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comparison agda/delta/monad.agda @ 90:55d11ce7e223

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Unify levels on data type. only use suc to proofs
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 19 Jan 2015 12:11:38 +0900
parents 526186c4f298
children bcd4fe52a504
comparison
equal deleted inserted replaced
89:5411ce26d525 90:55d11ce7e223
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 (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 (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) 135 bind (delta-fmap mu d) (n-tail n) ≡ bind (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 (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ 138 bind (delta-fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
139 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ 139 bind (delta (mu d) (delta-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) ⟩ 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) ⟩
141 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ 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) ⟩
142 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ 142 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
143 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ 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) 144 bind (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 (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ 147 bind (delta-fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
148 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ 148 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 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 ⟩ 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 ⟩ 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)) 153 bind (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 (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ 156 bind (delta-fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
157 bind (mono (mu (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 ⟩ 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)) ⟩ 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 ⟩ 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 ⟩ 161 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
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 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 ⟩ 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)) 165 bind (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 (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ 168 bind (delta-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 ⟩ 169 bind (delta (mu (mono d)) (delta-fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
170 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ 170 bind (delta d (delta-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 ⟩ 171 delta (headDelta ((n-tail (S n)) d)) (bind (delta-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) ⟩ 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) ⟩
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)) (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 ⟩ 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 ⟩ 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)) ⟩ 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 ⟩ 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 ⟩ 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)) 179 bind (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 (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ 182 bind (delta-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 ⟩ 183 bind (delta (mu (delta d dd)) (delta-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 ⟩ 184 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (delta-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)) ⟩ 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)) ⟩
186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ 186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (delta-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 ⟩ 187 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (delta-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) ⟩ 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) ⟩
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) (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 ⟩ 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 ⟩ 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 ⟩ 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) ⟩ 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) ⟩
195 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (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)) 196 bind (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 . 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 ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) 201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (delta-fmap mu)) d) ≡ ((mu ∙ 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 ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ 204 (mu ∙ delta-fmap mu) (delta x d) ≡⟨ refl ⟩
205 mu (fmap mu (delta x d)) ≡⟨ refl ⟩ 205 mu (delta-fmap mu (delta x d)) ≡⟨ refl ⟩
206 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ 206 mu (delta (mu x) (delta-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) ⟩ 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) ⟩
208 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ 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) ⟩
209 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ 209 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩
210 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ 210 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩
211 mu (mu (delta x d)) ≡⟨ refl ⟩ 211 mu (mu (delta x d)) ≡⟨ refl ⟩
212 (mu ∙ mu) (delta x d) 212 (mu ∙ 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 (fmap eta d) (n-tail n)) ≡ d 216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (delta-fmap 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 (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ 219 bind (delta-fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩
220 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ 220 bind (delta (eta x) (delta-fmap eta d)) id ≡⟨ refl ⟩
221 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ 221 delta (headDelta (eta x)) (bind (delta-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) ⟩ 222 delta x (bind (delta-fmap 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 (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ 225 bind (delta-fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
226 bind (mono (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 ⟩ 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 (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ 230 bind (delta-fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩
231 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ 231 bind (delta (eta x) (delta-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 ⟩ 232 delta (headDelta ((n-tail (S n) (eta x)))) (bind (delta-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) ⟩ 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) ⟩
234 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ 234 delta (headDelta (eta x)) (bind (delta-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) ⟩ 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) ⟩
236 delta x d 236 delta x d
237 ∎ 237 ∎
238 238
239 239
240 -- monad-law-2 : join . fmap return = join . return = id 240 -- monad-law-2 : join . delta-fmap return = join . return = id
241 -- monad-law-2 join . 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 ∙ fmap eta) d ≡ (mu ∙ eta) d 243 (mu ∙ delta-fmap eta) d ≡ (mu ∙ 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 ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ 246 (mu ∙ delta-fmap eta) (delta x d) ≡⟨ refl ⟩
247 mu (fmap eta (delta x d)) ≡⟨ refl ⟩ 247 mu (delta-fmap eta (delta x d)) ≡⟨ refl ⟩
248 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ 248 mu (delta (mono x) (delta-fmap eta d)) ≡⟨ refl ⟩
249 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ 249 delta (headDelta (mono x)) (bind (delta-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) ⟩ 250 delta x (bind (delta-fmap 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 mu (mono (delta x d)) ≡⟨ refl ⟩
253 mu (eta (delta x d)) ≡⟨ refl ⟩ 253 mu (eta (delta x d)) ≡⟨ refl ⟩
254 (mu ∙ eta) (delta x d) 254 (mu ∙ eta) (delta x d)
255 ∎ 255 ∎
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) -> (mu ∙ 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 = 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 ≡ (fmap f ∙ eta) x 264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (delta-fmap f ∙ 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 (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) 269 bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (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 (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ 272 bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩
273 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ 273 bind (delta (delta-fmap f d) (delta-fmap (delta-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) ⟩ 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) ⟩
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) ⟩ 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) ⟩
276 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ 276 delta (f (headDelta d)) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
277 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ 277 delta-fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
278 fmap f (bind (delta d ds) (n-tail O)) ∎ 278 delta-fmap f (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 (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ 280 bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
281 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ 281 bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩
282 n-tail (S n) (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 fmap f (n-tail (S n) d) ≡⟨ refl ⟩ 283 delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩
284 fmap f (bind (mono d) (n-tail (S n))) 284 delta-fmap f (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 (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ 287 bind (delta-fmap (delta-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 ⟩ 288 delta (headDelta (n-tail (S n) (delta-fmap f d))) (bind (delta-fmap (delta-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) ⟩ 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) ⟩
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)) ⟩ 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)) ⟩
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) ⟩ 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) ⟩
292 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ 292 delta (f (headDelta (n-tail (S n) d))) (delta-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 ⟩ 293 delta-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 ⟩ 294 delta-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))) ∎ 295 delta-fmap f (bind (delta d ds) (n-tail (S n))) ∎
296 296
297 297
298 -- monad-law-4 : join . fmap (fmap f) = fmap f . join 298 -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join
299 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (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 ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d 300 (mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ 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 ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ 303 (mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩
304 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ 304 mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩
305 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ 305 mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩
306 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ 306 delta (headDelta (mono (f x))) (bind (delta-fmap (delta-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) ⟩ 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) ⟩
308 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ 308 delta (f x) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
309 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ 309 delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
310 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ 310 delta-fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩
311 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ 311 delta-fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩
312 (fmap f ∙ mu) (delta (mono x) ds) ∎ 312 (delta-fmap f ∙ 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 ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ 314 (mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩
315 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ 315 mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩
316 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ 316 mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩
317 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ 317 delta (headDelta (delta (f x) (delta-fmap f d))) (bind (delta-fmap (delta-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) ⟩ 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) ⟩
319 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ 319 delta (f x) (delta-fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
320 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ 320 delta-fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
321 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ 321 delta-fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩
322 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ 322 delta-fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩
323 (fmap f ∙ mu) (delta (delta x d) ds) ∎ 323 (delta-fmap f ∙ 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 { mu = mu;
327 eta = eta; 327 eta = eta;
328 association-law = monad-law-1; 328 association-law = monad-law-1;