YES

Problem:
 app(app(append(),nil()),ys) -> ys
 app(app(append(),app(app(cons(),x),xs)),ys) -> app(app(cons(),x),app(app(append(),xs),ys))
 app(app(flatwith(),f),app(leaf(),x)) -> app(app(cons(),app(f,x)),nil())
 app(app(flatwith(),f),app(node(),xs)) -> app(app(flatwithsub(),f),xs)
 app(app(flatwithsub(),f),nil()) -> nil()
 app(app(flatwithsub(),f),app(app(cons(),x),xs)) ->
 app(app(append(),app(app(flatwith(),f),x)),app(app(flatwithsub(),f),xs))

Proof:
 Uncurry Processor:
  append2(nil(),ys) -> ys
  append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
  flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
  flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
  flatwithsub2(f,nil()) -> nil()
  flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
  app(append1(x4),x5) -> append2(x4,x5)
  app(append(),x5) -> append1(x5)
  app(cons1(x4),x5) -> cons2(x4,x5)
  app(cons(),x5) -> cons1(x5)
  app(flatwith1(x4),x5) -> flatwith2(x4,x5)
  app(flatwith(),x5) -> flatwith1(x5)
  app(leaf(),x5) -> leaf1(x5)
  app(node(),x5) -> node1(x5)
  app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
  app(flatwithsub(),x5) -> flatwithsub1(x5)
  DP Processor:
   DPs:
    append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
    flatwith{2,#}(f,leaf1(x)) -> app#(f,x)
    flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs)
    flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
    flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
    flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs))
    app#(append1(x4),x5) -> append{2,#}(x4,x5)
    app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5)
    app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5)
   TRS:
    append2(nil(),ys) -> ys
    append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
    flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
    flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
    flatwithsub2(f,nil()) -> nil()
    flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
    app(append1(x4),x5) -> append2(x4,x5)
    app(append(),x5) -> append1(x5)
    app(cons1(x4),x5) -> cons2(x4,x5)
    app(cons(),x5) -> cons1(x5)
    app(flatwith1(x4),x5) -> flatwith2(x4,x5)
    app(flatwith(),x5) -> flatwith1(x5)
    app(leaf(),x5) -> leaf1(x5)
    app(node(),x5) -> node1(x5)
    app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
    app(flatwithsub(),x5) -> flatwithsub1(x5)
   TDG Processor:
    DPs:
     append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x)
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs)
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
     flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs))
     app#(append1(x4),x5) -> append{2,#}(x4,x5)
     app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5)
     app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5)
    TRS:
     append2(nil(),ys) -> ys
     append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
     flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
     flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
     flatwithsub2(f,nil()) -> nil()
     flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
     app(append1(x4),x5) -> append2(x4,x5)
     app(append(),x5) -> append1(x5)
     app(cons1(x4),x5) -> cons2(x4,x5)
     app(cons(),x5) -> cons1(x5)
     app(flatwith1(x4),x5) -> flatwith2(x4,x5)
     app(flatwith(),x5) -> flatwith1(x5)
     app(leaf(),x5) -> leaf1(x5)
     app(node(),x5) -> node1(x5)
     app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
     app(flatwithsub(),x5) -> flatwithsub1(x5)
    graph:
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs))
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x) ->
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs)
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x) ->
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x)
     flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs)) ->
     append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
     app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs))
     app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
     app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
     app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5) ->
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs)
     app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5) ->
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x)
     app#(append1(x4),x5) -> append{2,#}(x4,x5) ->
     append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> append{2,#}(flatwith2(f,x),flatwithsub2(f,xs))
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
     flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs) ->
     flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x) ->
     app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5)
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x) ->
     app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5)
     flatwith{2,#}(f,leaf1(x)) -> app#(f,x) ->
     app#(append1(x4),x5) -> append{2,#}(x4,x5)
     append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys) ->
     append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
    SCC Processor:
     #sccs: 2
     #rules: 7
     #arcs: 19/81
     DPs:
      flatwithsub{2,#}(f,cons2(x,xs)) -> flatwithsub{2,#}(f,xs)
      flatwithsub{2,#}(f,cons2(x,xs)) -> flatwith{2,#}(f,x)
      flatwith{2,#}(f,leaf1(x)) -> app#(f,x)
      app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5)
      flatwith{2,#}(f,node1(xs)) -> flatwithsub{2,#}(f,xs)
      app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5)
     TRS:
      append2(nil(),ys) -> ys
      append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
      flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
      flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
      flatwithsub2(f,nil()) -> nil()
      flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
      app(append1(x4),x5) -> append2(x4,x5)
      app(append(),x5) -> append1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
      app(flatwith1(x4),x5) -> flatwith2(x4,x5)
      app(flatwith(),x5) -> flatwith1(x5)
      app(leaf(),x5) -> leaf1(x5)
      app(node(),x5) -> node1(x5)
      app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
      app(flatwithsub(),x5) -> flatwithsub1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(flatwith{2,#}) = 1
       pi(app#) = 1
       pi(flatwithsub{2,#}) = 1
      problem:
       DPs:
        app#(flatwith1(x4),x5) -> flatwith{2,#}(x4,x5)
        app#(flatwithsub1(x4),x5) -> flatwithsub{2,#}(x4,x5)
       TRS:
        append2(nil(),ys) -> ys
        append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
        flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
        flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
        flatwithsub2(f,nil()) -> nil()
        flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
        app(append1(x4),x5) -> append2(x4,x5)
        app(append(),x5) -> append1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
        app(flatwith1(x4),x5) -> flatwith2(x4,x5)
        app(flatwith(),x5) -> flatwith1(x5)
        app(leaf(),x5) -> leaf1(x5)
        app(node(),x5) -> node1(x5)
        app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
        app(flatwithsub(),x5) -> flatwithsub1(x5)
      SCC Processor:
       #sccs: 0
       #rules: 0
       #arcs: 12/4
       
