Mercurial > hg > Members > atton > delta_monad
comparison agda/delta/monad.agda @ 94:bcd4fe52a504
Rewrite monad definitions for delta/deltaM
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 19 Jan 2015 17:10:29 +0900 |
parents | 55d11ce7e223 |
children | dfe8c67390bd |
comparison
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93:8d92ed54a94f | 94:bcd4fe52a504 |
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13 | 13 |
14 | 14 |
15 -- Monad-laws (Category) | 15 -- Monad-laws (Category) |
16 | 16 |
17 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> | 17 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> |
18 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) | 18 n-tail n (delta-bind ds (n-tail m)) ≡ delta-bind (n-tail n ds) (n-tail (m + n)) |
19 monad-law-1-5 O O ds = refl | 19 monad-law-1-5 O O ds = refl |
20 monad-law-1-5 O (S n) (mono ds) = begin | 20 monad-law-1-5 O (S n) (mono ds) = begin |
21 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ | 21 n-tail (S n) (delta-bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ |
22 n-tail (S n) ds ≡⟨ refl ⟩ | 22 n-tail (S n) ds ≡⟨ refl ⟩ |
23 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ | 23 delta-bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ |
24 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 24 delta-bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
25 bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) | 25 delta-bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) |
26 ∎ | 26 ∎ |
27 monad-law-1-5 O (S n) (delta d ds) = begin | 27 monad-law-1-5 O (S n) (delta d ds) = begin |
28 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ | 28 n-tail (S n) (delta-bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ |
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)) ⟩ | 29 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ |
30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ | 30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ refl ⟩ |
31 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ | 31 (n-tail n) (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
32 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ | 32 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
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) ⟩ | 33 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ |
34 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 34 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
35 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) | 35 delta-bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) |
36 ∎ | 36 ∎ |
37 monad-law-1-5 (S m) n (mono (mono x)) = begin | 37 monad-law-1-5 (S m) n (mono (mono x)) = begin |
38 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ | 38 n-tail n (delta-bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ |
39 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ | 39 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ |
40 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ | 40 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ |
41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ | 41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ |
42 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ | 42 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ |
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))) ⟩ | 43 delta-bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ |
44 bind (n-tail n (mono (mono x))) (n-tail (S m + n)) | 44 delta-bind (n-tail n (mono (mono x))) (n-tail (S m + n)) |
45 ∎ | 45 ∎ |
46 monad-law-1-5 (S m) n (mono (delta x ds)) = begin | 46 monad-law-1-5 (S m) n (mono (delta x ds)) = begin |
47 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ | 47 n-tail n (delta-bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ |
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)) ⟩ | 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)) ⟩ |
49 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ | 49 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ |
50 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ | 50 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ |
51 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ | 51 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ |
52 n-tail (m + n) ds ≡⟨ refl ⟩ | 52 n-tail (m + n) ds ≡⟨ refl ⟩ |
53 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ | 53 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ |
54 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ | 54 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ |
55 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ | 55 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ |
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))) ⟩ | 56 delta-bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ |
57 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) | 57 delta-bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) |
58 ∎ | 58 ∎ |
59 monad-law-1-5 (S m) O (delta d ds) = begin | 59 monad-law-1-5 (S m) O (delta d ds) = begin |
60 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | 60 n-tail O (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
61 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | 61 (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
62 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ | 62 delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
63 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ | 63 delta-bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ |
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)) ⟩ | 64 delta-bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> delta-bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ |
65 bind (n-tail O (delta d ds)) (n-tail (S m + O)) | 65 delta-bind (n-tail O (delta d ds)) (n-tail (S m + O)) |
66 ∎ | 66 ∎ |
67 monad-law-1-5 (S m) (S n) (delta d ds) = begin | 67 monad-law-1-5 (S m) (S n) (delta d ds) = begin |
