id-meta : ℕ -> ℕ -> SingleLinkedStack ℕ -> Meta
id-meta n e s = record { context = record {n = n ; element = just e}
                       ; nextCS = (N.cs id) ; stack = s}

push-pop-type : ℕ -> ℕ  -> ℕ -> Element ℕ -> Set₁
push-pop-type n e x s = M.exec (M.csComp {meta} (M.cs popOnce) (M.cs pushOnce)) meta ≡ meta
  where
    meta = id-meta n e record {top = just (cons x (just s))}

push-pop : (n e x : ℕ) -> (s : Element ℕ) ->  push-pop-type n e x s
push-pop n e x s = refl