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)
  TDG 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)
   graph:
    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) -> mod#(minus(x,y),y) ->
    mod#(s(x),s(y)) -> le#(y,x)
    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) -> 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)
    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)
   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:
     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
    
    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