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)) ⟩ | 68 n-tail (S n) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((delta-bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ |
69 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | 69 ((n-tail n) ∙ tailDelta) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
70 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ | 70 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
71 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ | 71 (n-tail n) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
72 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ | 72 (n-tail n) (delta-bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ |
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)) ⟩ | 73 delta-bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> delta-bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ |
74 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ | 74 delta-bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ |
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) ⟩ | 75 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ |
76 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) | 76 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) |
77 ∎ | 77 ∎ |
78 | 78 |
79 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> | 79 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> |
80 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) | 80 headDelta ((n-tail n) (delta-bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) |
81 monad-law-1-4 O O (mono dd) = refl | 81 monad-law-1-4 O O (mono dd) = refl |
82 monad-law-1-4 O O (delta dd dd₁) = refl | 82 monad-law-1-4 O O (delta dd dd₁) = refl |
83 monad-law-1-4 O (S n) (mono dd) = begin | 83 monad-law-1-4 O (S n) (mono dd) = begin |
84 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ | 84 headDelta (n-tail (S n) (delta-bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ |
85 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ | 85 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ |
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)) ⟩ | 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)) ⟩ |
87 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ | 87 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ |
88 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) | 88 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) |
89 ∎ | 89 ∎ |
90 monad-law-1-4 O (S n) (delta d ds) = begin | 90 monad-law-1-4 O (S n) (delta d ds) = begin |
91 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ | 91 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ |
92 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ | 92 headDelta (n-tail (S n) (delta-bind (delta d ds) id)) ≡⟨ refl ⟩ |
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)) ⟩ | 93 headDelta (n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (delta-bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ |
94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ | 94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ refl ⟩ |
95 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ | 95 headDelta (n-tail n (delta-bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ |
96 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ | 96 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ |
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) ⟩ | 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) ⟩ |
98 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ | 98 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ |
99 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) | 99 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) |
100 ∎ | 100 ∎ |
101 monad-law-1-4 (S m) n (mono dd) = begin | 101 monad-law-1-4 (S m) n (mono dd) = begin |
102 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ | 102 headDelta (n-tail n (delta-bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ |
103 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ | 103 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ |
104 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ | 104 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ |
105 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ | 105 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ |
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)) ⟩ | 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)) ⟩ |
107 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) | 107 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) |
108 ∎ | 108 ∎ |
109 monad-law-1-4 (S m) O (delta d ds) = begin | 109 monad-law-1-4 (S m) O (delta d ds) = begin |
110 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ | 110 headDelta (n-tail O (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
111 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | 111 headDelta (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
112 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ | 112 headDelta (delta (headDelta ((n-tail (S m) d))) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
113 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ | 113 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ |
114 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ | 114 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ |
115 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ | 115 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ |
116 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) | 116 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) |
117 ∎ | 117 ∎ |
118 monad-law-1-4 (S m) (S n) (delta d ds) = begin | 118 monad-law-1-4 (S m) (S n) (delta d ds) = begin |
119 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ | 119 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
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)) ⟩ | 120 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ |
121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ | 121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ |
122 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ | 122 headDelta (n-tail n (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
123 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ | 123 headDelta (n-tail n (delta-bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ |
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)) ⟩ | 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)) ⟩ |
125 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ | 125 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ |
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) ⟩ | 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) ⟩ |
127 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) | 127 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) |
128 ∎ | 128 ∎ |
129 | 129 |
130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) | 130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (delta-mu d) ≡ (headDelta (headDelta d)) |
131 monad-law-1-2 (mono _) = refl | 131 monad-law-1-2 (mono _) = refl |
132 monad-law-1-2 (delta _ _) = refl | 132 monad-law-1-2 (delta _ _) = refl |
133 | 133 |
134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> | 134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> |
135 bind (delta-fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) | 135 delta-bind (delta-fmap delta-mu d) (n-tail n) ≡ delta-bind (delta-bind d (n-tail n)) (n-tail n) |
136 monad-law-1-3 O (mono d) = refl | 136 monad-law-1-3 O (mono d) = refl |
137 monad-law-1-3 O (delta d ds) = begin | 137 monad-law-1-3 O (delta d ds) = begin |
138 bind (delta-fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | 138 delta-bind (delta-fmap delta-mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
139 bind (delta (mu d) (delta-fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ | 139 delta-bind (delta (delta-mu d) (delta-fmap delta-mu ds)) (n-tail O) ≡⟨ refl ⟩ |
140 delta (headDelta (mu d)) (bind (delta-fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (delta-fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ | 140 delta (headDelta (delta-mu d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (delta-bind (delta-fmap delta-mu ds) tailDelta)) (monad-law-1-2 d) ⟩ |
141 delta (headDelta (headDelta d)) (bind (delta-fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ | 141 delta (headDelta (headDelta d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ |
142 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ | 142 delta (headDelta (headDelta d)) (delta-bind (delta-bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ |
143 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ | 143 delta-bind (delta (headDelta d) (delta-bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ |
144 bind (bind (delta d ds) (n-tail O)) (n-tail O) | 144 delta-bind (delta-bind (delta d ds) (n-tail O)) (n-tail O) |
145 ∎ | 145 ∎ |
146 monad-law-1-3 (S n) (mono (mono d)) = begin | 146 monad-law-1-3 (S n) (mono (mono d)) = begin |
147 bind (delta-fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ | 147 delta-bind (delta-fmap delta-mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ |
148 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ | 148 delta-bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ |
149 (n-tail (S n)) d ≡⟨ refl ⟩ | 149 (n-tail (S n)) d ≡⟨ refl ⟩ |
150 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ | 150 delta-bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ |
151 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | 151 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
152 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | 152 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
153 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) | 153 delta-bind (delta-bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) |
154 ∎ | 154 ∎ |
155 monad-law-1-3 (S n) (mono (delta d ds)) = begin | 155 monad-law-1-3 (S n) (mono (delta d ds)) = begin |
156 bind (delta-fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ | 156 delta-bind (delta-fmap delta-mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
157 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ | 157 delta-bind (mono (delta-mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
158 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ | 158 n-tail (S n) (delta-mu (delta d ds)) ≡⟨ refl ⟩ |
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)) ⟩ | 159 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ |
160 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | 160 (n-tail n ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
161 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ | 161 n-tail n (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
162 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ | 162 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
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) ⟩ | 163 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (delta-bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ |
164 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 164 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
165 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) | 165 delta-bind (delta-bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) |
166 ∎ | 166 ∎ |
167 monad-law-1-3 (S n) (delta (mono d) ds) = begin | 167 monad-law-1-3 (S n) (delta (mono d) ds) = begin |
168 bind (delta-fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 168 delta-bind (delta-fmap delta-mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
169 bind (delta (mu (mono d)) (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 169 delta-bind (delta (delta-mu (mono d)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
170 bind (delta d (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 170 delta-bind (delta d (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
171 delta (headDelta ((n-tail (S n)) d)) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 171 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
