Mercurial > hg > Members > atton > delta_monad
comparison agda/delta.agda @ 59:46b15f368905
Define bind and mu for Infinite Delta
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Sat, 22 Nov 2014 16:03:40 +0900 |
parents | dfcd72dc697e |
children | 73bb981cb1c6 |
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58:1229ee398567 | 59:46b15f368905 |
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31 fmap f (delta x d) = delta (f x) (fmap f d) | 31 fmap f (delta x d) = delta (f x) (fmap f d) |
32 | 32 |
33 | 33 |
34 | 34 |
35 -- Monad (Category) | 35 -- Monad (Category) |
36 | |
37 -- TODO: mu | |
38 -- TODO: bind | |
39 | |
40 | |
41 eta : {l : Level} {A : Set l} -> A -> Delta A | 36 eta : {l : Level} {A : Set l} -> A -> Delta A |
42 eta x = mono x | 37 eta x = mono x |
38 | |
39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B | |
40 bind (mono x) f = f x | |
41 bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f)) | |
42 | |
43 -- can not apply id. because different Level | |
44 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A | |
45 mu d = bind d id | |
46 | |
43 | 47 |
44 returnS : {l : Level} {A : Set l} -> A -> Delta A | 48 returnS : {l : Level} {A : Set l} -> A -> Delta A |
45 returnS x = mono x | 49 returnS x = mono x |
46 | 50 |
47 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A | 51 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A |
79 | 83 |
80 | 84 |
81 -- Monad-laws (Category) | 85 -- Monad-laws (Category) |
82 | 86 |
83 -- monad-law-1 : join . fmap join = join . join | 87 -- monad-law-1 : join . fmap join = join . join |
84 --monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) | 88 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) |
89 monad-law-1 (mono d) = refl | |
90 monad-law-1 (delta (mono (mono x)) (mono (mono (mono xx)))) = refl | |
91 monad-law-1 (delta (mono (mono x)) (mono (mono (delta xx d)))) = refl | |
92 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (mono (mono xxx))))) = refl | |
93 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (mono (delta xxx d))))) = refl | |
94 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (mono x₂)))))) = refl | |
95 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (delta x₂ (mono x₃))))))) = refl | |
96 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (mono (delta x₂ (delta x₃ d₂))))))) = refl | |
97 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (mono x₂) (mono (mono x₃))))))) = refl | |
98 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (delta x₂ (mono x₃)) (mono (mono x₄))))))) = refl | |
99 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta (delta x₂ (delta x₃ d₂)) (mono (mono x₄))))))) = refl | |
100 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta d₂ (mono (delta x₂ d₃))))))) = refl | |
101 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (mono x₁) (delta d₂ (delta d₃ d₄)))))) = refl | |
102 monad-law-1 (delta (mono (mono x)) (mono (delta (mono xx) (delta (delta x₁ d₁) d₂)))) = refl | |
103 monad-law-1 (delta (mono (mono x)) (mono (delta (delta x₁ d) d₁))) = refl | |
104 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (mono d₁)))) = refl | |
105 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (mono d₂))))) = refl | |
106 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (mono x₃) d₃))))) = refl | |
107 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (delta x₃ (mono x₄)) d₃))))) = refl | |
108 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (mono x₂) (delta (delta x₃ (delta x₄ d₂)) d₃))))) = refl | |
109 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (mono d₂))))) = refl | |
110 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (mono x₄) d₃))))) = refl | |
111 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (delta x₄ (mono x₅)) d₃))))) = refl | |
112 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (mono x₃)) (delta (delta x₄ (delta x₅ d₂)) d₃))))) = refl | |
113 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (mono d₂))))) = refl | |
114 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (mono x₅) d₃))))) = refl | |
115 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (mono x₆)) (mono d₃)))))) = refl | |
116 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (delta x₆ d₂)) (mono d₃)))))) = refl | |
117 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (mono x₆)) (delta d₃ d₄)))))) = refl | |
118 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (mono x₄))) (delta (delta x₅ (delta x₆ d₂)) (delta d₃ d₄)))))) = refl | |
119 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ d₁))) (mono d₂))))) = refl | |
120 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (mono x₆) d₃))))) = refl | |
121 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (delta x₆ (mono x₇)) d₃))))) = refl | |
