Mercurial > hg > Papers > 2018 > ryokka-thesis

comparison final_pre/src/AgdaNPushNPopProof.agda @ 7:28f900230c26

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add final_pre
author ryokka
date Mon, 19 Feb 2018 23:32:24 +0900
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6:d927f6b3d2b3 7:28f900230c26
1 pop-n-push-type : ℕ -> ℕ -> ℕ -> SingleLinkedStack ℕ -> Set₁
2 pop-n-push-type n cn ce s = M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) meta
3 ≡ M.exec (n-push n) meta
4 where
5 meta = id-meta cn ce s
6
7 pop-n-push : (n cn ce : ℕ) -> (s : SingleLinkedStack ℕ) -> pop-n-push-type n cn ce s
8 pop-n-push zero cn ce s = refl
9 pop-n-push (suc n) cn ce s = begin
10 M.exec (M.csComp (M.cs popOnce) (n-push (suc (suc n)))) (id-meta cn ce s)
11 ≡⟨ refl ⟩
12 M.exec (M.csComp (M.cs popOnce) (M.csComp (n-push (suc n)) (M.cs pushOnce))) (id-meta cn ce s)
13 ≡⟨ exec-comp (M.cs popOnce) (M.csComp (n-push (suc n)) (M.cs pushOnce)) (id-meta cn ce s) ⟩
14 M.exec (M.cs popOnce) (M.exec (M.csComp (n-push (suc n)) (M.cs pushOnce)) (id-meta cn ce s))
15 ≡⟨ cong (\x -> M.exec (M.cs popOnce) x) (exec-comp (n-push (suc n)) (M.cs pushOnce) (id-meta cn ce s)) ⟩
16 M.exec (M.cs popOnce) (M.exec (n-push (suc n))(M.exec (M.cs pushOnce) (id-meta cn ce s)))
17 ≡⟨ refl ⟩
18 M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})))
19 ≡⟨ sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))) ⟩
20 M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
21 ≡⟨ pop-n-push n cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}) ⟩
22 M.exec (n-push n) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
23 ≡⟨ refl ⟩
24 M.exec (n-push n) (pushOnce (id-meta cn ce s))
25 ≡⟨ refl ⟩
26 M.exec (n-push n) (M.exec (M.cs pushOnce) (id-meta cn ce s))
27 ≡⟨ refl ⟩
28 M.exec (n-push (suc n)) (id-meta cn ce s)
29 ∎
30
31
32
33 n-push-pop-type : ℕ -> ℕ -> ℕ -> SingleLinkedStack ℕ -> Set₁
34 n-push-pop-type n cn ce st = M.exec (M.csComp (n-pop n) (n-push n)) meta ≡ meta
35 where
36 meta = id-meta cn ce st
37
38 n-push-pop : (n cn ce : ℕ) -> (s : SingleLinkedStack ℕ) -> n-push-pop-type n cn ce s
39 n-push-pop zero cn ce s = refl
40 n-push-pop (suc n) cn ce s = begin
41 M.exec (M.csComp (n-pop (suc n)) (n-push (suc n))) (id-meta cn ce s)
42 ≡⟨ refl ⟩
43 M.exec (M.csComp (M.cs (\m -> M.exec (n-pop n) (popOnce m))) (n-push (suc n))) (id-meta cn ce s)
44 ≡⟨ exec-comp (M.cs (\m -> M.exec (n-pop n) (popOnce m))) (n-push (suc n)) (id-meta cn ce s) ⟩
45 M.exec (M.cs (\m -> M.exec (n-pop n) (popOnce m))) (M.exec (n-push (suc n)) (id-meta cn ce s))
46 ≡⟨ refl ⟩
47 M.exec (n-pop n) (popOnce (M.exec (n-push (suc n)) (id-meta cn ce s)))
48 ≡⟨ refl ⟩
49 M.exec (n-pop n) (M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce s)))
50 ≡⟨ cong (\x -> M.exec (n-pop n) x) (sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce s))) ⟩
51 M.exec (n-pop n) (M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce s))
52 ≡⟨ cong (\x -> M.exec (n-pop n) x) (pop-n-push n cn ce s) ⟩
53 M.exec (n-pop n) (M.exec (n-push n) (id-meta cn ce s))
54 ≡⟨ sym (exec-comp (n-pop n) (n-push n) (id-meta cn ce s)) ⟩
55 M.exec (M.csComp (n-pop n) (n-push n)) (id-meta cn ce s)
56 ≡⟨ n-push-pop n cn ce s ⟩
57 id-meta cn ce s
58 ∎