pop-n-push-type : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Set@$\text{1}$@
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@$\text{1}$@
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
  @$\blacksquare$@