Mercurial > hg > Members > atton > delta_monad
annotate agda/delta.agda @ 79:7307e43a3c76
Prove monad-law-4
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 01 Dec 2014 17:25:59 +0900 |
parents | f02391a7402f |
children | fc5cd8c50312 |
rev | line source |
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1 open import list |
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2 open import basic |
76 | 3 open import nat |
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4 |
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5 open import Level |
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Define Monad-law 1-4
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6 open import Relation.Binary.PropositionalEquality |
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Define Monad-law 1-4
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7 open ≡-Reasoning |
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8 |
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9 module delta where |
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12 data Delta {l : Level} (A : Set l) : (Set (suc l)) where |
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13 mono : A -> Delta A |
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14 delta : A -> Delta A -> Delta A |
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15 |
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16 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A |
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17 deltaAppend (mono x) d = delta x d |
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18 deltaAppend (delta x d) ds = delta x (deltaAppend d ds) |
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19 |
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20 headDelta : {l : Level} {A : Set l} -> Delta A -> A |
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21 headDelta (mono x) = x |
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22 headDelta (delta x _) = x |
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23 |
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24 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A |
79 | 25 tailDelta (mono x) = mono x |
26 tailDelta (delta _ d) = d | |
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27 |
76 | 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) | |
31 | |
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32 |
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33 -- Functor |
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34 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B) |
79 | 35 fmap f (mono x) = mono (f x) |
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36 fmap f (delta x d) = delta (f x) (fmap f d) |
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37 |
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38 |
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39 |
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40 -- Monad (Category) |
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41 eta : {l : Level} {A : Set l} -> A -> Delta A |
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42 eta x = mono x |
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43 |
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44 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B |
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45 bind (mono x) f = f x |
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46 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f)) |
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47 |
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48 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A |
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49 mu d = bind d id |
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50 |
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51 returnS : {l : Level} {A : Set l} -> A -> Delta A |
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52 returnS x = mono x |
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53 |
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54 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A |
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55 returnSS x y = deltaAppend (returnS x) (returnS y) |
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56 |
33 | 57 |
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58 -- Monad (Haskell) |
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59 return : {l : Level} {A : Set l} -> A -> Delta A |
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60 return = eta |
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61 |
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Proof monad-law-h-2, trying monad-law-h-3
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62 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} -> |
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63 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B) |
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64 (mono x) >>= f = f x |
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65 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f)) |
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66 |
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67 |
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68 |
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69 -- proofs |
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70 |
76 | 71 -- sub-proofs |
72 | |
73 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n) | |
79 | 74 n-tail-plus O = refl |
76 | 75 n-tail-plus (S n) = begin |
76 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩ | |
77 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩ | |
78 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩ | |
79 n-tail (S (S n)) | |
80 ∎ | |
81 | |
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) | |
83 n-tail-add O m = refl | |
84 n-tail-add (S n) O = begin | |
85 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩ | |
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86 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩ |
76 | 87 n-tail (S n + O) |
88 ∎ | |
89 n-tail-add {l} {A} {d} (S n) (S m) = begin | |
90 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩ | |
91 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩ | |
92 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩ | |
93 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩ | |
94 n-tail (S (n + S m)) ≡⟨ refl ⟩ | |
95 n-tail (S n + S m) ∎ | |
96 | |
97 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) -> | |
98 (n-tail n) (mono x) ≡ (mono x) | |
79 | 99 tail-delta-to-mono O x = refl |
76 | 100 tail-delta-to-mono (S n) x = begin |
101 n-tail (S 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) ⟩ | |
104 tailDelta (mono x) ≡⟨ refl ⟩ | |
105 mono x ∎ | |
106 | |
79 | 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) | |
109 head-delta-natural-transformation f (mono x) = refl | |
110 head-delta-natural-transformation f (delta x d) = refl | |
111 | |
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) | |
114 n-tail-natural-transformation O f d = refl | |
115 n-tail-natural-transformation (S n) f (mono x) = begin | |
116 n-tail (S n) (fmap f (mono x)) ≡⟨ refl ⟩ | |
117 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩ | |
118 (mono (f x)) ≡⟨ refl ⟩ | |
119 fmap f (mono x) ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩ | |
120 fmap f (n-tail (S n) (mono x)) ∎ | |
121 n-tail-natural-transformation (S n) f (delta x d) = begin | |
122 n-tail (S n) (fmap f (delta x d)) ≡⟨ refl ⟩ | |
123 n-tail (S n) (delta (f x) (fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩ | |
124 ((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩ | |
125 n-tail n (fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩ | |
126 fmap f (n-tail n d) ≡⟨ refl ⟩ | |
127 fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩ | |
128 fmap f (n-tail (S n) (delta x d)) ∎ | |
129 | |
130 | |
131 | |
132 | |
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133 -- Functor-laws |
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134 |
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135 -- Functor-law-1 : T(id) = id' |
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136 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d |
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137 functor-law-1 (mono x) = refl |
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138 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d) |
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139 |
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140 -- Functor-law-2 : T(f . g) = T(f) . T(g) |
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141 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> |
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142 (f : B -> C) -> (g : A -> B) -> (d : Delta A) -> |
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143 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d |
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144 functor-law-2 f g (mono x) = refl |
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145 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d) |
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146 |
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147 |
39 | 148 -- Monad-laws (Category) |
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149 {- |
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150 |
76 | 151 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)) | |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
153 monad-law-1-5 O O ds = refl |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
154 monad-law-1-5 O (S n) (mono ds) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
155 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
156 n-tail (S n) ds ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
157 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
158 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
159 bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
160 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
161 monad-law-1-5 O (S n) (delta d ds) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
162 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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)) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
164 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
165 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
166 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
168 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
169 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
170 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
171 monad-law-1-5 (S m) n (mono (mono x)) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
172 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
173 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
174 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
175 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
176 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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))) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
178 bind (n-tail n (mono (mono x))) (n-tail (S m + n)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
179 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
180 monad-law-1-5 (S m) n (mono (delta x ds)) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
