YES

Problem:
 f(f(a(),x),a()) -> f(f(f(x,a()),f(a(),a())),a())

Proof:
 Extended Uncurrying Processor:
  application symbol: f
  symbol table:
   a ==> a0/0 a1/1 a2/2
   
  uncurry-rules:
   f(a1(x1),x2) -> a2(x1,x2)
   f(a0(),x1) -> a1(x1)
  eta-rules:
   
  problem:
   a2(x,a0()) -> f(f(f(x,a0()),a1(a0())),a0())
   f(a1(x1),x2) -> a2(x1,x2)
   f(a0(),x1) -> a1(x1)
  DP Processor:
   DPs:
    a{2,#}(x,a0()) -> f#(x,a0())
    a{2,#}(x,a0()) -> f#(f(x,a0()),a1(a0()))
    a{2,#}(x,a0()) -> f#(f(f(x,a0()),a1(a0())),a0())
    f#(a1(x1),x2) -> a{2,#}(x1,x2)
   TRS:
    a2(x,a0()) -> f(f(f(x,a0()),a1(a0())),a0())
    f(a1(x1),x2) -> a2(x1,x2)
    f(a0(),x1) -> a1(x1)
   TDG Processor:
    DPs:
     a{2,#}(x,a0()) -> f#(x,a0())
     a{2,#}(x,a0()) -> f#(f(x,a0()),a1(a0()))
     a{2,#}(x,a0()) -> f#(f(f(x,a0()),a1(a0())),a0())
     f#(a1(x1),x2) -> a{2,#}(x1,x2)
    TRS:
     a2(x,a0()) -> f(f(f(x,a0()),a1(a0())),a0())
     f(a1(x1),x2) -> a2(x1,x2)
     f(a0(),x1) -> a1(x1)
    graph:
     f#(a1(x1),x2) -> a{2,#}(x1,x2) ->
     a{2,#}(x,a0()) -> f#(f(f(x,a0()),a1(a0())),a0())
     f#(a1(x1),x2) -> a{2,#}(x1,x2) ->
     a{2,#}(x,a0()) -> f#(f(x,a0()),a1(a0()))
     f#(a1(x1),x2) -> a{2,#}(x1,x2) ->
     a{2,#}(x,a0()) -> f#(x,a0())
     a{2,#}(x,a0()) -> f#(f(f(x,a0()),a1(a0())),a0()) ->
     f#(a1(x1),x2) -> a{2,#}(x1,x2)
     a{2,#}(x,a0()) -> f#(f(x,a0()),a1(a0())) ->
     f#(a1(x1),x2) -> a{2,#}(x1,x2)
     a{2,#}(x,a0()) -> f#(x,a0()) -> f#(a1(x1),x2) -> a{2,#}(x1,x2)
    Bounds Processor:
     bound: 0
     enrichment: match-dp
     automaton:
      final states: {9}
      transitions:
       a{2,0}(12,10) -> 13*
       a{2,0}(10,11) -> 14*
       f70() -> 12*
       a{1,0}(10) -> 11*
       f{#,0}(14,10) -> 9*
       f0(14,10) -> 13*
       f0(13,11) -> 14*
       f0(12,10) -> 13*
       a{0,0}() -> 10*
       11 -> 13*
     problem:
      DPs:
       a{2,#}(x,a0()) -> f#(x,a0())
       a{2,#}(x,a0()) -> f#(f(x,a0()),a1(a0()))
       f#(a1(x1),x2) -> a{2,#}(x1,x2)
      TRS:
       a2(x,a0()) -> f(f(f(x,a0()),a1(a0())),a0())
       f(a1(x1),x2) -> a2(x1,x2)
       f(a0(),x1) -> a1(x1)
     Bounds Processor:
      bound: 1
      enrichment: match
      automaton:
       final states: {11,10,8,7,4,1}
       transitions:
        a{2,1}(2,5) -> 9*
        f1(3,13) -> 6,3
        a{2,0}(2,5) -> 9*
        a{2,0}(3,3) -> 10*
        a{2,0}(3,2) -> 6*
        a{0,1}() -> 13*
        a{2,#,0}(3,3) -> 7*
        a{2,#,0}(2,5) -> 4,1
        a{2,#,0}(3,2) -> 1*
        a{1,0}(2) -> 5*
        a{1,0}(3) -> 11*
        f{#,0}(3,2) -> 1*
        f{#,0}(6,5) -> 1,4
        f0(9,2) -> 6,8
        f0(6,5) -> 9*
        f0(3,2) -> 6*
        a{0,0}() -> 2*
        a{2,#,1}(2,5) -> 4,1
        f180() -> 3*
        a{1,1}(13) -> 13*
        f{#,1}(3,13) -> 1,7
        4 -> 7*
        8 -> 10*
        1 -> 7*
        5 -> 6*
      problem:
       DPs:
        
       TRS:
        a2(x,a0()) -> f(f(f(x,a0()),a1(a0())),a0())
        f(a1(x1),x2) -> a2(x1,x2)
        f(a0(),x1) -> a1(x1)
      Qed