Mercurial > hg > Members > atton > delta_monad
annotate agda/delta/monad.agda @ 103:a271f3ff1922
Delte type dependencie in Monad record for escape implicit type conflict
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Mon, 26 Jan 2015 14:08:46 +0900 |
parents | dfe8c67390bd |
children | ebd0d6e2772c |
rev | line source |
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526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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1 open import basic |
526186c4f298
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2 open import delta |
526186c4f298
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3 open import delta.functor |
526186c4f298
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4 open import nat |
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5 open import laws |
526186c4f298
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6 |
526186c4f298
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7 |
526186c4f298
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8 open import Level |
526186c4f298
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9 open import Relation.Binary.PropositionalEquality |
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10 open ≡-Reasoning |
526186c4f298
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11 |
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12 module delta.monad where |
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13 |
526186c4f298
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14 |
526186c4f298
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15 -- Monad-laws (Category) |
526186c4f298
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16 |
526186c4f298
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17 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Nat) (n : Nat) -> (ds : Delta (Delta A)) -> |
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18 n-tail n (delta-bind ds (n-tail m)) ≡ delta-bind (n-tail n ds) (n-tail (m + n)) |
88
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19 monad-law-1-5 O O ds = refl |
526186c4f298
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20 monad-law-1-5 O (S n) (mono ds) = begin |
94
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21 n-tail (S n) (delta-bind (mono ds) (n-tail O)) ≡⟨ refl ⟩ |
88
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22 n-tail (S n) ds ≡⟨ refl ⟩ |
94
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23 delta-bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩ |
bcd4fe52a504
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24 delta-bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
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25 delta-bind (n-tail (S n) (mono ds)) (n-tail (O + S n)) |
88
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26 ∎ |
526186c4f298
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27 monad-law-1-5 O (S n) (delta d ds) = begin |
94
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28 n-tail (S n) (delta-bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩ |
bcd4fe52a504
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29 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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30 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta )) ≡⟨ refl ⟩ |
bcd4fe52a504
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31 (n-tail n) (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
bcd4fe52a504
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32 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
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33 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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34 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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35 delta-bind (n-tail (S n) (delta d ds)) (n-tail (O + S n)) |
88
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36 ∎ |
526186c4f298
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37 monad-law-1-5 (S m) n (mono (mono x)) = begin |
94
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Rewrite monad definitions for delta/deltaM
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38 n-tail n (delta-bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩ |
88
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39 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩ |
526186c4f298
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40 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩ |
526186c4f298
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41 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩ |
526186c4f298
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42 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩ |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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43 delta-bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
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44 delta-bind (n-tail n (mono (mono x))) (n-tail (S m + n)) |
88
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45 ∎ |
526186c4f298
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46 monad-law-1-5 (S m) n (mono (delta x ds)) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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47 n-tail n (delta-bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩ |
88
526186c4f298
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48 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ |
526186c4f298
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49 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ |
526186c4f298
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50 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ |
526186c4f298
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51 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ |
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52 n-tail (m + n) ds ≡⟨ refl ⟩ |
526186c4f298
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53 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ |
526186c4f298
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54 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ |
526186c4f298
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55 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩ |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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56 delta-bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> delta-bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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57 delta-bind (n-tail n (mono (delta x ds))) (n-tail (S m + n)) |
88
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58 ∎ |
526186c4f298
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59 monad-law-1-5 (S m) O (delta d ds) = begin |
94
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Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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60 n-tail O (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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61 (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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62 delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
