Mercurial > hg > Members > atton > delta_monad
annotate agda/delta/functor.agda @ 146:57601209eff3 default tip
Add an example used multi types on Delta
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Tue, 24 Mar 2015 17:04:00 +0900 |
parents | d205ff1e406f |
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rev | line source |
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1 open import Level |
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2 open import Relation.Binary.PropositionalEquality |
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3 open ≡-Reasoning |
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5 open import basic |
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6 open import delta |
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7 open import laws |
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8 open import nat |
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9 |
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10 module delta.functor where |
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11 |
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12 -- Functor-laws |
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13 |
131 | 14 -- functor-law 1 : T(id) = id' |
15 delta-preserve-id : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> (delta-fmap id) d ≡ id d | |
16 delta-preserve-id (mono x) = refl | |
17 delta-preserve-id (delta x d) = cong (delta x) (delta-preserve-id d) | |
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18 |
131 | 19 |
20 -- functor-law 2 : T(f . g) = T(f) . T(g) | |
21 delta-covariant : {l : Level} {n : Nat} {A B C : Set l} -> | |
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22 (f : B -> C) -> (g : A -> B) -> (d : Delta A (S n)) -> |
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23 (delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-fmap g)) d |
131 | 24 delta-covariant f g (mono x) = refl |
25 delta-covariant f g (delta x d) = cong (delta (f (g x))) (delta-covariant f g d) | |
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26 |
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131 | 28 delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat} {f g : A -> B} |
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29 (eq : (x : A) -> f x ≡ g x) -> (d : Delta A (S n)) -> |
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30 delta-fmap f d ≡ delta-fmap g d |
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31 delta-fmap-equiv {l} {A} {B} {O} {f} {g} eq (mono x) = begin |
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32 mono (f x) ≡⟨ cong mono (eq x) ⟩ |
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33 mono (g x) |
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34 ∎ |
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35 delta-fmap-equiv {l} {A} {B} {S n} {f} {g} eq (delta x d) = begin |
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36 delta (f x) (delta-fmap f d) ≡⟨ cong (\de -> delta de (delta-fmap f d)) (eq x) ⟩ |
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37 delta (g x) (delta-fmap f d) ≡⟨ cong (\de -> delta (g x) de) (delta-fmap-equiv eq d) ⟩ |
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38 delta (g x) (delta-fmap g d) |
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39 ∎ |
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40 |
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41 |
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42 |
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43 delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n)) |
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44 delta-is-functor = record { fmap = delta-fmap |
131 | 45 ; preserve-id = delta-preserve-id |
46 ; covariant = \f g -> delta-covariant g f | |
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47 ; fmap-equiv = delta-fmap-equiv |
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48 } |