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comparison agda/delta.agda @ 80:fc5cd8c50312 InfiniteDelta

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Adjust proofs
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 01 Dec 2014 17:30:49 +0900
parents 7307e43a3c76
children 6789c65a75bc
comparison
equal deleted inserted replaced
79:7307e43a3c76 80:fc5cd8c50312
102 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ 102 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
103 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩ 103 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
104 tailDelta (mono x) ≡⟨ refl ⟩ 104 tailDelta (mono x) ≡⟨ refl ⟩
105 mono x ∎ 105 mono x ∎
106 106
107 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} 107 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
108 -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d) 108 -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d)
109 head-delta-natural-transformation f (mono x) = refl 109 head-delta-natural-transformation f (mono x) = refl
110 head-delta-natural-transformation f (delta x d) = refl 110 head-delta-natural-transformation f (delta x d) = refl
111 111
112 n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} 112 n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
113 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d) 113 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d)
114 n-tail-natural-transformation O f d = refl 114 n-tail-natural-transformation O f d = refl
115 n-tail-natural-transformation (S n) f (mono x) = begin 115 n-tail-natural-transformation (S n) f (mono x) = begin
144 functor-law-2 f g (mono x) = refl 144 functor-law-2 f g (mono x) = refl
145 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d) 145 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
146 146
147 147
148 -- Monad-laws (Category) 148 -- Monad-laws (Category)
149 {-
150 149
151 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> 150 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
152 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) 151 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
153 monad-law-1-5 O O ds = refl 152 monad-law-1-5 O O ds = refl
154 monad-law-1-5 O (S n) (mono ds) = begin 153 monad-law-1-5 O (S n) (mono ds) = begin
155 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ 154 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
156 n-tail (S n) ds ≡⟨ refl ⟩ 155 n-tail (S n) ds ≡⟨ refl ⟩
157 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ 156 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
158 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ 157 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
159 bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) 158 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
160 ∎ 159 ∎
161 monad-law-1-5 O (S n) (delta d ds) = begin 160 monad-law-1-5 O (S n) (delta d ds) = begin
162 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ 161 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
163 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ 162 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
164 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ 163 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
165 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ 164 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
166 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ 165 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
167 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) ⟩ 166 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) ⟩
168 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ 167 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
169 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) 168 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
170 ∎ 169 ∎
171 monad-law-1-5 (S m) n (mono (mono x)) = begin 170 monad-law-1-5 (S m) n (mono (mono x)) = begin
172 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ 171 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
173 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ 172 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
174 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ 173 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
175 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ 174 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
176 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ 175 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
177 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))) ⟩ 176 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))) ⟩
178 bind (n-tail n (mono (mono x))) (n-tail (S m + n)) 177 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
179 ∎ 178 ∎
180 monad-law-1-5 (S m) n (mono (delta x ds)) = begin 179 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
181 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ 180 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
182 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ 181 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
183 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ 182 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
184 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ 183 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
185 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ 184 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩
186 n-tail (m + n) ds ≡⟨ refl ⟩ 185 n-tail (m + n) ds ≡⟨ refl ⟩
187 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ 186 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
188 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ 187 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
189 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ 188 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
190 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))) ⟩ 189 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))) ⟩
191 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) 190 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
192 ∎ 191 ∎
193 monad-law-1-5 (S m) O (delta d ds) = begin 192 monad-law-1-5 (S m) O (delta d ds) = begin
194 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ 193 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
195 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ 194 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
196 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ 195 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
197 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ 196 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
198 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)) ⟩ 197 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)) ⟩
199 bind (n-tail O (delta d ds)) (n-tail (S m + O)) 198 bind (n-tail O (delta d ds)) (n-tail (S m + O))
200 ∎ 199 ∎
201 monad-law-1-5 (S m) (S n) (delta d ds) = begin 200 monad-law-1-5 (S m) (S n) (delta d ds) = begin
202 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)) ⟩ 201 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)) ⟩
203 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ 202 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
204 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ 203 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
205 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ 204 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
206 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ 205 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
207 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)) ⟩ 206 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)) ⟩
208 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ 207 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
209 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) ⟩ 208 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) ⟩
210 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) 209 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
211 ∎ 210 ∎
212 211
213 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> 212 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
214 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) 213 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
215 monad-law-1-4 O O (mono dd) = refl 214 monad-law-1-4 O O (mono dd) = refl
216 monad-law-1-4 O O (delta dd dd₁) = refl 215 monad-law-1-4 O O (delta dd dd₁) = refl
217 monad-law-1-4 O (S n) (mono dd) = begin 216 monad-law-1-4 O (S n) (mono dd) = begin
218 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ 217 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
219 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ 218 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
220 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)) ⟩ 219 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)) ⟩
221 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ 220 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩
222 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) 221 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd))))
223 ∎ 222 ∎
224 monad-law-1-4 O (S n) (delta d ds) = begin 223 monad-law-1-4 O (S n) (delta d ds) = begin
225 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ 224 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩
226 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ 225 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩
