Mercurial > hg > Members > atton > delta_monad
annotate agda/deltaM/monad.agda @ 112:0a3b6cb91a05
Prove left-unity-law for DeltaM
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Fri, 30 Jan 2015 21:57:31 +0900 |
parents | 9fe3d0bd1149 |
children | 08403eb8db8b |
rev | line source |
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98
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Trying Monad-laws for DeltaM
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1 open import Level |
b7f0879e854e
Trying Monad-laws for DeltaM
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2 open import Relation.Binary.PropositionalEquality |
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Trying Monad-laws for DeltaM
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3 open ≡-Reasoning |
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Trying Monad-laws for DeltaM
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4 |
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Trying Monad-laws for DeltaM
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5 open import basic |
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Trying Monad-laws for DeltaM
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6 open import delta |
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7 open import delta.functor |
111
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Prove right-unity-law on DeltaM
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8 open import delta.monad |
98
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Trying Monad-laws for DeltaM
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9 open import deltaM |
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Trying Monad-laws for DeltaM
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10 open import deltaM.functor |
104
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Trying redenition Delta with length constraints
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11 open import nat |
98
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Trying Monad-laws for DeltaM
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12 open import laws |
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Trying Monad-laws for DeltaM
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13 |
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Trying Monad-laws for DeltaM
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14 module deltaM.monad where |
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Trying Monad-laws for DeltaM
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15 open Functor |
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Trying Monad-laws for DeltaM
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16 open NaturalTransformation |
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Trying Monad-laws for DeltaM
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17 open Monad |
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Trying Monad-laws for DeltaM
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18 |
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Trying Monad-laws for DeltaM
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19 |
102
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
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20 |
112
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Prove left-unity-law for DeltaM
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21 postulate deltaM-right-unity-law : {l : Level} {A : Set l} |
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Prove left-unity-law for DeltaM
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22 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} {n : Nat} |
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Prove left-unity-law for DeltaM
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23 (d : DeltaM M {functorM} {monadM} A (S n)) -> (deltaM-mu ∙ deltaM-eta) d ≡ id d |
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Prove left-unity-law for DeltaM
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24 {- |
111
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Prove right-unity-law on DeltaM
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25 deltaM-right-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x)) = begin |
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Prove right-unity-law on DeltaM
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26 deltaM-mu (deltaM-eta (deltaM (mono x))) ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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27 deltaM-mu (deltaM (mono (eta mm (deltaM (mono x))))) ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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28 deltaM (mono (mu mm (fmap fm (headDeltaM {M = M})(eta mm (deltaM (mono x)))))) |
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Prove right-unity-law on DeltaM
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29 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (eta-is-nt mm headDeltaM (deltaM (mono x)) )) ⟩ |
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Prove right-unity-law on DeltaM
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30 deltaM (mono (mu mm (eta mm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) (deltaM (mono x)))))) ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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31 deltaM (mono (mu mm (eta mm x))) ≡⟨ cong (\de -> deltaM (mono de)) (sym (right-unity-law mm x)) ⟩ |
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Prove right-unity-law on DeltaM
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32 deltaM (mono x) |
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Prove right-unity-law on DeltaM
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33 ∎ |
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Prove right-unity-law on DeltaM
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34 deltaM-right-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin |
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Prove right-unity-law on DeltaM
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35 deltaM-mu (deltaM-eta (deltaM (delta x d))) |
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Prove right-unity-law on DeltaM
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36 ≡⟨ refl ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
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37 deltaM-mu (deltaM (delta (eta mm (deltaM (delta x d))) (delta-eta (eta mm (deltaM (delta x d)))))) |
104
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Trying redenition Delta with length constraints
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38 ≡⟨ refl ⟩ |
112
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Prove left-unity-law for DeltaM
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39 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {monadM = mm}) (eta mm (deltaM (delta x d))))))) |
111
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Prove right-unity-law on DeltaM
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40 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
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Prove right-unity-law on DeltaM
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41 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
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Prove right-unity-law on DeltaM
