Take the abstract state to be a finite sequence of integers, initially empty. Let denote the entry state, the empty sequence, and concatenation. A Scala trait presents the interface; its comments give the mathematical preconditions and postconditions:
trait DoubleEndedQueue {
  // Pre: true. Post: result == s0.isEmpty and s == s0.
  def isEmpty: Boolean
  // Pre: true. Post: s == [x] ++ s0.
  def addLeft(x: Int): Unit
  // Pre: s0.nonEmpty. Post: result == s0.head, s == s0.tail.
  def getLeft(): Int
  // Pre: true. Post: s == s0 ++ [x].
  def addRight(x: Int): Unit
  // Pre: s0.nonEmpty. Post: result == s0.last, s == s0.init.
  def getRight(): Int
}
The double-ended queue has no abstract capacity limit in this first specification. Both removals require nonemptiness; the query leaves the state unchanged, an instance of command-query separation.

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