181 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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)) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
183 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ |
76 | 184 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
185 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
186 n-tail (m + n) ds ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
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187 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
188 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
189 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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))) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
191 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
192 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
193 monad-law-1-5 (S m) O (delta d ds) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
194 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
195 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
196 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
197 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
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)) ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
199 bind (n-tail O (delta d ds)) (n-tail (S m + O)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
200 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
201 monad-law-1-5 (S m) (S n) (delta d ds) = begin |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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)) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
203 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
204 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
205 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
206 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
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)) ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
208 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
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) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
210 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
211 ∎ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
212 |
76 | 213 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))) | |
74 | 215 monad-law-1-4 O O (mono dd) = refl |
216 monad-law-1-4 O O (delta dd dd₁) = refl | |
217 monad-law-1-4 O (S n) (mono dd) = begin | |
218 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ | |
219 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)) ⟩ | |
221 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)))) | |
223 ∎ | |
224 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 ⟩ | |
226 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)) ⟩ | |
228 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 ⟩ | |
230 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) ⟩ | |
232 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)))) | |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
234 ∎ |
74 | 235 monad-law-1-4 (S m) n (mono dd) = begin |
236 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ | |
76 | 237 headDelta (n-tail n ((n-tail (S m)) dd))≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
238 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ |
74 | 239 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)) ⟩ | |
241 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) | |
242 ∎ | |
243 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 ⟩ | |
245 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 ⟩ | |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
247 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ |
74 | 248 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ |
249 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)))) | |
251 ∎ | |
252 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 ⟩ | |
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)) ⟩ | |
255 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 ⟩ | |
257 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ | |
77
4b16b485a4b2
Split nat definition
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
76
diff
changeset
|
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)) ⟩ |
76 | 259 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ |
74 | 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) ⟩ |
261 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) | |
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262 ∎ |
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263 |
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264 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) |
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265 monad-law-1-2 (mono _) = refl |
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266 monad-law-1-2 (delta _ _) = refl |
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267 |
76 | 268 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> |
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269 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) |
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270 monad-law-1-3 O (mono d) = refl |
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271 monad-law-1-3 O (delta d ds) = begin |
76 | 272 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
273 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) ⟩ | |
275 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ | |
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276 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ |
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277 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ |
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278 bind (bind (delta d ds) (n-tail O)) (n-tail O) |
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279 ∎ |
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280 monad-law-1-3 (S n) (mono (mono d)) = begin |
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281 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ |
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282 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ |
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283 (n-tail (S n)) d ≡⟨ refl ⟩ |
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284 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ |
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285 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
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286 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
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287 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) |
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288 ∎ |
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289 monad-law-1-3 (S n) (mono (delta d ds)) = begin |
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290 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
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291 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
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292 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ |
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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)) ⟩ |
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Proving monad-law-1
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294 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ |
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295 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
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296 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
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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) ⟩ |
72
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298 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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299 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) |
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300 ∎ |
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301 monad-law-1-3 (S n) (delta (mono d) ds) = begin |
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302 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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303 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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304 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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Proving monad-law-1
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305 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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Proving monad-law-1
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306 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) ⟩ |
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Proving monad-law-1
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307 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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Proving monad-law-1
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308 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
72
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309 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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310 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)) ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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311 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 ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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312 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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313 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) |
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314 ∎ |
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315 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin |
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316 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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317 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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318 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
73
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Proving monad-law-1
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319 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)) ⟩ |
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Proving monad-law-1
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320 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
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321 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
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322 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) ⟩ |
74 | 323 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) ⟩ |
73
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Proving monad-law-1
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324 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 ⟩ |
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Proving monad-law-1
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325 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 ⟩ |
72
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Trying prove infinite-delta. but I think this definition was missed.
