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()
  Usable Rule 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:
    f12(x,y) -> x
    f12(x,y) -> y
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
   ADG 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:
     f12(x,y) -> x
     f12(x,y) -> y
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
    graph:
     if#(true(),x,y) -> int#(s(x),y) -> int#(x,y) -> le#(x,y)
     if#(true(),x,y) -> int#(s(x),y) -> int#(x,y) -> if#(le(x,y),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)
    Restore Modifier:
     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()
     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()
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [le#](x0, x1) = x0 + 1,
        
        [nil] = 0,
        
        [cons](x0, x1) = 0,
        
        [if](x0, x1, x2) = 0,
        
        [int](x0, x1) = 0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        le#(s(x),s(y)) = x + 2 >= x + 1 = le#(x,y)
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        int(x,y) = 0 >= 0 = if(le(x,y),x,y)
        
        if(true(),x,y) = 0 >= 0 = cons(x,int(s(x),y))
        
        if(false(),x,y) = 0 >= 0 = 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