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))
  EDG 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)) -> 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),y) -> le#(s(x),y)
    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#(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),y) -> le#(s(x),y)
    if_gcd#(true(),s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(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)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),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))
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [if_gcd#](x0, x1, x2) = x1 + x2,
      
      [gcd#](x0, x1) = x0 + x1,
      
      [if_gcd](x0, x1, x2) = x1 + x2,
      
      [gcd](x0, x1) = x0 + x1 + 1,
      
      [if_minus](x0, x1, x2) = x1,
      
      [minus](x0, x1) = x0,
      
      [false] = 0,
      
      [s](x0) = x0 + 1,
      
      [true] = 0,
      
      [le](x0, x1) = 1,
      
      [0] = 0
     orientation:
      if_gcd#(false(),s(x),s(y)) = x + y + 2 >= x + y + 1 = gcd#(minus(y,x),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))
      
      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) = x + 1 >= x + 1 = if_minus(le(s(x),y),s(x),y)
      
      if_minus(true(),s(x),y) = x + 1 >= 0 = 0()
      
      if_minus(false(),s(x),y) = x + 1 >= x + 1 = s(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:
       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(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:
      dimension: 1
      interpretation:
       [if_gcd#](x0, x1, x2) = 0,
       
       [gcd#](x0, x1) = 1,
       
       [if_gcd](x0, x1, x2) = 0,
       
       [gcd](x0, x1) = x1,
       
       [if_minus](x0, x1, x2) = 0,
       
       [minus](x0, x1) = 0,
       
       [false] = 0,
       
       [s](x0) = 0,
       
       [true] = 0,
       
       [le](x0, x1) = 0,
       
       [0] = 0
      orientation:
       gcd#(s(x),s(y)) = 1 >= 0 = if_gcd#(le(y,x),s(x),s(y))
       
       le(0(),y) = 0 >= 0 = true()
       
       le(s(x),0()) = 0 >= 0 = false()
       
       le(s(x),s(y)) = 0 >= 0 = le(x,y)
       
       minus(0(),y) = 0 >= 0 = 0()
       
       minus(s(x),y) = 0 >= 0 = if_minus(le(s(x),y),s(x),y)
       
       if_minus(true(),s(x),y) = 0 >= 0 = 0()
       
       if_minus(false(),s(x),y) = 0 >= 0 = s(minus(x,y))
       
       gcd(0(),y) = y >= y = y
       
       gcd(s(x),0()) = 0 >= 0 = s(x)
       
       gcd(s(x),s(y)) = 0 >= 0 = if_gcd(le(y,x),s(x),s(y))
       
       if_gcd(true(),s(x),s(y)) = 0 >= 0 = gcd(minus(x,y),s(y))
       
       if_gcd(false(),s(x),s(y)) = 0 >= 0 = 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:
     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))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if_minus#) = 1
     problem:
      DPs:
       minus#(s(x),y) -> if_minus#(le(s(x),y),s(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:
      dimension: 1
      interpretation:
       [if_minus#](x0, x1, x2) = 0,
       
       [minus#](x0, x1) = x0,
       
       [if_gcd](x0, x1, x2) = x0,
       
       [gcd](x0, x1) = x1,
       
       [if_minus](x0, x1, x2) = x0,
       
       [minus](x0, x1) = 1,
       
       [false] = 1,
       
       [s](x0) = 1,
       
       [true] = 1,
       
       [le](x0, x1) = 1,
       
       [0] = 1
      orientation:
       minus#(s(x),y) = 1 >= 0 = if_minus#(le(s(x),y),s(x),y)
       
       le(0(),y) = 1 >= 1 = true()
       
       le(s(x),0()) = 1 >= 1 = false()
       
       le(s(x),s(y)) = 1 >= 1 = le(x,y)
       
       minus(0(),y) = 1 >= 1 = 0()
       
       minus(s(x),y) = 1 >= 1 = if_minus(le(s(x),y),s(x),y)
       
       if_minus(true(),s(x),y) = 1 >= 1 = 0()
       
       if_minus(false(),s(x),y) = 1 >= 1 = s(minus(x,y))
       
       gcd(0(),y) = y >= y = y
       
       gcd(s(x),0()) = 1 >= 1 = s(x)
       
       gcd(s(x),s(y)) = 1 >= 1 = if_gcd(le(y,x),s(x),s(y))
       
       if_gcd(true(),s(x),s(y)) = 1 >= 1 = gcd(minus(x,y),s(y))
       
       if_gcd(false(),s(x),s(y)) = 1 >= 1 = 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))
    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(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