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comparison agda/delta.agda @ 88:526186c4f298

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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 6789c65a75bc
children 5411ce26d525
comparison
equal deleted inserted replaced
87:6789c65a75bc 88:526186c4f298
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135 {-
136 -- Monad-laws (Category)
137
138 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
139 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
140 monad-law-1-5 O O ds = refl
141 monad-law-1-5 O (S n) (mono ds) = begin
142 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
143 n-tail (S n) ds ≡⟨ refl ⟩
144 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
145 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
146 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
147 ∎
148 monad-law-1-5 O (S n) (delta d ds) = begin
149 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
150 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
151 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
152 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
153 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
154 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) ⟩
155 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
156 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
157 ∎
158 monad-law-1-5 (S m) n (mono (mono x)) = begin
159 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
160 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
161 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
162 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
163 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
164 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))) ⟩
165 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
166 ∎
167 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
168 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
169 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
170 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
171 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
172 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩
173 n-tail (m + n) ds ≡⟨ refl ⟩
174 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
175 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
176 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
177 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))) ⟩
178 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
179 ∎
180 monad-law-1-5 (S m) O (delta d ds) = begin
181 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
182 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
183 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
184 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
185 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)) ⟩
186 bind (n-tail O (delta d ds)) (n-tail (S m + O))
187 ∎
188 monad-law-1-5 (S m) (S n) (delta d ds) = begin
189 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)) ⟩
190 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
191 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
192 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
193 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
194 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)) ⟩
195 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
196 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) ⟩
197 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
198 ∎
199
200 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
201 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
202 monad-law-1-4 O O (mono dd) = refl
203 monad-law-1-4 O O (delta dd dd₁) = refl
204 monad-law-1-4 O (S n) (mono dd) = begin
205 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
206 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
207 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)) ⟩
208 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩
209 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd))))
210 ∎
211 monad-law-1-4 O (S n) (delta d ds) = begin
212 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩
213 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩
214 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)) ⟩
215 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩
216 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩
217 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
218 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) ⟩
219 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
220 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
221 ∎
222 monad-law-1-4 (S m) n (mono dd) = begin
223 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
224 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
225 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩
226 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
227 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)) ⟩
228 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
229 ∎
230 monad-law-1-4 (S m) O (delta d ds) = begin
231 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
232 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
233 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
234 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩
235 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩
236 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩
237 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds))))
238 ∎
239 monad-law-1-4 (S m) (S n) (delta d ds) = begin
240 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
241 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)) ⟩
242 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
243 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
244 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
245 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)) ⟩
246 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
247 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) ⟩
248 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
249 ∎
250
251 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
252 monad-law-1-2 (mono _) = refl
253 monad-law-1-2 (delta _ _) = refl
254
255 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
256 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
257 monad-law-1-3 O (mono d) = refl
258 monad-law-1-3 O (delta d ds) = begin
259 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
260 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
261 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
262 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
263 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
264 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
265 bind (bind (delta d ds) (n-tail O)) (n-tail O)
266 ∎
267 monad-law-1-3 (S n) (mono (mono d)) = begin
268 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
269 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
270 (n-tail (S n)) d ≡⟨ refl ⟩
271 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
272 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
273 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
274 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
275 ∎
276 monad-law-1-3 (S n) (mono (delta d ds)) = begin
277 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
278 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
279 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
280 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
281 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
282 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
283 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
284 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) ⟩
285 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
286 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
287 ∎
288 monad-law-1-3 (S n) (delta (mono d) ds) = begin
289 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