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326 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 ⟩ |
73
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Proving monad-law-1
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327 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) ⟩ |
72
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Trying prove infinite-delta. but I think this definition was missed.
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328 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 ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
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329 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
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Trying prove infinite-delta. but I think this definition was missed.
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330 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) |
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331 ∎ |
70
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Change prove method. use Int ...
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332 |
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Change prove method. use Int ...
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333 |
39 | 334 -- monad-law-1 : join . fmap join = join . join |
59
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Define bind and mu for Infinite Delta
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335 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) |
72
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336 monad-law-1 (mono d) = refl |
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337 monad-law-1 (delta x d) = begin |
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338 (mu ∙ fmap mu) (delta x d) |
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339 ≡⟨ refl ⟩ |
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|
340 mu (fmap mu (delta x d)) |
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341 ≡⟨ refl ⟩ |
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Change prove method. use Int ...
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342 mu (delta (mu x) (fmap mu d)) |
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343 ≡⟨ refl ⟩ |
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344 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) |
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345 ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ |
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Change prove method. use Int ...
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346 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) |
72
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
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71
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347 ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ |
70
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348 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) |
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Change prove method. use Int ...
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349 ≡⟨ refl ⟩ |
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Change prove method. use Int ...
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350 mu (delta (headDelta x) (bind d tailDelta)) |
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Change prove method. use Int ...
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351 ≡⟨ refl ⟩ |
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Change prove method. use Int ...
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352 mu (mu (delta x d)) |
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353 ≡⟨ refl ⟩ |
18a20a14c4b2
Change prove method. use Int ...
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354 (mu ∙ mu) (delta x d) |
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355 ∎ |
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Change prove method. use Int ...
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356 |
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357 |
78
f02391a7402f
Prove monad-law-2, 3
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358 -} |
70
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Change prove method. use Int ...
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359 |
78
f02391a7402f
Prove monad-law-2, 3
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77
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360 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (bind (fmap eta d) (n-tail n)) ≡ d |
f02391a7402f
Prove monad-law-2, 3
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77
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361 monad-law-2-1 O (mono x) = refl |
f02391a7402f
Prove monad-law-2, 3
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77
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362 monad-law-2-1 O (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
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77
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363 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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364 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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365 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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366 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ |
f02391a7402f
Prove monad-law-2, 3
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367 delta x d ∎ |
f02391a7402f
Prove monad-law-2, 3
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368 monad-law-2-1 (S n) (mono x) = begin |
f02391a7402f
Prove monad-law-2, 3
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369 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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370 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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371 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ |
f02391a7402f
Prove monad-law-2, 3
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372 mono x ∎ |
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Prove monad-law-2, 3
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373 monad-law-2-1 (S n) (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
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374 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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375 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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376 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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377 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) ⟩ |
f02391a7402f
Prove monad-law-2, 3
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378 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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379 delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ |
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Prove monad-law-2, 3
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380 delta x d |
f02391a7402f
Prove monad-law-2, 3
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381 ∎ |
63
474ed34e4f02
proving monad-law-1 ...
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382 |
474ed34e4f02
proving monad-law-1 ...