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42 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
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43 (sym (eta-is-nt mm headDeltaM (deltaM (delta x d)))) ⟩ |
9fe3d0bd1149
Prove right-unity-law on DeltaM
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44 appendDeltaM (deltaM (mono (mu mm (eta mm ((headDeltaM {monadM = mm}) (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
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45 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
9fe3d0bd1149
Prove right-unity-law on DeltaM
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46 ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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47 appendDeltaM (deltaM (mono (mu mm (eta mm x)))) |
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Prove right-unity-law on DeltaM
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48 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
112
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Prove left-unity-law for DeltaM
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49 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d)))))))) |
111
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Prove right-unity-law on DeltaM
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50 (sym (right-unity-law mm x)) ⟩ |
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Prove right-unity-law on DeltaM
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51 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-eta (eta mm (deltaM (delta x d))))))) |
103
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Delte type dependencie in Monad record for escape implicit type conflict
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52 ≡⟨ refl ⟩ |
111
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Prove right-unity-law on DeltaM
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53 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-fmap (fmap fm tailDeltaM) (delta-eta (eta mm (deltaM (delta x d))))))) |
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Prove right-unity-law on DeltaM
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54 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM de))) (sym (eta-is-nt delta-is-monad (fmap fm tailDeltaM) (eta mm (deltaM (delta x d))))) ⟩ |
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Prove right-unity-law on DeltaM
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55 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (fmap fm tailDeltaM (eta mm (deltaM (delta x d))))))) |
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Prove right-unity-law on DeltaM
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56 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta de)))) (sym (eta-is-nt mm tailDeltaM (deltaM (delta x d)))) ⟩ |
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Prove right-unity-law on DeltaM
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57 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (tailDeltaM (deltaM (delta x d))))))) |
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Prove right-unity-law on DeltaM
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58 ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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59 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM (delta-eta (eta mm (deltaM d))))) |
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Prove right-unity-law on DeltaM
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60 ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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61 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-eta (deltaM d))) |
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Prove right-unity-law on DeltaM
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62 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-right-unity-law (deltaM d)) ⟩ |
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Prove right-unity-law on DeltaM
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63 appendDeltaM (deltaM (mono x)) (deltaM d) |
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Prove right-unity-law on DeltaM
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64 ≡⟨ refl ⟩ |
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Prove right-unity-law on DeltaM
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65 deltaM (delta x d) |
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Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
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66 ∎ |
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67 -} |
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68 |
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Prove left-unity-law for DeltaM
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69 |
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70 fmap-headDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat} |
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Prove left-unity-law for DeltaM
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71 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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Prove left-unity-law for DeltaM
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72 (x : M A) -> (fmap functorM ((headDeltaM {l} {A} {n} {M} {functorM} {monadM}) ∙ deltaM-eta) x) ≡ fmap functorM (eta monadM) x |
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Prove left-unity-law for DeltaM
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73 fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm} x = refl |
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Prove left-unity-law for DeltaM
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74 fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x = refl |
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Prove left-unity-law for DeltaM
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75 |
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76 fmap-tailDeltaM-with-deltaM-eta : {l : Level} {A : Set l} {n : Nat} |
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Prove left-unity-law for DeltaM
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77 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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Prove left-unity-law for DeltaM
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78 (d : DeltaM M {functorM} {monadM} A (S n)) -> |
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Prove left-unity-law for DeltaM
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79 deltaM-fmap ((tailDeltaM {n = n} {monadM = monadM} ) ∙ deltaM-eta) d ≡ deltaM-fmap (deltaM-eta) d |
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Prove left-unity-law for DeltaM
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80 fmap-tailDeltaM-with-deltaM-eta {n = O} d = refl |
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Prove left-unity-law for DeltaM
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81 fmap-tailDeltaM-with-deltaM-eta {n = S n} d = refl |
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Prove left-unity-law for DeltaM
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82 |
104
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Trying redenition Delta with length constraints
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83 |