172 delta (headDelta ((n-tail (S n)) d)) (bind (delta-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) ⟩ | 172 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-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) ⟩ |
173 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ | 173 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
174 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ | 174 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
175 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 175 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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)) ⟩ | 176 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ |
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 ⟩ | 177 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
178 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ | 178 delta-bind (delta (headDelta ((n-tail (S n)) (mono d))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
179 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) | 179 delta-bind (delta-bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) |
180 ∎ | 180 ∎ |
181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin | 181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin |
182 bind (delta-fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 182 delta-bind (delta-fmap delta-mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
183 bind (delta (mu (delta d dd)) (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 183 delta-bind (delta (delta-mu (delta d dd)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
184 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 184 delta (headDelta ((n-tail (S n)) (delta-mu (delta d dd)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
185 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ | 185 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ |
186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
187 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (delta-fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 187 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
188 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (delta-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) ⟩ | 188 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ |
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) ⟩ | 189 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ |
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 ⟩ | 190 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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 ⟩ | 191 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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 ⟩ | 192 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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) ⟩ | 193 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (delta-bind (delta-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))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ |
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 ⟩ | 194 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
195 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ | 195 delta-bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
196 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) | 196 delta-bind (delta-bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) |
197 ∎ | 197 ∎ |
198 | 198 |
199 | 199 |
200 -- monad-law-1 : join . delta-fmap join = join . join | 200 -- monad-law-1 : join . delta-fmap join = join . join |
201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (delta-fmap mu)) d) ≡ ((mu ∙ mu) d) | 201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) |
202 monad-law-1 (mono d) = refl | 202 monad-law-1 (mono d) = refl |
203 monad-law-1 (delta x d) = begin | 203 monad-law-1 (delta x d) = begin |
204 (mu ∙ delta-fmap mu) (delta x d) ≡⟨ refl ⟩ | 204 (delta-mu ∙ delta-fmap delta-mu) (delta x d) ≡⟨ refl ⟩ |
205 mu (delta-fmap mu (delta x d)) ≡⟨ refl ⟩ | 205 delta-mu (delta-fmap delta-mu (delta x d)) ≡⟨ refl ⟩ |
206 mu (delta (mu x) (delta-fmap mu d)) ≡⟨ refl ⟩ | 206 delta-mu (delta (delta-mu x) (delta-fmap delta-mu d)) ≡⟨ refl ⟩ |
207 delta (headDelta (mu x)) (bind (delta-fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (delta-fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ | 207 delta (headDelta (delta-mu x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\x -> delta x (delta-bind (delta-fmap delta-mu d) tailDelta)) (monad-law-1-2 x) ⟩ |
208 delta (headDelta (headDelta x)) (bind (delta-fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ | 208 delta (headDelta (headDelta x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ |
209 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ | 209 delta (headDelta (headDelta x)) (delta-bind (delta-bind d tailDelta) tailDelta) ≡⟨ refl ⟩ |
210 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ | 210 delta-mu (delta (headDelta x) (delta-bind d tailDelta)) ≡⟨ refl ⟩ |
211 mu (mu (delta x d)) ≡⟨ refl ⟩ | 211 delta-mu (delta-mu (delta x d)) ≡⟨ refl ⟩ |
212 (mu ∙ mu) (delta x d) | 212 (delta-mu ∙ delta-mu) (delta x d) |
213 ∎ | 213 ∎ |
214 | 214 |
215 | 215 |
216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (delta-fmap eta d) (n-tail n)) ≡ d | 216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (delta-bind (delta-fmap delta-eta d) (n-tail n)) ≡ d |
217 monad-law-2-1 O (mono x) = refl | 217 monad-law-2-1 O (mono x) = refl |
218 monad-law-2-1 O (delta x d) = begin | 218 monad-law-2-1 O (delta x d) = begin |
219 bind (delta-fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ | 219 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ |
220 bind (delta (eta x) (delta-fmap eta d)) id ≡⟨ refl ⟩ | 220 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) id ≡⟨ refl ⟩ |
221 delta (headDelta (eta x)) (bind (delta-fmap eta d) tailDelta) ≡⟨ refl ⟩ | 221 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ |
222 delta x (bind (delta-fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ | 222 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ |
223 delta x d ∎ | 223 delta x d ∎ |
224 monad-law-2-1 (S n) (mono x) = begin | 224 monad-law-2-1 (S n) (mono x) = begin |
225 bind (delta-fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ | 225 delta-bind (delta-fmap delta-eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
226 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ | 226 delta-bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ | 227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ |
228 mono x ∎ | 228 mono x ∎ |
229 monad-law-2-1 (S n) (delta x d) = begin | 229 monad-law-2-1 (S n) (delta x d) = begin |
230 bind (delta-fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ | 230 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ |
231 bind (delta (eta x) (delta-fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ | 231 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) (n-tail (S n)) ≡⟨ refl ⟩ |
232 delta (headDelta ((n-tail (S n) (eta x)))) (bind (delta-fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 232 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
233 delta (headDelta ((n-tail (S n) (eta x)))) (bind (delta-fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (delta-fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ | 233 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ |
234 delta (headDelta (eta x)) (bind (delta-fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ | 234 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
235 delta x (bind (delta-fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ | 235 delta x (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ |
236 delta x d | 236 delta x d |
237 ∎ | 237 ∎ |
238 | 238 |
239 | 239 |
240 -- monad-law-2 : join . delta-fmap return = join . return = id | 240 -- monad-law-2 : join . delta-fmap return = join . return = id |
241 -- monad-law-2 join . delta-fmap return = join . return | 241 -- monad-law-2 join . delta-fmap return = join . return |
242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> | 242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> |
243 (mu ∙ delta-fmap eta) d ≡ (mu ∙ eta) d | 243 (delta-mu ∙ delta-fmap delta-eta) d ≡ (delta-mu ∙ delta-eta) d |
244 monad-law-2 (mono x) = refl | 244 monad-law-2 (mono x) = refl |
245 monad-law-2 (delta x d) = begin | 245 monad-law-2 (delta x d) = begin |
246 (mu ∙ delta-fmap eta) (delta x d) ≡⟨ refl ⟩ | 246 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩ |
247 mu (delta-fmap eta (delta x d)) ≡⟨ refl ⟩ | 247 delta-mu (delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩ |
248 mu (delta (mono x) (delta-fmap eta d)) ≡⟨ refl ⟩ | 248 delta-mu (delta (mono x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ |
249 delta (headDelta (mono x)) (bind (delta-fmap eta d) tailDelta) ≡⟨ refl ⟩ | 249 delta (headDelta (mono x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ |
250 delta x (bind (delta-fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ | 250 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ |
251 (delta x d) ≡⟨ refl ⟩ | 251 (delta x d) ≡⟨ refl ⟩ |
252 mu (mono (delta x d)) ≡⟨ refl ⟩ | 252 delta-mu (mono (delta x d)) ≡⟨ refl ⟩ |
253 mu (eta (delta x d)) ≡⟨ refl ⟩ | 253 delta-mu (delta-eta (delta x d)) ≡⟨ refl ⟩ |
254 (mu ∙ eta) (delta x d) | 254 (delta-mu ∙ delta-eta) (delta x d) |
255 ∎ | 255 ∎ |
256 | 256 |
257 | 257 |
258 -- monad-law-2' : join . return = id | 258 -- monad-law-2' : join . return = id |
259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d | 259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (delta-mu ∙ delta-eta) d ≡ id d |
260 monad-law-2' d = refl | 260 monad-law-2' d = refl |
261 | 261 |
262 | 262 |
263 -- monad-law-3 : return . f = delta-fmap f . return | 263 -- monad-law-3 : return . f = delta-fmap f . return |
264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (delta-fmap f ∙ eta) x | 264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (delta-eta ∙ f) x ≡ (delta-fmap f ∙ delta-eta) x |
265 monad-law-3 f x = refl | 265 monad-law-3 f x = refl |
266 | 266 |
267 | 267 |
268 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> | 268 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> |
269 bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (bind ds (n-tail n)) | 269 delta-bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (delta-bind ds (n-tail n)) |
270 monad-law-4-1 O f (mono d) = refl | 270 monad-law-4-1 O f (mono d) = refl |
271 monad-law-4-1 O f (delta d ds) = begin | 271 monad-law-4-1 O f (delta d ds) = begin |
272 bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | 272 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
273 bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ | 273 delta-bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ |
274 delta (headDelta (delta-fmap f d)) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ | 274 delta (headDelta (delta-fmap f d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ |
275 delta (f (headDelta d)) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ | 275 delta (f (headDelta d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ |
276 delta (f (headDelta d)) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | 276 delta (f (headDelta d)) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
277 delta-fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | 277 delta-fmap f (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
278 delta-fmap f (bind (delta d ds) (n-tail O)) ∎ | 278 delta-fmap f (delta-bind (delta d ds) (n-tail O)) ∎ |
279 monad-law-4-1 (S n) f (mono d) = begin | 279 monad-law-4-1 (S n) f (mono d) = begin |
280 bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | 280 delta-bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
281 bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ | 281 delta-bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ |
