Mercurial > hg > Members > atton > delta_monad
annotate agda/delta.agda @ 87:6789c65a75bc
Split functor-proofs into delta.functor
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 19 Jan 2015 11:00:34 +0900 |
parents | fc5cd8c50312 |
children | 526186c4f298 |
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 open import laws |
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5 |
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6 open import Level |
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7 open import Relation.Binary.PropositionalEquality |
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8 open ≡-Reasoning |
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9 |
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10 module delta where |
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11 |
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13 data Delta {l : Level} (A : Set l) : (Set l) where |
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14 mono : A -> Delta A |
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15 delta : A -> Delta A -> Delta A |
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16 |
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17 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A |
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18 deltaAppend (mono x) d = delta x d |
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19 deltaAppend (delta x d) ds = delta x (deltaAppend d ds) |
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20 |
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21 headDelta : {l : Level} {A : Set l} -> Delta A -> A |
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22 headDelta (mono x) = x |
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23 headDelta (delta x _) = x |
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24 |
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25 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A |
79 | 26 tailDelta (mono x) = mono x |
27 tailDelta (delta _ d) = d | |
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28 |
76 | 29 n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A)) |
30 n-tail O = id | |
31 n-tail (S n) = tailDelta ∙ (n-tail n) | |
32 | |
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33 |
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34 -- Functor |
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35 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B) |
79 | 36 fmap f (mono x) = mono (f x) |
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37 fmap f (delta x d) = delta (f x) (fmap f d) |
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38 |
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39 |
38
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40 |
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41 -- Monad (Category) |
43
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42 eta : {l : Level} {A : Set l} -> A -> Delta A |
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43 eta x = mono x |
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44 |
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45 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B |
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46 bind (mono x) f = f x |
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47 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f)) |
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48 |
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49 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A |
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50 mu d = bind d id |
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51 |
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52 returnS : {l : Level} {A : Set l} -> A -> Delta A |
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53 returnS x = mono x |
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54 |
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55 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A |
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56 returnSS x y = deltaAppend (returnS x) (returnS y) |
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57 |
33 | 58 |
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59 -- Monad (Haskell) |
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60 return : {l : Level} {A : Set l} -> A -> Delta A |
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61 return = eta |
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62 |
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Proof monad-law-h-2, trying monad-law-h-3
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63 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} -> |
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64 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B) |
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65 (mono x) >>= f = f x |
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66 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f)) |
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67 |
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68 |
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69 |
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70 -- proofs |
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71 |
76 | 72 -- sub-proofs |
73 | |
74 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n) | |
79 | 75 n-tail-plus O = refl |
76 | 76 n-tail-plus (S n) = begin |
77 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩ | |
78 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩ | |
79 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩ | |
80 n-tail (S (S n)) | |
81 ∎ | |
82 | |
83 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) | |
84 n-tail-add O m = refl | |
85 n-tail-add (S n) O = begin | |
86 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩ | |
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87 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩ |
76 | 88 n-tail (S n + O) |
89 ∎ | |
90 n-tail-add {l} {A} {d} (S n) (S m) = begin | |
91 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩ | |
92 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩ | |
93 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩ | |
94 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩ | |
95 n-tail (S (n + S m)) ≡⟨ refl ⟩ | |
96 n-tail (S n + S m) ∎ | |
97 | |
98 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) -> | |
99 (n-tail n) (mono x) ≡ (mono x) | |
79 | 100 tail-delta-to-mono O x = refl |
76 | 101 tail-delta-to-mono (S n) x = begin |
102 n-tail (S n) (mono x) ≡⟨ refl ⟩ | |
103 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ | |
104 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩ | |
105 tailDelta (mono x) ≡⟨ refl ⟩ | |
106 mono x ∎ | |
107 | |
80 | 108 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} |
79 | 109 -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d) |
110 head-delta-natural-transformation f (mono x) = refl | |
80 | 111 head-delta-natural-transformation f (delta x d) = refl |
79 | 112 |
113 n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} | |
114 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d) | |
115 n-tail-natural-transformation O f d = refl | |
116 n-tail-natural-transformation (S n) f (mono x) = begin | |
117 n-tail (S n) (fmap f (mono x)) ≡⟨ refl ⟩ | |
118 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩ | |
119 (mono (f x)) ≡⟨ refl ⟩ | |
120 fmap f (mono x) ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩ | |
121 fmap f (n-tail (S n) (mono x)) ∎ | |
122 n-tail-natural-transformation (S n) f (delta x d) = begin | |
123 n-tail (S n) (fmap f (delta x d)) ≡⟨ refl ⟩ | |
124 n-tail (S n) (delta (f x) (fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩ | |
125 ((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩ | |
126 n-tail n (fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩ | |
127 fmap f (n-tail n d) ≡⟨ refl ⟩ | |
128 fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩ | |
129 fmap f (n-tail (S n) (delta x d)) ∎ | |
130 | |
131 | |
132 | |
133 | |
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134 |
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135 {- |
39 | 136 -- Monad-laws (Category) |
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137 |