227 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)) ⟩ 226 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)) ⟩
228 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ 227 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩
229 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ 228 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩
230 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ 229 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
231 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) ⟩ 230 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) ⟩
232 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ 231 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
233 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) 232 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
234 ∎ 233 ∎
235 monad-law-1-4 (S m) n (mono dd) = begin 234 monad-law-1-4 (S m) n (mono dd) = begin
236 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ 235 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
237 headDelta (n-tail n ((n-tail (S m)) dd))≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ 236 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
238 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ 237 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩
239 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ 238 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
240 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)) ⟩ 239 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)) ⟩
241 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) 240 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
242 ∎ 241 ∎
243 monad-law-1-4 (S m) O (delta d ds) = begin 242 monad-law-1-4 (S m) O (delta d ds) = begin
244 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ 243 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
245 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ 244 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
246 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ 245 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
247 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ 246 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩
248 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ 247 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩
249 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ 248 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩
250 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) 249 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds))))
251 ∎ 250 ∎
252 monad-law-1-4 (S m) (S n) (delta d ds) = begin 251 monad-law-1-4 (S m) (S n) (delta d ds) = begin
253 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ 252 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
254 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)) ⟩ 253 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)) ⟩
255 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ 254 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
256 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ 255 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
257 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ 256 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
258 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)) ⟩ 257 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)) ⟩
259 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ 258 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
260 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) ⟩ 259 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) ⟩
261 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) 260 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
262 ∎ 261 ∎
263 262
264 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) 263 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
265 monad-law-1-2 (mono _) = refl 264 monad-law-1-2 (mono _) = refl
267 266
268 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> 267 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
269 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) 268 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
270 monad-law-1-3 O (mono d) = refl 269 monad-law-1-3 O (mono d) = refl
271 monad-law-1-3 O (delta d ds) = begin 270 monad-law-1-3 O (delta d ds) = begin
272 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ 271 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
273 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ 272 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
274 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ 273 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
275 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ 274 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
276 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ 275 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
277 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ 276 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
278 bind (bind (delta d ds) (n-tail O)) (n-tail O) 277 bind (bind (delta d ds) (n-tail O)) (n-tail O)
279 ∎ 278 ∎
280 monad-law-1-3 (S n) (mono (mono d)) = begin 279 monad-law-1-3 (S n) (mono (mono d)) = begin
281 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ 280 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
282 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ 281 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
283 (n-tail (S n)) d ≡⟨ refl ⟩ 282 (n-tail (S n)) d ≡⟨ refl ⟩
284 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ 283 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
285 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ 284 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
286 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ 285 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
287 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) 286 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
288 ∎ 287 ∎
289 monad-law-1-3 (S n) (mono (delta d ds)) = begin 288 monad-law-1-3 (S n) (mono (delta d ds)) = begin
290 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ 289 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
291 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ 290 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
292 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ 291 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
293 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ 292 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
294 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ 293 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
295 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ 294 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
296 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ 295 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
297 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) ⟩ 296 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) ⟩
298 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ 297 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
299 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) 298 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
300 ∎ 299 ∎
301 monad-law-1-3 (S n) (delta (mono d) ds) = begin 300 monad-law-1-3 (S n) (delta (mono d) ds) = begin
302 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ 301 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
303 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ 302 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
333 332
334 -- monad-law-1 : join . fmap join = join . join 333 -- monad-law-1 : join . fmap join = join . join
335 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) 334 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
336 monad-law-1 (mono d) = refl 335 monad-law-1 (mono d) = refl
337 monad-law-1 (delta x d) = begin 336 monad-law-1 (delta x d) = begin
338 (mu ∙ fmap mu) (delta x d) 337 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩
339 ≡⟨ refl ⟩ 338 mu (fmap mu (delta x d)) ≡⟨ refl ⟩
340 mu (fmap mu (delta x d)) 339 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩
341 ≡⟨ refl ⟩ 340 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
342 mu (delta (mu x) (fmap mu d)) 341 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
343 ≡⟨ refl ⟩ 342 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩
344 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) 343 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩
345 ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ 344 mu (mu (delta x d)) ≡⟨ refl ⟩
346 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta)
347 ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
348 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta)
349 ≡⟨ refl ⟩
350 mu (delta (headDelta x) (bind d tailDelta))
351 ≡⟨ refl ⟩
352 mu (mu (delta x d))
353 ≡⟨ refl ⟩
354 (mu ∙ mu) (delta x d) 345 (mu ∙ mu) (delta x d)
355 ∎ 346 ∎
356 347
357
358 -}
359 348
360 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d 349 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d
361 monad-law-2-1 O (mono x) = refl 350 monad-law-2-1 O (mono x) = refl
362 monad-law-2-1 O (delta x d) = begin 351 monad-law-2-1 O (delta x d) = begin
363 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ 352 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