YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(x,0()) -> x
 minus(s(x),s(y)) -> 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),s(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(x,0()) -> x
   minus(s(x),s(y)) -> 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))
  EDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(s(x),s(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(x,0()) -> x
    minus(s(x),s(y)) -> 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)) -> le#(y,x)
    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) ->
    minus#(s(x),s(y)) -> minus#(x,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)) -> 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) ->
    minus#(s(x),s(y)) -> minus#(x,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)) -> 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#(false(),s(x),s(y)) -> minus#(y,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) -> le#(s(x),s(y)) -> le#(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: 3
    #rules: 5
    #arcs: 13/64
    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(x,0()) -> x
     minus(s(x),s(y)) -> 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:
     dimension: 1
     interpretation:
      [if_gcd#](x0, x1, x2) = x0 + x1 + x2,
      
      [gcd#](x0, x1) = x0 + x1 + 1,
      
      [if_gcd](x0, x1, x2) = x1 + x2 + 1,
      
      [gcd](x0, x1) = x0 + x1 + 1,
      
      [minus](x0, x1) = x0,
      
      [false] = 0,
      
      [s](x0) = x0 + 1,
      
      [true] = 1,
      
      [le](x0, x1) = 1,
      
      [0] = 0
     orientation:
      if_gcd#(false(),s(x),s(y)) = x + y + 2 >= x + y + 2 = gcd#(minus(y,x),s(x))
      
      gcd#(s(x),s(y)) = x + y + 3 >= x + y + 3 = if_gcd#(le(y,x),s(x),s(y))
      
      if_gcd#(true(),s(x),s(y)) = x + y + 3 >= x + y + 2 = gcd#(minus(x,y),s(y))
      
      le(0(),y) = 1 >= 1 = true()
      
      le(s(x),0()) = 1 >= 0 = false()
      
      le(s(x),s(y)) = 1 >= 1 = le(x,y)
      
      minus(x,0()) = x >= x = x
      
      minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
      
      gcd(0(),y) = y + 1 >= y = y
      
      gcd(s(x),0()) = x + 2 >= x + 1 = s(x)
      
      gcd(s(x),s(y)) = x + y + 3 >= x + y + 3 = if_gcd(le(y,x),s(x),s(y))
      
      if_gcd(true(),s(x),s(y)) = x + y + 3 >= x + y + 2 = gcd(minus(x,y),s(y))
      
      if_gcd(false(),s(x),s(y)) = x + y + 3 >= x + y + 2 = gcd(minus(y,x),s(x))
     problem:
      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))
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> 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:
      dimension: 1
      interpretation:
       [if_gcd#](x0, x1, x2) = x1 + x2,
       
       [gcd#](x0, x1) = x0 + x1 + 1,
       
       [if_gcd](x0, x1, x2) = x1 + x2,
       
       [gcd](x0, x1) = x0 + x1,
       
       [minus](x0, x1) = x0,
       
       [false] = 0,
       
       [s](x0) = x0 + 1,
       
       [true] = 0,
       
       [le](x0, x1) = x0,
       
       [0] = 0
      orientation:
       if_gcd#(false(),s(x),s(y)) = x + y + 2 >= x + y + 2 = gcd#(minus(y,x),s(x))
       
       gcd#(s(x),s(y)) = x + y + 3 >= x + y + 2 = if_gcd#(le(y,x),s(x),s(y))
       
       le(0(),y) = 0 >= 0 = true()
       
       le(s(x),0()) = x + 1 >= 0 = false()
       
       le(s(x),s(y)) = x + 1 >= x = le(x,y)
       
       minus(x,0()) = x >= x = x
       
       minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
       
       gcd(0(),y) = y >= y = y
       
       gcd(s(x),0()) = x + 1 >= x + 1 = s(x)
       
       gcd(s(x),s(y)) = x + y + 2 >= x + y + 2 = if_gcd(le(y,x),s(x),s(y))
       
       if_gcd(true(),s(x),s(y)) = x + y + 2 >= x + y + 1 = gcd(minus(x,y),s(y))
       
       if_gcd(false(),s(x),s(y)) = x + y + 2 >= x + y + 1 = gcd(minus(y,x),s(x))
      problem:
       DPs:
        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(x,0()) -> x
        minus(s(x),s(y)) -> 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:
       dimension: 1
       interpretation:
        [if_gcd#](x0, x1, x2) = x2,
        
        [gcd#](x0, x1) = x0,
        
        [if_gcd](x0, x1, x2) = x1 + x2,
        
        [gcd](x0, x1) = x0 + x1 + 1,
        
        [minus](x0, x1) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        if_gcd#(false(),s(x),s(y)) = y + 1 >= y = gcd#(minus(y,x),s(x))
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        gcd(0(),y) = y + 1 >= y = y
        
        gcd(s(x),0()) = x + 2 >= x + 1 = s(x)
        
        gcd(s(x),s(y)) = x + y + 3 >= x + y + 2 = if_gcd(le(y,x),s(x),s(y))
        
        if_gcd(true(),s(x),s(y)) = x + y + 2 >= x + y + 2 = gcd(minus(x,y),s(y))
        
        if_gcd(false(),s(x),s(y)) = x + y + 2 >= x + y + 2 = 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(x,0()) -> x
         minus(s(x),s(y)) -> 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(x,0()) -> x
     minus(s(x),s(y)) -> 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)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> 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),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(s(x),s(y)) -> 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)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> 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