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

annotate agda/delta.agda @ 65:6d0193011f89

Trying prove monad-law-1 by another pattern

author	Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date	Wed, 26 Nov 2014 17:11:33 +0900
parents	15eec529dfc4
children	472b4cbb3dcf

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
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	110 monad-law-1-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) ->
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	111 (fmap mu d) ≡ (bind d tailDelta)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	112 monad-law-1-1 (mono (mono d)) = refl
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	113 monad-law-1-1 (mono (delta (mono x) ds)) = begin
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	114 fmap mu (mono (delta (mono x) ds))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	115 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	116 mono (mu (delta (mono x) ds))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	117 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	118 mono (deltaAppend (headDelta (mono x)) (bind ds tailDelta))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	119 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	120 mono (delta x (bind ds tailDelta))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	121 ≡⟨ {!!} ⟩ --?
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	122 ds
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	123 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	124 tailDelta (delta (mono x) ds)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	125 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	126 bind (mono (delta (mono x) ds)) tailDelta
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	127 ∎
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	128
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	129 monad-law-1-1 (mono (delta (delta x d) ds)) = {!!}
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	130 monad-law-1-1 (delta d ds) = {!!}
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	131
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	132 -- 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	133 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	134 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	135 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	136 (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	137 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	138 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	139 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	140 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	141 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	142 mu (delta x (fmap mu d))
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	143 ≡⟨ cong (\d -> mu (delta x d)) (monad-law-1-1 d) ⟩ -- ?
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	144 mu (delta x (bind d tailDelta))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	145 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	146 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	147 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	148 mu (mu (delta (mono x) d))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	149 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	150 (mu ∙ mu) (delta (mono x) d)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	151 ∎
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	152 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	153 (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	154 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	155 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	156 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	157 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	158 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	159 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	160 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	161 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	162 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	163 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	164 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	165 delta x (bind (fmap mu d) tailDelta)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	166 ≡⟨ cong (\d -> delta x (bind d tailDelta)) (monad-law-1-1 d) ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	167 delta x (bind (bind d tailDelta) tailDelta)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	168 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	169 deltaAppend (headDelta (mono x)) (bind (bind d tailDelta) tailDelta)
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	170 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	171 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	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 (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	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 (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	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 (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	178 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	179 (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	180 ∎
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	181 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	182 (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	183 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	184 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	185 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	186 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	187 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	188 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	189 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	190 mu (delta (delta x (bind xs tailDelta)) (fmap mu ds))
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	191 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	192 deltaAppend (headDelta (delta x (bind xs tailDelta))) (bind (fmap mu ds) tailDelta)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	193 ≡⟨ refl ⟩
65 6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	194 delta x (bind (fmap mu ds) tailDelta)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	195 ≡⟨ cong (\d -> delta x (bind d tailDelta)) (monad-law-1-1 ds) ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	196 delta x (bind (bind ds tailDelta) tailDelta)
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	197 ≡⟨ refl ⟩
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 d)) (bind (bind ds tailDelta) tailDelta)
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 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 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	201 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	202 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	203 ≡⟨ refl ⟩
6d0193011f89 Trying prove monad-law-1 by another pattern Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 64 diff changeset	204 (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	205 ∎
29 e0ba1bf564dd Apply level to some functions Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 28 diff changeset	206
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	207 {-
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	208 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	209 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	210 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	211 (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	212 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	213 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	214 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	215 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	216 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	217 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	218 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	219 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	220 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	221 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	222 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	223 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	224 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	225 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	226 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	227 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	228 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	229 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	230 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	231 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	232 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	233 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	234 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	235 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	236 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	237 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	238 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	239 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	240 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	241 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	242 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	243 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	244 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	245 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	246 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	247 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	248 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	249 (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	250 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	251 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	252
62 0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	253 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	254 (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	255 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	256 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	257 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	258 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	259 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	260 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	261 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	262 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	263 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	264 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	265 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 60 diff changeset	266 (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	267 ∎
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	268
34 b7c4e6276bcf Proof Monad-law-2-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 33 diff changeset	269
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	270
474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	271
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	272 -- 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	273 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	274 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	275
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	276 -- 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	277 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	278 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	279
39 b9b26b470cc2 Add Comments Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 38 diff changeset	280 -- 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	281 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	282 (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	283 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	284
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	285
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	286 -- 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	287 -- 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	288 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	289 (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	290 (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	291 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	292
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	293
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	294
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 39 diff changeset	295 -- 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	296 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	297 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	298 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	299
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	300
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	301
41 23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	302
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3 Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 40 diff changeset	303 -- 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	304 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	305 (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	306 (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	307 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	308 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	309 (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	310 ≡⟨ refl ⟩
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	311 -- (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	312 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	313 ≡⟨ refl ⟩
59 46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	314 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	315 ≡⟨ {!!} ⟩
46b15f368905 Define bind and mu for Infinite Delta Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 57 diff changeset	316 ((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	317 ∎
63 474ed34e4f02 proving monad-law-1 ... Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> parents: 62 diff changeset	318 -}

Mercurial > hg > Members > atton > delta_monad

annotate agda/delta.agda @ 65:6d0193011f89