Members/atton/delta_monad: agda/delta.agda annotate

annotate agda/delta.agda @ 68:f9c9207c40b7

Trying prove monad-law-1 by another pattern ....

author	Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date	Thu, 27 Nov 2014 14:46:39 +0900
parents	e70be6a2bf72
children	295e8ed39c0c

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
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	8 module delta where
26 5ba82f107a95 Define Similar in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: diff changeset	9
54 9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	10
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	12 mono : A -> Delta A
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	13 delta : A -> Delta A -> Delta A
54 9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	14
9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	16 deltaAppend (mono x) d = delta x d
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
54 9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	18
9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	19 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	20 headDelta (mono x) = mono x
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	21 headDelta (delta x _) = mono x
54 9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	22
9bb7c9bee94f Trying redefine delta for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 43 diff changeset	23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	24 tailDelta (mono x) = mono x
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	25 tailDelta (delta _ d) = d
26 5ba82f107a95 Define Similar in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: diff changeset	26
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	27
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	28 -- Functor
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	30 fmap f (mono x) = mono (f x)
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	31 fmap f (delta x d) = delta (f x) (fmap f d)
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	32
26 5ba82f107a95 Define Similar in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: diff changeset	33
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	34
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	35 -- Monad (Category)
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	36 eta : {l : Level} {A : Set l} -> A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	37 eta x = mono x
27 742e62fc63e4 Define Monad-law 1-4 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 26 diff changeset	38
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	39 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 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	40 bind (mono x) f = f x
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	41 bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f))
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	42
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	44 mu d = bind d id
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	45
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	46
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	47 returnS : {l : Level} {A : Set l} -> A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	48 returnS x = mono x
26 5ba82f107a95 Define Similar in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: diff changeset	49
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	50 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	51 returnSS x y = deltaAppend (returnS x) (returnS y)
26 5ba82f107a95 Define Similar in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: diff changeset	52
33 0bc402f970b3 Proof Monad-law 1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 32 diff changeset	53
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	54 -- Monad (Haskell)
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	55 return : {l : Level} {A : Set l} -> A -> Delta 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	56 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	57
41 23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	58 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	60 (mono x) >>= f = f x
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	61 (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
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	62
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	63
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	64
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	65 -- proofs
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	66
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	67 -- sub proofs
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	68
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	69 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} ->
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	70 (f : A -> B) (d : Delta A) -> (headDelta (fmap f d)) ≡ fmap f (headDelta d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	71 head-delta-natural-transformation f (mono x) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	72 head-delta-natural-transformation f (delta x d) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	73
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	74 tail-delta-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	75 (f : A -> B) (d : Delta A) -> (tailDelta (fmap f d)) ≡ fmap f (tailDelta d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	76 tail-delta-natural-transfomation f (mono x) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	77 tail-delta-natural-transfomation f (delta x d) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	78
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	79 delta-append-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	80 (f : A -> B) (d : Delta A) (dd : Delta A) ->
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	81 deltaAppend (fmap f d) (fmap f dd) ≡ fmap f (deltaAppend d dd)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	82 delta-append-natural-transfomation f (mono x) dd = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	83 delta-append-natural-transfomation f (delta x d) dd = begin
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	84 deltaAppend (fmap f (delta x d)) (fmap f dd)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	85 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	86 deltaAppend (delta (f x) (fmap f d)) (fmap f dd)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	87 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	88 delta (f x) (deltaAppend (fmap f d) (fmap f dd))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	89 ≡⟨ cong (\d -> delta (f x) d) (delta-append-natural-transfomation f d dd) ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	90 delta (f x) (fmap f (deltaAppend d dd))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	91 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	92 fmap f (deltaAppend (delta x d) dd)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	93 ∎
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	94 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	95
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	96 -- Functor-law-1 : T(id) = id'
55 9c8c09334e32 Redefine Delta for infinite changes in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 54 diff changeset	97 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	98 functor-law-1 (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	99 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	100
6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	101 -- 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	102 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
55 9c8c09334e32 Redefine Delta for infinite changes in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 54 diff changeset	103 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
9c8c09334e32 Redefine Delta for infinite changes in Agda Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 54 diff changeset	104 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
57 dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	105 functor-law-2 f g (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 56 diff changeset	106 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
38 6ce83b2c9e59 Proof Functor-laws Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 36 diff changeset	107
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	108 -- Monad-laws (Category)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	109
68 f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	110 monad-law-1-2 : {l : Level} {A : Set l} -> (ds : (Delta (Delta (Delta A)))) ->
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	111 bind (fmap mu ds) tailDelta ≡ bind (bind ds tailDelta) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	112 monad-law-1-2 (mono (mono ds)) = refl
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	113 monad-law-1-2 (mono (delta (mono x) ds₁)) = refl