76 | 138 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> |
139 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n)) | |
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140 monad-law-1-5 O O ds = refl |
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141 monad-law-1-5 O (S n) (mono ds) = begin |
80 | 142 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ |
143 n-tail (S n) ds ≡⟨ refl ⟩ | |
144 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ | |
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145 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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146 bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) |
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147 ∎ |
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148 monad-law-1-5 O (S n) (delta d ds) = begin |
80 | 149 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ |
150 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ | |
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151 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩ |
80 | 152 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
153 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ | |
154 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ | |
155 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
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156 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) |
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157 ∎ |
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158 monad-law-1-5 (S m) n (mono (mono x)) = begin |
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159 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ |
80 | 160 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ |
161 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ | |
162 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ | |
163 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ | |
164 bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ | |
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165 bind (n-tail n (mono (mono x))) (n-tail (S m + n)) |
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166 ∎ |
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167 monad-law-1-5 (S m) n (mono (delta x ds)) = begin |
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168 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ |
80 | 169 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ |
170 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ | |
171 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ | |
172 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ | |
173 n-tail (m + n) ds ≡⟨ refl ⟩ | |
174 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ | |
175 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ | |
176 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ | |
177 bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ | |
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178 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) |
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179 ∎ |
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180 monad-law-1-5 (S m) O (delta d ds) = begin |
80 | 181 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
182 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | |
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183 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
80 | 184 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ |
185 bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ | |
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186 bind (n-tail O (delta d ds)) (n-tail (S m + O)) |
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187 ∎ |
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188 monad-law-1-5 (S m) (S n) (delta d ds) = begin |
80 | 189 n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ |
190 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | |
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191 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
80 | 192 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
193 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ | |
194 bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ | |
195 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ | |
196 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ | |
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197 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) |
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198 ∎ |
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199 |
76 | 200 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> |
201 headDelta ((n-tail n) (bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) | |
74 | 202 monad-law-1-4 O O (mono dd) = refl |
203 monad-law-1-4 O O (delta dd dd₁) = refl | |
204 monad-law-1-4 O (S n) (mono dd) = begin | |
80 | 205 headDelta (n-tail (S n) (bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ |
206 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ | |
207 headDelta (n-tail (S n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S n) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩ | |
74 | 208 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ |
209 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) | |
210 ∎ | |
211 monad-law-1-4 O (S n) (delta d ds) = begin | |
80 | 212 headDelta (n-tail (S n) (bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ |
213 headDelta (n-tail (S n) (bind (delta d ds) id)) ≡⟨ refl ⟩ | |
214 headDelta (n-tail (S n) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ | |
74 | 215 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta))) ≡⟨ refl ⟩ |
80 | 216 headDelta (n-tail n (bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ |
217 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ | |
218 headDelta (n-tail (S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ | |
219 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ | |
74 | 220 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) |
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221 ∎ |
74 | 222 monad-law-1-4 (S m) n (mono dd) = begin |
223 headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ | |
80 | 224 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ |
225 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ | |
226 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ | |
227 headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩ | |
74 | 228 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) |
229 ∎ | |
230 monad-law-1-4 (S m) O (delta d ds) = begin | |
80 | 231 headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
232 headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ | |
74 | 233 headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
80 | 234 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ |
235 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ | |
236 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ | |
74 | 237 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) |
238 ∎ | |
239 monad-law-1-4 (S m) (S n) (delta d ds) = begin | |
80 | 240 headDelta (n-tail (S n) (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
241 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ | |
74 | 242 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ |
80 | 243 headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
244 headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ | |
245 headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩ | |
246 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ | |
247 headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ | |
74 | 248 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) |
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249 ∎ |
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250 |
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251 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d)) |
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252 monad-law-1-2 (mono _) = refl |
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253 monad-law-1-2 (delta _ _) = refl |
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254 |
76 | 255 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> |
