MAYBE

Problem:
 times(x,y) -> help(x,y,0())
 help(x,y,c) -> if(lt(c,y),x,y,c)
 if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
 if(false(),x,y,c) -> 0()
 lt(0(),s(x)) -> true()
 lt(s(x),0()) -> false()
 lt(s(x),s(y)) -> lt(x,y)
 plus(x,0()) -> x
 plus(0(),x) -> x
 plus(x,s(y)) -> s(plus(x,y))
 plus(s(x),y) -> s(plus(x,y))

Proof:
 DP Processor:
  DPs:
   times#(x,y) -> help#(x,y,0())
   help#(x,y,c) -> lt#(c,y)
   help#(x,y,c) -> if#(lt(c,y),x,y,c)
   if#(true(),x,y,c) -> help#(x,y,s(c))
   if#(true(),x,y,c) -> plus#(x,help(x,y,s(c)))
   lt#(s(x),s(y)) -> lt#(x,y)
   plus#(x,s(y)) -> plus#(x,y)
   plus#(s(x),y) -> plus#(x,y)
  TRS:
   times(x,y) -> help(x,y,0())
   help(x,y,c) -> if(lt(c,y),x,y,c)
   if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
   if(false(),x,y,c) -> 0()
   lt(0(),s(x)) -> true()
   lt(s(x),0()) -> false()
   lt(s(x),s(y)) -> lt(x,y)
   plus(x,0()) -> x
   plus(0(),x) -> x
   plus(x,s(y)) -> s(plus(x,y))
   plus(s(x),y) -> s(plus(x,y))
  TDG Processor:
   DPs:
    times#(x,y) -> help#(x,y,0())
    help#(x,y,c) -> lt#(c,y)
    help#(x,y,c) -> if#(lt(c,y),x,y,c)
    if#(true(),x,y,c) -> help#(x,y,s(c))
    if#(true(),x,y,c) -> plus#(x,help(x,y,s(c)))
    lt#(s(x),s(y)) -> lt#(x,y)
    plus#(x,s(y)) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y)
   TRS:
    times(x,y) -> help(x,y,0())
    help(x,y,c) -> if(lt(c,y),x,y,c)
    if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
    if(false(),x,y,c) -> 0()
    lt(0(),s(x)) -> true()
    lt(s(x),0()) -> false()
    lt(s(x),s(y)) -> lt(x,y)
    plus(x,0()) -> x
    plus(0(),x) -> x
    plus(x,s(y)) -> s(plus(x,y))
    plus(s(x),y) -> s(plus(x,y))
   graph:
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(x,s(y)) -> plus#(x,y)
    plus#(x,s(y)) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
    plus#(x,s(y)) -> plus#(x,y) ->
    plus#(x,s(y)) -> plus#(x,y)
    if#(true(),x,y,c) -> plus#(x,help(x,y,s(c))) ->
    plus#(s(x),y) -> plus#(x,y)
    if#(true(),x,y,c) -> plus#(x,help(x,y,s(c))) ->
    plus#(x,s(y)) -> plus#(x,y)
    if#(true(),x,y,c) -> help#(x,y,s(c)) ->
    help#(x,y,c) -> if#(lt(c,y),x,y,c)
    if#(true(),x,y,c) -> help#(x,y,s(c)) -> help#(x,y,c) -> lt#(c,y)
    lt#(s(x),s(y)) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    help#(x,y,c) -> if#(lt(c,y),x,y,c) ->
    if#(true(),x,y,c) -> plus#(x,help(x,y,s(c)))
    help#(x,y,c) -> if#(lt(c,y),x,y,c) ->
    if#(true(),x,y,c) -> help#(x,y,s(c))
    help#(x,y,c) -> lt#(c,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    times#(x,y) -> help#(x,y,0()) ->
    help#(x,y,c) -> if#(lt(c,y),x,y,c)
    times#(x,y) -> help#(x,y,0()) -> help#(x,y,c) -> lt#(c,y)
   SCC Processor:
    #sccs: 3
    #rules: 5
    #arcs: 14/64
    DPs:
     if#(true(),x,y,c) -> help#(x,y,s(c))
     help#(x,y,c) -> if#(lt(c,y),x,y,c)
    TRS:
     times(x,y) -> help(x,y,0())
     help(x,y,c) -> if(lt(c,y),x,y,c)
     if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
     if(false(),x,y,c) -> 0()
     lt(0(),s(x)) -> true()
     lt(s(x),0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     plus(x,0()) -> x
     plus(0(),x) -> x
     plus(x,s(y)) -> s(plus(x,y))
     plus(s(x),y) -> s(plus(x,y))
    Open
    
    DPs:
     lt#(s(x),s(y)) -> lt#(x,y)
    TRS:
     times(x,y) -> help(x,y,0())
     help(x,y,c) -> if(lt(c,y),x,y,c)
     if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
     if(false(),x,y,c) -> 0()
     lt(0(),s(x)) -> true()
     lt(s(x),0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     plus(x,0()) -> x
     plus(0(),x) -> x
     plus(x,s(y)) -> s(plus(x,y))
     plus(s(x),y) -> s(plus(x,y))
    Subterm Criterion Processor:
     simple projection:
      pi(lt#) = 1
     problem:
      DPs:
       
      TRS:
       times(x,y) -> help(x,y,0())
       help(x,y,c) -> if(lt(c,y),x,y,c)
       if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
       if(false(),x,y,c) -> 0()
       lt(0(),s(x)) -> true()
       lt(s(x),0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       plus(x,0()) -> x
       plus(0(),x) -> x
       plus(x,s(y)) -> s(plus(x,y))
       plus(s(x),y) -> s(plus(x,y))
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
     plus#(x,s(y)) -> plus#(x,y)
    TRS:
     times(x,y) -> help(x,y,0())
     help(x,y,c) -> if(lt(c,y),x,y,c)
     if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
     if(false(),x,y,c) -> 0()
     lt(0(),s(x)) -> true()
     lt(s(x),0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     plus(x,0()) -> x
     plus(0(),x) -> x
     plus(x,s(y)) -> s(plus(x,y))
     plus(s(x),y) -> s(plus(x,y))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 1
     problem:
      DPs:
       plus#(s(x),y) -> plus#(x,y)
      TRS:
       times(x,y) -> help(x,y,0())
       help(x,y,c) -> if(lt(c,y),x,y,c)
       if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
       if(false(),x,y,c) -> 0()
       lt(0(),s(x)) -> true()
       lt(s(x),0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       plus(x,0()) -> x
       plus(0(),x) -> x
       plus(x,s(y)) -> s(plus(x,y))
       plus(s(x),y) -> s(plus(x,y))
     Subterm Criterion Processor:
      simple projection:
       pi(plus#) = 0
      problem:
       DPs:
        
       TRS:
        times(x,y) -> help(x,y,0())
        help(x,y,c) -> if(lt(c,y),x,y,c)
        if(true(),x,y,c) -> plus(x,help(x,y,s(c)))
        if(false(),x,y,c) -> 0()
        lt(0(),s(x)) -> true()
        lt(s(x),0()) -> false()
        lt(s(x),s(y)) -> lt(x,y)
        plus(x,0()) -> x
        plus(0(),x) -> x
        plus(x,s(y)) -> s(plus(x,y))
        plus(s(x),y) -> s(plus(x,y))
      Qed