122 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (mono x₅)))) (delta (delta x₆ (delta x₇ d₂)) d₃))))) = refl | |
123 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (mono x₇) d₃))))) = refl | |
124 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (delta x₇ (mono x₈)) d₃))))) = refl | |
125 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (mono x₆))))) (delta (delta x₇ (delta x₈ d₂)) d₃))))) = refl | |
126 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (mono x₈) d₃))))) = refl | |
127 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (delta x₈ (mono x₉)) d₃))))) = refl | |
128 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (mono x₇)))))) (delta (delta x₈ (delta x₉ d₂)) d₃))))) = refl | |
129 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (mono x₉) d₃))))) = refl | |
130 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (delta x₉ (mono x₁₀)) d₃))))) = refl | |
131 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (mono x₈))))))) (delta (delta x₉ (delta x₁₀ d₂)) d₃))))) = refl | |
132 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (mono x₁₀) d₃))))) = refl | |
133 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (delta x₁₀ (mono x₁₁)) d₃))))) = refl | |
134 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (mono x₉)))))))) (delta (delta x₁₀ (delta x₁₁ d₂)) d₃))))) = refl | |
135 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (mono x₁₁) d₃))))) = refl | |
136 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (delta x₁₁ (mono x₁₂)) d₃))))) = refl | |
137 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (mono x₁₀))))))))) (delta (delta x₁₁ (delta x₁₂ d₂)) d₃))))) = refl | |
138 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (mono x₁₂) d₃))))) = refl | |
139 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (delta x₁₂ (mono x₁₃)) d₃))))) = refl | |
140 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (mono x₁₁)))))))))) (delta (delta x₁₂ (delta x₁₃ d₂)) d₃))))) = refl | |
141 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (mono x₁₃) d₃))))) = refl | |
142 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (delta x₁₃ (mono x₁₄)) d₃))))) = refl | |
143 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (mono x₁₂))))))))))) (delta (delta x₁₃ (delta x₁₄ d₂)) d₃))))) = refl | |
144 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (mono x₁₄) d₃))))) = refl | |
145 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (delta x₁₄ (mono x₁₅)) d₃))))) = refl | |
146 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (mono x₁₃)))))))))))) (delta (delta x₁₄ (delta x₁₅ d₂)) d₃))))) = refl | |
147 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (mono x₁₅) d₃))))) = refl | |
148 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (delta x₁₅ (mono x₁₆)) d₃))))) = refl | |
149 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (mono x₁₄))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) d₃))))) = refl | |
150 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (mono x₁₅) d₃))))) = refl | |
151 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (mono x₁₆)) d₃))))) = refl | |
152 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) (mono d₃)))))) = refl | |
153 monad-law-1 (delta (mono (mono x)) (delta (mono (mono x₁)) (mono (delta (delta x₂ (delta x₃ (delta x₄ (delta x₅ (delta x₆ (delta x₇ (delta x₈ (delta x₉ (delta x₁₀ (delta x₁₁ (delta x₁₂ (delta x₁₃ (delta x₁₄ d₁))))))))))))) (delta (delta x₁₅ (delta x₁₆ d₂)) (delta d₃ d₄)))))) = refl | |
154 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ d)) (mono (mono (mono x₂))))) = refl | |
155 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ d)) (mono (mono (delta x₂ d₁))))) = refl | |
156 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (mono d₂))))) = refl | |
157 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (mono x₄) d₃))))) = refl | |
158 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (delta x₄ (mono x₅)) d₃))))) = refl | |
159 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (mono x₃) (delta (delta x₄ (delta x₅ d₂)) d₃))))) = refl | |
160 | |
161 -- 6 goals | |
162 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (mono (mono x₅)))))) = refl | |
163 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (mono (delta x₅ d₂)))))) = refl | |
164 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (mono x₅) d₃))))) = refl | |
165 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (delta x₅ (mono x₆)) d₃))))) = refl | |
166 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (mono x₄)) (delta (delta x₅ (delta x₆ d₂)) d₃))))) = refl | |
167 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (mono d₂))))) = refl | |
168 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (mono x₆) d₃))))) = refl | |