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	114 monad-law-1-2 (mono (delta (delta x ds) ds₁)) = refl
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	115 monad-law-1-2 (delta (mono x) (mono d)) = begin
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	116 bind (fmap mu (delta (mono x) (mono d))) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	117 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	118 bind (delta (mu (mono x)) (fmap mu (mono d))) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	119 ≡⟨ {!!} ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	120 bind (delta x (bind (mono d) tailDelta)) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	121 ≡⟨ {!!} ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	122 bind (delta x (bind (mono d) (tailDelta ∙ tailDelta))) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	123 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	124 bind (deltaAppend (mono x) (bind (mono d) (tailDelta ∙ tailDelta))) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	125 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	126 bind (bind (delta (mono x) (mono d)) tailDelta) tailDelta
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	127 ∎
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	128 monad-law-1-2 (delta (mono x) (delta d d₁)) = {!!}
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	129 monad-law-1-2 (delta (delta x x₁) (mono d)) = {!!}
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	130 monad-law-1-2 (delta (delta x x₁) (delta d d₁)) = {!!}
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	131
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	132
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	133
66 472b4cbb3dcf Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 65 diff changeset	134 monad-law-1-1 : {l : Level} {A : Set l} -> (x : Delta A) -> (d : Delta (Delta (Delta A))) ->
472b4cbb3dcf Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 65 diff changeset	135 mu (delta x (fmap mu d)) ≡ mu (delta x (bind d tailDelta))
68 f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	136 monad-law-1-1 (mono x) ds = begin
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	137 mu (delta (mono x) (fmap mu ds))
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	138 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	139 deltaAppend (headDelta (mono x)) (bind (fmap mu ds) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	140 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	141 delta x (bind (fmap mu ds) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	142 ≡⟨ cong (\d -> delta x d) (monad-law-1-2 ds) ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	143 delta x (bind (bind ds tailDelta) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	144 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	145 deltaAppend (headDelta (mono x)) (bind (bind ds tailDelta) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	146 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	147 mu (delta (mono x) (bind ds tailDelta))
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	148 ∎
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	149 monad-law-1-1 (delta x d) ds = begin
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	150 mu (delta (delta x d) (fmap mu ds))
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	151 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	152 deltaAppend (mono x) (bind (fmap mu ds) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	153 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	154 delta x (bind (fmap mu ds) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	155 ≡⟨ cong (\d -> delta x d) (monad-law-1-2 ds) ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	156 delta x (bind (bind ds tailDelta) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	157 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	158 deltaAppend (mono x) (bind (bind ds tailDelta) tailDelta)
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	159 ≡⟨ refl ⟩
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	160 mu (delta (delta x d) (bind ds tailDelta))
f9c9207c40b7 Trying prove monad-law-1 by another pattern .... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 67 diff changeset	161 ∎
67 e70be6a2bf72 Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 66 diff changeset	162
e70be6a2bf72 Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 66 diff changeset	163
e70be6a2bf72 Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 66 diff changeset	164
e70be6a2bf72 Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 66 diff changeset	165
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	166
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	167 -- monad-law-1 : join . fmap join = join . join
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	168 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	169 monad-law-1 (mono d) = refl
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	170 monad-law-1 (delta (mono x) d) = begin
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	171 (mu ∙ fmap mu) (delta (mono x) d)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	172 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	173 mu (fmap mu (delta (mono x) d))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	174 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	175 mu (delta (mu (mono x)) (fmap mu d))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	176 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	177 mu (delta x (fmap mu d))
66 472b4cbb3dcf Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 65 diff changeset	178 ≡⟨ monad-law-1-1 x d ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	179 mu (delta x (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	180 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	181 mu (deltaAppend (headDelta (mono x)) (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	182 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	183 mu (mu (delta (mono x) d))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	184 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	185 (mu ∙ mu) (delta (mono x) d)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	186 ∎
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	187 monad-law-1 (delta (delta (mono x) xs) d) = begin
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	188 (mu ∙ fmap mu) (delta (delta (mono x) xs) d)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	189 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	190 mu (fmap mu (delta (delta (mono x) xs) d))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	191 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	192 mu (delta (mu (delta (mono x) xs)) (fmap mu d))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	193 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	194 mu (delta (deltaAppend (headDelta (mono x)) (bind xs tailDelta)) (fmap mu d))
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	195 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	196 mu (delta (delta x (bind xs tailDelta)) (fmap mu d))
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	197 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	198 deltaAppend (headDelta (delta x (bind xs tailDelta))) (bind (fmap mu d) tailDelta)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	199 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	200 delta x (bind (fmap mu d) tailDelta)
66 472b4cbb3dcf Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 65 diff changeset	201 ≡⟨ monad-law-1-1 (mono x) d ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	202 mu (delta (mono x) (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	203 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	204 mu (deltaAppend (mono (mono x)) (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	205 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	206 mu (deltaAppend (headDelta (delta (mono x) xs)) (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	207 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	208 mu (mu (delta (delta (mono x) xs) d))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	209 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	210 (mu ∙ mu) (delta (delta (mono x) xs) d)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	211 ∎
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	212 monad-law-1 (delta (delta (delta x d) xs) ds) = begin
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	213 (mu ∙ fmap mu) (delta (delta (delta x d) xs) ds)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	214 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	215 mu (fmap mu (delta (delta (delta x d) xs) ds))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	216 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	217 mu (delta (mu (delta (delta x d) xs)) (fmap mu ds))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	218 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	219 mu (delta (deltaAppend (headDelta (delta x d)) (bind xs tailDelta)) (fmap mu ds))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	220 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	221 mu (delta (delta x (bind xs tailDelta)) (fmap mu ds))
