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Split monad-proofs into delta.monad
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 19 Jan 2015 11:10:58 +0900
parents 6789c65a75bc
children 5411ce26d525
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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5ba82f107a95 Define Similar in Agda
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diff changeset
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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diff changeset
8 open ≡-Reasoning
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5ba82f107a95 Define Similar in Agda
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9
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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
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15 delta : A -> Delta A -> Delta A
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16
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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
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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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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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35 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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36 fmap f (mono x) = mono (f x)
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37 fmap f (delta x d) = delta (f x) (fmap f d)
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40
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41 -- Monad (Category)
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42 eta : {l : Level} {A : Set l} -> A -> Delta A
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43 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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44
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46b15f368905 Define bind and mu for Infinite Delta
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45 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
46b15f368905 Define bind and mu for Infinite Delta
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46 bind (mono x) f = f x
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47 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f))
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46b15f368905 Define bind and mu for Infinite Delta
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48
46b15f368905 Define bind and mu for Infinite Delta
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49 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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6d0193011f89 Trying prove monad-law-1 by another pattern
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50 mu d = bind d id
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51
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90b171e3a73e Rename to Delta from Similar
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52 returnS : {l : Level} {A : Set l} -> A -> Delta A
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53 returnS x = mono x
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55 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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56 returnSS x y = deltaAppend (returnS x) (returnS y)
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0bc402f970b3 Proof Monad-law 1
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58
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59 -- Monad (Haskell)
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60 return : {l : Level} {A : Set l} -> A -> Delta A
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61 return = eta
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62
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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63 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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64 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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65 (mono x) >>= f = f x
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66 (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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6ce83b2c9e59 Proof Functor-laws
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69
6ce83b2c9e59 Proof Functor-laws
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70 -- proofs
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72 -- sub-proofs
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73
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74 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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75 n-tail-plus O = refl
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76 n-tail-plus (S n) = begin
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77 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
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78 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
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79 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
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80 n-tail (S (S n))
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81 ∎
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82
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83 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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84 n-tail-add O m = refl
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85 n-tail-add (S n) O = begin
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86 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
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4b16b485a4b2 Split nat definition
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87 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
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88 n-tail (S n + O)
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89 ∎
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90 n-tail-add {l} {A} {d} (S n) (S m) = begin
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91 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩
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92 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
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93 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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94 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
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95 n-tail (S (n + S m)) ≡⟨ refl ⟩
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96 n-tail (S n + S m) ∎
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97
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98 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
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99 (n-tail n) (mono x) ≡ (mono x)
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100 tail-delta-to-mono O x = refl
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101 tail-delta-to-mono (S n) x = begin
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102 n-tail (S n) (mono x) ≡⟨ refl ⟩
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103 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
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104 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
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105 tailDelta (mono x) ≡⟨ refl ⟩
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106 mono x ∎
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107
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fc5cd8c50312 Adjust proofs
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108 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
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7307e43a3c76 Prove monad-law-4
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109 -> (f : A -> B) -> (d : Delta A) -> headDelta (fmap f d) ≡ f (headDelta d)
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110 head-delta-natural-transformation f (mono x) = refl
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111 head-delta-natural-transformation f (delta x d) = refl
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112
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113 n-tail-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll}
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114 -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (fmap f d) ≡ fmap f (n-tail n d)
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diff changeset
115 n-tail-natural-transformation O f d = refl
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
116 n-tail-natural-transformation (S n) f (mono x) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
117 n-tail (S n) (fmap f (mono x)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
118 n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
119 (mono (f x)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
120 fmap f (mono x) ≡⟨ cong (\d -> fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
121 fmap f (n-tail (S n) (mono x)) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
122 n-tail-natural-transformation (S n) f (delta x d) = begin
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
123 n-tail (S n) (fmap f (delta x d)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
124 n-tail (S n) (delta (f x) (fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (fmap f d))) (sym (n-tail-plus n)) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
125 ((n-tail n) ∙ tailDelta) (delta (f x) (fmap f d)) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
126 n-tail n (fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
127 fmap f (n-tail n d) ≡⟨ refl ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
128 fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> fmap f (t (delta x d))) (n-tail-plus n) ⟩
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
129 fmap f (n-tail (S n) (delta x d)) ∎
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
130
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
131
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
132
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
133
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
134