MAYBE

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

Proof:
 DP Processor:
  DPs:
   lt#(s(x),s(y)) -> lt#(x,y)
   fac#(x) -> help#(x,0())
   help#(x,c) -> lt#(c,x)
   help#(x,c) -> if#(lt(c,x),x,c)
   if#(true(),x,c) -> help#(x,s(c))
  TRS:
   lt(0(),s(x)) -> true()
   lt(x,0()) -> false()
   lt(s(x),s(y)) -> lt(x,y)
   fac(x) -> help(x,0())
   help(x,c) -> if(lt(c,x),x,c)
   if(true(),x,c) -> times(s(c),help(x,s(c)))
   if(false(),x,c) -> s(0())
  Usable Rule Processor:
   DPs:
    lt#(s(x),s(y)) -> lt#(x,y)
    fac#(x) -> help#(x,0())
    help#(x,c) -> lt#(c,x)
    help#(x,c) -> if#(lt(c,x),x,c)
    if#(true(),x,c) -> help#(x,s(c))
   TRS:
    f13(x,y) -> x
    f13(x,y) -> y
    lt(0(),s(x)) -> true()
    lt(x,0()) -> false()
    lt(s(x),s(y)) -> lt(x,y)
   TDG Processor:
    DPs:
     lt#(s(x),s(y)) -> lt#(x,y)
     fac#(x) -> help#(x,0())
     help#(x,c) -> lt#(c,x)
     help#(x,c) -> if#(lt(c,x),x,c)
     if#(true(),x,c) -> help#(x,s(c))
    TRS:
     f13(x,y) -> x
     f13(x,y) -> y
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
    graph:
     if#(true(),x,c) -> help#(x,s(c)) ->
     help#(x,c) -> if#(lt(c,x),x,c)
     if#(true(),x,c) -> help#(x,s(c)) -> help#(x,c) -> lt#(c,x)
     help#(x,c) -> if#(lt(c,x),x,c) -> if#(true(),x,c) -> help#(x,s(c))
     help#(x,c) -> lt#(c,x) -> lt#(s(x),s(y)) -> lt#(x,y)
     fac#(x) -> help#(x,0()) -> help#(x,c) -> if#(lt(c,x),x,c)
     fac#(x) -> help#(x,0()) -> help#(x,c) -> lt#(c,x)
     lt#(s(x),s(y)) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    Restore Modifier:
     DPs:
      lt#(s(x),s(y)) -> lt#(x,y)
      fac#(x) -> help#(x,0())
      help#(x,c) -> lt#(c,x)
      help#(x,c) -> if#(lt(c,x),x,c)
      if#(true(),x,c) -> help#(x,s(c))
     TRS:
      lt(0(),s(x)) -> true()
      lt(x,0()) -> false()
      lt(s(x),s(y)) -> lt(x,y)
      fac(x) -> help(x,0())
      help(x,c) -> if(lt(c,x),x,c)
      if(true(),x,c) -> times(s(c),help(x,s(c)))
      if(false(),x,c) -> s(0())
     SCC Processor:
      #sccs: 2
      #rules: 3
      #arcs: 7/25
      DPs:
       if#(true(),x,c) -> help#(x,s(c))
       help#(x,c) -> if#(lt(c,x),x,c)
      TRS:
       lt(0(),s(x)) -> true()
       lt(x,0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       fac(x) -> help(x,0())
       help(x,c) -> if(lt(c,x),x,c)
       if(true(),x,c) -> times(s(c),help(x,s(c)))
       if(false(),x,c) -> s(0())
      Open
      
      DPs:
       lt#(s(x),s(y)) -> lt#(x,y)
      TRS:
       lt(0(),s(x)) -> true()
       lt(x,0()) -> false()
       lt(s(x),s(y)) -> lt(x,y)
       fac(x) -> help(x,0())
       help(x,c) -> if(lt(c,x),x,c)
       if(true(),x,c) -> times(s(c),help(x,s(c)))
       if(false(),x,c) -> s(0())
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [lt#](x0, x1) = x1 + 1,
        
        [times](x0, x1) = 1,
        
        [if](x0, x1, x2) = 1,
        
        [help](x0, x1) = 1,
        
        [fac](x0) = 1,
        
        [false] = 0,
        
        [true] = 0,
        
        [lt](x0, x1) = 0,
        
        [s](x0) = x0 + 1,
        
        [0] = 0
       orientation:
        lt#(s(x),s(y)) = y + 2 >= y + 1 = lt#(x,y)
        
        lt(0(),s(x)) = 0 >= 0 = true()
        
        lt(x,0()) = 0 >= 0 = false()
        
        lt(s(x),s(y)) = 0 >= 0 = lt(x,y)
        
        fac(x) = 1 >= 1 = help(x,0())
        
        help(x,c) = 1 >= 1 = if(lt(c,x),x,c)
        
        if(true(),x,c) = 1 >= 1 = times(s(c),help(x,s(c)))
        
        if(false(),x,c) = 1 >= 1 = s(0())
       problem:
        DPs:
         
        TRS:
         lt(0(),s(x)) -> true()
         lt(x,0()) -> false()
         lt(s(x),s(y)) -> lt(x,y)
         fac(x) -> help(x,0())
         help(x,c) -> if(lt(c,x),x,c)
         if(true(),x,c) -> times(s(c),help(x,s(c)))
         if(false(),x,c) -> s(0())
       Qed