YES

Problem:
 a__first(0(),X) -> nil()
 a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
 a__from(X) -> cons(mark(X),from(s(X)))
 mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
 mark(from(X)) -> a__from(mark(X))
 mark(0()) -> 0()
 mark(nil()) -> nil()
 mark(s(X)) -> s(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 a__first(X1,X2) -> first(X1,X2)
 a__from(X) -> from(X)

Proof:
 DP Processor:
  DPs:
   a__first#(s(X),cons(Y,Z)) -> mark#(Y)
   a__from#(X) -> mark#(X)
   mark#(first(X1,X2)) -> mark#(X2)
   mark#(first(X1,X2)) -> mark#(X1)
   mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
   mark#(from(X)) -> mark#(X)
   mark#(from(X)) -> a__from#(mark(X))
   mark#(s(X)) -> mark#(X)
   mark#(cons(X1,X2)) -> mark#(X1)
  TRS:
   a__first(0(),X) -> nil()
   a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
   a__from(X) -> cons(mark(X),from(s(X)))
   mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0()) -> 0()
   mark(nil()) -> nil()
   mark(s(X)) -> s(mark(X))
   mark(cons(X1,X2)) -> cons(mark(X1),X2)
   a__first(X1,X2) -> first(X1,X2)
   a__from(X) -> from(X)
  EDG Processor:
   DPs:
    a__first#(s(X),cons(Y,Z)) -> mark#(Y)
    a__from#(X) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> a__from#(mark(X))
    mark#(s(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1)
   TRS:
    a__first(0(),X) -> nil()
    a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
    a__from(X) -> cons(mark(X),from(s(X)))
    mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
    mark(from(X)) -> a__from(mark(X))
    mark(0()) -> 0()
    mark(nil()) -> nil()
    mark(s(X)) -> s(mark(X))
    mark(cons(X1,X2)) -> cons(mark(X1),X2)
    a__first(X1,X2) -> first(X1,X2)
    a__from(X) -> from(X)
   graph:
    a__from#(X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2)
    a__from#(X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    a__from#(X) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2)
    mark#(from(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    mark#(from(X)) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(first(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(first(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X2) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(first(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(first(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(first(X1,X2)) -> mark#(X1) ->
    mark#(cons(X1,X2)) -> mark#(X1)
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) ->
    a__first#(s(X),cons(Y,Z)) -> mark#(Y)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X2)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> mark#(X1)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) ->
    mark#(from(X)) -> a__from#(mark(X))
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X)
    mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2)
    mark#(s(X)) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1)
    mark#(s(X)) -> mark#(X) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X))
    mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X)
    mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(first(X1,X2)) -> mark#(X2)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(first(X1,X2)) -> mark#(X1)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(from(X)) -> mark#(X)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(from(X)) -> a__from#(mark(X))
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) ->
    mark#(s(X)) -> mark#(X)
    a__first#(s(X),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1)
   SCC Processor:
    #sccs: 1
    #rules: 9
    #arcs: 51/81
    DPs:
     a__from#(X) -> mark#(X)
     mark#(cons(X1,X2)) -> mark#(X1)
     mark#(s(X)) -> mark#(X)
     mark#(from(X)) -> a__from#(mark(X))
     mark#(from(X)) -> mark#(X)
     mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
     a__first#(s(X),cons(Y,Z)) -> mark#(Y)
     mark#(first(X1,X2)) -> mark#(X1)
     mark#(first(X1,X2)) -> mark#(X2)
    TRS:
     a__first(0(),X) -> nil()
     a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
     a__from(X) -> cons(mark(X),from(s(X)))
     mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
     mark(from(X)) -> a__from(mark(X))
     mark(0()) -> 0()
     mark(nil()) -> nil()
     mark(s(X)) -> s(mark(X))
     mark(cons(X1,X2)) -> cons(mark(X1),X2)
     a__first(X1,X2) -> first(X1,X2)
     a__from(X) -> from(X)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [a__from#](x0) = x0 + 1,
      