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63 delta-bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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64 delta-bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> delta-bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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65 delta-bind (n-tail O (delta d ds)) (n-tail (S m + O)) |
88
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66 ∎ |
526186c4f298
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67 monad-law-1-5 (S m) (S n) (delta d ds) = begin |
94
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Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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68 n-tail (S n) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((delta-bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
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69 ((n-tail n) ∙ tailDelta) (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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90
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70 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
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71 (n-tail n) (delta-bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
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72 (n-tail n) (delta-bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
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73 delta-bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> delta-bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
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74 delta-bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
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75 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> delta-bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
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76 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) |
88
526186c4f298
Split monad-proofs into delta.monad
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77 ∎ |
526186c4f298
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78 |
526186c4f298
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79 monad-law-1-4 : {l : Level} {A : Set l} -> (m n : Nat) -> (dd : Delta (Delta A)) -> |
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Rewrite monad definitions for delta/deltaM
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80 headDelta ((n-tail n) (delta-bind dd (n-tail m))) ≡ headDelta ((n-tail (m + n)) (headDelta (n-tail n dd))) |
88
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Split monad-proofs into delta.monad
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81 monad-law-1-4 O O (mono dd) = refl |
526186c4f298
Split monad-proofs into delta.monad
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parents:
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82 monad-law-1-4 O O (delta dd dd₁) = refl |
526186c4f298
Split monad-proofs into delta.monad
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83 monad-law-1-4 O (S n) (mono dd) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
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changeset
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84 headDelta (n-tail (S n) (delta-bind (mono dd) (n-tail O))) ≡⟨ refl ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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85 headDelta (n-tail (S n) dd) ≡⟨ refl ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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changeset
|
86 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)) ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
87 headDelta (n-tail (S n) (headDelta (n-tail (S n) (mono dd)))) ≡⟨ refl ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
88 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (mono dd)))) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
89 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
90 monad-law-1-4 O (S n) (delta d ds) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
91 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail O))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
92 headDelta (n-tail (S n) (delta-bind (delta d ds) id)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
93 headDelta (n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta d) (delta-bind ds tailDelta)))) (sym (n-tail-plus n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
94 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
95 headDelta (n-tail n (delta-bind ds tailDelta)) ≡⟨ monad-law-1-4 (S O) n ds ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
96 headDelta (n-tail (S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
97 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) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
98 headDelta (n-tail (S n) (headDelta (n-tail (S n) (delta d ds)))) ≡⟨ refl ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
99 headDelta (n-tail (O + S n) (headDelta (n-tail (S n) (delta d ds)))) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
100 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
101 monad-law-1-4 (S m) n (mono dd) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
102 headDelta (n-tail n (delta-bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
103 headDelta (n-tail n ((n-tail (S m)) dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
104 headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
105 headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
106 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)) ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
107 headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
108 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
109 monad-law-1-4 (S m) O (delta d ds) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
110 headDelta (n-tail O (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
111 headDelta (delta-bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
112 headDelta (delta (headDelta ((n-tail (S m) d))) (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
113 headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
114 headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
115 headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
116 headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
117 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
118 monad-law-1-4 (S m) (S n) (delta d ds) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
119 headDelta (n-tail (S n) (delta-bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
120 headDelta (n-tail (S n) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m)))))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) (sym (n-tail-plus n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
121 headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (delta-bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
122 headDelta (n-tail n (delta-bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
123 headDelta (n-tail n (delta-bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
124 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)) ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
