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Temporary commit : Proving association-law ...
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 01 Feb 2015 17:06:55 +0900
parents 0a3b6cb91a05
children f02c5ad4a327
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1 open import Level
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3 open import basic
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4 open import delta
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5 open import delta.functor
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6 open import nat
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7 open import laws
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8
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9 module deltaM where
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10
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11 -- DeltaM definitions
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12
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13 data DeltaM {l : Level}
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14 (M : Set l -> Set l)
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15 {functorM : Functor M}
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16 {monadM : Monad M functorM}
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17 (A : Set l)
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18 : (Nat -> Set l) where
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19 deltaM : {n : Nat} -> Delta (M A) (S n) -> DeltaM M {functorM} {monadM} A (S n)
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20
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21
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22 -- DeltaM utils
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24 headDeltaM : {l : Level} {A : Set l} {n : Nat}
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25 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
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26 -> DeltaM M {functorM} {monadM} A (S n) -> M A
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27 headDeltaM (deltaM d) = headDelta d
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30 tailDeltaM : {l : Level} {A : Set l} {n : Nat}
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31 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
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32 -> DeltaM {l} M {functorM} {monadM} A (S (S n)) -> DeltaM M {functorM} {monadM} A (S n)
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33 tailDeltaM {_} {n} (deltaM d) = deltaM (tailDelta d)
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36 appendDeltaM : {l : Level} {A : Set l} {n m : Nat}
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37 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
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38 DeltaM M {functorM} {monadM} A (S n) ->
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39 DeltaM M {functorM} {monadM} A (S m) ->
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40 DeltaM M {functorM} {monadM} A ((S n) + (S m))
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41 appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)
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44 dmap : {l : Level} {A B : Set l} {n : Nat}
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45 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
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46 (M A -> B) -> DeltaM M {functorM} {monadM} A (S n) -> Delta B (S n)
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47 dmap f (deltaM d) = delta-fmap f d
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50 -- functor definitions
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51 open Functor
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52 deltaM-fmap : {l : Level} {A B : Set l} {n : Nat}
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53 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
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54 -> (A -> B) -> DeltaM M {functorM} {monadM} A (S n) -> DeltaM M {functorM} {monadM} B (S n)
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55 deltaM-fmap {l} {A} {B} {n} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)
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60 -- monad definitions
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61 open Monad
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63 deltaM-eta : {l : Level} {A : Set l} {n : Nat}
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64 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
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65 A -> (DeltaM M {functorM} {monadM} A (S n))
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66 deltaM-eta {n = n} {monadM = mm} x = deltaM (delta-eta {n = n} (eta mm x))
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68 deltaM-mu : {l : Level} {A : Set l} {n : Nat}
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69 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
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70 DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A (S n)) (S n) ->
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71 DeltaM M {functorM} {monadM} A (S n)
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72 deltaM-mu {n = O} {functorM = fm} {monadM = mm} (deltaM (mono x)) = deltaM (mono (mu mm (fmap fm headDeltaM x)))
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73 deltaM-mu {n = S n} {functorM = fm} {monadM = mm} (deltaM (delta x d)) = appendDeltaM (deltaM (mono (mu mm (fmap fm headDeltaM x))))
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74 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM d)))
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77 deltaM-bind : {l : Level} {A B : Set l}
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78 {n : Nat}
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79 {M : Set l -> Set l}
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80 {functorM : Functor M}
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81 {monadM : Monad M functorM} ->
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82 (DeltaM M {functorM} {monadM} A (S n)) ->
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83 (A -> DeltaM M {functorM} {monadM} B (S n))
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84 -> DeltaM M {functorM} {monadM} B (S n)
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85 deltaM-bind {n = O} {monadM = mm} (deltaM (mono x)) f = deltaM (mono (bind mm x (headDeltaM ∙ f)))
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86 deltaM-bind {n = S n} {monadM = mm} (deltaM (delta x d)) f = appendDeltaM (deltaM (mono (bind mm x (headDeltaM ∙ f))))
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87 (deltaM-bind (deltaM d) (tailDeltaM ∙ f))
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