YES
* Step 1: Looptree YES
    + Considered Problem:
        Rules:
          0. f1(A,B)   -> f2(-1 + A,B)     [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && A >= 1] (?,1)
          1. f2(A,B)   -> f2(-1 + A,1 + B) [B >= 0 && A + B >= 0 && A >= 0 && A >= 1]           (?,1)
          2. f999(A,B) -> f1(1,-1 + B)     [B >= 1 && A = 0]                                    (1,1)
          3. f1(A,B)   -> f1(1 + A,-1 + B) [B >= 0 && -1 + A + B >= 0 && -1 + A >= 0 && B >= 1] (?,1)
          4. f2(A,B)   -> f1(1 + A,-1 + B) [B >= 0 && A + B >= 0 && A >= 0 && B >= 1]           (?,1)
        Signature:
          {(f1,2);(f2,2);(f999,2)}
        Flow Graph:
          [0->{1,4},1->{1,4},2->{0,3},3->{0,3},4->{0,3}]
        
    + Applied Processor:
        Looptree
    + Details:
        We construct a looptree:
          P: [0,1,2,3,4]
          |
          `- p:[0,3,4,1] c: [4]
             |
             +- p:[3] c: [3]
             |
             `- p:[1] c: [1]

YES