comparison agda/delta/monad.agda @ 88:526186c4f298

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
comparison
equal deleted inserted replaced
87:6789c65a75bc 88:526186c4f298
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