66 472b4cbb3dcf Trying prove monad-law-1 by another pattern ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 65 diff changeset	222 ≡⟨ monad-law-1-1 (delta x d) ds ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	223 mu (delta (delta x d) (bind ds tailDelta))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	224 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	225 mu (deltaAppend (headDelta (delta (delta x d) xs)) (bind ds tailDelta))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	226 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	227 (mu ∙ mu) (delta (delta (delta x d) xs) ds)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	228 ∎
29 e0ba1bf564dd Apply level to some functions Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	229
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	230 {-
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	231 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	232 monad-law-1 (mono d) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	233 monad-law-1 (delta x (mono d)) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	234 (mu ∙ fmap mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	235 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	236 mu ((fmap mu) (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	237 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	238 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	239 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	240 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	241 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	242 mu (delta (mu x) (mono (mu d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	243 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	244 bind (delta (mu x) (mono (mu d))) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	245 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	246 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	247 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	248 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	249 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	250 deltaAppend (headDelta (mu x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	251 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	252 deltaAppend (headDelta (mu x)) ((tailDelta ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	253 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	254 deltaAppend (headDelta (mu x)) (bind (mono d) (tailDelta ∙ mu))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	255 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	256 bind (delta x (mono d)) mu
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	257 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	258 mu (deltaAppend (headDelta x) (tailDelta d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	259 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	260 mu (deltaAppend (headDelta x) (bind (mono d) tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	261 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	262 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	263 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	264 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	265 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	266 mu (bind (delta x (mono d)) id)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	267 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	268 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	269 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	270 mu (mu (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	271 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	272 (mu ∙ mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	273 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	274 monad-law-1 (delta x (delta xx d)) = {!!}
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	275
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	276 monad-law-1 (delta x d) = begin
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	277 (mu ∙ fmap mu) (delta x d)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	278 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	279 mu ((fmap mu) (delta x d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	280 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	281 mu (delta (mu x) (fmap mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	282 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	283 bind (delta (mu x) (fmap mu d)) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	284 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	285 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	286 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	287 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	288 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	289 (mu ∙ mu) (delta x d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	290 ∎
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	291
34 b7c4e6276bcf Proof Monad-law-2-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 33 diff changeset	292
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	293
474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	294
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	295 -- monad-law-2-2 : join . return = id
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	296 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta 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	297 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	298
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	299 -- 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	300 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	301 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	302
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	303 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	304 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) ->
36 169ec60fcd36 Proof Monad-law-4 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 35 diff changeset	305 (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	306 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	307
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	308
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	309 -- 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	310 -- 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	311 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	312 (a : A) -> (k : A -> (Delta B)) ->
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	313 (return a >>= k) ≡ (k a)
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	314 monad-law-h-1 a k = refl
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	315
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	316
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	317
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	318 -- monad-law-h-2 : m >>= return = m
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	319 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	320 monad-law-h-2 (mono x) = refl
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	321 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	322
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	323
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	324
41 23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	325
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	326 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	327 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
43 90b171e3a73e Rename to Delta from Similar Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 42 diff changeset	328 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
41 23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	329 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	330 monad-law-h-3 (mono x) k h = refl
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	331 monad-law-h-3 (delta x d) k h = begin
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	332 (delta x d) >>= (\x -> k x >>= h)
42 1df4f9d88025 Proof Monad-law-3 (haskell) Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 41 diff changeset	333 ≡⟨ refl ⟩
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	334 -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	335 deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
42 1df4f9d88025 Proof Monad-law-3 (haskell) Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 41 diff changeset	336 ≡⟨ refl ⟩
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	337 deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	338 ≡⟨ {!!} ⟩
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	339 ((delta x d) >>= k) >>= h
41 23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	340 ∎
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	341 -}

Mercurial > hg > Members > atton > delta_monad

annotate agda/delta.agda @ 68:f9c9207c40b7