125 headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
126 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) ⟩ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
127 headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds)))) |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
128 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
129 |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
130 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (delta-mu d) ≡ (headDelta (headDelta d)) |
96
dfe8c67390bd
Unify Levels in delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
94
diff
changeset
|
131 monad-law-1-2 (mono _) = refl |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
132 monad-law-1-2 (delta _ _) = refl |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
133 |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
134 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta (Delta (Delta A))) -> |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
135 delta-bind (delta-fmap delta-mu d) (n-tail n) ≡ delta-bind (delta-bind d (n-tail n)) (n-tail n) |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
136 monad-law-1-3 O (mono d) = refl |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
137 monad-law-1-3 O (delta d ds) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
138 delta-bind (delta-fmap delta-mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
139 delta-bind (delta (delta-mu d) (delta-fmap delta-mu ds)) (n-tail O) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
140 delta (headDelta (delta-mu d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (delta-bind (delta-fmap delta-mu ds) tailDelta)) (monad-law-1-2 d) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
141 delta (headDelta (headDelta d)) (delta-bind (delta-fmap delta-mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
142 delta (headDelta (headDelta d)) (delta-bind (delta-bind ds tailDelta) tailDelta) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
143 delta-bind (delta (headDelta d) (delta-bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
144 delta-bind (delta-bind (delta d ds) (n-tail O)) (n-tail O) |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
145 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
146 monad-law-1-3 (S n) (mono (mono d)) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
147 delta-bind (delta-fmap delta-mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
148 delta-bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩ |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
149 (n-tail (S n)) d ≡⟨ refl ⟩ |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
150 delta-bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> delta-bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
151 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
152 delta-bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
153 delta-bind (delta-bind (mono (mono d)) (n-tail (S n))) (n-tail (S n)) |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
154 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
155 monad-law-1-3 (S n) (mono (delta d ds)) = begin |
94
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
156 delta-bind (delta-fmap delta-mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
157 delta-bind (mono (delta-mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
158 n-tail (S n) (delta-mu (delta d ds)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
159 n-tail (S n) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (delta-bind ds tailDelta))) (sym (n-tail-plus n)) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
160 (n-tail n ∙ tailDelta) (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
161 n-tail n (delta-bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
162 delta-bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
163 delta-bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (delta-bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
164 delta-bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
90
diff
changeset
|
165 delta-bind (delta-bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n)) |
88
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
166 ∎ |
526186c4f298
Split monad-proofs into delta.monad
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
167 monad-law-1-3 (S n) (delta (mono d) ds) = begin |
94
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168 delta-bind (delta-fmap delta-mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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169 delta-bind (delta (delta-mu (mono d)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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170 delta-bind (delta d (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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171 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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172 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-fmap delta-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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173 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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174 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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175 delta (headDelta ((n-tail (S n)) d)) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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176 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩ |
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177 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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178 delta-bind (delta (headDelta ((n-tail (S n)) (mono d))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
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179 delta-bind (delta-bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n)) |
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180 ∎ |
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181 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin |
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182 delta-bind (delta-fmap delta-mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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183 delta-bind (delta (delta-mu (delta d dd)) (delta-fmap delta-mu ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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184 delta (headDelta ((n-tail (S n)) (delta-mu (delta d dd)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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185 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩ |
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186 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (delta-bind dd tailDelta)))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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187 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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188 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-fmap delta-mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩ |
