YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 pred(s(x)) -> x
 minus(x,0()) -> x
 minus(x,s(y)) -> pred(minus(x,y))
 gcd(0(),y) -> y
 gcd(s(x),0()) -> s(x)
 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
 if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
 if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(x,s(y)) -> minus#(x,y)
   minus#(x,s(y)) -> pred#(minus(x,y))
   gcd#(s(x),s(y)) -> le#(y,x)
   gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
   if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
   if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
   if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
   if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   pred(s(x)) -> x
   minus(x,0()) -> x
   minus(x,s(y)) -> pred(minus(x,y))
   gcd(0(),y) -> y
   gcd(s(x),0()) -> s(x)
   gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
   if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
   if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> minus#(x,y)
    minus#(x,s(y)) -> pred#(minus(x,y))
    gcd#(s(x),s(y)) -> le#(y,x)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    pred(s(x)) -> x
    minus(x,0()) -> x
    minus(x,s(y)) -> pred(minus(x,y))
    gcd(0(),y) -> y
    gcd(s(x),0()) -> s(x)
    gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
    if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
    if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
   graph:
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) ->
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x)) ->
    gcd#(s(x),s(y)) -> le#(y,x)
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x) ->
    minus#(x,s(y)) -> pred#(minus(x,y))
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x) ->
    minus#(x,s(y)) -> minus#(x,y)
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) ->
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y)) ->
    gcd#(s(x),s(y)) -> le#(y,x)
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> pred#(minus(x,y))
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> minus#(x,y)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x)
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y)) ->
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y)
    gcd#(s(x),s(y)) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y)
    minus#(x,s(y)) -> minus#(x,y) ->
    minus#(x,s(y)) -> pred#(minus(x,y))
    minus#(x,s(y)) -> minus#(x,y) -> minus#(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: 3
    #rules: 5
    #arcs: 16/81
    DPs:
     if_gcd#(false(),s(x),s(y)) -> gcd#(minus(y,x),s(x))
     gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
     if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [if_gcd#](x0, x1, x2) = 3x0 + 1x1 + x2 + 0,
      
      [gcd#](x0, x1) = 3x0 + x1 + 0,
      
      [if_gcd](x0, x1, x2) = x0 + 3x1 + x2 + 0,
      
      [gcd](x0, x1) = 3x0 + x1 + 2,
      
      [minus](x0, x1) = x0 + 0,
      
      [pred](x0) = x0 + 0,
      
      [false] = 1,
      
      [s](x0) = 4x0 + 4,
      
      [true] = 2,
      
      [le](x0, x1) = 3x1 + 2,
      
      [0] = 0
     orientation:
      if_gcd#(false(),s(x),s(y)) = 5x + 4y + 5 >= 4x + 3y + 4 = gcd#(minus(y,x),s(x))
      
      gcd#(s(x),s(y)) = 7x + 4y + 7 >= 6x + 4y + 5 = if_gcd#(le(y,x),s(x),s(y))
      
      if_gcd#(true(),s(x),s(y)) = 5x + 4y + 5 >= 3x + 4y + 4 = gcd#(minus(x,y),s(y))
      
      le(0(),y) = 3y + 2 >= 2 = true()
      
      le(s(x),0()) = 3 >= 1 = false()
      
      le(s(x),s(y)) = 7y + 7 >= 3y + 2 = le(x,y)
      
      pred(s(x)) = 4x + 4 >= x = x
      
      minus(x,0()) = x + 0 >= x = x
      
      minus(x,s(y)) = x + 0 >= x + 0 = pred(minus(x,y))
      
      gcd(0(),y) = y + 3 >= y = y
      
      gcd(s(x),0()) = 7x + 7 >= 4x + 4 = s(x)
      
      gcd(s(x),s(y)) = 7x + 4y + 7 >= 7x + 4y + 7 = if_gcd(le(y,x),s(x),s(y))
      
      if_gcd(true(),s(x),s(y)) = 7x + 4y + 7 >= 3x + 4y + 4 = gcd(minus(x,y),s(y))
      
      if_gcd(false(),s(x),s(y)) = 7x + 4y + 7 >= 4x + 3y + 4 = gcd(minus(y,x),s(x))
     problem:
      DPs:
       gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
       if_gcd#(true(),s(x),s(y)) -> gcd#(minus(x,y),s(y))
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       gcd(0(),y) -> y
       gcd(s(x),0()) -> s(x)
       gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
       if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
       if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
     KBO Processor:
      argument filtering:
       pi(0) = []
       pi(le) = []
       pi(true) = []
       pi(s) = [0]
       pi(false) = []
       pi(pred) = 0
       pi(minus) = 0
       pi(gcd) = [0,1]
       pi(if_gcd) = [0,1,2]
       pi(gcd#) = 0
       pi(if_gcd#) = 1
      weight function:
       w0 = 1
       w(gcd) = w(false) = w(s) = w(true) = w(le) = w(0) = 1
       w(if_gcd#) = w(gcd#) = w(if_gcd) = w(minus) = w(pred) = 0
      precedence:
       le > gcd > if_gcd# ~ gcd# ~ if_gcd ~ minus ~ pred ~ false ~ s ~ true ~ 0
      problem:
       DPs:
        gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
       TRS:
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
        pred(s(x)) -> x
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        gcd(0(),y) -> y
        gcd(s(x),0()) -> s(x)
        gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
        if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
        if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 4/1
       
    
    DPs:
     minus#(x,s(y)) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       gcd(0(),y) -> y
       gcd(s(x),0()) -> s(x)
       gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
       if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
       if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
     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)
     pred(s(x)) -> x
     minus(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     gcd(0(),y) -> y
     gcd(s(x),0()) -> s(x)
     gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
     if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
     if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       pred(s(x)) -> x
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       gcd(0(),y) -> y
       gcd(s(x),0()) -> s(x)
       gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
       if_gcd(true(),s(x),s(y)) -> gcd(minus(x,y),s(y))
       if_gcd(false(),s(x),s(y)) -> gcd(minus(y,x),s(x))
     Qed