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@$\text{1}$@
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