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comparison agda/delta.agda @ 76:c7076f9bbaed

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Refactors
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 01 Dec 2014 11:47:52 +0900
parents a4eb68476766
children 4b16b485a4b2
comparison
equal deleted inserted replaced
75:a4eb68476766 76:c7076f9bbaed
1 open import list 1 open import list
2 open import basic 2 open import basic
3 open import nat
3 4
4 open import Level 5 open import Level
5 open import Relation.Binary.PropositionalEquality 6 open import Relation.Binary.PropositionalEquality
6 open ≡-Reasoning 7 open ≡-Reasoning
7 8
21 headDelta (delta x _) = x 22 headDelta (delta x _) = x
22 23
23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A 24 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
24 tailDelta (mono x) = mono x 25 tailDelta (mono x) = mono x
25 tailDelta (delta _ d) = d 26 tailDelta (delta _ d) = d
27
28 n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A))
29 n-tail O = id
30 n-tail (S n) = tailDelta ∙ (n-tail n)
26 31
27 32
28 -- Functor 33 -- Functor
29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B) 34 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
30 fmap f (mono x) = mono (f x) 35 fmap f (mono x) = mono (f x)
61 66
62 67
63 68
64 -- proofs 69 -- proofs
65 70
66 -- Functor-laws 71 -- sub-proofs
67 72
68 -- Functor-law-1 : T(id) = id' 73 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
69 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
70 functor-law-1 (mono x) = refl
71 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
72
73 -- Functor-law-2 : T(f . g) = T(f) . T(g)
74 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
75 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
76 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
77 functor-law-2 f g (mono x) = refl
78 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
79
80
81 -- Monad-laws (Category)
82
83 data Int : Set where
84 O : Int
85 S : Int -> Int
86
87 _+_ : Int -> Int -> Int
88 O + n = n
89 (S m) + n = S (m + n)
90 postulate int-add-assoc : (n m : Int) -> n + m ≡ m + n
91 postulate int-add-right-zero : (n : Int) -> n ≡ n + O
92 postulate int-add-right : (n m : Int) -> S n + S m ≡ S (S (n + m))
93
94
95
96 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
97 n-tail O = id
98 n-tail (S n) = tailDelta ∙ (n-tail n)
99
100 n-tail-plus : {l : Level} {A : Set l} -> (n : Int) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
101 n-tail-plus O = refl 74 n-tail-plus O = refl
102 n-tail-plus (S n) = begin 75 n-tail-plus (S n) = begin
103 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩ 76 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
104 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩ 77 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
105 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩ 78 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
106 n-tail (S (S n)) 79 n-tail (S (S n))
107 ∎ 80 ∎
108 81
109 n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Int) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m) 82 n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
110 n-tail-add O m = refl 83 n-tail-add O m = refl
111 n-tail-add (S n) O = begin 84 n-tail-add (S n) O = begin
112 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩ 85 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
113 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (int-add-right-zero (S n))⟩ 86 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (int-add-right-zero (S n))⟩
114 n-tail (S n + O) 87 n-tail (S n + O)
119 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩ 92 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩
120 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩ 93 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
121 n-tail (S (n + S m)) ≡⟨ refl ⟩ 94 n-tail (S (n + S m)) ≡⟨ refl ⟩
122 n-tail (S n + S m) ∎ 95 n-tail (S n + S m) ∎
123 96
124 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) -> 97 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
125 (n-tail n) (mono x) ≡ (mono x) 98 (n-tail n) (mono x) ≡ (mono x)
126 tail-delta-to-mono O x = refl 99 tail-delta-to-mono O x = refl
127 tail-delta-to-mono (S n) x = begin 100 tail-delta-to-mono (S n) x = begin
128 n-tail (S n) (mono x) ≡⟨ refl ⟩ 101 n-tail (S n) (mono x) ≡⟨ refl ⟩
129 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ 102 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
130 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) ⟩
131 tailDelta (mono x) ≡⟨ refl ⟩ 104 tailDelta (mono x) ≡⟨ refl ⟩
132 mono x ∎ 105 mono x ∎
133 106
134 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Int) (n : Int) -> (ds : Delta (Delta A)) -> 107 -- Functor-laws
135 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) 108
109 -- Functor-law-1 : T(id) = id'
110 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
111 functor-law-1 (mono x) = refl
112 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
113
114 -- Functor-law-2 : T(f . g) = T(f) . T(g)
115 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
116 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
117 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
