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

push-pop-type : !$\mathbb{N}$! !$\rightarrow$! !$\mathbb{N}$!  !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! Element !$\mathbb{N}$! !$\rightarrow$! Set!$\_{1}$!
push-pop-type n e x s = M.exec (M.csComp {meta} (M.cs popOnce) (M.cs pushOnce)) meta !$\equiv$! meta
  where
    meta = id-meta n e record {top = just (cons x (just s))}

push-pop : (n e x : !$\mathbb{N}$!) !$\rightarrow$! (s : Element !$\mathbb{N}$!) !$\rightarrow$!  push-pop-type n e x s
push-pop n e x s = refl