MAYBE

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 int(x,y) -> if(le(x,y),x,y)
 if(true(),x,y) -> cons(x,int(s(x),y))
 if(false(),x,y) -> nil()

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   int#(x,y) -> le#(x,y)
   int#(x,y) -> if#(le(x,y),x,y)
   if#(true(),x,y) -> int#(s(x),y)
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   int(x,y) -> if(le(x,y),x,y)
   if(true(),x,y) -> cons(x,int(s(x),y))
   if(false(),x,y) -> nil()
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    int#(x,y) -> le#(x,y)
    int#(x,y) -> if#(le(x,y),x,y)
    if#(true(),x,y) -> int#(s(x),y)
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    int(x,y) -> if(le(x,y),x,y)
    if(true(),x,y) -> cons(x,int(s(x),y))
    if(false(),x,y) -> nil()
   graph:
    if#(true(),x,y) -> int#(s(x),y) -> int#(x,y) -> if#(le(x,y),x,y)
    if#(true(),x,y) -> int#(s(x),y) -> int#(x,y) -> le#(x,y)
    int#(x,y) -> if#(le(x,y),x,y) -> if#(true(),x,y) -> int#(s(x),y)
    int#(x,y) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 2
    #rules: 3
    #arcs: 5/16
    DPs:
     if#(true(),x,y) -> int#(s(x),y)
     int#(x,y) -> if#(le(x,y),x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     int(x,y) -> if(le(x,y),x,y)
     if(true(),x,y) -> cons(x,int(s(x),y))
     if(false(),x,y) -> nil()
    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)
     int(x,y) -> if(le(x,y),x,y)
     if(true(),x,y) -> cons(x,int(s(x),y))
     if(false(),x,y) -> nil()
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [le#](x0, x1) = 1x0,
      
      [nil] = 1,
      
      [cons](x0, x1) = -11x0 + -9x1 + 8,
      
      [if](x0, x1, x2) = -7x1 + 2x2 + 10,
      
      [int](x0, x1) = x0 + 9x1 + 12,
      
      [false] = 2,
      
      [s](x0) = 1x0 + -1,
      
      [true] = 0,
      
      [le](x0, x1) = -5x0 + 1x1 + -5,
      
      [0] = 6
     orientation:
      le#(s(x),s(y)) = 2x + 0 >= 1x = le#(x,y)
      
      le(0(),y) = 1y + 1 >= 0 = true()
      
      le(s(x),0()) = -4x + 7 >= 2 = false()
      
      le(s(x),s(y)) = -4x + 2y + 0 >= -5x + 1y + -5 = le(x,y)
      
      int(x,y) = x + 9y + 12 >= -7x + 2y + 10 = if(le(x,y),x,y)
      
      if(true(),x,y) = -7x + 2y + 10 >= -8x + y + 8 = cons(x,int(s(x),y))
      
      if(false(),x,y) = -7x + 2y + 10 >= 1 = nil()
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       int(x,y) -> if(le(x,y),x,y)
       if(true(),x,y) -> cons(x,int(s(x),y))
       if(false(),x,y) -> nil()
     Qed