Mercurial > hg > Members > atton > delta_monad

annotate agda/similar.agda @ 40:a7cd7740f33e

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add Haskell style Monad-laws and Proof Monad-laws-h-1
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 19 Oct 2014 16:17:46 +0900
parents b9b26b470cc2
children 23474bf242c6
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import list
28
6e6d646d7722 Split basic functions to file
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
2 open import basic
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
3
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
4 open import Level
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
5 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
6 open ≡-Reasoning
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 module similar where
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
10 data Similar {l : Level} (A : Set l) : (Set (suc l)) where
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 similar : List String -> A -> List String -> A -> Similar A
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
13
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
14 -- Functor
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
15 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Similar A) -> (Similar B)
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 fmap f (similar xs x ys y) = similar xs (f x) ys (f y)
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
18
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
19 -- Monad (Category)
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
20 mu : {l : Level} {A : Set l} -> Similar (Similar A) -> Similar A
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 mu (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = similar (lx ++ llx) x (ly ++ lly) y
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
23 eta : {l : Level} {A : Set l} -> A -> Similar A
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
24 eta x = similar [] x [] x
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
25
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
26 returnS : {l : Level} {A : Set l} -> A -> Similar A
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 returnS x = similar [[ (show x) ]] x [[ (show x) ]] x
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
29 returnSS : {l : Level} {A : Set l} -> A -> A -> Similar A
26
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 returnSS x y = similar [[ (show x) ]] x [[ (show y) ]] y
5ba82f107a95 Define Similar in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31
33
0bc402f970b3 Proof Monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 32
diff changeset
32
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
33 -- Monad (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
34 return : {l : Level} {A : Set l} -> A -> Similar A
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
35 return = eta
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
36
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
37 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
38 (x : Similar A) -> (f : A -> (Similar B)) -> (Similar B)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
39 x >>= f = mu (fmap f x)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
41
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
42
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
43 -- proofs
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
44
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
45
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
46 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
47
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
48 -- Functor-law-1 : T(id) = id'
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
49 functor-law-1 : {l : Level} {A : Set l} -> (s : Similar A) -> (fmap id) s ≡ id s
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
50 functor-law-1 (similar lx x ly y) = refl
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
51
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
52 -- Functor-law-2 : T(f . g) = T(f) . T(g)
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
53 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
54 (f : B -> C) -> (g : A -> B) -> (s : Similar A) ->
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
55 (fmap (f ∙ g)) s ≡ ((fmap f) ∙ (fmap g)) s
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
56 functor-law-2 f g (similar lx x ly y) = refl
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
57
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
58
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
59
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
60 -- Monad-laws (Category)
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
61
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
62 -- monad-law-1 : join . fmap join = join . join
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
63 monad-law-1 : {l : Level} {A : Set l} -> (s : Similar (Similar (Similar A))) -> ((mu ∙ (fmap mu)) s) ≡ ((mu ∙ mu) s)
32
71906644d206 Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 30
diff changeset
64 monad-law-1 (similar lx (similar llx (similar lllx x _ _) _ (similar _ _ _ _))
71906644d206 Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 30
diff changeset
65 ly (similar _ (similar _ _ _ _) lly (similar _ _ llly y))) = begin
71906644d206 Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 30
diff changeset
66 similar (lx ++ (llx ++ lllx)) x (ly ++ (lly ++ llly)) y
33
0bc402f970b3 Proof Monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 32
diff changeset
67 ≡⟨ cong (\left-list -> similar left-list x (ly ++ (lly ++ llly)) y) (list-associative lx llx lllx) ⟩
0bc402f970b3 Proof Monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 32
diff changeset
68 similar (lx ++ llx ++ lllx) x (ly ++ (lly ++ llly)) y
0bc402f970b3 Proof Monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 32
diff changeset
69 ≡⟨ cong (\right-list -> similar (lx ++ llx ++ lllx) x right-list y ) (list-associative ly lly llly) ⟩
32
71906644d206 Expand monad-law 1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 30
diff changeset
70 similar (lx ++ llx ++ lllx) x (ly ++ lly ++ llly) y
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
71 ∎
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
72
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
73
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
74 -- monad-law-2 : join . fmap return = join . return = id
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
75 -- monad-law-2-1 join . fmap return = join . return
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
76 monad-law-2-1 : {l : Level} {A : Set l} -> (s : Similar A) ->
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
77 (mu ∙ fmap eta) s ≡ (mu ∙ eta) s
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
78 monad-law-2-1 (similar lx x ly y) = begin
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
79 similar (lx ++ []) x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
80 ≡⟨ cong (\left-list -> similar left-list x (ly ++ []) y) (empty-append lx)⟩
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
81 similar lx x (ly ++ []) y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
82 ≡⟨ cong (\right-list -> similar lx x right-list y) (empty-append ly) ⟩
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
83 similar lx x ly y
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
84 ∎
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
85
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
86 -- monad-law-2-2 : join . return = id
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
87 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Similar A) -> (mu ∙ eta) s ≡ id s
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
88 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
89
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
90 -- monad-law-3 : return . f = fmap f . return
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
91 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
92 monad-law-3 f x = refl
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
93
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
94 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
95 monad-law-4 : {l : Level} {A B : Set l} (f : A -> B) (s : Similar (Similar A)) ->
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
96 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
97 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
98
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
99
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
100 -- Monad-laws (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
101 -- monad-law-h-1 : return a >>= k = k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
102 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
103 (a : A) -> (k : A -> (Similar B)) ->
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
104 (return a >>= k) ≡ (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
105 monad-law-h-1 a k = begin
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
106 return a >>= k
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
107 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
108 mu (fmap k (return a))
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
109 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
110 mu (return (k a))
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
111 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
112 (mu ∙ return) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
113 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
114 (mu ∙ eta) (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
115 ≡⟨ (monad-law-2-2 (k a)) ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
116 id (k a)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
117 ≡⟨ refl ⟩
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
118 k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
119 ∎
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
120
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
121 -- monad-law-h-2 : m >>= return = m
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
122 -- monad-law-h-3 : m >>= (× -> k x >>= h) = (m >>= k) >>= h