YES

Problem:
 a(a(y,0()),0()) -> y
 c(c(y)) -> y
 c(a(c(c(y)),x)) -> a(c(c(c(a(x,0())))),y)

Proof:
 DP Processor:
  DPs:
   c#(a(c(c(y)),x)) -> a#(x,0())
   c#(a(c(c(y)),x)) -> c#(a(x,0()))
   c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
   c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
   c#(a(c(c(y)),x)) -> a#(c(c(c(a(x,0())))),y)
  TRS:
   a(a(y,0()),0()) -> y
   c(c(y)) -> y
   c(a(c(c(y)),x)) -> a(c(c(c(a(x,0())))),y)
  TDG Processor:
   DPs:
    c#(a(c(c(y)),x)) -> a#(x,0())
    c#(a(c(c(y)),x)) -> c#(a(x,0()))
    c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
    c#(a(c(c(y)),x)) -> a#(c(c(c(a(x,0())))),y)
   TRS:
    a(a(y,0()),0()) -> y
    c(c(y)) -> y
    c(a(c(c(y)),x)) -> a(c(c(c(a(x,0())))),y)
   graph:
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0())))) ->
    c#(a(c(c(y)),x)) -> a#(c(c(c(a(x,0())))),y)
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0())))) ->
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0())))) ->
    c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0())))) ->
    c#(a(c(c(y)),x)) -> c#(a(x,0()))
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0())))) ->
    c#(a(c(c(y)),x)) -> a#(x,0())
    c#(a(c(c(y)),x)) -> c#(c(a(x,0()))) ->
    c#(a(c(c(y)),x)) -> a#(c(c(c(a(x,0())))),y)
    c#(a(c(c(y)),x)) -> c#(c(a(x,0()))) ->
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
    c#(a(c(c(y)),x)) -> c#(c(a(x,0()))) ->
    c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
    c#(a(c(c(y)),x)) -> c#(c(a(x,0()))) ->
    c#(a(c(c(y)),x)) -> c#(a(x,0()))
    c#(a(c(c(y)),x)) -> c#(c(a(x,0()))) ->
    c#(a(c(c(y)),x)) -> a#(x,0())
    c#(a(c(c(y)),x)) -> c#(a(x,0())) ->
    c#(a(c(c(y)),x)) -> a#(c(c(c(a(x,0())))),y)
    c#(a(c(c(y)),x)) -> c#(a(x,0())) ->
    c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
    c#(a(c(c(y)),x)) -> c#(a(x,0())) ->
    c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
    c#(a(c(c(y)),x)) -> c#(a(x,0())) ->
    c#(a(c(c(y)),x)) -> c#(a(x,0()))
    c#(a(c(c(y)),x)) -> c#(a(x,0())) -> c#(a(c(c(y)),x)) -> a#(x,0())
   SCC Processor:
    #sccs: 1
    #rules: 3
    #arcs: 15/25
    DPs:
     c#(a(c(c(y)),x)) -> c#(c(c(a(x,0()))))
     c#(a(c(c(y)),x)) -> c#(a(x,0()))
     c#(a(c(c(y)),x)) -> c#(c(a(x,0())))
    TRS:
     a(a(y,0()),0()) -> y
     c(c(y)) -> y
     c(a(c(c(y)),x)) -> a(c(c(c(a(x,0())))),y)
    Matrix Interpretation Processor: dim=3
     
     interpretation:
      [c#](x0) = [2 0 0]x0,
      
                [1 0 1]     [0]
      [c](x0) = [2 0 2]x0 + [2]
                [0 1 0]     [0],
      
                    [0 0 1]     [3 1 2]  
      [a](x0, x1) = [0 1 0]x0 + [3 1 3]x1
                    [1 0 0]     [0 1 1]  ,
      
            [0]
      [0] = [0]
            [0]
     orientation:
      c#(a(c(c(y)),x)) = [6 2 4]x + [4 0 4]y + [4] >= [2 2 2]x = c#(c(c(a(x,0()))))
      
      c#(a(c(c(y)),x)) = [6 2 4]x + [4 0 4]y + [4] >= [0 0 2]x = c#(a(x,0()))
      
      c#(a(c(c(y)),x)) = [6 2 4]x + [4 0 4]y + [4] >= [2 0 2]x = c#(c(a(x,0())))
      
                                  
      a(a(y,0()),0()) = y >= y = y
                                  
      
                [1 1 1]    [0]         
      c(c(y)) = [2 2 2]y + [2] >= y = y
                [2 0 2]    [2]         
      
                        [3 2 3]    [3 1 3]    [2]    [2 2 2]    [3 1 2]    [2]                         
      c(a(c(c(y)),x)) = [6 4 6]x + [6 2 6]y + [6] >= [6 2 6]x + [3 1 3]y + [6] = a(c(c(c(a(x,0())))),y)
                        [3 1 3]    [2 2 2]    [2]    [3 1 3]    [0 1 1]    [2]                         
     problem:
      DPs:
       
      TRS:
       a(a(y,0()),0()) -> y
       c(c(y)) -> y
       c(a(c(c(y)),x)) -> a(c(c(c(a(x,0())))),y)
     Qed