YES

Proof:
This system is quasi-decreasing.
By \cite{O02}, p. 214, Proposition 7.2.50.
This system is of type 3 or smaller.
This system is deterministic.
System R transformed to U(R).
Call external tool:
ttt2 - trs 30
Input:
  size(empty) -> 0
  size(push(x, y)) -> s(size(x))
  m -> s(s(s(s(0))))
  pop(empty) -> empty
  ?1(true, x, y) -> x
  pop(push(x, y)) -> ?1(le(size(x), m), x, y)
  top(empty) -> eentry
  ?2(true, x, y) -> y
  top(push(x, y)) -> ?2(le(size(x), m), x, y)
  le(x, 0) -> false
  le(0, s(x)) -> true
  le(s(x), s(y)) -> le(x, y)

 DP Processor:
  DPs:
   size#(push(x,y)) -> size#(x)
   pop#(push(x,y)) -> m#()
   pop#(push(x,y)) -> size#(x)
   pop#(push(x,y)) -> le#(size(x),m())
   pop#(push(x,y)) -> ?1#(le(size(x),m()),x,y)
   top#(push(x,y)) -> m#()
   top#(push(x,y)) -> size#(x)
   top#(push(x,y)) -> le#(size(x),m())
   top#(push(x,y)) -> ?2#(le(size(x),m()),x,y)
   le#(s(x),s(y)) -> le#(x,y)
  TRS:
   size(empty()) -> 0()
   size(push(x,y)) -> s(size(x))
   m() -> s(s(s(s(0()))))
   pop(empty()) -> empty()
   ?1(true(),x,y) -> x
   pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
   top(empty()) -> eentry()
   ?2(true(),x,y) -> y
   top(push(x,y)) -> ?2(le(size(x),m()),x,y)
   le(x,0()) -> false()
   le(0(),s(x)) -> true()
   le(s(x),s(y)) -> le(x,y)
  TDG Processor:
   DPs:
    size#(push(x,y)) -> size#(x)
    pop#(push(x,y)) -> m#()
    pop#(push(x,y)) -> size#(x)
    pop#(push(x,y)) -> le#(size(x),m())
    pop#(push(x,y)) -> ?1#(le(size(x),m()),x,y)
    top#(push(x,y)) -> m#()
    top#(push(x,y)) -> size#(x)
    top#(push(x,y)) -> le#(size(x),m())
    top#(push(x,y)) -> ?2#(le(size(x),m()),x,y)
    le#(s(x),s(y)) -> le#(x,y)
   TRS:
    size(empty()) -> 0()
    size(push(x,y)) -> s(size(x))
    m() -> s(s(s(s(0()))))
    pop(empty()) -> empty()
    ?1(true(),x,y) -> x
    pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
    top(empty()) -> eentry()
    ?2(true(),x,y) -> y
    top(push(x,y)) -> ?2(le(size(x),m()),x,y)
    le(x,0()) -> false()
    le(0(),s(x)) -> true()
    le(s(x),s(y)) -> le(x,y)
   graph:
    top#(push(x,y)) -> le#(size(x),m()) -> le#(s(x),s(y)) -> le#(x,y)
    top#(push(x,y)) -> size#(x) -> size#(push(x,y)) -> size#(x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    pop#(push(x,y)) -> le#(size(x),m()) -> le#(s(x),s(y)) -> le#(x,y)
    pop#(push(x,y)) -> size#(x) -> size#(push(x,y)) -> size#(x)
    size#(push(x,y)) -> size#(x) -> size#(push(x,y)) -> size#(x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 6/100
    DPs:
     size#(push(x,y)) -> size#(x)
    TRS:
     size(empty()) -> 0()
     size(push(x,y)) -> s(size(x))
     m() -> s(s(s(s(0()))))
     pop(empty()) -> empty()
     ?1(true(),x,y) -> x
     pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
     top(empty()) -> eentry()
     ?2(true(),x,y) -> y
     top(push(x,y)) -> ?2(le(size(x),m()),x,y)
     le(x,0()) -> false()
     le(0(),s(x)) -> true()
     le(s(x),s(y)) -> le(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(size#) = 0
     problem:
      DPs:
       
      TRS:
       size(empty()) -> 0()
       size(push(x,y)) -> s(size(x))
       m() -> s(s(s(s(0()))))
       pop(empty()) -> empty()
       ?1(true(),x,y) -> x
       pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
       top(empty()) -> eentry()
       ?2(true(),x,y) -> y
       top(push(x,y)) -> ?2(le(size(x),m()),x,y)
       le(x,0()) -> false()
       le(0(),s(x)) -> true()
       le(s(x),s(y)) -> le(x,y)
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     size(empty()) -> 0()
     size(push(x,y)) -> s(size(x))
     m() -> s(s(s(s(0()))))
     pop(empty()) -> empty()
     ?1(true(),x,y) -> x
     pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
     top(empty()) -> eentry()
     ?2(true(),x,y) -> y
     top(push(x,y)) -> ?2(le(size(x),m()),x,y)
     le(x,0()) -> false()
     le(0(),s(x)) -> true()
     le(s(x),s(y)) -> le(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 0
     problem:
      DPs:
       
      TRS:
       size(empty()) -> 0()
       size(push(x,y)) -> s(size(x))
       m() -> s(s(s(s(0()))))
       pop(empty()) -> empty()
       ?1(true(),x,y) -> x
       pop(push(x,y)) -> ?1(le(size(x),m()),x,y)
       top(empty()) -> eentry()
       ?2(true(),x,y) -> y
       top(push(x,y)) -> ?2(le(size(x),m()),x,y)
       le(x,0()) -> false()
       le(0(),s(x)) -> true()
       le(s(x),s(y)) -> le(x,y)
     Qed