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Split monad-proofs into delta.monad
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 19 Jan 2015 11:10:58 +0900
parents
children 55d11ce7e223
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88
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import basic
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import delta
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 open import delta.functor
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 open import nat
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 open import laws
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 open import Level
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Relation.Binary.PropositionalEquality
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open ≡-Reasoning
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 module delta.monad where
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 -- Monad-laws (Category)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 monad-law-1-5 O O ds = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 monad-law-1-5 O (S n) (mono ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 n-tail (S n) ds ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 monad-law-1-5 O (S n) (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
34 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37 monad-law-1-5 (S m) n (mono (mono x)) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
43 bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
44 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
46 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
48 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
49 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
52 n-tail (m + n) ds ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
53 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
56 bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
57 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
58 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59 monad-law-1-5 (S m) O (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
60 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
61 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
62 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
63 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
64 bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
65 bind (n-tail O (delta d ds)) (n-tail (S m + O))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
66 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
67 monad-law-1-5 (S m) (S n) (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
68 n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
69 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
70 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
72 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
73 bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
74 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
75 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
76 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
77 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
78
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
79 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
80 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
81 monad-law-1-4 O O (mono dd) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
82 monad-law-1-4 O O (delta dd dd₁) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83 monad-law-1-4 O (S n) (mono dd) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
84 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
85 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
86 headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
87 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
88 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
89 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90 monad-law-1-4 O (S n) (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
91 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
92 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
93 headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
95 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97 headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
98 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
99 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101 monad-law-1-4 (S m) n (mono dd) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
102 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
106 headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109 monad-law-1-4 (S m) O (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
110 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
111 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
112 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
113 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
114 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
116 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
117 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
118 monad-law-1-4 (S m) (S n) (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
119 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
120 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
122 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
123 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
124 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
125 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
126 headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
127 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
128 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
129
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
131 monad-law-1-2 (mono _) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
132 monad-law-1-2 (delta _ _) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
133
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
135 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
136 monad-law-1-3 O (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
137 monad-law-1-3 O (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
138 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
139 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
140 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
141 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
142 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
143 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
144 bind (bind (delta d ds) (n-tail O)) (n-tail O)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
145 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
146 monad-law-1-3 (S n) (mono (mono d)) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
147 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
148 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
149 (n-tail (S n)) d ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
150 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
151 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
152 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
153 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
154 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155 monad-law-1-3 (S n) (mono (delta d ds)) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
156 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
157 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
158 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
159 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
160 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
161 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
162 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
163 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
164 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
165 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
166 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
167 monad-law-1-3 (S n) (delta (mono d) ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
168 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
169 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
170 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
171 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
172 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
173 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
174 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
175 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
176 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
177 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
178 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
179 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
180 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
182 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
183 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
184 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
185 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
187 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
188 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
189 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
190 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
191 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
192 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
193 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
194 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
195 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
196 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
197 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
198
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
199
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
200 -- monad-law-1 : join . fmap join = join . join
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
202 monad-law-1 (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
203 monad-law-1 (delta x d) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
204 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
205 mu (fmap mu (delta x d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
206 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
207 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
208 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
209 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
210 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
211 mu (mu (delta x d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
212 (mu ∙ mu) (delta x d)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
213 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
214
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
215
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
217 monad-law-2-1 O (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
218 monad-law-2-1 O (delta x d) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
219 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
220 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
221 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
222 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
223 delta x d ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
224 monad-law-2-1 (S n) (mono x) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
225 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
226 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
228 mono x ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
229 monad-law-2-1 (S n) (delta x d) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
230 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
231 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
232 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
233 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
234 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
235 delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
236 delta x d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
237 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
238
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
239
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
240 -- monad-law-2 : join . fmap return = join . return = id
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
241 -- monad-law-2 join . fmap return = join . return
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
243 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
244 monad-law-2 (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
245 monad-law-2 (delta x d) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
246 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
247 mu (fmap eta (delta x d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
248 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
249 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
250 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
251 (delta x d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
252 mu (mono (delta x d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
253 mu (eta (delta x d)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
254 (mu ∙ eta) (delta x d)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
255 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
256
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
257
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
258 -- monad-law-2' : join . return = id
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
260 monad-law-2' d = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
261
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
262
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
263 -- monad-law-3 : return . f = fmap f . return
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
265 monad-law-3 f x = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
266
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
267
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
268 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
269 bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
270 monad-law-4-1 O f (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
271 monad-law-4-1 O f (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
272 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
273 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
274 delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
275 delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
276 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
277 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
278 fmap f (bind (delta d ds) (n-tail O)) ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
279 monad-law-4-1 (S n) f (mono d) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
280 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
281 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
282 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
283 fmap f (n-tail (S n) d) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
284 fmap f (bind (mono d) (n-tail (S n)))
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
285 ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
286 monad-law-4-1 (S n) f (delta d ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
287 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
288 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
289 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
290 delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
291 delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
292 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
293 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
294 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
295 fmap f (bind (delta d ds) (n-tail (S n))) ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
296
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
297
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
298 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
299 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
300 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
301 monad-law-4 f (mono d) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
302 monad-law-4 f (delta (mono x) ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
303 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
304 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
305 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
306 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
307 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
308 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
309 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
310 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
311 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
312 (fmap f ∙ mu) (delta (mono x) ds) ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
313 monad-law-4 f (delta (delta x d) ds) = begin
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
314 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
315 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
316 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
317 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
318 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
319 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
320 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
321 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
322 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
323 (fmap f ∙ mu) (delta (delta x d) ds) ∎
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
324
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
325 delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
326 delta-is-monad = record { mu = mu;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
327 eta = eta;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
328 association-law = monad-law-1;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
329 left-unity-law = monad-law-2;
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
330 right-unity-law = monad-law-2' }
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
331
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
332
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
333 {-
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
334 -- Monad-laws (Haskell)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
335 -- monad-law-h-1 : return a >>= k = k a
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
336 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
337 (a : A) -> (k : A -> (Delta B)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
338 (return a >>= k) ≡ (k a)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
339 monad-law-h-1 a k = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
340
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
341
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
342
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
343 -- monad-law-h-2 : m >>= return = m
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
344 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
345 monad-law-h-2 (mono x) = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
346 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
347
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
348
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
349 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
350 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
351 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
352 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
353 monad-law-h-3 (mono x) k h = refl
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
354 monad-law-h-3 (delta x d) k h = {!!}
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
355
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
356 -}
526186c4f298 Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
357