Mercurial > hg > Members > atton > delta_monad
comparison agda/delta/monad.agda @ 88:526186c4f298
Split monad-proofs into delta.monad
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 19 Jan 2015 11:10:58 +0900 |
parents | |
children | 55d11ce7e223 |
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87:6789c65a75bc | 88:526186c4f298 |
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1 open import basic | |
2 open import delta | |
3 open import delta.functor | |
4 open import nat | |
5 open import laws | |
6 | |
7 | |
8 open import Level | |
9 open import Relation.Binary.PropositionalEquality | |
10 open ≡-Reasoning | |
11 | |
12 module delta.monad where | |
13 | |
14 | |
15 -- Monad-laws (Category) | |
16 | |
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)) | |
19 monad-law-1-5 O O ds = refl | |
20 monad-law-1-5 O (S n) (mono ds) = begin | |
21 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ 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)) ⟩ | |
24 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)) | |
26 ∎ | |
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 ⟩ | |
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)) ⟩ | |
30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ | |
31 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ | |
32 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) ⟩ | |
34 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)) | |
36 ∎ | |
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 ⟩ | |
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 ⟩ | |
41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ | |
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))) ⟩ | |
44 bind (n-tail n (mono (mono x))) (n-tail (S m + n)) | |
45 ∎ | |
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 ⟩ | |
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 ⟩ | |
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) ⟩ | |
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))⟩ | |
54 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))) ⟩ | |
57 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) | |
58 ∎ | |
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 ⟩ | |
61 (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 ⟩ | |
63 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)) ⟩ | |
65 bind (n-tail O (delta d ds)) (n-tail (S m + O)) | |
66 ∎ | |
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)) ⟩ | |
69 ((n-tail n) ∙ tailDelta) (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 ⟩ | |
71 (n-tail n) (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 ⟩ | |
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)) ⟩ | |
74 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) ⟩ | |
76 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) | |
77 ∎ | |
78 | |
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))) | |
81 monad-law-1-4 O O (mono dd) = refl | |
82 monad-law-1-4 O O (delta dd dd₁) = refl | |
83 monad-law-1-4 O (S n) (mono dd) = begin | |
84 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ 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)) ⟩ | |
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)))) | |
89 ∎ | |
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 ⟩ | |
92 headDelta (n-tail (S n) (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)) ⟩ | |
94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ | |
95 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ | |
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) ⟩ | |
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)))) | |
100 ∎ | |
101 monad-law-1-4 (S m) n (mono dd) = begin | |
102 headDelta (n-tail n (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)) ⟩ | |
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 ⟩ | |
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)))) | |
108 ∎ | |
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 ⟩ | |
111 headDelta (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 ⟩ | |
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 ⟩ | |
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)))) | |
117 ∎ | |
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 ⟩ | |
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)) ⟩ | |
121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ | |
122 headDelta (n-tail n (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 ⟩ | |
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 ⟩ | |
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)))) | |
128 ∎ | |
129 | |
130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) | |
131 monad-law-1-2 (mono _) = refl | |
132 monad-law-1-2 (delta _ _) = refl | |
133 | |
134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> | |
135 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) | |
136 monad-law-1-3 O (mono d) = refl | |
137 monad-law-1-3 O (delta d ds) = begin | |
138 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | |
139 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ | |
140 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ | |
141 delta (headDelta (headDelta d)) (bind (fmap 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 ⟩ | |
143 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ | |
144 bind (bind (delta d ds) (n-tail O)) (n-tail O) | |
145 ∎ | |
146 monad-law-1-3 (S n) (mono (mono d)) = begin | |
147 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ | |
148 bind (mono d) (n-tail (S n)) ≡⟨ 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))⟩ | |
151 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 ⟩ | |
153 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) | |
154 ∎ | |
155 monad-law-1-3 (S n) (mono (delta d ds)) = begin | |
156 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ | |
157 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ | |
158 n-tail (S n) (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)) ⟩ | |
160 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
161 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ | |
162 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) ⟩ | |
164 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)) | |
166 ∎ | |
167 monad-law-1-3 (S n) (delta (mono d) ds) = begin | |
168 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
169 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
170 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
171 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
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) ⟩ | |
173 delta (headDelta ((n-tail (S n)) d)) (bind (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 ⟩ | |
175 delta (headDelta ((n-tail (S n)) d)) (bind (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)) ⟩ | |
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 ⟩ | |
178 bind (delta (headDelta ((n-tail (S n)) (mono d))) (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)) | |
180 ∎ | |
181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin | |
182 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
183 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
184 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
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)) ⟩ | |
186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
187 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
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) ⟩ | |
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) ⟩ | |