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256 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n) |
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257 monad-law-1-3 O (mono d) = refl |
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258 monad-law-1-3 O (delta d ds) = begin |
80 | 259 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
260 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩ | |
261 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩ | |
262 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ | |
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263 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ |
80 | 264 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ |
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265 bind (bind (delta d ds) (n-tail O)) (n-tail O) |
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266 ∎ |
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267 monad-law-1-3 (S n) (mono (mono d)) = begin |
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268 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ |
80 | 269 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ |
270 (n-tail (S n)) d ≡⟨ refl ⟩ | |
271 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ | |
272 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
273 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
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274 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) |
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275 ∎ |
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276 monad-law-1-3 (S n) (mono (delta d ds)) = begin |
80 | 277 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
278 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ | |
279 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩ | |
280 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ | |
281 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
282 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ | |
283 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ | |
284 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ | |
285 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
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286 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) |
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287 ∎ |
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288 monad-law-1-3 (S n) (delta (mono d) ds) = begin |
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289 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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290 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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291 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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292 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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293 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩ |
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294 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
295 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
72
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
296 delta (headDelta ((n-tail (S n)) d)) (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.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
297 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
298 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
299 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
300 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
301 ∎ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
302 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
303 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
304 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
305 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
306 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
307 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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
308 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
309 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ |
74 | 310 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
311 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
312 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
72
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
313 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
73
0ad0ae7a3cbe
Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
72
diff
changeset
|
314 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ |
72
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
315 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
316 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
317 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) |
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
318 ∎ |
70
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
319 |
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
320 |
39 | 321 -- monad-law-1 : join . fmap join = join . join |
59
46b15f368905
Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
57
diff
changeset
|
322 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d) |
72
e95f15af3f8b
Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
71
diff
changeset
|
323 monad-law-1 (mono d) = refl |
70
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
324 monad-law-1 (delta x d) = begin |
80 | 325 (mu ∙ fmap mu) (delta x d) ≡⟨ refl ⟩ |
326 mu (fmap mu (delta x d)) ≡⟨ refl ⟩ | |
327 mu (delta (mu x) (fmap mu d)) ≡⟨ refl ⟩ | |
328 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩ | |
329 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ | |
330 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta) ≡⟨ refl ⟩ | |
331 mu (delta (headDelta x) (bind d tailDelta)) ≡⟨ refl ⟩ | |
332 mu (mu (delta x d)) ≡⟨ refl ⟩ | |
70
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
333 (mu ∙ mu) (delta x d) |
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
334 ∎ |
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
335 |
18a20a14c4b2
Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
69
diff
changeset
|
336 |
78
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
337 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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
338 monad-law-2-1 O (mono x) = refl |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
339 monad-law-2-1 O (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
340 bind (fmap eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
341 bind (delta (eta x) (fmap eta d)) id ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
342 delta (headDelta (eta x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
343 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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
344 delta x d ∎ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
345 monad-law-2-1 (S n) (mono x) = begin |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
346 bind (fmap eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
347 bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
348 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
349 mono x ∎ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
350 monad-law-2-1 (S n) (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
351 bind (fmap eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
352 bind (delta (eta x) (fmap eta d)) (n-tail (S n)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
353 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
354 delta (headDelta ((n-tail (S n) (eta x)))) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (bind (fmap eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
355 delta (headDelta (eta x)) (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
356 delta x (bind (fmap eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
357 delta x d |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
358 ∎ |
63
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
359 |
474ed34e4f02
proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
62
diff
changeset
|
360 |
78
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
361 -- monad-law-2 : join . fmap return = join . return = id |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