282 n-tail (S n) (delta-fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ | 282 n-tail (S n) (delta-fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ |
283 delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩ | 283 delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩ |
284 delta-fmap f (bind (mono d) (n-tail (S n))) | 284 delta-fmap f (delta-bind (mono d) (n-tail (S n))) |
285 ∎ | 285 ∎ |
286 monad-law-4-1 (S n) f (delta d ds) = begin | 286 monad-law-4-1 (S n) f (delta d ds) = begin |
287 bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | 287 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
288 delta (headDelta (n-tail (S n) (delta-fmap f d))) (bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | 288 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
289 delta (headDelta (n-tail (S n) (delta-fmap f d))) (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ | 289 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ |
290 delta (headDelta (delta-fmap f ((n-tail (S n) d)))) (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ | 290 delta (headDelta (delta-fmap f ((n-tail (S n) d)))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ |
291 delta (f (headDelta (n-tail (S n) d))) (bind (delta-fmap (delta-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) ⟩ | 291 delta (f (headDelta (n-tail (S n) d))) (delta-bind (delta-fmap (delta-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) ⟩ |
292 delta (f (headDelta (n-tail (S n) d))) (delta-fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | 292 delta (f (headDelta (n-tail (S n) d))) (delta-fmap f (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ |
293 delta-fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | 293 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ |
294 delta-fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ | 294 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ |
295 delta-fmap f (bind (delta d ds) (n-tail (S n))) ∎ | 295 delta-fmap f (delta-bind (delta d ds) (n-tail (S n))) ∎ |
296 | 296 |
297 | 297 |
298 -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join | 298 -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join |
299 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (d : Delta (Delta A)) -> | 299 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (d : Delta (Delta A)) -> |
300 (mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ mu) d | 300 (delta-mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ delta-mu) d |
301 monad-law-4 f (mono d) = refl | 301 monad-law-4 f (mono d) = refl |
302 monad-law-4 f (delta (mono x) ds) = begin | 302 monad-law-4 f (delta (mono x) ds) = begin |
303 (mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ | 303 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ |
304 mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ | 304 delta-mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ |
305 mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ | 305 delta-mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ |
306 delta (headDelta (mono (f x))) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ | 306 delta (headDelta (mono (f x))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ |
307 delta (f x) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | 307 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ |
308 delta (f x) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | 308 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
309 delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | 309 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
310 delta-fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ | 310 delta-fmap f (delta (headDelta (mono x)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
311 delta-fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ | 311 delta-fmap f (delta-mu (delta (mono x) ds)) ≡⟨ refl ⟩ |
312 (delta-fmap f ∙ mu) (delta (mono x) ds) ∎ | 312 (delta-fmap f ∙ delta-mu) (delta (mono x) ds) ∎ |
313 monad-law-4 f (delta (delta x d) ds) = begin | 313 monad-law-4 f (delta (delta x d) ds) = begin |
314 (mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ | 314 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ |
315 mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ | 315 delta-mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ |
316 mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ | 316 delta-mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ |
317 delta (headDelta (delta (f x) (delta-fmap f d))) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ | 317 delta (headDelta (delta (f x) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ |
318 delta (f x) (bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | 318 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ |
319 delta (f x) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | 319 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
320 delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | 320 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
321 delta-fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ | 321 delta-fmap f (delta (headDelta (delta x d)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
322 delta-fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ | 322 delta-fmap f (delta-mu (delta (delta x d) ds)) ≡⟨ refl ⟩ |
323 (delta-fmap f ∙ mu) (delta (delta x d) ds) ∎ | 323 (delta-fmap f ∙ delta-mu) (delta (delta x d) ds) ∎ |
324 | 324 |
325 delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor | 325 delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor |
326 delta-is-monad = record { mu = mu; | 326 delta-is-monad = record { eta = delta-eta; |
327 eta = eta; | 327 mu = delta-mu; |
328 return = delta-eta; | |
329 bind = delta-bind; | |
328 association-law = monad-law-1; | 330 association-law = monad-law-1; |
329 left-unity-law = monad-law-2; | 331 left-unity-law = monad-law-2; |
330 right-unity-law = monad-law-2' } | 332 right-unity-law = monad-law-2' } |
331 | 333 |
332 | 334 |