--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/AgdaNPushNPopProof.agda	Tue Sep 08 18:38:08 2020 +0900
@@ -0,0 +1,58 @@
+pop-n-push-type : ℕ -> ℕ -> ℕ -> SingleLinkedStack ℕ -> Set₁
+pop-n-push-type n cn ce s = M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) meta
+                         ≡ M.exec (n-push n) meta
+  where
+    meta = id-meta cn ce s
+
+pop-n-push : (n cn ce : ℕ) -> (s : SingleLinkedStack ℕ) -> 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)
+  ≡⟨ refl ⟩
+  M.exec (M.csComp (M.cs popOnce) (M.csComp  (n-push  (suc n)) (M.cs pushOnce))) (id-meta cn ce s)
+  ≡⟨ exec-comp (M.cs popOnce) (M.csComp  (n-push  (suc n)) (M.cs pushOnce)) (id-meta cn ce s) ⟩
+  M.exec (M.cs popOnce) (M.exec (M.csComp  (n-push  (suc n)) (M.cs pushOnce)) (id-meta cn ce s))
+  ≡⟨ cong (\x -> M.exec (M.cs popOnce) x) (exec-comp (n-push (suc n)) (M.cs pushOnce) (id-meta cn ce s)) ⟩
+  M.exec (M.cs popOnce) (M.exec (n-push  (suc n))(M.exec (M.cs pushOnce) (id-meta cn ce s)))
+  ≡⟨ refl ⟩
+  M.exec (M.cs popOnce) (M.exec (n-push  (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})))
+  ≡⟨ sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))) ⟩
+  M.exec (M.csComp  (M.cs popOnce) (n-push  (suc n))) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
+  ≡⟨ pop-n-push n cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}) ⟩
+  M.exec (n-push n) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
+  ≡⟨ refl ⟩
+  M.exec (n-push n) (pushOnce (id-meta cn ce s))
+  ≡⟨ refl ⟩
+  M.exec (n-push n) (M.exec (M.cs pushOnce) (id-meta cn ce s))
+  ≡⟨ refl ⟩
+  M.exec (n-push (suc n)) (id-meta cn ce s)
+  ∎
+
+
+
+n-push-pop-type : ℕ ->  ℕ  -> ℕ -> SingleLinkedStack ℕ -> Set₁
+n-push-pop-type n cn ce st = M.exec (M.csComp (n-pop n) (n-push n)) meta ≡ meta
+  where
+    meta = id-meta cn ce st
+
+n-push-pop : (n cn ce : ℕ) -> (s : SingleLinkedStack ℕ) -> 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)
+  ≡⟨ refl ⟩
+  M.exec (M.csComp (M.cs (\m -> M.exec (n-pop n) (popOnce m))) (n-push (suc n))) (id-meta cn ce s)
+  ≡⟨ exec-comp (M.cs (\m -> M.exec (n-pop n) (popOnce m))) (n-push (suc n))  (id-meta cn ce s) ⟩
+  M.exec (M.cs (\m -> M.exec (n-pop  n) (popOnce m))) (M.exec (n-push (suc n)) (id-meta cn ce s))
+  ≡⟨ refl ⟩
+  M.exec (n-pop n) (popOnce (M.exec (n-push (suc n)) (id-meta cn ce s)))
+  ≡⟨ refl ⟩
+  M.exec (n-pop n) (M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce s)))
+  ≡⟨ cong (\x -> M.exec (n-pop n) x) (sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce s))) ⟩
+  M.exec (n-pop n) (M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce s))
+  ≡⟨ cong (\x -> M.exec (n-pop n) x) (pop-n-push n cn ce s) ⟩
+  M.exec (n-pop n) (M.exec (n-push n) (id-meta cn ce s))
+  ≡⟨ sym (exec-comp (n-pop n) (n-push n) (id-meta cn ce s)) ⟩
+  M.exec (M.csComp (n-pop n) (n-push n)) (id-meta cn ce s)
+  ≡⟨ n-push-pop n cn ce s ⟩
+  id-meta cn ce s
+  ∎