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))
    Usable Rule Processor:
     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:
      minus(x,0()) -> x
      minus(x,s(y)) -> pred(minus(x,y))
      pred(s(x)) -> x
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       minus(x,0()) -> x
       minus(x,s(y)) -> pred(minus(x,y))
       pred(s(x)) -> x
      interpretation:
       [if_gcd#](x0, x1, x2) = 1x1 + x2,
       
       [gcd#](x0, x1) = 1x0 + x1,
       
       [minus](x0, x1) = x0 + x1,
       
       [pred](x0) = x0,
       
       [false] = 1,
       
       [s](x0) = 2x0,
       
       [true] = 1,
       
       [le](x0, x1) = 2x1 + 0,
       
       [0] = 1
      orientation:
       if_gcd#(false(),s(x),s(y)) = 3x + 2y >= 2x + 1y = gcd#(minus(y,x),s(x))
       
       gcd#(s(x),s(y)) = 3x + 2y >= 3x + 2y = if_gcd#(le(y,x),s(x),s(y))
       
       if_gcd#(true(),s(x),s(y)) = 3x + 2y >= 1x + 2y = gcd#(minus(x,y),s(y))
       
       minus(x,0()) = x + 1 >= x = x
       
       minus(x,s(y)) = x + 2y >= x + y = pred(minus(x,y))
       
       pred(s(x)) = 2x >= x = x
       
       le(0(),y) = 2y + 0 >= 1 = true()
       
       le(s(x),0()) = 3 >= 1 = false()
       
       le(s(x),s(y)) = 4y + 0 >= 2y + 0 = le(x,y)
      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:
        minus(x,0()) -> x
        minus(x,s(y)) -> pred(minus(x,y))
        pred(s(x)) -> x
        le(0(),y) -> true()
        le(s(x),0()) -> false()
        le(s(x),s(y)) -> le(x,y)
      Restore Modifier:
       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))
       Usable Rule Processor:
        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)
         minus(x,0()) -> x
         minus(x,s(y)) -> pred(minus(x,y))
         pred(s(x)) -> x
        Arctic Interpretation Processor:
         dimension: 1
         usable rules:
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          minus(x,0()) -> x
          minus(x,s(y)) -> pred(minus(x,y))
          pred(s(x)) -> x
         interpretation:
          [if_gcd#](x0, x1, x2) = x0 + x1 + 0,
          
          [gcd#](x0, x1) = x0 + 0,
          
          [minus](x0, x1) = x0 + 0,
          
          [pred](x0) = x0 + 0,
          
          [false] = 0,
          
          [s](x0) = 1x0 + 2,
          
          [true] = 1,
          
          [le](x0, x1) = x1 + 1,
          
          [0] = 0
         orientation:
          gcd#(s(x),s(y)) = 1x + 2 >= 1x + 2 = if_gcd#(le(y,x),s(x),s(y))
          
          if_gcd#(true(),s(x),s(y)) = 1x + 2 >= x + 0 = gcd#(minus(x,y),s(y))
          
          le(0(),y) = y + 1 >= 1 = true()
          
          le(s(x),0()) = 1 >= 0 = false()
          
          le(s(x),s(y)) = 1y + 2 >= y + 1 = le(x,y)
          
          minus(x,0()) = x + 0 >= x = x
          
          minus(x,s(y)) = x + 0 >= x + 0 = pred(minus(x,y))
          
          pred(s(x)) = 1x + 2 >= x = 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(x,0()) -> x
           minus(x,s(y)) -> pred(minus(x,y))
           pred(s(x)) -> x
         Restore Modifier:
          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