YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(0(),y) -> 0()
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
 if_minus(true(),s(x),y) -> 0()
 if_minus(false(),s(x),y) -> s(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#(s(x),y) -> le#(s(x),y)
   minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
   if_minus#(false(),s(x),y) -> 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)
   minus(0(),y) -> 0()
   minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
   if_minus(true(),s(x),y) -> 0()
   if_minus(false(),s(x),y) -> s(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#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> 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)
    minus(0(),y) -> 0()
    minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
    if_minus(true(),s(x),y) -> 0()
    if_minus(false(),s(x),y) -> s(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#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_gcd#(false(),s(x),s(y)) -> minus#(y,x) ->
    minus#(s(x),y) -> le#(s(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#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(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)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) ->
    if_minus#(false(),s(x),y) -> minus#(x,y)
    minus#(s(x),y) -> le#(s(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)
   SCC Processor:
    #sccs: 3
    #rules: 6
    #arcs: 18/100
    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)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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) = 0
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(minus) = 0
      pi(if_minus) = 1
      pi(gcd) = [0,1]
      pi(if_gcd) = [1,2]
      pi(gcd#) = [0,1]
      pi(if_gcd#) = [1,2]
     weight function:
      w0 = 1
      w(gcd#) = w(false) = w(s) = w(true) = w(0) = 1
      w(if_gcd#) = w(if_gcd) = w(gcd) = w(if_minus) = w(minus) = w(le) = 0
     precedence:
      gcd > if_gcd# ~ 0 > gcd# ~ if_gcd ~ if_minus ~ minus ~ false ~ s ~ true ~ le
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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:
     minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
     if_minus#(false(),s(x),y) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [if_minus#](x0, x1, x2) = x0 + 4x1 + 3,
      
      [minus#](x0, x1) = 4x0 + 6,
      
      [if_gcd](x0, x1, x2) = 2x0 + x1 + x2 + 1,
      
      [gcd](x0, x1) = 4x0 + x1 + 2,
      
      [if_minus](x0, x1, x2) = x1,
      
      [minus](x0, x1) = x0,
      
      [false] = 0,
      
      [s](x0) = 4x0 + 1,
      
      [true] = 0,
      
      [le](x0, x1) = 1,
      
      [0] = 0
     orientation:
      minus#(s(x),y) = 16x + 10 >= 16x + 8 = if_minus#(le(s(x),y),s(x),y)
      
      if_minus#(false(),s(x),y) = 16x + 7 >= 4x + 6 = minus#(x,y)
      
      le(0(),y) = 1 >= 0 = true()
      
      le(s(x),0()) = 1 >= 0 = false()
      
      le(s(x),s(y)) = 1 >= 1 = le(x,y)
      
      minus(0(),y) = 0 >= 0 = 0()
      
      minus(s(x),y) = 4x + 1 >= 4x + 1 = if_minus(le(s(x),y),s(x),y)
      
      if_minus(true(),s(x),y) = 4x + 1 >= 0 = 0()
      
      if_minus(false(),s(x),y) = 4x + 1 >= 4x + 1 = s(minus(x,y))
      
      gcd(0(),y) = y + 2 >= y = y
      
      gcd(s(x),0()) = 16x + 6 >= 4x + 1 = s(x)
      
      gcd(s(x),s(y)) = 16x + 4y + 7 >= 4x + 4y + 5 = if_gcd(le(y,x),s(x),s(y))
      
      if_gcd(true(),s(x),s(y)) = 4x + 4y + 3 >= 4x + 4y + 3 = gcd(minus(x,y),s(y))
      
      if_gcd(false(),s(x),s(y)) = 4x + 4y + 3 >= 4x + 4y + 3 = gcd(minus(y,x),s(x))
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(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(minus) = 0
      pi(if_minus) = 1
      pi(gcd) = [0,1]
      pi(if_gcd) = [1,2]
      pi(le#) = 1
     weight function:
      w0 = 1
      w(le#) = w(gcd) = w(if_minus) = w(false) = w(s) = w(true) = w(le) = w(
      0) = 1
      w(if_gcd) = w(minus) = 0
     precedence:
      le# ~ if_gcd ~ le ~ 0 > gcd ~ if_minus ~ minus ~ false ~ s ~ true
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(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