118 functor-law-2 f g (mono x) = refl
119 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
120
121
122 -- Monad-laws (Category)
123
124
125 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) ->
126 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
136 monad-law-1-5 O O ds = refl 127 monad-law-1-5 O O ds = refl
137 monad-law-1-5 O (S n) (mono ds) = begin 128 monad-law-1-5 O (S n) (mono ds) = begin
138 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ 129 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
139 n-tail (S n) ds ≡⟨ refl ⟩ 130 n-tail (S n) ds ≡⟨ refl ⟩
140 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ 131 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
162 ∎ 153 ∎
163 monad-law-1-5 (S m) n (mono (delta x ds)) = begin 154 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
164 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ 155 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
165 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ 156 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
166 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ 157 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
167 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add n m) ⟩ 158 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩
168 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩ 159 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩
169 n-tail (m + n) ds ≡⟨ refl ⟩ 160 n-tail (m + n) ds ≡⟨ refl ⟩
170 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ 161 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
171 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ 162 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
172 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ 163 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
191 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ 182 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
192 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) ⟩ 183 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) ⟩
193 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) 184 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
194 ∎ 185 ∎
195 186
196 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Int) -> (dd : Delta (Delta A)) -> 187 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) ->
197 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) 188 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd)))
198 monad-law-1-4 O O (mono dd) = refl 189 monad-law-1-4 O O (mono dd) = refl
199 monad-law-1-4 O O (delta dd dd₁) = refl 190 monad-law-1-4 O O (delta dd dd₁) = refl
200 monad-law-1-4 O (S n) (mono dd) = begin 191 monad-law-1-4 O (S n) (mono dd) = begin
201 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ 192 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩
202 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ 193 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩
215 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ 206 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩
216 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) 207 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds))))
217 ∎ 208 ∎
218 monad-law-1-4 (S m) n (mono dd) = begin 209 monad-law-1-4 (S m) n (mono dd) = begin
219 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ 210 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩
220 headDelta (n-tail n ((n-tail (S m)) dd))≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add n (S m)) ⟩ 211 headDelta (n-tail n ((n-tail (S m)) dd))≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩
221 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (int-add-assoc n (S m)) ⟩ 212 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (int-add-assoc n (S m)) ⟩
222 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ 213 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩
223 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)) ⟩ 214 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)) ⟩
224 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) 215 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd))))
225 ∎ 216 ∎
236 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ 227 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩
237 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)) ⟩ 228 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)) ⟩
238 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ 229 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩
239 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ 230 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
240 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ 231 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩
241 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (int-add-right m n)) ⟩ 232 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (int-add-right m n)) ⟩
242 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ 233 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩
243 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) ⟩ 234 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) ⟩
244 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) 235 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
