MAYBE

Problem:
 f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
 trunc(0()) -> 0()
 trunc(s(0())) -> 0()
 trunc(s(s(x))) -> s(s(trunc(x)))
 gt(0(),v) -> false()
 gt(s(u),0()) -> true()
 gt(s(u),s(v)) -> gt(u,v)

Proof:
 DP Processor:
  DPs:
   f#(true(),x,y) -> trunc#(x)
   f#(true(),x,y) -> gt#(x,y)
   f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
   trunc#(s(s(x))) -> trunc#(x)
   gt#(s(u),s(v)) -> gt#(u,v)
  TRS:
   f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
   trunc(0()) -> 0()
   trunc(s(0())) -> 0()
   trunc(s(s(x))) -> s(s(trunc(x)))
   gt(0(),v) -> false()
   gt(s(u),0()) -> true()
   gt(s(u),s(v)) -> gt(u,v)
  Usable Rule Processor:
   DPs:
    f#(true(),x,y) -> trunc#(x)
    f#(true(),x,y) -> gt#(x,y)
    f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
    trunc#(s(s(x))) -> trunc#(x)
    gt#(s(u),s(v)) -> gt#(u,v)
   TRS:
    f10(x,y) -> x
    f10(x,y) -> y
    trunc(0()) -> 0()
    trunc(s(0())) -> 0()
    trunc(s(s(x))) -> s(s(trunc(x)))
    gt(0(),v) -> false()
    gt(s(u),0()) -> true()
    gt(s(u),s(v)) -> gt(u,v)
   CDG Processor:
    DPs:
     f#(true(),x,y) -> trunc#(x)
     f#(true(),x,y) -> gt#(x,y)
     f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
     trunc#(s(s(x))) -> trunc#(x)
     gt#(s(u),s(v)) -> gt#(u,v)
    TRS:
     f10(x,y) -> x
     f10(x,y) -> y
     trunc(0()) -> 0()
     trunc(s(0())) -> 0()
     trunc(s(s(x))) -> s(s(trunc(x)))
     gt(0(),v) -> false()
     gt(s(u),0()) -> true()
     gt(s(u),s(v)) -> gt(u,v)
    graph:
     gt#(s(u),s(v)) -> gt#(u,v) -> gt#(s(u),s(v)) -> gt#(u,v)
     trunc#(s(s(x))) -> trunc#(x) -> trunc#(s(s(x))) -> trunc#(x)
     f#(true(),x,y) -> gt#(x,y) -> gt#(s(u),s(v)) -> gt#(u,v)
     f#(true(),x,y) -> trunc#(x) ->
     trunc#(s(s(x))) -> trunc#(x)
     f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y)) ->
     f#(true(),x,y) -> trunc#(x)
     f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y)) ->
     f#(true(),x,y) -> gt#(x,y)
     f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y)) -> f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
    Restore Modifier:
     DPs:
      f#(true(),x,y) -> trunc#(x)
      f#(true(),x,y) -> gt#(x,y)
      f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
      trunc#(s(s(x))) -> trunc#(x)
      gt#(s(u),s(v)) -> gt#(u,v)
     TRS:
      f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
      trunc(0()) -> 0()
      trunc(s(0())) -> 0()
      trunc(s(s(x))) -> s(s(trunc(x)))
      gt(0(),v) -> false()
      gt(s(u),0()) -> true()
      gt(s(u),s(v)) -> gt(u,v)
     SCC Processor:
      #sccs: 3
      #rules: 3
      #arcs: 7/25
      DPs:
       f#(true(),x,y) -> f#(gt(x,y),trunc(x),s(y))
      TRS:
       f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
       trunc(0()) -> 0()
       trunc(s(0())) -> 0()
       trunc(s(s(x))) -> s(s(trunc(x)))
       gt(0(),v) -> false()
       gt(s(u),0()) -> true()
       gt(s(u),s(v)) -> gt(u,v)
      Open
      
      DPs:
       trunc#(s(s(x))) -> trunc#(x)
      TRS:
       f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
       trunc(0()) -> 0()
       trunc(s(0())) -> 0()
       trunc(s(s(x))) -> s(s(trunc(x)))
       gt(0(),v) -> false()
       gt(s(u),0()) -> true()
       gt(s(u),s(v)) -> gt(u,v)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [trunc#](x0) = x0,
        
        [false] = 0,
        
        [0] = 1,
        
        [s](x0) = x0 + 1,
        
        [trunc](x0) = x0,
        
        [gt](x0, x1) = x0,
        
        [f](x0, x1, x2) = 0,
        
        [true] = 1
       orientation:
        trunc#(s(s(x))) = x + 2 >= x = trunc#(x)
        
        f(true(),x,y) = 0 >= 0 = f(gt(x,y),trunc(x),s(y))
        
        trunc(0()) = 1 >= 1 = 0()
        
        trunc(s(0())) = 2 >= 1 = 0()
        
        trunc(s(s(x))) = x + 2 >= x + 2 = s(s(trunc(x)))
        
        gt(0(),v) = 1 >= 0 = false()
        
        gt(s(u),0()) = u + 1 >= 1 = true()
        
        gt(s(u),s(v)) = u + 1 >= u = gt(u,v)
       problem:
        DPs:
         
        TRS:
         f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
         trunc(0()) -> 0()
         trunc(s(0())) -> 0()
         trunc(s(s(x))) -> s(s(trunc(x)))
         gt(0(),v) -> false()
         gt(s(u),0()) -> true()
         gt(s(u),s(v)) -> gt(u,v)
       Qed
      
      DPs:
       gt#(s(u),s(v)) -> gt#(u,v)
      TRS:
       f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
       trunc(0()) -> 0()
       trunc(s(0())) -> 0()
       trunc(s(s(x))) -> s(s(trunc(x)))
       gt(0(),v) -> false()
       gt(s(u),0()) -> true()
       gt(s(u),s(v)) -> gt(u,v)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [gt#](x0, x1) = x0 + 1,
        
        [false] = 0,
        
        [0] = 0,
        
        [s](x0) = x0 + 1,
        
        [trunc](x0) = x0,
        
        [gt](x0, x1) = x1 + 1,
        
        [f](x0, x1, x2) = 0,
        
        [true] = 0
       orientation:
        gt#(s(u),s(v)) = u + 2 >= u + 1 = gt#(u,v)
        
        f(true(),x,y) = 0 >= 0 = f(gt(x,y),trunc(x),s(y))
        
        trunc(0()) = 0 >= 0 = 0()
        
        trunc(s(0())) = 1 >= 0 = 0()
        
        trunc(s(s(x))) = x + 2 >= x + 2 = s(s(trunc(x)))
        
        gt(0(),v) = v + 1 >= 0 = false()
        
        gt(s(u),0()) = 1 >= 0 = true()
        
        gt(s(u),s(v)) = v + 2 >= v + 1 = gt(u,v)
       problem:
        DPs:
         
        TRS:
         f(true(),x,y) -> f(gt(x,y),trunc(x),s(y))
         trunc(0()) -> 0()
         trunc(s(0())) -> 0()
         trunc(s(s(x))) -> s(s(trunc(x)))
         gt(0(),v) -> false()
         gt(s(u),0()) -> true()
         gt(s(u),s(v)) -> gt(u,v)
       Qed