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Unify Levels in delta
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 20 Jan 2015 16:25:53 +0900
parents bcd4fe52a504
children ebd0d6e2772c
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26
5ba82f107a95 Define Similar in Agda
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1 open import list
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6e6d646d7722 Split basic functions to file
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2 open import basic
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c7076f9bbaed Refactors
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3 open import nat
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6789c65a75bc Split functor-proofs into delta.functor
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4 open import laws
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e0ba1bf564dd Apply level to some functions
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5
e0ba1bf564dd Apply level to some functions
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6 open import Level
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742e62fc63e4 Define Monad-law 1-4
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7 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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8 open ≡-Reasoning
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5ba82f107a95 Define Similar in Agda
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9
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90b171e3a73e Rename to Delta from Similar
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10 module delta where
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12
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6789c65a75bc Split functor-proofs into delta.functor
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13 data Delta {l : Level} (A : Set l) : (Set l) where
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dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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14 mono : A -> Delta A
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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15 delta : A -> Delta A -> Delta A
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9bb7c9bee94f Trying redefine delta for infinite changes
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16
9bb7c9bee94f Trying redefine delta for infinite changes
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17 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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18 deltaAppend (mono x) d = delta x d
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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19 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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20
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21 headDelta : {l : Level} {A : Set l} -> Delta A -> A
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22 headDelta (mono x) = x
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23 headDelta (delta x _) = x
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9bb7c9bee94f Trying redefine delta for infinite changes
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24
9bb7c9bee94f Trying redefine delta for infinite changes
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25 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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7307e43a3c76 Prove monad-law-4
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26 tailDelta (mono x) = mono x
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27 tailDelta (delta _ d) = d
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5ba82f107a95 Define Similar in Agda
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29 n-tail : {l : Level} {A : Set l} -> Nat -> ((Delta A) -> (Delta A))
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30 n-tail O = id
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31 n-tail (S n) = tailDelta ∙ (n-tail n)
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32
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6ce83b2c9e59 Proof Functor-laws
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33
6ce83b2c9e59 Proof Functor-laws
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34 -- Functor
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dfe8c67390bd Unify Levels in delta
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35 delta-fmap : {l : Level} {A B : Set l} -> (A -> B) -> (Delta A) -> (Delta B)
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36 delta-fmap f (mono x) = mono (f x)
5411ce26d525 Defining DeltaM in Agda...
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37 delta-fmap f (delta x d) = delta (f x) (delta-fmap f d)
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dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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38
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5ba82f107a95 Define Similar in Agda
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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41 -- Monad (Category)
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bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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42 delta-eta : {l : Level} {A : Set l} -> A -> Delta A
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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43 delta-eta x = mono x
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46b15f368905 Define bind and mu for Infinite Delta
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44
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bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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45 delta-bind : {l : Level} {A B : Set l} -> (Delta A) -> (A -> Delta B) -> Delta B
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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46 delta-bind (mono x) f = f x
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47 delta-bind (delta x d) f = delta (headDelta (f x)) (delta-bind d (tailDelta ∙ f))
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46b15f368905 Define bind and mu for Infinite Delta
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48
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49 delta-mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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50 delta-mu d = delta-bind d id
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5ba82f107a95 Define Similar in Agda
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51
5ba82f107a95 Define Similar in Agda
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0bc402f970b3 Proof Monad-law 1
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53
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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54 -- Monad (Haskell)
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bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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55 delta-return : {l : Level} {A : Set l} -> A -> Delta A
bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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56 delta-return = delta-eta
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57
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55d11ce7e223 Unify levels on data type. only use suc to proofs
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58 _>>=_ : {l : Level} {A B : Set l} ->
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90b171e3a73e Rename to Delta from Similar
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59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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60 (mono x) >>= f = f x
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61 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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63
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6ce83b2c9e59 Proof Functor-laws
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64
6ce83b2c9e59 Proof Functor-laws
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65 -- proofs
6ce83b2c9e59 Proof Functor-laws
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67 -- sub-proofs
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68
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69 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
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7307e43a3c76 Prove monad-law-4
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70 n-tail-plus O = refl
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71 n-tail-plus (S n) = begin
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72 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
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73 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
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74 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
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75 n-tail (S (S n))
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76 ∎
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77
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78 n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
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79 n-tail-add O m = refl
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80 n-tail-add (S n) O = begin
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81 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
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4b16b485a4b2 Split nat definition
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82 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
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83 n-tail (S n + O)
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84 ∎
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85 n-tail-add {l} {A} {d} (S n) (S m) = begin
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86 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩
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87 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
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88 tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩
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89 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
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90 n-tail (S (n + S m)) ≡⟨ refl ⟩
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91 n-tail (S n + S m) ∎
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92
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93 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
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94 (n-tail n) (mono x) ≡ (mono x)
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95 tail-delta-to-mono O x = refl
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96 tail-delta-to-mono (S n) x = begin
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97 n-tail (S n) (mono x) ≡⟨ refl ⟩
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98 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
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99 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
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100 tailDelta (mono x) ≡⟨ refl ⟩
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101 mono x ∎
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102
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dfe8c67390bd Unify Levels in delta
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103 head-delta-natural-transformation : {l : Level} {A B : Set l}
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5411ce26d525 Defining DeltaM in Agda...
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104 -> (f : A -> B) -> (d : Delta A) -> headDelta (delta-fmap f d) ≡ f (headDelta d)
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7307e43a3c76 Prove monad-law-4
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105 head-delta-natural-transformation f (mono x) = refl
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fc5cd8c50312 Adjust proofs
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106 head-delta-natural-transformation f (delta x d) = refl
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7307e43a3c76 Prove monad-law-4
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107
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dfe8c67390bd Unify Levels in delta
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108 n-tail-natural-transformation : {l : Level} {A B : Set l}
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109 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (delta-fmap f d) ≡ delta-fmap f (n-tail n d)
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7307e43a3c76 Prove monad-law-4
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110 n-tail-natural-transformation O f d = refl
7307e43a3c76 Prove monad-law-4
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parents: 78
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111 n-tail-natural-transformation (S n) f (mono x) = begin
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112 n-tail (S n) (delta-fmap f (mono x)) ≡⟨ refl ⟩
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7307e43a3c76 Prove monad-law-4
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parents: 78
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113 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩
7307e43a3c76 Prove monad-law-4
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114 (mono (f x)) ≡⟨ refl ⟩
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Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
115 delta-fmap f (mono x) ≡⟨ cong (\d -> delta-fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
116 delta-fmap f (n-tail (S n) (mono x)) ∎
79
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
117 n-tail-natural-transformation (S n) f (delta x d) = begin
89
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
118 n-tail (S n) (delta-fmap f (delta x d)) ≡⟨ refl ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
119 n-tail (S n) (delta (f x) (delta-fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (delta-fmap f d))) (sym (n-tail-plus n)) ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
120 ((n-tail n) ∙ tailDelta) (delta (f x) (delta-fmap f d)) ≡⟨ refl ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
121 n-tail n (delta-fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
122 delta-fmap f (n-tail n d) ≡⟨ refl ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
123 delta-fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> delta-fmap f (t (delta x d))) (n-tail-plus n) ⟩
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
124 delta-fmap f (n-tail (S n) (delta x d)) ∎