MAYBE

Problem:
 battle(H(0(),x),n) -> battle(x,s(n))
 battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
 battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
 f(x,y,o()) -> y
 f(x,y,s(n)) -> H(x,f(x,y,n))

Proof:
 DP Processor:
  DPs:
   battle#(H(0(),x),n) -> battle#(x,s(n))
   battle#(H(H(0(),x),y),n) -> f#(x,y,n)
   battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
   battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
   battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
   f#(x,y,s(n)) -> f#(x,y,n)
  TRS:
   battle(H(0(),x),n) -> battle(x,s(n))
   battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
   battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
   f(x,y,o()) -> y
   f(x,y,s(n)) -> H(x,f(x,y,n))
  Usable Rule Processor:
   DPs:
    battle#(H(0(),x),n) -> battle#(x,s(n))
    battle#(H(H(0(),x),y),n) -> f#(x,y,n)
    battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
    battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
    battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
    f#(x,y,s(n)) -> f#(x,y,n)
   TRS:
    f8(x,y) -> x
    f8(x,y) -> y
    f(x,y,o()) -> y
    f(x,y,s(n)) -> H(x,f(x,y,n))
   ADG Processor:
    DPs:
     battle#(H(0(),x),n) -> battle#(x,s(n))
     battle#(H(H(0(),x),y),n) -> f#(x,y,n)
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
     battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
     f#(x,y,s(n)) -> f#(x,y,n)
    TRS:
     f8(x,y) -> x
     f8(x,y) -> y
     f(x,y,o()) -> y
     f(x,y,s(n)) -> H(x,f(x,y,n))
    graph:
     f#(x,y,s(n)) -> f#(x,y,n) ->
     f#(x,y,s(n)) -> f#(x,y,n)
     battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n) ->
     f#(x,y,s(n)) -> f#(x,y,n)
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n)) ->
     battle#(H(0(),x),n) -> battle#(x,s(n))
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n)) ->
     battle#(H(H(0(),x),y),n) -> f#(x,y,n)
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n)) ->
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
     battle#(H(H(0(),x),y),n) -> f#(x,y,n) ->
     f#(x,y,s(n)) -> f#(x,y,n)
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n)) ->
     battle#(H(0(),x),n) -> battle#(x,s(n))
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n)) ->
     battle#(H(H(0(),x),y),n) -> f#(x,y,n)
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n)) ->
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
     battle#(H(0(),x),n) -> battle#(x,s(n)) ->
     battle#(H(0(),x),n) -> battle#(x,s(n))
     battle#(H(0(),x),n) -> battle#(x,s(n)) ->
     battle#(H(H(0(),x),y),n) -> f#(x,y,n)
     battle#(H(0(),x),n) -> battle#(x,s(n)) ->
     battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
     battle#(H(0(),x),n) -> battle#(x,s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
     battle#(H(0(),x),n) -> battle#(x,s(n)) ->
     battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
    Restore Modifier:
     DPs:
      battle#(H(0(),x),n) -> battle#(x,s(n))
      battle#(H(H(0(),x),y),n) -> f#(x,y,n)
      battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
      battle#(H(H(H(0(),x),y),z),n) -> f#(x,y,n)
      battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
      f#(x,y,s(n)) -> f#(x,y,n)
     TRS:
      battle(H(0(),x),n) -> battle(x,s(n))
      battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
      battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
      f(x,y,o()) -> y
      f(x,y,s(n)) -> H(x,f(x,y,n))
     SCC Processor:
      #sccs: 2
      #rules: 4
      #arcs: 18/36
      DPs:
       battle#(H(H(H(0(),x),y),z),n) -> battle#(H(f(x,y,n),z),s(n))
       battle#(H(H(0(),x),y),n) -> battle#(f(x,y,n),s(n))
       battle#(H(0(),x),n) -> battle#(x,s(n))
      TRS:
       battle(H(0(),x),n) -> battle(x,s(n))
       battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
       battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
       f(x,y,o()) -> y
       f(x,y,s(n)) -> H(x,f(x,y,n))
      Open
      
      DPs:
       f#(x,y,s(n)) -> f#(x,y,n)
      TRS:
       battle(H(0(),x),n) -> battle(x,s(n))
       battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
       battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
       f(x,y,o()) -> y
       f(x,y,s(n)) -> H(x,f(x,y,n))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [f#](x0, x1, x2) = x2 + 1,
        
        [o] = 0,
        
        [f](x0, x1, x2) = x1,
        
        [s](x0) = x0 + 1,
        
        [battle](x0, x1) = 0,
        
        [H](x0, x1) = x1,
        
        [0] = 0
       orientation:
        f#(x,y,s(n)) = n + 2 >= n + 1 = f#(x,y,n)
        
        battle(H(0(),x),n) = 0 >= 0 = battle(x,s(n))
        
        battle(H(H(0(),x),y),n) = 0 >= 0 = battle(f(x,y,n),s(n))
        
        battle(H(H(H(0(),x),y),z),n) = 0 >= 0 = battle(H(f(x,y,n),z),s(n))
        
        f(x,y,o()) = y >= y = y
        
        f(x,y,s(n)) = y >= y = H(x,f(x,y,n))
       problem:
        DPs:
         
        TRS:
         battle(H(0(),x),n) -> battle(x,s(n))
         battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
         battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
         f(x,y,o()) -> y
         f(x,y,s(n)) -> H(x,f(x,y,n))
       Qed