     
     DPs:
      append{2,#}(cons2(x,xs),ys) -> append{2,#}(xs,ys)
     TRS:
      append2(nil(),ys) -> ys
      append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
      flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
      flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
      flatwithsub2(f,nil()) -> nil()
      flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
      app(append1(x4),x5) -> append2(x4,x5)
      app(append(),x5) -> append1(x5)
      app(cons1(x4),x5) -> cons2(x4,x5)
      app(cons(),x5) -> cons1(x5)
      app(flatwith1(x4),x5) -> flatwith2(x4,x5)
      app(flatwith(),x5) -> flatwith1(x5)
      app(leaf(),x5) -> leaf1(x5)
      app(node(),x5) -> node1(x5)
      app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
      app(flatwithsub(),x5) -> flatwithsub1(x5)
     Subterm Criterion Processor:
      simple projection:
       pi(append{2,#}) = 0
      problem:
       DPs:
        
       TRS:
        append2(nil(),ys) -> ys
        append2(cons2(x,xs),ys) -> cons2(x,append2(xs,ys))
        flatwith2(f,leaf1(x)) -> cons2(app(f,x),nil())
        flatwith2(f,node1(xs)) -> flatwithsub2(f,xs)
        flatwithsub2(f,nil()) -> nil()
        flatwithsub2(f,cons2(x,xs)) -> append2(flatwith2(f,x),flatwithsub2(f,xs))
        app(append1(x4),x5) -> append2(x4,x5)
        app(append(),x5) -> append1(x5)
        app(cons1(x4),x5) -> cons2(x4,x5)
        app(cons(),x5) -> cons1(x5)
        app(flatwith1(x4),x5) -> flatwith2(x4,x5)
        app(flatwith(),x5) -> flatwith1(x5)
        app(leaf(),x5) -> leaf1(x5)
        app(node(),x5) -> node1(x5)
        app(flatwithsub1(x4),x5) -> flatwithsub2(x4,x5)
        app(flatwithsub(),x5) -> flatwithsub1(x5)
      Qed