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189 delta (headDelta ((n-tail n) (delta-bind dd tailDelta))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 (S O) n dd) ⟩ |
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190 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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191 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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192 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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193 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (delta-bind (delta-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))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩ |
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194 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (delta-bind (delta-bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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195 delta-bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩ |
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196 delta-bind (delta-bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n)) |
88
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197 ∎ |
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198 |
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199 |
90
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200 -- monad-law-1 : join . delta-fmap join = join . join |
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201 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((delta-mu ∙ (delta-fmap delta-mu)) d) ≡ ((delta-mu ∙ delta-mu) d) |
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202 monad-law-1 (mono d) = refl |
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203 monad-law-1 (delta x d) = begin |
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204 (delta-mu ∙ delta-fmap delta-mu) (delta x d) ≡⟨ refl ⟩ |
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205 delta-mu (delta-fmap delta-mu (delta x d)) ≡⟨ refl ⟩ |
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206 delta-mu (delta (delta-mu x) (delta-fmap delta-mu d)) ≡⟨ refl ⟩ |
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207 delta (headDelta (delta-mu x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\x -> delta x (delta-bind (delta-fmap delta-mu d) tailDelta)) (monad-law-1-2 x) ⟩ |
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208 delta (headDelta (headDelta x)) (delta-bind (delta-fmap delta-mu d) tailDelta) ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩ |
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209 delta (headDelta (headDelta x)) (delta-bind (delta-bind d tailDelta) tailDelta) ≡⟨ refl ⟩ |
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210 delta-mu (delta (headDelta x) (delta-bind d tailDelta)) ≡⟨ refl ⟩ |
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211 delta-mu (delta-mu (delta x d)) ≡⟨ refl ⟩ |
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212 (delta-mu ∙ delta-mu) (delta x d) |
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213 ∎ |
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214 |
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215 |
94
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216 monad-law-2-1 : {l : Level} {A : Set l} -> (n : Nat) -> (d : Delta A) -> (delta-bind (delta-fmap delta-eta d) (n-tail n)) ≡ d |
88
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217 monad-law-2-1 O (mono x) = refl |
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218 monad-law-2-1 O (delta x d) = begin |
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219 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail O) ≡⟨ refl ⟩ |
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220 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) id ≡⟨ refl ⟩ |
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221 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ |
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222 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\de -> delta x de) (monad-law-2-1 (S O) d) ⟩ |
88
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223 delta x d ∎ |
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224 monad-law-2-1 (S n) (mono x) = begin |
94
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225 delta-bind (delta-fmap delta-eta (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
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226 delta-bind (mono (mono x)) (n-tail (S n)) ≡⟨ refl ⟩ |
88
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227 n-tail (S n) (mono x) ≡⟨ tail-delta-to-mono (S n) x ⟩ |
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228 mono x ∎ |
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229 monad-law-2-1 (S n) (delta x d) = begin |
94
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230 delta-bind (delta-fmap delta-eta (delta x d)) (n-tail (S n)) ≡⟨ refl ⟩ |
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231 delta-bind (delta (delta-eta x) (delta-fmap delta-eta d)) (n-tail (S n)) ≡⟨ refl ⟩ |
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232 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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233 delta (headDelta ((n-tail (S n) (delta-eta x)))) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta (de)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n))))) (tail-delta-to-mono (S n) x) ⟩ |
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Rewrite monad definitions for delta/deltaM
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234 delta (headDelta (delta-eta x)) (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ refl ⟩ |
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235 delta x (delta-bind (delta-fmap delta-eta d) (n-tail (S (S n)))) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S (S n)) d) ⟩ |
88
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236 delta x d |
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237 ∎ |
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238 |
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239 |
90
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240 -- monad-law-2 : join . delta-fmap return = join . return = id |
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241 -- monad-law-2 join . delta-fmap return = join . return |
88
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242 monad-law-2 : {l : Level} {A : Set l} -> (d : Delta A) -> |
94
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243 (delta-mu ∙ delta-fmap delta-eta) d ≡ (delta-mu ∙ delta-eta) d |
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244 monad-law-2 (mono x) = refl |
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245 monad-law-2 (delta x d) = begin |
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246 (delta-mu ∙ delta-fmap delta-eta) (delta x d) ≡⟨ refl ⟩ |
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247 delta-mu (delta-fmap delta-eta (delta x d)) ≡⟨ refl ⟩ |
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248 delta-mu (delta (mono x) (delta-fmap delta-eta d)) ≡⟨ refl ⟩ |
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249 delta (headDelta (mono x)) (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ refl ⟩ |
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250 delta x (delta-bind (delta-fmap delta-eta d) tailDelta) ≡⟨ cong (\d -> delta x d) (monad-law-2-1 (S O) d) ⟩ |