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383 |
78
f02391a7402f
Prove monad-law-2, 3
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384 -- monad-law-2 : join . fmap return = join . return = id |
f02391a7402f
Prove monad-law-2, 3
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385 -- monad-law-2 join . fmap return = join . return |
f02391a7402f
Prove monad-law-2, 3
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386 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> |
f02391a7402f
Prove monad-law-2, 3
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387 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d |
f02391a7402f
Prove monad-law-2, 3
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388 monad-law-2 (mono x) = refl |
f02391a7402f
Prove monad-law-2, 3
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389 monad-law-2 (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
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77
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390 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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391 mu (fmap eta (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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392 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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393 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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394 delta x (bind (fmap eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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395 (delta x d) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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396 mu (mono (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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397 mu (eta (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
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77
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398 (mu ∙ eta) (delta x d) |
f02391a7402f
Prove monad-law-2, 3
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|
399 ∎ |
f02391a7402f
Prove monad-law-2, 3
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|
400 |
f02391a7402f
Prove monad-law-2, 3
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changeset
|
401 |
f02391a7402f
Prove monad-law-2, 3
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77
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402 -- monad-law-2' : join . return = id |
f02391a7402f
Prove monad-law-2, 3
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77
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403 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d |
f02391a7402f
Prove monad-law-2, 3
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parents:
77
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|
404 monad-law-2' d = refl |
70
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|
405 |
35
c5cdbedc68ad
Proof Monad-law-2-2
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34
diff
changeset
|
406 |
39 | 407 -- monad-law-3 : return . f = fmap f . return |
40
a7cd7740f33e
Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
39
diff
changeset
|
408 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x |
36 | 409 monad-law-3 f x = refl |
27
742e62fc63e4
Define Monad-law 1-4
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parents:
26
diff
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|
410 |
79 | 411 |
412 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> | |
413 bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) | |
414 monad-law-4-1 O f (mono d) = refl | |
415 monad-law-4-1 O f (delta d ds) = begin | |
416 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | |
417 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ | |
418 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) ⟩ | |
419 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) ⟩ | |
420 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
421 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
422 fmap f (bind (delta d ds) (n-tail O)) ∎ | |
423 monad-law-4-1 (S n) f (mono d) = begin | |
424 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
425 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
426 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ | |
427 fmap f (n-tail (S n) d) ≡⟨ refl ⟩ | |
428 fmap f (bind (mono d) (n-tail (S n))) | |
429 ∎ | |
430 monad-law-4-1 (S n) f (delta d ds) = begin | |
431 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
432 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
433 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) ⟩ | |
434 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)) ⟩ | |
435 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) ⟩ | |
436 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
437 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
438 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ | |
439 fmap f (bind (delta d ds) (n-tail (S n))) ∎ | |
440 | |
441 | |
39 | 442 -- monad-law-4 : join . fmap (fmap f) = fmap f . join |
70
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443 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) -> |
18a20a14c4b2
Change prove method. use Int ...
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|
444 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d |
79 | 445 monad-law-4 f (mono d) = refl |
446 monad-law-4 f (delta (mono x) ds) = begin | |
447 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ | |
448 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ | |
449 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
450 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
451 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
452 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
453 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
454 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
455 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ | |
456 (fmap f ∙ mu) (delta (mono x) ds) ∎ | |
457 monad-law-4 f (delta (delta x d) ds) = begin | |
458 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ | |
459 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
460 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
461 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
462 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
463 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
464 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
465 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
466 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
467 (fmap f ∙ mu) (delta (delta x d) ds) ∎ | |
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468 |
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469 |
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470 |
79 | 471 {- |
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472 -- Monad-laws (Haskell) |
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473 -- monad-law-h-1 : return a >>= k = k a |
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474 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> |
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475 (a : A) -> (k : A -> (Delta B)) -> |
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476 (return a >>= k) ≡ (k a) |
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477 monad-law-h-1 a k = refl |
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478 |
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479 |
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480 |
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481 -- monad-law-h-2 : m >>= return = m |
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482 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m |
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483 monad-law-h-2 (mono x) = refl |
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484 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
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485 |
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486 |
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487 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h |
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488 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> |
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489 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> |
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490 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) |
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491 monad-law-h-3 (mono x) k h = refl |
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492 monad-law-h-3 (delta x d) k h = {!!} |
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493 |
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494 -} |