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Prove association-law for DeltaM
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Mon, 02 Feb 2015 11:54:23 +0900
parents e6499a50ccbd
children d205ff1e406f
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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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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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10 module delta where
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12 data Delta {l : Level} (A : Set l) : (Nat -> (Set l)) where
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13 mono : A -> Delta A (S O)
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14 delta : {n : Nat} -> A -> Delta A (S n) -> Delta A (S (S n))
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16 deltaAppend : {l : Level} {A : Set l} {n m : Nat} -> Delta A (S n) -> Delta A (S m) -> Delta A ((S n) + (S m))
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dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
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17 deltaAppend (mono x) d = delta x d
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18 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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20 headDelta : {l : Level} {A : Set l} {n : Nat} -> Delta A (S n) -> A
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21 headDelta (mono x) = x
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22 headDelta (delta x _) = x
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24 tailDelta : {l : Level} {A : Set l} {n : Nat} -> Delta A (S (S n)) -> Delta A (S n)
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25 tailDelta (delta _ d) = d
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6ce83b2c9e59 Proof Functor-laws
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29 -- Functor
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30 delta-fmap : {l : Level} {A B : Set l} {n : Nat} -> (A -> B) -> (Delta A (S n)) -> (Delta B (S n))
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31 delta-fmap f (mono x) = mono (f x)
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32 delta-fmap f (delta x d) = delta (f x) (delta-fmap f d)
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36 -- Monad (Category)
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37 delta-eta : {l : Level} {A : Set l} {n : Nat} -> A -> Delta A (S n)
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38 delta-eta {n = O} x = mono x
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39 delta-eta {n = (S n)} x = delta x (delta-eta {n = n} x)
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46b15f368905 Define bind and mu for Infinite Delta
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44 delta-mu : {l : Level} {A : Set l} {n : Nat} -> (Delta (Delta A (S n)) (S n)) -> Delta A (S n)
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45 delta-mu (mono x) = x
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46 delta-mu (delta x d) = delta (headDelta x) (delta-mu (delta-fmap tailDelta d))
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46b15f368905 Define bind and mu for Infinite Delta
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48 delta-bind : {l : Level} {A B : Set l} {n : Nat} -> (Delta A (S n)) -> (A -> Delta B (S n)) -> Delta B (S n)
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49 delta-bind d f = delta-mu (delta-fmap f d)
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51 --delta-bind (mono x) f = f x
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52 --delta-bind (delta x d) f = delta (headDelta (f x)) (tailDelta (f x))
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0bc402f970b3 Proof Monad-law 1
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55 {-
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56 -- Monad (Haskell)
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57 delta-return : {l : Level} {A : Set l} -> A -> Delta A (S O)
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bcd4fe52a504 Rewrite monad definitions for delta/deltaM
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58 delta-return = delta-eta
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60 _>>=_ : {l : Level} {A B : Set l} {n : Nat} ->
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61 (x : Delta A n) -> (f : A -> (Delta B n)) -> (Delta B n)
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62 d >>= f = delta-bind d f
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64 -}
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66 {-
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67 -- proofs
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69 -- sub-proofs
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71 n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
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72 n-tail-plus O = refl
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73 n-tail-plus (S n) = begin
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74 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
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75 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
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76 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
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77 n-tail (S (S n))
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78 ∎
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79
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80 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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81 n-tail-add O m = refl
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82 n-tail-add (S n) O = begin
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83 n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩
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4b16b485a4b2 Split nat definition
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84 n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩
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85 n-tail (S n + O)
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86 ∎
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87 n-tail-add {l} {A} {d} (S n) (S m) = begin
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88 n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩
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89 (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩
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90 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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91 tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩
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92 n-tail (S (n + S m)) ≡⟨ refl ⟩
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93 n-tail (S n + S m) ∎
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94
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95 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) ->
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96 (n-tail n) (mono x) ≡ (mono x)
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97 tail-delta-to-mono O x = refl
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98 tail-delta-to-mono (S n) x = begin
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99 n-tail (S n) (mono x) ≡⟨ refl ⟩
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100 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
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101 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
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102 tailDelta (mono x) ≡⟨ refl ⟩
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103 mono x ∎
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104
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dfe8c67390bd Unify Levels in delta
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105 head-delta-natural-transformation : {l : Level} {A B : Set l}
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106 -> (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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107 head-delta-natural-transformation f (mono x) = refl
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fc5cd8c50312 Adjust proofs
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108 head-delta-natural-transformation f (delta x d) = refl
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109
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110 n-tail-natural-transformation : {l : Level} {A B : Set l}
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111 -> (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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112 n-tail-natural-transformation O f d = refl
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113 n-tail-natural-transformation (S n) f (mono x) = begin
89
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
114 n-tail (S n) (delta-fmap f (mono x)) ≡⟨ refl ⟩
79
7307e43a3c76 Prove monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
115 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
116 (mono (f x)) ≡⟨ refl ⟩
89
5411ce26d525 Defining DeltaM in Agda...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 88
diff changeset
117 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
118 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
119 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
120 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
121 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
122 ((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
123 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
124 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
125 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
126 delta-fmap f (n-tail (S n) (delta x d)) ∎
104
ebd0d6e2772c Trying redenition Delta with length constraints
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 96
diff changeset
127 -}