MAYBE

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,0()) -> x
 minus(0(),s(y)) -> 0()
 minus(s(x),s(y)) -> minus(x,y)
 plus(x,0()) -> x
 plus(x,s(y)) -> s(plus(x,y))
 mod(s(x),0()) -> 0()
 mod(x,s(y)) -> help(x,s(y),0())
 help(x,s(y),c) -> if(le(c,x),x,s(y),c)
 if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
 if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(s(x),s(y)) -> minus#(x,y)
   plus#(x,s(y)) -> plus#(x,y)
   mod#(x,s(y)) -> help#(x,s(y),0())
   help#(x,s(y),c) -> le#(c,x)
   help#(x,s(y),c) -> if#(le(c,x),x,s(y),c)
   if#(true(),x,s(y),c) -> plus#(c,s(y))
   if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
   if#(false(),x,s(y),c) -> minus#(c,s(y))
   if#(false(),x,s(y),c) -> minus#(x,minus(c,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(0(),s(y)) -> 0()
   minus(s(x),s(y)) -> minus(x,y)
   plus(x,0()) -> x
   plus(x,s(y)) -> s(plus(x,y))
   mod(s(x),0()) -> 0()
   mod(x,s(y)) -> help(x,s(y),0())
   help(x,s(y),c) -> if(le(c,x),x,s(y),c)
   if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
   if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y)
    plus#(x,s(y)) -> plus#(x,y)
    mod#(x,s(y)) -> help#(x,s(y),0())
    help#(x,s(y),c) -> le#(c,x)
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c)
    if#(true(),x,s(y),c) -> plus#(c,s(y))
    if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
    if#(false(),x,s(y),c) -> minus#(c,s(y))
    if#(false(),x,s(y),c) -> minus#(x,minus(c,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(0(),s(y)) -> 0()
    minus(s(x),s(y)) -> minus(x,y)
    plus(x,0()) -> x
    plus(x,s(y)) -> s(plus(x,y))
    mod(s(x),0()) -> 0()
    mod(x,s(y)) -> help(x,s(y),0())
    help(x,s(y),c) -> if(le(c,x),x,s(y),c)
    if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
    if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
   graph:
    if#(false(),x,s(y),c) -> minus#(x,minus(c,s(y))) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    if#(false(),x,s(y),c) -> minus#(c,s(y)) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) ->
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c)
    if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y))) ->
    help#(x,s(y),c) -> le#(c,x)
    if#(true(),x,s(y),c) -> plus#(c,s(y)) ->
    plus#(x,s(y)) -> plus#(x,y)
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c) ->
    if#(false(),x,s(y),c) -> minus#(x,minus(c,s(y)))
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c) ->
    if#(false(),x,s(y),c) -> minus#(c,s(y))
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c) ->
    if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c) ->
    if#(true(),x,s(y),c) -> plus#(c,s(y))
    help#(x,s(y),c) -> le#(c,x) -> le#(s(x),s(y)) -> le#(x,y)
    mod#(x,s(y)) -> help#(x,s(y),0()) ->
    help#(x,s(y),c) -> if#(le(c,x),x,s(y),c)
    mod#(x,s(y)) -> help#(x,s(y),0()) -> help#(x,s(y),c) -> le#(c,x)
    plus#(x,s(y)) -> plus#(x,y) -> plus#(x,s(y)) -> plus#(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: 4
    #rules: 5
    #arcs: 15/100
    DPs:
     if#(true(),x,s(y),c) -> help#(x,s(y),plus(c,s(y)))
     help#(x,s(y),c) -> if#(le(c,x),x,s(y),c)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(x,0()) -> x
     minus(0(),s(y)) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     mod(s(x),0()) -> 0()
     mod(x,s(y)) -> help(x,s(y),0())
     help(x,s(y),c) -> if(le(c,x),x,s(y),c)
     if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
     if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
    Open
    
    DPs:
     plus#(x,s(y)) -> plus#(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(0(),s(y)) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     mod(s(x),0()) -> 0()
     mod(x,s(y)) -> help(x,s(y),0())
     help(x,s(y),c) -> if(le(c,x),x,s(y),c)
     if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
     if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = [1]
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(minus) = 0
      pi(plus) = [0,1]
      pi(mod) = [0]
      pi(help) = 0
      pi(if) = 1
      pi(plus#) = 1
     weight function:
      w0 = 1
      w(plus#) = w(if) = w(help) = w(mod) = w(minus) = w(false) = w(s) = w(
      true) = w(le) = w(0) = 1
      w(plus) = 0
     precedence:
      plus# ~ mod ~ false ~ true > plus > if ~ help ~ minus ~ s ~ le ~ 0
     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(0(),s(y)) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       mod(s(x),0()) -> 0()
       mod(x,s(y)) -> help(x,s(y),0())
       help(x,s(y),c) -> if(le(c,x),x,s(y),c)
       if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
       if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
     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(0(),s(y)) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     mod(s(x),0()) -> 0()
     mod(x,s(y)) -> help(x,s(y),0())
     help(x,s(y),c) -> if(le(c,x),x,s(y),c)
     if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
     if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = 0
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(minus) = 0
      pi(plus) = [0,1]
      pi(mod) = 0
      pi(help) = 0
      pi(if) = 1
      pi(le#) = 1
     weight function:
      w0 = 1
      w(le#) = w(if) = w(help) = w(plus) = w(minus) = w(false) = w(s) = w(
      true) = w(le) = w(0) = 1
      w(mod) = 0
     precedence:
      le# ~ plus ~ minus ~ false ~ 0 > if ~ help ~ mod ~ s ~ true ~ le
     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(0(),s(y)) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       mod(s(x),0()) -> 0()
       mod(x,s(y)) -> help(x,s(y),0())
       help(x,s(y),c) -> if(le(c,x),x,s(y),c)
       if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
       if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
     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(0(),s(y)) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     plus(x,0()) -> x
     plus(x,s(y)) -> s(plus(x,y))
     mod(s(x),0()) -> 0()
     mod(x,s(y)) -> help(x,s(y),0())
     help(x,s(y),c) -> if(le(c,x),x,s(y),c)
     if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
     if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = 0
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(minus) = 0
      pi(plus) = [0,1]
      pi(mod) = 0
      pi(help) = 0
      pi(if) = 1
      pi(minus#) = 1
     weight function:
      w0 = 1
      w(minus#) = w(if) = w(help) = w(plus) = w(minus) = w(false) = w(
      s) = w(true) = w(le) = w(0) = 1
      w(mod) = 0
     precedence:
      minus# ~ plus ~ minus ~ false ~ 0 > if ~ help ~ mod ~ s ~ true ~ le
     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(0(),s(y)) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       plus(x,0()) -> x
       plus(x,s(y)) -> s(plus(x,y))
       mod(s(x),0()) -> 0()
       mod(x,s(y)) -> help(x,s(y),0())
       help(x,s(y),c) -> if(le(c,x),x,s(y),c)
       if(true(),x,s(y),c) -> help(x,s(y),plus(c,s(y)))
       if(false(),x,s(y),c) -> minus(x,minus(c,s(y)))
     Qed