362 -- monad-law-2 join . fmap return = join . return |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
363 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
364 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
365 monad-law-2 (mono x) = refl |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
366 monad-law-2 (delta x d) = begin |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
367 (mu ∙ fmap eta) (delta x d) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
368 mu (fmap eta (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
369 mu (delta (mono x) (fmap eta d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
370 delta (headDelta (mono x)) (bind (fmap eta d) tailDelta) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
371 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
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
372 (delta x d) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
373 mu (mono (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
374 mu (eta (delta x d)) ≡⟨ refl ⟩ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
375 (mu ∙ eta) (delta x d) |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
376 ∎ |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
377 |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
378 |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
379 -- monad-law-2' : join . return = id |
f02391a7402f
Prove monad-law-2, 3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
77
diff
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380 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (mu ∙ eta) d ≡ id d |
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381 monad-law-2' d = refl |
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382 |
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Proof Monad-law-2-2
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383 |
39 | 384 -- monad-law-3 : return . f = fmap f . return |
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385 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x |
36 | 386 monad-law-3 f x = refl |
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387 |
79 | 388 |
389 monad-law-4-1 : {l ll : Level} {A : Set l} {B : Set ll} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> | |
390 bind (fmap (fmap f) ds) (n-tail n) ≡ fmap f (bind ds (n-tail n)) | |
391 monad-law-4-1 O f (mono d) = refl | |
392 monad-law-4-1 O f (delta d ds) = begin | |
393 bind (fmap (fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ | |
394 bind (delta (fmap f d) (fmap (fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ | |
395 delta (headDelta (fmap f d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ | |
396 delta (f (headDelta d)) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ | |
397 delta (f (headDelta d)) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
398 fmap f (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
399 fmap f (bind (delta d ds) (n-tail O)) ∎ | |
400 monad-law-4-1 (S n) f (mono d) = begin | |
401 bind (fmap (fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
402 bind (mono (fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ | |
403 n-tail (S n) (fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ | |
404 fmap f (n-tail (S n) d) ≡⟨ refl ⟩ | |
405 fmap f (bind (mono d) (n-tail (S n))) | |
406 ∎ | |
407 monad-law-4-1 (S n) f (delta d ds) = begin | |
408 bind (fmap (fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ | |
409 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ | |
410 delta (headDelta (n-tail (S n) (fmap f d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ | |
411 delta (headDelta (fmap f ((n-tail (S n) d)))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (bind (fmap (fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ | |
412 delta (f (headDelta (n-tail (S n) d))) (bind (fmap (fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (f (headDelta (n-tail (S n) d))) de) (monad-law-4-1 (S (S n)) f ds) ⟩ | |
413 delta (f (headDelta (n-tail (S n) d))) (fmap f (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
414 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ | |
415 fmap f (delta (headDelta (n-tail (S n) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ | |
416 fmap f (bind (delta d ds) (n-tail (S n))) ∎ | |
417 | |
418 | |
39 | 419 -- monad-law-4 : join . fmap (fmap f) = fmap f . join |
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420 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) -> |
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421 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d |
79 | 422 monad-law-4 f (mono d) = refl |
423 monad-law-4 f (delta (mono x) ds) = begin | |
424 (mu ∙ fmap (fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ | |
425 mu ( fmap (fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ | |
426 mu (delta (mono (f x)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
427 delta (headDelta (mono (f x))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
428 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
429 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
430 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
431 fmap f (delta (headDelta (mono x)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
432 fmap f (mu (delta (mono x) ds)) ≡⟨ refl ⟩ | |
433 (fmap f ∙ mu) (delta (mono x) ds) ∎ | |
434 monad-law-4 f (delta (delta x d) ds) = begin | |
435 (mu ∙ fmap (fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ | |
436 mu (fmap (fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
437 mu (delta (delta (f x) (fmap f d)) (fmap (fmap f) ds)) ≡⟨ refl ⟩ | |
438 delta (headDelta (delta (f x) (fmap f d))) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ refl ⟩ | |
439 delta (f x) (bind (fmap (fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ | |
440 delta (f x) (fmap f (bind ds tailDelta)) ≡⟨ refl ⟩ | |
441 fmap f (delta x (bind ds tailDelta)) ≡⟨ refl ⟩ | |
442 fmap f (delta (headDelta (delta x d)) (bind ds tailDelta)) ≡⟨ refl ⟩ | |
443 fmap f (mu (delta (delta x d) ds)) ≡⟨ refl ⟩ | |
444 (fmap f ∙ mu) (delta (delta x d) ds) ∎ | |
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445 |
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446 |
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447 |
79 | 448 {- |
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449 -- Monad-laws (Haskell) |
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450 -- monad-law-h-1 : return a >>= k = k a |
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451 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} -> |
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452 (a : A) -> (k : A -> (Delta B)) -> |
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453 (return a >>= k) ≡ (k a) |
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454 monad-law-h-1 a k = refl |
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455 |
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456 |
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457 |
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458 -- monad-law-h-2 : m >>= return = m |
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459 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m |
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460 monad-law-h-2 (mono x) = refl |
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461 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
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462 |
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463 |
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464 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h |
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465 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} -> |
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466 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) -> |
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467 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h) |
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468 monad-law-h-3 (mono x) k h = refl |
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469 monad-law-h-3 (delta x d) k h = {!!} |
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470 |
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471 -} |
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472 -} |