  [A >= B && 0 >= B && A >= 1 && A + B >= 2]
          f2(A,B) -> f2(A,-1 + B)  [A >= B && B >= 2 && A >= 1 && A + B >= 2]
          f0(A,B) -> f2(A,B)       [0 >= 1 + A && 0 >= 1 + B]
          f0(A,B) -> f2(A,B)       [0 >= 1 + A && B >= 1]
          f0(A,B) -> f2(A,B)       [A >= 1 && 0 >= 1 + B]
          f0(A,B) -> f2(A,B)       [A >= 1 && B >= 1]
        Signature:
          {(f0,2);(f2,2)}
        Rule Graph:
          [0->{0},1->{1,7},2->{2},3->{0,3},4->{2,4},5->{5},6->{5,6},7->{7},8->{2,4},9->{0,3},10->{5,6},11->{1,7}]
    + Applied Processor:
        Decompose NoGreedy
    + Details:
        We construct a looptree:
          P: [0,1,2,3,4,5,6,7,8,9,10,11]
          |
          +- p:[6] c: [6]
          |
          +- p:[5] c: [5]
          |
          +- p:[4] c: [4]
          |
          +- p:[3] c: [3]
          |
          +- p:[2] c: [2]
          |
          +- p:[1] c: [1]
          |
          +- p:[7] c: [7]
          |
          `- p:[0] c: [0]
* Step 4: CloseWith YES
    + Considered Problem:
        (Rules:
          f2(A,B) -> f2(1 + A,B)   [A + B >= 0 && B >= 1 + A && 0 >= 2 + A && B >= 1]
          f2(A,B) -> f2(1 + A,B)   [A + B >= 0 && B >= 1 + A && A >= 0 && B >= 1]
          f2(A,B) -> f2(A,1 + B)   [B >= A && 0 >= 2 + B && 0 >= 1 + A && 0 >= 1 + A + B]
          f2(A,B) -> f2(A,1 + B)   [B >= A && B >= 0 && 0 >= 1 + A && 0 >= 1 + A + B]
          f2(A,B) -> f2(-1 + A,B)  [1 >= A + B && A >= 1 + B && 0 >= A && 0 >= 1 + B]
          f2(A,B) -> f2(-1 + A,B)  [1 >= A + B && A >= 1 + B && A >= 2 && 0 >= 1 + B]
          f2(A,B) -> f2(A,-1 + B)  [A >= B && 0 >= B && A >= 1 && A + B >= 2]
          f2(A,B) -> f2(A,-1 + B)  [A >= B && B >= 2 && A >= 1 && A + B >= 2]
          f0(A,B) -> f2(A,B)       [0 >= 1 + A && 0 >= 1 + B]
          f0(A,B) -> f2(A,B)       [0 >= 1 + A && B >= 1]
          f0(A,B) -> f2(A,B)       [A >= 1 && 0 >= 1 + B]
          f0(A,B) -> f2(A,B)       [A >= 1 && B >= 1]
        Signature:
          {(f0,2);(f2,2)}
        Rule Graph:
          [0->{0},1->{1,7},2->{2},3->{0,3},4->{2,4},5->{5},6->{5,6},7->{7},8->{2,4},9->{0,3},10->{5,6},11->{1,7}]
        ,We construct a looptree:
          P: [0,1,2,3,4,5,6,7,8,9,10,11]
          |
          +- p:[6] c: [6]
          |
          +- p:[5] c: [5]
          |
          +- p:[4] c: [4]
          |
          +- p:[3] c: [3]
          |
          +- p:[2] c: [2]
          |
          +- p:[1] c: [1]
          |
          +- p:[7] c: [7]
          |
          `- p:[0] c: [0])
    + Applied Processor:
        CloseWith True
    + Details:
        ()

YES