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Trying redenition Delta with length constraints
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84 |
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Prove left-unity-law for DeltaM
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85 deltaM-left-unity-law : {l : Level} {A : Set l} |
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Prove left-unity-law for DeltaM
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86 {M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} |
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Prove left-unity-law for DeltaM
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87 {n : Nat} |
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88 (d : DeltaM M {functorM} {monadM} A (S n)) -> |
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89 (deltaM-mu ∙ (deltaM-fmap deltaM-eta)) d ≡ id d |
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0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
90 deltaM-left-unity-law {l} {A} {M} {fm} {mm} {O} (deltaM (mono x)) = begin |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
91 deltaM-mu (deltaM-fmap deltaM-eta (deltaM (mono x))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
92 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
93 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (mono x))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
94 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
95 deltaM-mu (deltaM (mono (fmap fm deltaM-eta x))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
96 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
97 deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {O} {M}) (fmap fm deltaM-eta x)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
98 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (sym (covariant fm deltaM-eta headDeltaM x)) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
99 deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {O} {M} {fm} {mm}) ∙ deltaM-eta) x))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
100 ≡⟨ cong (\de -> deltaM (mono (mu mm de))) (fmap-headDeltaM-with-deltaM-eta {l} {A} {O} {M} {fm} {mm} x) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
101 deltaM (mono (mu mm (fmap fm (eta mm) x))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
102 ≡⟨ cong (\de -> deltaM (mono de)) (left-unity-law mm x) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
103 deltaM (mono x) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
104 ∎ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
105 deltaM-left-unity-law {l} {A} {M} {fm} {mm} {S n} (deltaM (delta x d)) = begin |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
106 deltaM-mu (deltaM-fmap deltaM-eta (deltaM (delta x d))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
107 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
108 deltaM-mu (deltaM (delta-fmap (fmap fm deltaM-eta) (delta x d))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
109 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
110 deltaM-mu (deltaM (delta (fmap fm deltaM-eta x) (delta-fmap (fmap fm deltaM-eta) d))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
111 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
112 appendDeltaM (deltaM (mono (mu mm (fmap fm (headDeltaM {l} {A} {S n} {M} {fm} {mm}) (fmap fm deltaM-eta x))))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
113 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
114 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
115 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
116 (sym (covariant fm deltaM-eta headDeltaM x)) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
117 appendDeltaM (deltaM (mono (mu mm (fmap fm ((headDeltaM {l} {A} {S n} {M} {fm} {mm}) ∙ deltaM-eta) x)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
118 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
119 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono (mu mm de))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
120 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d))))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
121 (fmap-headDeltaM-with-deltaM-eta {l} {A} {S n} {M} {fm} {mm} x) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
122 appendDeltaM (deltaM (mono (mu mm (fmap fm (eta mm) x)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
123 (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
124 |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
125 ≡⟨ cong (\de -> (appendDeltaM (deltaM (mono de)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
126 (left-unity-law mm x) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
127 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap tailDeltaM (deltaM (delta-fmap (fmap fm deltaM-eta) d)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
128 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
129 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap (tailDeltaM {n = n})(deltaM-fmap deltaM-eta (deltaM d)))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
130 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (sym (covariant deltaM-is-functor deltaM-eta tailDeltaM (deltaM d))) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
131 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap ((tailDeltaM {n = n}) ∙ deltaM-eta) (deltaM d))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
132 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) (deltaM-mu de)) (fmap-tailDeltaM-with-deltaM-eta (deltaM d)) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
133 appendDeltaM (deltaM (mono x)) (deltaM-mu (deltaM-fmap deltaM-eta (deltaM d))) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
134 ≡⟨ cong (\de -> appendDeltaM (deltaM (mono x)) de) (deltaM-left-unity-law (deltaM d)) ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
135 appendDeltaM (deltaM (mono x)) (deltaM d) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
136 ≡⟨ refl ⟩ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
137 deltaM (delta x d) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
138 ∎ |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
139 |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
140 |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
141 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
142 deltaM-is-monad : {l : Level} {A : Set l} {n : Nat} |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
143 {M : Set l -> Set l} |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
144 (functorM : Functor M) |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
145 (monadM : Monad M functorM) -> |
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
146 Monad {l} (\A -> DeltaM M {functorM} {monadM} A (S n)) (deltaM-is-functor {l} {n}) |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
147 deltaM-is-monad functorM monadM = record |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
148 { mu = deltaM-mu; |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
149 eta = deltaM-eta; |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
150 return = deltaM-eta; |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
151 bind = deltaM-bind; |
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
152 association-law = {!!}; |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
153 left-unity-law = deltaM-left-unity-law; |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
154 right-unity-law = (\x -> (sym (deltaM-right-unity-law x))) ; |