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251 (delta x d) ≡⟨ refl ⟩ |
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252 delta-mu (mono (delta x d)) ≡⟨ refl ⟩ |
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253 delta-mu (delta-eta (delta x d)) ≡⟨ refl ⟩ |
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254 (delta-mu ∙ delta-eta) (delta x d) |
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255 ∎ |
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256 |
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257 |
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258 -- monad-law-2' : join . return = id |
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259 monad-law-2' : {l : Level} {A : Set l} -> (d : Delta A) -> (delta-mu ∙ delta-eta) d ≡ id d |
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260 monad-law-2' d = refl |
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261 |
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262 |
90
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263 -- monad-law-3 : return . f = delta-fmap f . return |
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264 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (delta-eta ∙ f) x ≡ (delta-fmap f ∙ delta-eta) x |
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265 monad-law-3 f x = refl |
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266 |
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267 |
96
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268 monad-law-4-1 : {l : Level} {A B : Set l} -> (n : Nat) -> (f : A -> B) -> (ds : Delta (Delta A)) -> |
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269 delta-bind (delta-fmap (delta-fmap f) ds) (n-tail n) ≡ delta-fmap f (delta-bind ds (n-tail n)) |
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270 monad-law-4-1 O f (mono d) = refl |
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271 monad-law-4-1 O f (delta d ds) = begin |
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272 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail O) ≡⟨ refl ⟩ |
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273 delta-bind (delta (delta-fmap f d) (delta-fmap (delta-fmap f) ds)) (n-tail O) ≡⟨ refl ⟩ |
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274 delta (headDelta (delta-fmap f d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta)) (head-delta-natural-transformation f d) ⟩ |
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275 delta (f (headDelta d)) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f (headDelta d)) de) (monad-law-4-1 (S O) f ds) ⟩ |
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276 delta (f (headDelta d)) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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277 delta-fmap f (delta (headDelta d) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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278 delta-fmap f (delta-bind (delta d ds) (n-tail O)) ∎ |
88
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279 monad-law-4-1 (S n) f (mono d) = begin |
94
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280 delta-bind (delta-fmap (delta-fmap f) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩ |
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281 delta-bind (mono (delta-fmap f d)) (n-tail (S n)) ≡⟨ refl ⟩ |
90
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282 n-tail (S n) (delta-fmap f d) ≡⟨ n-tail-natural-transformation (S n) f d ⟩ |
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283 delta-fmap f (n-tail (S n) d) ≡⟨ refl ⟩ |
94
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284 delta-fmap f (delta-bind (mono d) (n-tail (S n))) |
88
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|
285 ∎ |
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286 monad-law-4-1 (S n) f (delta d ds) = begin |
94
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287 delta-bind (delta-fmap (delta-fmap f) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩ |
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288 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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289 delta (headDelta (n-tail (S n) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta de) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (n-tail-natural-transformation (S n) f d) ⟩ |
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Rewrite monad definitions for delta/deltaM
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290 delta (headDelta (delta-fmap f ((n-tail (S n) d)))) (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de (delta-bind (delta-fmap (delta-fmap f) ds) (n-tail (S (S n))))) (head-delta-natural-transformation f (n-tail (S n) d)) ⟩ |
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Rewrite monad definitions for delta/deltaM
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|
291 delta (f (headDelta (n-tail (S n) d))) (delta-bind (delta-fmap (delta-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) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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changeset
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292 delta (f (headDelta (n-tail (S n) d))) (delta-fmap f (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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|
293 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (n-tail (S (S n))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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294 delta-fmap f (delta (headDelta (n-tail (S n) d)) (delta-bind ds (tailDelta ∙ (n-tail (S n))))) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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90
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changeset
|
295 delta-fmap f (delta-bind (delta d ds) (n-tail (S n))) ∎ |
88
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|
296 |
526186c4f298
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|
297 |
90
55d11ce7e223
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|
298 -- monad-law-4 : join . delta-fmap (delta-fmap f) = delta-fmap f . join |
55d11ce7e223
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|
299 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (d : Delta (Delta A)) -> |
94
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Rewrite monad definitions for delta/deltaM
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|
300 (delta-mu ∙ delta-fmap (delta-fmap f)) d ≡ (delta-fmap f ∙ delta-mu) d |
88
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changeset
|
301 monad-law-4 f (mono d) = refl |
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|
302 monad-law-4 f (delta (mono x) ds) = begin |
94
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Rewrite monad definitions for delta/deltaM
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|
303 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (mono x) ds) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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304 delta-mu ( delta-fmap (delta-fmap f) (delta (mono x) ds)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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|
