MAYBE

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 mod(0(),y) -> 0()
 mod(s(x),0()) -> 0()
 mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
 if_mod(true(),x,y) -> mod(minus(x,y),y)
 if_mod(false(),s(x),s(y)) -> s(x)

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(s(x),s(y)) -> minus#(x,y)
   mod#(s(x),s(y)) -> le#(y,x)
   mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
   if_mod#(true(),x,y) -> minus#(x,y)
   if_mod#(true(),x,y) -> mod#(minus(x,y),y)
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   mod(0(),y) -> 0()
   mod(s(x),0()) -> 0()
   mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
   if_mod(true(),x,y) -> mod(minus(x,y),y)
   if_mod(false(),s(x),s(y)) -> s(x)
  Usable Rule Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y)
    mod#(s(x),s(y)) -> le#(y,x)
    mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
    if_mod#(true(),x,y) -> minus#(x,y)
    if_mod#(true(),x,y) -> mod#(minus(x,y),y)
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
   CDG Processor:
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
     minus#(s(x),s(y)) -> minus#(x,y)
     mod#(s(x),s(y)) -> le#(y,x)
     mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
     if_mod#(true(),x,y) -> minus#(x,y)
     if_mod#(true(),x,y) -> mod#(minus(x,y),y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
    graph:
     if_mod#(true(),x,y) -> mod#(minus(x,y),y) ->
     mod#(s(x),s(y)) -> le#(y,x)
     if_mod#(true(),x,y) -> mod#(minus(x,y),y) ->
     mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
     if_mod#(true(),x,y) -> minus#(x,y) ->
     minus#(s(x),s(y)) -> minus#(x,y)
     mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) ->
     if_mod#(true(),x,y) -> minus#(x,y)
     mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) ->
     if_mod#(true(),x,y) -> mod#(minus(x,y),y)
     mod#(s(x),s(y)) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y)
     minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
     le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    Restore Modifier:
     DPs:
      le#(s(x),s(y)) -> le#(x,y)
      minus#(s(x),s(y)) -> minus#(x,y)
      mod#(s(x),s(y)) -> le#(y,x)
      mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
      if_mod#(true(),x,y) -> minus#(x,y)
      if_mod#(true(),x,y) -> mod#(minus(x,y),y)
     TRS:
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      mod(0(),y) -> 0()
      mod(s(x),0()) -> 0()
      mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
      if_mod(true(),x,y) -> mod(minus(x,y),y)
      if_mod(false(),s(x),s(y)) -> s(x)
     SCC Processor:
      #sccs: 3
      #rules: 4
      #arcs: 8/36
      DPs:
       if_mod#(true(),x,y) -> mod#(minus(x,y),y)
       mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),x,y) -> mod(minus(x,y),y)
       if_mod(false(),s(x),s(y)) -> s(x)
      Open
      
      DPs:
       le#(s(x),s(y)) -> le#(x,y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),x,y) -> mod(minus(x,y),y)
       if_mod(false(),s(x),s(y)) -> s(x)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [le#](x0, x1) = x0 + x1,
        
        [if_mod](x0, x1, x2) = x0 + x1,
        
        [mod](x0, x1) = x0 + 1,
        
        [minus](x0, x1) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 1,
        
        [le](x0, x1) = 1,
        
        [0] = 1
       orientation:
        le#(s(x),s(y)) = x + y + 2 >= x + y = le#(x,y)
        
        le(0(),y) = 1 >= 1 = true()
        
        le(s(x),0()) = 1 >= 0 = false()
        
        le(s(x),s(y)) = 1 >= 1 = le(x,y)
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        mod(0(),y) = 2 >= 1 = 0()
        
        mod(s(x),0()) = x + 2 >= 1 = 0()
        
        mod(s(x),s(y)) = x + 2 >= x + 2 = if_mod(le(y,x),s(x),s(y))
        
        if_mod(true(),x,y) = x + 1 >= x + 1 = mod(minus(x,y),y)
        
        if_mod(false(),s(x),s(y)) = x + 1 >= x + 1 = s(x)
       problem:
        DPs:
         
        TRS:
         le(0(),y) -> true()
         le(s(x),0()) -> false()
         le(s(x),s(y)) -> le(x,y)
         minus(x,0()) -> x
         minus(s(x),s(y)) -> minus(x,y)
         mod(0(),y) -> 0()
         mod(s(x),0()) -> 0()
         mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
         if_mod(true(),x,y) -> mod(minus(x,y),y)
         if_mod(false(),s(x),s(y)) -> s(x)
       Qed
      
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       mod(0(),y) -> 0()
       mod(s(x),0()) -> 0()
       mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
       if_mod(true(),x,y) -> mod(minus(x,y),y)
       if_mod(false(),s(x),s(y)) -> s(x)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [minus#](x0, x1) = x0 + x1,
        
        [if_mod](x0, x1, x2) = x0 + x1,
        
        [mod](x0, x1) = x0 + 1,
        
        [minus](x0, x1) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 1,
        
        [le](x0, x1) = 1,
        
        [0] = 1
       orientation:
        minus#(s(x),s(y)) = x + y + 2 >= x + y = minus#(x,y)
        
        le(0(),y) = 1 >= 1 = true()
        
        le(s(x),0()) = 1 >= 0 = false()
        
        le(s(x),s(y)) = 1 >= 1 = le(x,y)
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        mod(0(),y) = 2 >= 1 = 0()
        
        mod(s(x),0()) = x + 2 >= 1 = 0()
        
        mod(s(x),s(y)) = x + 2 >= x + 2 = if_mod(le(y,x),s(x),s(y))
        
        if_mod(true(),x,y) = x + 1 >= x + 1 = mod(minus(x,y),y)
        
        if_mod(false(),s(x),s(y)) = x + 1 >= x + 1 = s(x)
       problem:
        DPs:
         
        TRS:
         le(0(),y) -> true()
         le(s(x),0()) -> false()
         le(s(x),s(y)) -> le(x,y)
         minus(x,0()) -> x
         minus(s(x),s(y)) -> minus(x,y)
         mod(0(),y) -> 0()
         mod(s(x),0()) -> 0()
         mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
         if_mod(true(),x,y) -> mod(minus(x,y),y)
         if_mod(false(),s(x),s(y)) -> s(x)
       Qed