112
0a3b6cb91a05
Prove left-unity-law for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
111
diff
changeset
|
155 eta-is-nt = {!!} |
111
9fe3d0bd1149
Prove right-unity-law on DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
104
diff
changeset
|
156 } |
104
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
157 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
158 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
159 |
ebd0d6e2772c
Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
160 |
102
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
161 |
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
162 |
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
163 {- |
98
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
164 deltaM-association-law : {l : Level} {A : Set l} |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
165 {M : {l' : Level} -> Set l' -> Set l'} |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
166 (functorM : {l' : Level} -> Functor {l'} M) |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
167 (monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM) |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
168 -> (d : DeltaM M (DeltaM M (DeltaM M {functorM} {monadM} A))) |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
169 -> ((deltaM-mu ∙ (deltaM-fmap deltaM-mu)) d) ≡ ((deltaM-mu ∙ deltaM-mu) d) |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
170 deltaM-association-law functorM monadM (deltaM (mono x)) = begin |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
171 (deltaM-mu ∙ deltaM-fmap deltaM-mu) (deltaM (mono x)) ≡⟨ refl ⟩ |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
172 deltaM-mu (deltaM-fmap deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ |
102
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
173 deltaM-mu (deltaM (delta-fmap (fmap functorM deltaM-mu) (mono x))) ≡⟨ {!!} ⟩ |
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
174 deltaM-mu (deltaM (mono (bind monadM x headDeltaM))) ≡⟨ refl ⟩ |
9c62373bd474
Trying right-unity-law on DeltaM. but do not fit implicit type in eta...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
98
diff
changeset
|
175 deltaM-mu (deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ |
98
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
176 deltaM-mu (deltaM-mu (deltaM (mono x))) ≡⟨ refl ⟩ |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
177 (deltaM-mu ∙ deltaM-mu) (deltaM (mono x)) ∎ |
b7f0879e854e
Trying Monad-laws for DeltaM
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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178 deltaM-association-law functorM monadM (deltaM (delta x d)) = {!!} |
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179 -} |
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180 |
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181 {- |
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182 |
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183 nya : {l : Level} {A B C : Set l} -> |
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184 {M : {l' : Level} -> Set l' -> Set l'} |
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185 {functorM : {l' : Level} -> Functor {l'} M } |
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186 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} |
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187 (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M {functorM} {monadM} B)) -> (g : B -> (DeltaM M C)) -> |
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188 (x : M A) -> |
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189 (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡ |
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190 (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g) |
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191 nya = {!!} |
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192 |
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193 deltaM-monad-law-h-3 : {l : Level} {A B C : Set l} -> |
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194 {M : {l' : Level} -> Set l' -> Set l'} |
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195 {functorM : {l' : Level} -> Functor {l'} M } |
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196 {monadM : {l' : Level} {A : Set l'} -> Monad {l'} {A} M functorM} |
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197 (m : DeltaM M {functorM} {monadM} A) -> (f : A -> (DeltaM M B)) -> (g : B -> (DeltaM M C)) -> |
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198 (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡ (deltaM-bind (deltaM-bind m f) g) |
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199 {- |
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200 deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin |
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201 (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩ |
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202 |
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203 (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩ |
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204 (deltaM-bind (deltaM (fmap delta-is-functor (\x -> (bind {l} {A} monadM x (headDeltaM ∙ f))) (mono x))) g) ≡⟨ refl ⟩ |
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205 (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩ |
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206 (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g) ∎ |
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207 -} |
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208 |
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209 deltaM-monad-law-h-3 {l} {A} {B} {C} {M} {functorM} {monadM} (deltaM (mono x)) f g = begin |
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210 (deltaM-bind (deltaM (mono x)) (\x -> deltaM-bind (f x) g)) ≡⟨ refl ⟩ |
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211 (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))))) ≡⟨ {!!} ⟩ |
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212 -- (deltaM (fmap delta-is-functor (\x -> bind {l} {A} monadM x (headDeltaM ∙ (\x -> deltaM-bind (f x) g))) (mono x))) ≡⟨ {!!} ⟩ |
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213 deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩ |
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214 deltaM (mono (bind {l} {B} monadM (bind {_} {A} monadM x (headDeltaM ∙ f)) (headDeltaM ∙ g))) ≡⟨ {!!} ⟩ |
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215 (deltaM-bind (deltaM (mono (bind {l} {A} monadM x (headDeltaM ∙ f)))) g) ≡⟨ refl ⟩ |
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216 (deltaM-bind (deltaM-bind (deltaM (mono x)) f) g) |
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217 ∎ |
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218 deltaM-monad-law-h-3 (deltaM (delta x d)) f g = {!!} |
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219 {- |
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220 begin |
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221 (deltaM-bind m (\x -> deltaM-bind (f x) g)) ≡⟨ {!!} ⟩ |
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222 (deltaM-bind (deltaM-bind m f) g) |
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223 ∎ |
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224 -} |
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225 -} |