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Prove mu-is-nt for DeltaM with fmap-equiv
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 03 Feb 2015 11:45:33 +0900
parents 0f9ecd118a03
children
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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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9 module deltaM where
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11 -- DeltaM definitions
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13 data DeltaM {l : Level} {T : Set l -> Set l} {F : Functor T}
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14 (M : Monad T F) (A : Set l) : (Nat -> Set l) where
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15 deltaM : {n : Nat} -> Delta (T A) (S n) -> DeltaM M A (S n)
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18 -- DeltaM utils
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20 unDeltaM : {l : Level} {A : Set l} {n : Nat}
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21 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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22 (DeltaM M A (S n)) -> Delta (T A) (S n)
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23 unDeltaM (deltaM d) = d
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27 headDeltaM : {l : Level} {A : Set l} {n : Nat}
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28 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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29 DeltaM M A (S n) -> T A
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30 headDeltaM (deltaM d) = headDelta d
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34 tailDeltaM : {l : Level} {A : Set l} {n : Nat}
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35 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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36 DeltaM M A (S (S n)) -> DeltaM M A (S n)
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37 tailDeltaM {_} {n} (deltaM d) = deltaM (tailDelta d)
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41 appendDeltaM : {l : Level} {A : Set l} {n m : Nat}
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42 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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43 DeltaM M A (S n) -> DeltaM M A (S m) -> DeltaM M A ((S n) + (S m))
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44 appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)
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47 dmap : {l : Level} {A B : Set l} {n : Nat}
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48 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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49 (T A -> B) -> DeltaM M A (S n) -> Delta B (S n)
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50 dmap f (deltaM d) = delta-fmap f d
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55 -- functor definitions
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56 open Functor
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57 deltaM-fmap : {l : Level} {A B : Set l} {n : Nat}
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58 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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59 (A -> B) -> DeltaM M A (S n) -> DeltaM M B (S n)
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60 deltaM-fmap {l} {A} {B} {n} {M} {functorM} f d = deltaM (fmap delta-is-functor (fmap functorM f) (unDeltaM d))
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65 -- monad definitions
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66 open Monad
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68 deltaM-eta : {l : Level} {A : Set l} {n : Nat}
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69 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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70 A -> (DeltaM M A (S n))
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71 deltaM-eta {n = n} {M = M} x = deltaM (delta-eta {n = n} (eta M x))
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75 deltaM-mu : {l : Level} {A : Set l} {n : Nat}
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76 {T : Set l -> Set l} {F : Functor T} {M : Monad T F} ->
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77 DeltaM M (DeltaM M A (S n)) (S n) -> DeltaM M A (S n)
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78 deltaM-mu {n = O} {F = F} {M = M} d = deltaM (mono (mu M (fmap F headDeltaM (headDeltaM d))))
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79 deltaM-mu {n = S n} {F = F} {M = M} d = deltaM (delta (mu M (fmap F headDeltaM (headDeltaM d)))
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80 (unDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM d)))))