annotate agda/delta/functor.agda @ 112:0a3b6cb91a05

Prove left-unity-law for DeltaM
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Fri, 30 Jan 2015 21:57:31 +0900
parents e6499a50ccbd
children 47f144540d51
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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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10 module delta.functor where
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11
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12 -- Functor-laws
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13
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14 -- Functor-law-1 : T(id) = id'
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15 functor-law-1 : {l : Level} {A : Set l} {n : Nat} -> (d : Delta A (S n)) -> (delta-fmap id) d ≡ id d
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16 functor-law-1 (mono x) = refl
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17 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
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18
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19 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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20 functor-law-2 : {l : Level} {n : Nat} {A B C : Set l} ->
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21 (f : B -> C) -> (g : A -> B) -> (d : Delta A (S n)) ->
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8d92ed54a94f Prove functor-laws for deltaM
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22 (delta-fmap (f ∙ g)) d ≡ ((delta-fmap f) ∙ (delta-fmap g)) d
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23 functor-law-2 f g (mono x) = refl
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24 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
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26 delta-fmap-equiv : {l : Level} {A B : Set l} {n : Nat}
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27 {f g : A -> B} -> (eq : (x : A) -> f x ≡ g x) (d : Delta A (S n)) ->
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28 delta-fmap f d ≡ delta-fmap g d
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29 delta-fmap-equiv {f = f} {g = g} eq (mono x) = begin
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30 mono (f x) ≡⟨ cong (\he -> (mono he)) (eq x) ⟩
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31 mono (g x) ∎
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32 delta-fmap-equiv {f = f} {g = g} eq (delta x d) = begin
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33 delta (f x) (delta-fmap f d) ≡⟨ cong (\hx -> (delta hx (delta-fmap f d))) (eq x) ⟩
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34 delta (g x) (delta-fmap f d) ≡⟨ cong (\fx -> (delta (g x) fx)) (delta-fmap-equiv {f = f} {g = g} eq d) ⟩
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35 delta (g x) (delta-fmap g d) ∎
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39 delta-is-functor : {l : Level} {n : Nat} -> Functor {l} (\A -> Delta A (S n))
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5411ce26d525 Defining DeltaM in Agda...
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40 delta-is-functor = record { fmap = delta-fmap ;
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41 preserve-id = functor-law-1;
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42 covariant = \f g -> functor-law-2 g f;
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43 fmap-equiv = delta-fmap-equiv }