290 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
291 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
292 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
293 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) ⟩
294 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
295 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
296 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
297 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)) ⟩
298 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 ⟩
299 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
300 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
301 ∎
302 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
303 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
304 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
305 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
306 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)) ⟩
307 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
308 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
309 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) ⟩
310 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) ⟩
311 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 ⟩
312 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 ⟩
313 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 ⟩
314 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) ⟩
315 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 ⟩
316 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
317 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
318 ∎
319
320
321 -- monad-law-1 : join . fmap join = join . join
322 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
323 monad-law-1 (mono d) = refl
324 monad-law-1 (delta x d) = begin
325 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩
326 mu (fmap mu (delta x d)) ≡⟨ refl ⟩
327 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩
328 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
329 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
330 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩
331 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩
332 mu (mu (delta x d)) ≡⟨ refl ⟩
333 (mu ∙ mu) (delta x d)
334 ∎
335
336
337 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d
338 monad-law-2-1 O (mono x) = refl
339 monad-law-2-1 O (delta x d) = begin
340 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩
341 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩
342 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
343 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩
344 delta x d ∎
345 monad-law-2-1 (S n) (mono x) = begin
346 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
347 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩
348 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩
349 mono x ∎
350 monad-law-2-1 (S n) (delta x d) = begin
351 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩
352 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩
353 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
354 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) ⟩
355 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩
356 delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩
357 delta x d
358 ∎
359
360
361 -- monad-law-2 : join . fmap return = join . return = id
362 -- monad-law-2 join . fmap return = join . return
363 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) ->
364 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d
365 monad-law-2 (mono x) = refl
366 monad-law-2 (delta x d) = begin
367 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩
368 mu (fmap eta (delta x d)) ≡⟨ refl ⟩
369 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩
370 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩
371 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩
372 (delta x d) ≡⟨ refl ⟩
373 mu (mono (delta x d)) ≡⟨ refl ⟩
374 mu (eta (delta x d)) ≡⟨ refl ⟩
375 (mu ∙ eta) (delta x d)
376 ∎
377
378
379 -- monad-law-2' : join . return = id
380 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d
381 monad-law-2' d = refl
382
383
384 -- monad-law-3 : return . f = fmap f . return
385 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
386 monad-law-3 f x = refl
387
388
389 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) ->
390 bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n))
391 monad-law-4-1 O f (mono d) = refl
392 monad-law-4-1 O f (delta d ds) = begin
393 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩
394 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩
395 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) ⟩
396 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) ⟩
397 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
398 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
399 fmap f (bind (delta d ds) (n-tail O)) ∎
400 monad-law-4-1 (S n) f (mono d) = begin
401 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
402 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩
403 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩
404 fmap f (n-tail (S n) d) ≡⟨ refl ⟩
405 fmap f (bind (mono d) (n-tail (S n)))
406 ∎
407 monad-law-4-1 (S n) f (delta d ds) = begin
408 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
409 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
410 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) ⟩
411 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)) ⟩
412 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) ⟩
413 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
414 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩
415 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩
416 fmap f (bind (delta d ds) (n-tail (S n))) ∎
417
418
419 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
420 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) ->
421 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d
422 monad-law-4 f (mono d) = refl
423 monad-law-4 f (delta (mono x) ds) = begin
424 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩
425 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩
426 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
427 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
428 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
429 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
430 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
431 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩
432 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩
433 (fmap f ∙ mu) (delta (mono x) ds) ∎
434 monad-law-4 f (delta (delta x d) ds) = begin
435 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩
436 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩
437 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩
438 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩
439 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩
440 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩
441 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩
442 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩
443 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩
444 (fmap f ∙ mu) (delta (delta x d) ds) ∎
445
446
447
448 {-
449 -- Monad-laws (Haskell)
450 -- monad-law-h-1 : return a >>= k = k a
451 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
452 (a : A) -> (k : A -> (Delta B)) ->
453 (return a >>= k) ≡ (k a)
454 monad-law-h-1 a k = refl
455
456
457
458 -- monad-law-h-2 : m >>= return = m
459 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
460 monad-law-h-2 (mono x) = refl
461 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
462
463
464 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
465 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
466 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
467 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
468 monad-law-h-3 (mono x) k h = refl
469 monad-law-h-3 (delta x d) k h = {!!}
470
471 -}
472 -}