245 ∎ 236 ∎
246 237
247 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) 238 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
248 monad-law-1-2 (mono _) = refl 239 monad-law-1-2 (mono _) = refl
249 monad-law-1-2 (delta _ _) = refl 240 monad-law-1-2 (delta _ _) = refl
250 241
251 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Int) -> (d : Delta (Delta (Delta A))) -> 242 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) ->
252 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) 243 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
253 monad-law-1-3 O (mono d) = refl 244 monad-law-1-3 O (mono d) = refl
254 monad-law-1-3 O (delta d ds) = begin 245 monad-law-1-3 O (delta d ds) = begin
255 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ 246 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
256 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ 247 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
257 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ 248 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
258 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ 249 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
259 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ 250 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
260 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ 251 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
261 bind (bind (delta d ds) (n-tail O)) (n-tail O) 252 bind (bind (delta d ds) (n-tail O)) (n-tail O)
262 ∎ 253 ∎
263 monad-law-1-3 (S n) (mono (mono d)) = begin 254 monad-law-1-3 (S n) (mono (mono d)) = begin
311 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 ⟩ 302 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 ⟩
312 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ 303 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
313 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) 304 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
314 ∎ 305 ∎
315 306
316 {-
317 monad-law-1-3 (S n) (mono d) = begin
318 bind (fmap mu (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
319 bind (mono (mu d)) (n-tail (S n)) ≡⟨ refl ⟩
320 n-tail (S n) (mu d) ≡⟨ {!!} ⟩
321 bind (n-tail (S n) d) (n-tail (S n)) ≡⟨ refl ⟩
322 bind (bind (mono d) (n-tail (S n))) (n-tail (S n))
323 ∎
324 monad-law-1-3 (S n) (delta d ds) = begin
325 bind (fmap mu (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
326 bind (delta (mu d) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
327 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
328 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩
329 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
330 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
331 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
332 bind (delta (headDelta ((n-tail (S n)) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
333 bind (bind (delta d ds) (n-tail (S n))) (n-tail (S n))
334 ∎
335 -}
336 307
337 -- monad-law-1 : join . fmap join = join . join 308 -- monad-law-1 : join . fmap join = join . join
338 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) 309 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
339 monad-law-1 (mono d) = refl 310 monad-law-1 (mono d) = refl
340 {-
341 monad-law-1 (delta x (mono d)) = begin
342 (mu ∙ fmap mu) (delta x (mono d)) ≡⟨ refl ⟩
343 mu (fmap mu (delta x (mono d))) ≡⟨ refl ⟩
344 mu (delta (mu x) (mono (mu d))) ≡⟨ refl ⟩
345 delta (headDelta (mu x)) (bind (mono (mu d)) tailDelta) ≡⟨ refl ⟩
346 delta (headDelta (mu x)) (tailDelta (mu d)) ≡⟨ cong (\dx -> delta dx (tailDelta (mu d))) (monad-law-1-2 x) ⟩
347 delta (headDelta (headDelta x)) (tailDelta (mu d)) ≡⟨ {!!} ⟩
348 delta (headDelta (headDelta x)) (bind (tailDelta d) tailDelta) ≡⟨ refl ⟩
349 mu (delta (headDelta x) (tailDelta d)) ≡⟨ refl ⟩
350 mu (delta (headDelta x) (bind (mono d) tailDelta)) ≡⟨ refl ⟩
351 mu (mu (delta x (mono d))) ≡⟨ refl ⟩
352 (mu ∙ mu) (delta x (mono d))
353 ∎
354 monad-law-1 (delta x (delta d ds)) = begin
355 (mu ∙ fmap mu) (delta x (delta d ds)) ≡⟨ refl ⟩
356 mu (fmap mu (delta x (delta d ds))) ≡⟨ refl ⟩
357 mu (delta (mu x) (delta (mu d) (fmap mu ds))) ≡⟨ refl ⟩
358 delta (headDelta (mu x)) (bind (delta (mu d) (fmap mu ds)) tailDelta) ≡⟨ refl ⟩
359 delta (headDelta (mu x)) (delta (headDelta (tailDelta (mu d))) (bind (fmap mu ds) (tailDelta ∙ tailDelta))) ≡⟨ {!!} ⟩
360 delta (headDelta (headDelta x)) (delta (headDelta (tailDelta (headDelta (tailDelta d)))) (bind (bind ds (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))) ≡⟨ refl ⟩
361 delta (headDelta (headDelta x)) (bind (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta))) tailDelta) ≡⟨ refl ⟩
362 delta (headDelta (headDelta x)) (bind (bind (delta d ds) tailDelta) tailDelta) ≡⟨ refl ⟩
363 mu (delta (headDelta x) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
364 mu (mu (delta x (delta d ds))) ≡⟨ refl ⟩
365 (mu ∙ mu) (delta x (delta d ds))
366 ∎
367 -}
368
369 monad-law-1 (delta x d) = begin 311 monad-law-1 (delta x d) = begin
370 (mu ∙ fmap mu) (delta x d) 312 (mu ∙ fmap mu) (delta x d)
371 ≡⟨ refl ⟩ 313 ≡⟨ refl ⟩
372 mu (fmap mu (delta x d)) 314 mu (fmap mu (delta x d))
373 ≡⟨ refl ⟩ 315 ≡⟨ refl ⟩