pop-n-push-type : !$\mathbb{N}$! !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! SingleLinkedStack !$\mathbb{N}$! !$\rightarrow$! Set!$\_{1}$!
pop-n-push-type n cn ce s = M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) meta
                         !$\equiv$! M.exec (n-push n) meta
  where
    meta = id-meta cn ce s

pop-n-push : (n cn ce : !$\mathbb{N}$!) !$\rightarrow$! (s : SingleLinkedStack !$\mathbb{N}$!) !$\rightarrow$! pop-n-push-type n cn ce s
pop-n-push zero cn ce s    = refl
pop-n-push (suc n) cn ce s = begin
  M.exec (M.csComp  (M.cs popOnce) (n-push  (suc (suc n)))) (id-meta cn ce s)
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (M.csComp (M.cs popOnce) (M.csComp  (n-push  (suc n)) (M.cs pushOnce))) (id-meta cn ce s)
  !$\equiv$!!$\langle$! exec-comp (M.cs popOnce) (M.csComp  (n-push  (suc n)) (M.cs pushOnce)) (id-meta cn ce s) !$\rangle$!
  M.exec (M.cs popOnce) (M.exec (M.csComp  (n-push  (suc n)) (M.cs pushOnce)) (id-meta cn ce s))
  !$\equiv$!!$\langle$! cong (\x !$\rightarrow$! M.exec (M.cs popOnce) x) (exec-comp (n-push (suc n)) (M.cs pushOnce) (id-meta cn ce s)) !$\rangle$!
  M.exec (M.cs popOnce) (M.exec (n-push  (suc n))(M.exec (M.cs pushOnce) (id-meta cn ce s)))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (M.cs popOnce) (M.exec (n-push  (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})))
  !$\equiv$!!$\langle$! sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))) !$\rangle$!
  M.exec (M.csComp  (M.cs popOnce) (n-push  (suc n))) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
  !$\equiv$!!$\langle$! pop-n-push n cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}) !$\rangle$!
  M.exec (n-push n) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (n-push n) (pushOnce (id-meta cn ce s))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (n-push n) (M.exec (M.cs pushOnce) (id-meta cn ce s))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (n-push (suc n)) (id-meta cn ce s)
  !$\blacksquare$!



n-push-pop-type : !$\mathbb{N}$! !$\rightarrow$!  !$\mathbb{N}$!  !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! SingleLinkedStack !$\mathbb{N}$! !$\rightarrow$! Set!$\_{1}$!
n-push-pop-type n cn ce st = M.exec (M.csComp (n-pop n) (n-push n)) meta !$\equiv$! meta
  where
    meta = id-meta cn ce st

n-push-pop : (n cn ce : !$\mathbb{N}$!) !$\rightarrow$! (s : SingleLinkedStack !$\mathbb{N}$!) !$\rightarrow$! n-push-pop-type n cn ce s
n-push-pop zero    cn ce s = refl
n-push-pop (suc n) cn ce s = begin
  M.exec (M.csComp (n-pop (suc n)) (n-push (suc n))) (id-meta cn ce s)
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (M.csComp (M.cs (\m !$\rightarrow$! M.exec (n-pop n) (popOnce m))) (n-push (suc n))) (id-meta cn ce s)
  !$\equiv$!!$\langle$! exec-comp (M.cs (\m !$\rightarrow$! M.exec (n-pop n) (popOnce m))) (n-push (suc n))  (id-meta cn ce s) !$\rangle$!
  M.exec (M.cs (\m !$\rightarrow$! M.exec (n-pop  n) (popOnce m))) (M.exec (n-push (suc n)) (id-meta cn ce s))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (n-pop n) (popOnce (M.exec (n-push (suc n)) (id-meta cn ce s)))
  !$\equiv$!!$\langle$! refl !$\rangle$!
  M.exec (n-pop n) (M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce s)))
  !$\equiv$!!$\langle$! cong (\x !$\rightarrow$! M.exec (n-pop n) x) (sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce s))) !$\rangle$!
  M.exec (n-pop n) (M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce s))
  !$\equiv$!!$\langle$! cong (\x !$\rightarrow$! M.exec (n-pop n) x) (pop-n-push n cn ce s) !$\rangle$!
  M.exec (n-pop n) (M.exec (n-push n) (id-meta cn ce s))
  !$\equiv$!!$\langle$! sym (exec-comp (n-pop n) (n-push n) (id-meta cn ce s)) !$\rangle$!
  M.exec (M.csComp (n-pop n) (n-push n)) (id-meta cn ce s)
  !$\equiv$!!$\langle$! n-push-pop n cn ce s !$\rangle$!
  id-meta cn ce s
  !$\blacksquare$!