      [mark#](x0) = x0,
      
      [a__first#](x0, x1) = x0 + x1,
      
      [from](x0) = x0 + 1,
      
      [a__from](x0) = x0 + 1,
      
      [first](x0, x1) = x0 + x1,
      
      [mark](x0) = x0,
      
      [cons](x0, x1) = x0 + 1,
      
      [s](x0) = x0 + 1,
      
      [nil] = 0,
      
      [a__first](x0, x1) = x0 + x1,
      
      [0] = 0
     orientation:
      a__from#(X) = X + 1 >= X = mark#(X)
      
      mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1)
      
      mark#(s(X)) = X + 1 >= X = mark#(X)
      
      mark#(from(X)) = X + 1 >= X + 1 = a__from#(mark(X))
      
      mark#(from(X)) = X + 1 >= X = mark#(X)
      
      mark#(first(X1,X2)) = X1 + X2 >= X1 + X2 = a__first#(mark(X1),mark(X2))
      
      a__first#(s(X),cons(Y,Z)) = X + Y + 2 >= Y = mark#(Y)
      
      mark#(first(X1,X2)) = X1 + X2 >= X1 = mark#(X1)
      
      mark#(first(X1,X2)) = X1 + X2 >= X2 = mark#(X2)
      
      a__first(0(),X) = X >= 0 = nil()
      
      a__first(s(X),cons(Y,Z)) = X + Y + 2 >= Y + 1 = cons(mark(Y),first(X,Z))
      
      a__from(X) = X + 1 >= X + 1 = cons(mark(X),from(s(X)))
      
      mark(first(X1,X2)) = X1 + X2 >= X1 + X2 = a__first(mark(X1),mark(X2))
      
      mark(from(X)) = X + 1 >= X + 1 = a__from(mark(X))
      
      mark(0()) = 0 >= 0 = 0()
      
      mark(nil()) = 0 >= 0 = nil()
      
      mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
      
      mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2)
      
      a__first(X1,X2) = X1 + X2 >= X1 + X2 = first(X1,X2)
      
      a__from(X) = X + 1 >= X + 1 = from(X)
     problem:
      DPs:
       mark#(from(X)) -> a__from#(mark(X))
       mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
       mark#(first(X1,X2)) -> mark#(X1)
       mark#(first(X1,X2)) -> mark#(X2)
      TRS:
       a__first(0(),X) -> nil()
       a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
       a__from(X) -> cons(mark(X),from(s(X)))
       mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
       mark(from(X)) -> a__from(mark(X))
       mark(0()) -> 0()
       mark(nil()) -> nil()
       mark(s(X)) -> s(mark(X))
       mark(cons(X1,X2)) -> cons(mark(X1),X2)
       a__first(X1,X2) -> first(X1,X2)
       a__from(X) -> from(X)
     Matrix Interpretation Processor:
      dimension: 1
      interpretation:
       [a__from#](x0) = 0,
       
       [mark#](x0) = x0,
       
       [a__first#](x0, x1) = 0,
       
       [from](x0) = 1,
       
       [a__from](x0) = 1,
       
       [first](x0, x1) = x0 + x1,
       
       [mark](x0) = x0,
       
       [cons](x0, x1) = 0,
       
       [s](x0) = 0,
       
       [nil] = 0,
       
       [a__first](x0, x1) = x0 + x1,
       
       [0] = 0
      orientation:
       mark#(from(X)) = 1 >= 0 = a__from#(mark(X))
       
       mark#(first(X1,X2)) = X1 + X2 >= 0 = a__first#(mark(X1),mark(X2))
       
       mark#(first(X1,X2)) = X1 + X2 >= X1 = mark#(X1)
       
       mark#(first(X1,X2)) = X1 + X2 >= X2 = mark#(X2)
       
       a__first(0(),X) = X >= 0 = nil()
       
       a__first(s(X),cons(Y,Z)) = 0 >= 0 = cons(mark(Y),first(X,Z))
       
       a__from(X) = 1 >= 0 = cons(mark(X),from(s(X)))
       
       mark(first(X1,X2)) = X1 + X2 >= X1 + X2 = a__first(mark(X1),mark(X2))
       
       mark(from(X)) = 1 >= 1 = a__from(mark(X))
       
       mark(0()) = 0 >= 0 = 0()
       
       mark(nil()) = 0 >= 0 = nil()
       
       mark(s(X)) = 0 >= 0 = s(mark(X))
       
       mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2)
       
       a__first(X1,X2) = X1 + X2 >= X1 + X2 = first(X1,X2)
       
       a__from(X) = 1 >= 1 = from(X)
      problem:
       DPs:
        mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2))
        mark#(first(X1,X2)) -> mark#(X1)
        mark#(first(X1,X2)) -> mark#(X2)
       TRS:
        a__first(0(),X) -> nil()
        a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
        a__from(X) -> cons(mark(X),from(s(X)))
        mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
        mark(from(X)) -> a__from(mark(X))
        mark(0()) -> 0()
        mark(nil()) -> nil()
        mark(s(X)) -> s(mark(X))
        mark(cons(X1,X2)) -> cons(mark(X1),X2)
        a__first(X1,X2) -> first(X1,X2)
        a__from(X) -> from(X)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [mark#](x0) = 1,
        