305 delta-mu (delta (mono (f x)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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changeset
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306 delta (headDelta (mono (f x))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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90
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changeset
|
307 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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changeset
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308 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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changeset
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309 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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changeset
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310 delta-fmap f (delta (headDelta (mono x)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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311 delta-fmap f (delta-mu (delta (mono x) ds)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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changeset
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312 (delta-fmap f ∙ delta-mu) (delta (mono x) ds) ∎ |
88
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|
313 monad-law-4 f (delta (delta x d) ds) = begin |
94
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Rewrite monad definitions for delta/deltaM
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314 (delta-mu ∙ delta-fmap (delta-fmap f)) (delta (delta x d) ds) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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315 delta-mu (delta-fmap (delta-fmap f) (delta (delta x d) ds)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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316 delta-mu (delta (delta (f x) (delta-fmap f d)) (delta-fmap (delta-fmap f) ds)) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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317 delta (headDelta (delta (f x) (delta-fmap f d))) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ refl ⟩ |
bcd4fe52a504
Rewrite monad definitions for delta/deltaM
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318 delta (f x) (delta-bind (delta-fmap (delta-fmap f) ds) tailDelta) ≡⟨ cong (\de -> delta (f x) de) (monad-law-4-1 (S O) f ds) ⟩ |
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Rewrite monad definitions for delta/deltaM
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changeset
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319 delta (f x) (delta-fmap f (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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320 delta-fmap f (delta x (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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changeset
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321 delta-fmap f (delta (headDelta (delta x d)) (delta-bind ds tailDelta)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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changeset
|
322 delta-fmap f (delta-mu (delta (delta x d) ds)) ≡⟨ refl ⟩ |
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Rewrite monad definitions for delta/deltaM
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90
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changeset
|
323 (delta-fmap f ∙ delta-mu) (delta (delta x d) ds) ∎ |
88
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|
324 |
103
a271f3ff1922
Delte type dependencie in Monad record for escape implicit type conflict
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96
diff
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325 delta-is-monad : {l : Level} -> Monad {l} Delta delta-is-functor |
94
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326 delta-is-monad = record { eta = delta-eta; |
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327 mu = delta-mu; |
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328 return = delta-eta; |
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329 bind = delta-bind; |
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330 association-law = monad-law-1; |
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331 left-unity-law = monad-law-2; |
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332 right-unity-law = monad-law-2' } |
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333 |
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334 |
96
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335 |
88
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336 {- |
96
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337 |
88
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338 -- Monad-laws (Haskell) |
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339 -- monad-law-h-1 : return a >>= k = k a |
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340 monad-law-h-1 : {l : Level} {A B : Set l} -> |
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341 (a : A) -> (k : A -> (Delta B)) -> |
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342 (delta-return a >>= k) ≡ (k a) |
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343 monad-law-h-1 a k = refl |
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344 |
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345 |
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346 |
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347 -- monad-law-h-2 : m >>= return = m |
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348 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= delta-return) ≡ m |
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349 monad-law-h-2 (mono x) = refl |
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350 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d) |
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351 |
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352 |
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353 |
96
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354 -- monad-law-h-3 : m >>= (\x -> f x >>= g) = (m >>= f) >>= g |
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355 monad-law-h-3 : {l : Level} {A B C : Set l} -> |
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356 (m : Delta A) -> (f : A -> (Delta B)) -> (g : B -> (Delta C)) -> |
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357 (delta-bind m (\x -> delta-bind (f x) g)) ≡ (delta-bind (delta-bind m f) g) |
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358 monad-law-h-3 (mono x) f g = refl |
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359 monad-law-h-3 (delta x d) f g = begin |
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360 (delta-bind (delta x d) (\x -> delta-bind (f x) g)) ≡⟨ {!!} ⟩ |
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361 (delta-bind (delta-bind (delta x d) f) g) ∎ |
88
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362 |
96
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363 |
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364 |
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365 |
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366 -} |