169 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (delta x₆ (mono x₇)) d₃))))) = refl | |
170 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (mono x₅))) (delta (delta x₆ (delta x₇ d₂)) d₃))))) = refl | |
171 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (delta x₅ d₁))) (mono d₂))))) = {!!} | |
172 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (mono x₂))) (mono (delta (delta x₃ (delta x₄ (delta x₅ d₁))) (delta d₂ d₃))))) = {!!} | |
173 monad-law-1 (delta (mono (mono x)) (delta (mono (delta x₁ (delta x₂ d))) (mono (delta d₁ d₂)))) = {!!} | |
174 monad-law-1 (delta (mono (mono x)) (delta (mono d) (delta d₁ d₂))) = {!!} | |
175 monad-law-1 (delta (mono (mono x)) (delta (delta d d₁) d₂)) = {!!} | |
176 -- | |
177 monad-law-1 (delta (mono (delta x x₁)) d) = {!!} | |
178 monad-law-1 (delta (delta x x₁) d) = {!!} | |
179 | |
85 | 180 |
86 | 181 |
87 {- | 182 {- |
88 -- monad-law-2-2 : join . return = id | 183 -- monad-law-2-2 : join . return = id |
89 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s | 184 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s |
102 -- Monad-laws (Haskell) | 197 -- Monad-laws (Haskell) |
103 -- monad-law-h-1 : return a >>= k = k a | 198 -- monad-law-h-1 : return a >>= k = k a |
104 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> | 199 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> |
105 (a : A) -> (k : A -> (Delta B)) -> | 200 (a : A) -> (k : A -> (Delta B)) -> |
106 (return a >>= k) ≡ (k a) | 201 (return a >>= k) ≡ (k a) |
107 monad-law-h-1 a k = begin | 202 monad-law-h-1 a k = refl |
108 return a >>= k | 203 |
109 ≡⟨ refl ⟩ | 204 |
110 mu (fmap k (return a)) | |
111 ≡⟨ refl ⟩ | |
112 mu (return (k a)) | |
113 ≡⟨ refl ⟩ | |
114 (mu ∙ return) (k a) | |
115 ≡⟨ refl ⟩ | |
116 (mu ∙ eta) (k a) | |
117 ≡⟨ (monad-law-2-2 (k a)) ⟩ | |
118 id (k a) | |
119 ≡⟨ refl ⟩ | |
120 k a | |
121 ∎ | |
122 | 205 |
123 -- monad-law-h-2 : m >>= return = m | 206 -- monad-law-h-2 : m >>= return = m |
124 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m | 207 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m |
125 monad-law-h-2 (similar lx x ly y) = monad-law-2-1 (similar lx x ly y) | 208 monad-law-h-2 (mono x) = refl |
209 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) | |
210 | |
211 | |
212 | |
126 | 213 |
127 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h | 214 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h |
128 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> | 215 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> |
129 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> | 216 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> |
130 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) | 217 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) |
131 monad-law-h-3 (similar lx x ly y) k h = begin | 218 monad-law-h-3 (mono x) k h = refl |
132 ((similar lx x ly y) >>= (\x -> (k x) >>= h)) | 219 monad-law-h-3 (delta x d) k h = begin |
220 (delta x d) >>= (\x -> k x >>= h) | |
133 ≡⟨ refl ⟩ | 221 ≡⟨ refl ⟩ |
134 mu (fmap (\x -> k x >>= h) (similar lx x ly y)) | 222 -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f)) |
223 deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) | |
135 ≡⟨ refl ⟩ | 224 ≡⟨ refl ⟩ |
136 (mu ∙ fmap (\x -> k x >>= h)) (similar lx x ly y) | 225 deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h))) |
137 ≡⟨ refl ⟩ | 226 ≡⟨ {!!} ⟩ |
138 (mu ∙ fmap (\x -> mu (fmap h (k x)))) (similar lx x ly y) | 227 ((delta x d) >>= k) >>= h |
139 ≡⟨ refl ⟩ | |
140 (mu ∙ fmap (mu ∙ (\x -> fmap h (k x)))) (similar lx x ly y) | |
141 ≡⟨ refl ⟩ | |
142 (mu ∙ (fmap mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y) | |
143 ≡⟨ refl ⟩ | |
144 (mu ∙ (fmap mu)) ((fmap (\x -> fmap h (k x))) (similar lx x ly y)) | |
145 ≡⟨ monad-law-1 (((fmap (\x -> fmap h (k x))) (similar lx x ly y))) ⟩ | |
146 (mu ∙ mu) ((fmap (\x -> fmap h (k x))) (similar lx x ly y)) | |
147 ≡⟨ refl ⟩ | |
148 (mu ∙ (mu ∙ (fmap (\x -> fmap h (k x))))) (similar lx x ly y) | |
149 ≡⟨ refl ⟩ | |
150 (mu ∙ (mu ∙ (fmap ((fmap h) ∙ k)))) (similar lx x ly y) | |
151 ≡⟨ refl ⟩ | |
152 (mu ∙ (mu ∙ ((fmap (fmap h)) ∙ (fmap k)))) (similar lx x ly y) | |
153 ≡⟨ refl ⟩ | |
154 (mu ∙ (mu ∙ (fmap (fmap h)))) (fmap k (similar lx x ly y)) | |
155 ≡⟨ refl ⟩ | |
156 mu ((mu ∙ (fmap (fmap h))) (fmap k (similar lx x ly y))) | |
157 ≡⟨ cong (\fx -> mu fx) (monad-law-4 h (fmap k (similar lx x ly y))) ⟩ | |
158 mu (fmap h (mu (similar lx (k x) ly (k y)))) | |
159 ≡⟨ refl ⟩ | |
160 (mu ∙ fmap h) (mu (fmap k (similar lx x ly y))) | |
161 ≡⟨ refl ⟩ | |
162 mu (fmap h (mu (fmap k (similar lx x ly y)))) | |
163 ≡⟨ refl ⟩ | |
164 (mu (fmap k (similar lx x ly y))) >>= h | |
165 ≡⟨ refl ⟩ | |
166 ((similar lx x ly y) >>= k) >>= h | |
167 ∎ | 228 ∎ |
168 | |
169 -} | 229 -} |