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 ⟩ | |
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 ⟩ | |
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 ⟩ | |
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) ⟩ | |
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 ⟩ | |
195 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (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)) | |
197 ∎ | |
198 | |
199 | |
200 -- monad-law-1 : join . fmap join = join . join | |
201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) | |
202 monad-law-1 (mono d) = refl | |
203 monad-law-1 (delta x d) = begin | |
204 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ | |
205 mu (fmap mu (delta x d)) ≡⟨ refl ⟩ | |
206 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ | |
207 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ | |
208 delta (headDelta (headDelta x)) (bind (fmap 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 ⟩ | |
210 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ | |
211 mu (mu (delta x d)) ≡⟨ refl ⟩ | |
212 (mu ∙ mu) (delta x d) | |
213 ∎ | |
214 | |
215 | |
216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d | |
217 monad-law-2-1 O (mono x) = refl | |
218 monad-law-2-1 O (delta x d) = begin | |
219 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ | |
220 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ | |
221 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ | |
222 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ | |
223 delta x d ∎ | |
224 monad-law-2-1 (S n) (mono x) = begin | |
225 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ | |
226 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ | |
227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ | |
228 mono x ∎ | |
229 monad-law-2-1 (S n) (delta x d) = begin | |
230 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
231 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
232 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
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) ⟩ | |
234 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ | |
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) ⟩ | |
236 delta x d | |
237 ∎ | |
238 | |
239 | |
240 -- monad-law-2 : join . fmap return = join . return = id | |
241 -- monad-law-2 join . fmap return = join . return | |
242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> | |
243 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d | |
244 monad-law-2 (mono x) = refl | |
245 monad-law-2 (delta x d) = begin | |
246 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ | |
247 mu (fmap eta (delta x d)) ≡⟨ refl ⟩ | |
248 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ | |
249 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ | |
250 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ | |
251 (delta x d) ≡⟨ refl ⟩ | |
252 mu (mono (delta x d)) ≡⟨ refl ⟩ | |
253 mu (eta (delta x d)) ≡⟨ refl ⟩ | |
254 (mu ∙ eta) (delta x d) | |
255 ∎ | |
256 | |
257 | |
258 -- monad-law-2' : join . return = id | |
259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d | |
260 monad-law-2' d = refl | |
261 | |
262 | |
263 -- monad-law-3 : return . f = fmap f . return | |
264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x | |
265 monad-law-3 f x = refl | |
266 | |
267 | |
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 (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) | |
270 monad-law-4-1 O f (mono d) = refl | |
271 monad-law-4-1 O f (delta d ds) = begin | |
272 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | |
273 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ | |
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) ⟩ | |
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) ⟩ | |
276 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
277 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
278 fmap f (bind (delta d ds) (n-tail O)) ∎ | |
279 monad-law-4-1 (S n) f (mono d) = begin | |
280 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
281 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
282 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ | |
283 fmap f (n-tail (S n) d) ≡⟨ refl ⟩ | |
284 fmap f (bind (mono d) (n-tail (S n))) | |
285 ∎ | |
286 monad-law-4-1 (S n) f (delta d ds) = begin | |
287 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
288 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
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) ⟩ | |
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)) ⟩ | |
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) ⟩ | |
292 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
293 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
294 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ | |
295 fmap f (bind (delta d ds) (n-tail (S n))) ∎ | |
296 | |
297 | |
298 -- monad-law-4 : join . fmap (fmap f) = fmap f . join | |
299 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) -> | |
300 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d | |
301 monad-law-4 f (mono d) = refl | |
302 monad-law-4 f (delta (mono x) ds) = begin | |
303 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ | |
304 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ | |
305 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
306 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
307 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
308 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
309 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
310 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
311 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ | |
312 (fmap f ∙ mu) (delta (mono x) ds) ∎ | |
313 monad-law-4 f (delta (delta x d) ds) = begin | |
314 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ | |
315 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
316 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
317 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
318 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
319 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
320 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
321 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
322 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
323 (fmap f ∙ mu) (delta (delta x d) ds) ∎ | |
324 | |
325 delta-is-monad : {l : Level} {A : Set l} -> Monad {l} {A} Delta delta-is-functor | |
326 delta-is-monad = record { mu = mu; | |
327 eta = eta; | |
328 association-law = monad-law-1; | |
329 left-unity-law = monad-law-2; | |
330 right-unity-law = monad-law-2' } | |
331 | |
332 | |
333 {- | |
334 -- Monad-laws (Haskell) | |
335 -- monad-law-h-1 : return a >>= k = k a | |
336 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> | |
337 (a : A) -> (k : A -> (Delta B)) -> | |
338 (return a >>= k) ≡ (k a) | |
339 monad-law-h-1 a k = refl | |
340 | |
341 | |
342 | |
343 -- monad-law-h-2 : m >>= return = m | |
344 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m | |
345 monad-law-h-2 (mono x) = refl | |
346 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) | |
347 | |
348 | |
349 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h | |
350 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> | |
351 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> | |
352 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) | |
353 monad-law-h-3 (mono x) k h = refl | |
354 monad-law-h-3 (delta x d) k h = {!!} | |
355 | |
356 -} | |
357 |