MAYBE

Problem:
 p(0()) -> 0()
 p(s(x)) -> x
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,y) -> if(le(x,y),x,y)
 if(true(),x,y) -> 0()
 if(false(),x,y) -> s(minus(p(x),y))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,y) -> le#(x,y)
   minus#(x,y) -> if#(le(x,y),x,y)
   if#(false(),x,y) -> p#(x)
   if#(false(),x,y) -> minus#(p(x),y)
  TRS:
   p(0()) -> 0()
   p(s(x)) -> x
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(x,y) -> if(le(x,y),x,y)
   if(true(),x,y) -> 0()
   if(false(),x,y) -> s(minus(p(x),y))
  Usable Rule Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,y) -> le#(x,y)
    minus#(x,y) -> if#(le(x,y),x,y)
    if#(false(),x,y) -> p#(x)
    if#(false(),x,y) -> minus#(p(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)
    p(0()) -> 0()
    p(s(x)) -> x
   TDG Processor:
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
     minus#(x,y) -> le#(x,y)
     minus#(x,y) -> if#(le(x,y),x,y)
     if#(false(),x,y) -> p#(x)
     if#(false(),x,y) -> minus#(p(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)
     p(0()) -> 0()
     p(s(x)) -> x
    graph:
     if#(false(),x,y) -> minus#(p(x),y) ->
     minus#(x,y) -> if#(le(x,y),x,y)
     if#(false(),x,y) -> minus#(p(x),y) -> minus#(x,y) -> le#(x,y)
     minus#(x,y) -> if#(le(x,y),x,y) ->
     if#(false(),x,y) -> minus#(p(x),y)
     minus#(x,y) -> if#(le(x,y),x,y) -> if#(false(),x,y) -> p#(x)
     minus#(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)
      minus#(x,y) -> le#(x,y)
      minus#(x,y) -> if#(le(x,y),x,y)
      if#(false(),x,y) -> p#(x)
      if#(false(),x,y) -> minus#(p(x),y)
     TRS:
      p(0()) -> 0()
      p(s(x)) -> x
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      minus(x,y) -> if(le(x,y),x,y)
      if(true(),x,y) -> 0()
      if(false(),x,y) -> s(minus(p(x),y))
     SCC Processor:
      #sccs: 2
      #rules: 3
      #arcs: 6/25
      DPs:
       if#(false(),x,y) -> minus#(p(x),y)
       minus#(x,y) -> if#(le(x,y),x,y)
      TRS:
       p(0()) -> 0()
       p(s(x)) -> x
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,y) -> if(le(x,y),x,y)
       if(true(),x,y) -> 0()
       if(false(),x,y) -> s(minus(p(x),y))
      Open
      
      DPs:
       le#(s(x),s(y)) -> le#(x,y)
      TRS:
       p(0()) -> 0()
       p(s(x)) -> x
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,y) -> if(le(x,y),x,y)
       if(true(),x,y) -> 0()
       if(false(),x,y) -> s(minus(p(x),y))
      Open