        [a__first#](x0, x1) = 0,
        
        [from](x0) = 0,
        
        [a__from](x0) = 0,
        
        [first](x0, x1) = 0,
        
        [mark](x0) = 0,
        
        [cons](x0, x1) = 0,
        
        [s](x0) = 0,
        
        [nil] = 0,
        
        [a__first](x0, x1) = 0,
        
        [0] = 0
       orientation:
        mark#(first(X1,X2)) = 1 >= 0 = a__first#(mark(X1),mark(X2))
        
        mark#(first(X1,X2)) = 1 >= 1 = mark#(X1)
        
        mark#(first(X1,X2)) = 1 >= 1 = mark#(X2)
        
        a__first(0(),X) = 0 >= 0 = nil()
        
        a__first(s(X),cons(Y,Z)) = 0 >= 0 = cons(mark(Y),first(X,Z))
        
        a__from(X) = 0 >= 0 = cons(mark(X),from(s(X)))
        
        mark(first(X1,X2)) = 0 >= 0 = a__first(mark(X1),mark(X2))
        
        mark(from(X)) = 0 >= 0 = a__from(mark(X))
        
        mark(0()) = 0 >= 0 = 0()
        
        mark(nil()) = 0 >= 0 = nil()
        
        mark(s(X)) = 0 >= 0 = s(mark(X))
        
        mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2)
        
        a__first(X1,X2) = 0 >= 0 = first(X1,X2)
        
        a__from(X) = 0 >= 0 = from(X)
       problem:
        DPs:
         mark#(first(X1,X2)) -> mark#(X1)
         mark#(first(X1,X2)) -> mark#(X2)
        TRS:
         a__first(0(),X) -> nil()
         a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
         a__from(X) -> cons(mark(X),from(s(X)))
         mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
         mark(from(X)) -> a__from(mark(X))
         mark(0()) -> 0()
         mark(nil()) -> nil()
         mark(s(X)) -> s(mark(X))
         mark(cons(X1,X2)) -> cons(mark(X1),X2)
         a__first(X1,X2) -> first(X1,X2)
         a__from(X) -> from(X)
       Matrix Interpretation Processor:
        dimension: 1
        interpretation:
         [mark#](x0) = x0 + 1,
         
         [from](x0) = 0,
         
         [a__from](x0) = 0,
         
         [first](x0, x1) = x0 + x1 + 1,
         
         [mark](x0) = x0,
         
         [cons](x0, x1) = 0,
         
         [s](x0) = 0,
         
         [nil] = 0,
         
         [a__first](x0, x1) = x0 + x1 + 1,
         
         [0] = 1
        orientation:
         mark#(first(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1)
         
         mark#(first(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2)
         
         a__first(0(),X) = X + 2 >= 0 = nil()
         
         a__first(s(X),cons(Y,Z)) = 1 >= 0 = cons(mark(Y),first(X,Z))
         
         a__from(X) = 0 >= 0 = cons(mark(X),from(s(X)))
         
         mark(first(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__first(mark(X1),mark(X2))
         
         mark(from(X)) = 0 >= 0 = a__from(mark(X))
         
         mark(0()) = 1 >= 1 = 0()
         
         mark(nil()) = 0 >= 0 = nil()
         
         mark(s(X)) = 0 >= 0 = s(mark(X))
         
         mark(cons(X1,X2)) = 0 >= 0 = cons(mark(X1),X2)
         
         a__first(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = first(X1,X2)
         
         a__from(X) = 0 >= 0 = from(X)
        problem:
         DPs:
          
         TRS:
          a__first(0(),X) -> nil()
          a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
          a__from(X) -> cons(mark(X),from(s(X)))
          mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
          mark(from(X)) -> a__from(mark(X))
          mark(0()) -> 0()
          mark(nil()) -> nil()
          mark(s(X)) -> s(mark(X))
          mark(cons(X1,X2)) -> cons(mark(X1),X2)
          a__first(X1,X2) -> first(X1,X2)
          a__from(X) -> from(X)
        Qed