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Proving monad-law-1
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 30 Nov 2014 22:26:50 +0900
parents e95f15af3f8b
children 1f4ea5cb153d
comparison
equal deleted inserted replaced
72:e95f15af3f8b 73:0ad0ae7a3cbe
89 O + n = n 89 O + n = n
90 (S m) + n = S (m + n) 90 (S m) + n = S (m + n)
91 91
92 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A)) 92 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
93 n-tail O = id 93 n-tail O = id
94 n-tail (S n) = (n-tail n) ∙ tailDelta 94 n-tail (S n) = tailDelta ∙ (n-tail n)
95 95
96 postulate n-tail-plus : (n : Int) -> (tailDelta ∙ (n-tail n)) ≡ n-tail (S n) 96 flip : {l : Level} {A : Set l} -> (f : A -> A) -> f ∙ (f ∙ f) ≡ (f ∙ f) ∙ f
97 flip f = refl
98
99 n-tail-plus : {l : Level} {A : Set l} -> (n : Int) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
100 n-tail-plus O = refl
101 n-tail-plus (S n) = begin
102 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
103 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
104 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
105 n-tail (S (S n))
106 ∎
107
108 postulate n-tail-add : {l : Level} {A : Set l} -> (n m : Int) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
109 postulate int-add-assoc : (n m : Int) -> n + m ≡ m + n
110 postulate int-add-right-zero : (n : Int) -> n ≡ n + O
111 postulate int-add-right : (n m : Int) -> S n + S m ≡ S (S (n + m))
112
97 113
98 114
99 115
100 116
101 117
102 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) -> 118 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) ->
103 (n-tail n) (mono x) ≡ (mono x) 119 (n-tail n) (mono x) ≡ (mono x)
104 tail-delta-to-mono O x = refl 120 tail-delta-to-mono O x = refl
105 tail-delta-to-mono (S n) x = begin 121 tail-delta-to-mono (S n) x = begin
106 n-tail (S n) (mono x) ≡⟨ refl ⟩
107 ((n-tail n) ∙ tailDelta) (mono x) ≡⟨ refl ⟩
108 (n-tail n) (tailDelta (mono x)) ≡⟨ refl ⟩
109 (n-tail n) (mono x) ≡⟨ tail-delta-to-mono n x ⟩
110 mono x
111 ∎
112 {- begin
113 n-tail (S n) (mono x) ≡⟨ refl ⟩ 122 n-tail (S n) (mono x) ≡⟨ refl ⟩
114 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ 123 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
115 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩ 124 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
116 tailDelta (mono x) ≡⟨ refl ⟩ 125 tailDelta (mono x) ≡⟨ refl ⟩
117 mono x 126 mono x
118 ∎ 127 ∎
119 -} 128
129 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Int) (n : Int) -> (ds : Delta (Delta A)) ->
130 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
131 monad-law-1-5 O O ds = refl
132 monad-law-1-5 O (S n) (mono ds) = begin
133 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
134 n-tail (S n) ds ≡⟨ refl ⟩
135 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
136 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
137 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
138 ∎
139 monad-law-1-5 O (S n) (delta d ds) = begin
140 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
141 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
142 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
143 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
144 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
145 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) ⟩
146 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
147 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
148 ∎
149 monad-law-1-5 (S m) n (mono (mono x)) = begin
150 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
151 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
152 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
153 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
154 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
155 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))) ⟩
156 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
157 ∎
158 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
159 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
160 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
161 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
162 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add n m) ⟩
163 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩
164 n-tail (m + n) ds ≡⟨ refl ⟩
165 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
166 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
167 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
168 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))) ⟩
169 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
170 ∎
171 monad-law-1-5 (S m) O (delta d ds) = begin
172 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
173 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
174 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
175 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
176 bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (int-add-right-zero (S m)) ⟩
177 bind (n-tail O (delta d ds)) (n-tail (S m + O))
178 ∎
179 monad-law-1-5 (S m) (S n) (delta d ds) = begin
180 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)) ⟩
181 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
182 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
183 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
184 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
185 bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (int-add-right m n)) ⟩
186 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
187 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) ⟩
188 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
189 ∎
190
191 monad-law-1-4 : {l : Level} {A : Set l} -> (n : Int) -> (dd : Delta (Delta A)) ->
192 headDelta ((n-tail n) (bind dd tailDelta)) ≡ headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))
193 monad-law-1-4 O (mono dd) = refl
194 monad-law-1-4 O (delta dd dd₁) = refl
195 monad-law-1-4 (S n) (mono dd) = begin
196 headDelta (n-tail (S n) (bind (mono dd) tailDelta)) ≡⟨ refl ⟩
197 headDelta (n-tail (S n) (tailDelta dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-plus (S n)) ⟩
198 headDelta (n-tail (S (S n)) dd) ≡⟨ refl ⟩
199 headDelta (n-tail (S (S n)) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S (S n)) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
200 headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (mono dd))))
201 ∎
202 monad-law-1-4 (S n) (delta d ds) = begin
203 headDelta (n-tail (S n) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
204 headDelta (n-tail (S n) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta)))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta))))) (sym (n-tail-plus n)) ⟩
205 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta)))) ≡⟨ refl ⟩
206 headDelta (n-tail n (bind ds (tailDelta ∙ tailDelta))) ≡⟨ {!!} ⟩
207 headDelta (n-tail (S (S n)) (headDelta ((n-tail n ds)))) ≡⟨ refl ⟩
208 headDelta (n-tail (S (S n)) (headDelta ((n-tail n ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S (S n)) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
209 headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (delta d ds))))
210 ∎
211
120 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) 212 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
121 monad-law-1-2 (mono _) = refl 213 monad-law-1-2 (mono _) = refl
122 monad-law-1-2 (delta _ _) = refl 214 monad-law-1-2 (delta _ _) = refl
123 215
124 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Int) -> (d : Delta (Delta (Delta A))) -> 216 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Int) -> (d : Delta (Delta (Delta A))) ->
144 ∎ 236 ∎
145 monad-law-1-3 (S n) (mono (delta d ds)) = begin 237 monad-law-1-3 (S n) (mono (delta d ds)) = begin
146 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ 238 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
147 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ 239 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
148 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ 240 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
149 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ 241 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
150 n-tail n (bind ds tailDelta) ≡⟨ {!!} ⟩ 242 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
243 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
151 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ 244 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
245 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) ⟩
152 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ 246 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
153 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) 247 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
154 ∎ 248 ∎
155 monad-law-1-3 (S n) (delta (mono d) ds) = begin 249 monad-law-1-3 (S n) (delta (mono d) ds) = begin
156 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ 250 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
157 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ 251 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
158 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ 252 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
159 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩ 253 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
160 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩ 254 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) ⟩
255 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
256 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
161 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ 257 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
162 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)) ⟩ 258 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)) ⟩
163 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 ⟩ 259 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 ⟩
164 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ 260 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
165 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) 261 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
166 ∎ 262 ∎
167 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin 263 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
168 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ 264 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
169 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ 265 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
170 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ 266 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
171 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ 267 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)) ⟩
172 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ {!!} ⟩ 268 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
173 269 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
270 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) ⟩
271 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 n dd) ⟩
272 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 ⟩
273 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 ⟩
174 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 ⟩ 274 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 ⟩
275 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) ⟩
175 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 ⟩ 276 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 ⟩
176 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ 277 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
177 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) 278 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
178 ∎ 279 ∎
179 280