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annotate code-data.agda @ 441:61550782be4a

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preinital full subcategory done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 30 Aug 2016 15:11:17 +0900
parents 71c817f28bc6
children 3d41a8edbf63
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Category -- https://github.com/konn/category-agda
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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3 --open import Category.HomReasoning
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 open import HomReasoning
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 open import cat-utility
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 open import Category.Cat
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 module code-data { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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9
356
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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10 -- DataObj is a set of code segment with reverse computation
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 record DataObj : Set (c₁ ⊔ c₂ ⊔ ℓ) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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13 dom : Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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14 codom : Obj A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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15 code : Hom A dom codom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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16 rev-code : Hom A codom dom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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17 id-left : A [ A [ code o rev-code ] ≈ id1 A codom ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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18 id-right : A [ A [ rev-code o code ] ≈ id1 A dom ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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20 open DataObj
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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21
356
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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22 -- DataHom is a set of data segment with computational continuation
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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23 record isDataHom (a : DataObj ) (b : DataObj ) : Set (c₁ ⊔ c₂ ⊔ ℓ) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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24 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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25 continuation : Hom A (codom a) (dom b)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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26 data-dom = a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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27 data-codom = b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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28
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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29 open isDataHom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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30
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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31 DataHom : (a : DataObj ) (b : DataObj ) → Set (c₁ ⊔ c₂ ⊔ ℓ)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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32 DataHom = λ a b → isDataHom a b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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33
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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34 DataId : { a : DataObj } → DataHom a a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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35 DataId {a} = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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36 continuation = rev-code a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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38
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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39 _∙_ : {a b c : DataObj } → DataHom b c → DataHom a b → DataHom a c
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40 _∙_ {a} {b} {c} g f = record { continuation = A [ continuation g o A [ code b o continuation f ] ] }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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41
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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42 _≗_ : {a : DataObj } {b : DataObj } (f g : DataHom a b ) → Set ℓ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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43 _≗_ {a} {b} f g = A [ continuation f ≈ continuation g ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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44
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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45 open import Relation.Binary.Core
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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46
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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47 isDataCategory : IsCategory DataObj DataHom _≗_ _∙_ DataId
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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48 isDataCategory = record { isEquivalence = isEquivalence
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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49 ; identityL = \{a} {b} {f} -> identityL a b f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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50 ; identityR = \{a} {b} {f} -> identityR a b f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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51 ; o-resp-≈ = \{a} {b} {c} {f} {g} {h} {i} -> o-resp {a} {b} {c} {f} {g} {h} {i}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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52 ; associative = \{a} {b} {c} {d} {f} {g} {h} -> associative a b c d f g h
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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53 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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54 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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55 open ≈-Reasoning (A)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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56 o-resp : {A B C : DataObj} {f g : DataHom A B} {h i : DataHom B C} → f ≗ g → h ≗ i → (h ∙ f) ≗ (i ∙ g)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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57 o-resp {a} {b} {c} {f} {g} {h} {i} f≗g h≗i = begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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58 continuation (h ∙ f)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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60 continuation h o code b o continuation f
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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61 ≈⟨ cdr ( cdr ( f≗g )) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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62 continuation h o code b o continuation g
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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63 ≈⟨ car h≗i ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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64 continuation i o code b o continuation g
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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65 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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66 continuation (i ∙ g)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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67 ∎
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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68 associative : (a b c d : DataObj) (f : DataHom c d) (g : DataHom b c) (h : DataHom a b) → (f ∙ (g ∙ h)) ≗ ((f ∙ g) ∙ h)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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69 associative a b c d f g h = begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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70 continuation (f ∙ (g ∙ h))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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71 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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72 continuation f o code c o continuation g o code b o continuation h
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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73 ≈⟨ cdr assoc ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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74 continuation f o (code c o continuation g) o code b o continuation h
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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75 ≈⟨ assoc ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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76 (continuation f o code c o continuation g) o code b o continuation h
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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77 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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78 continuation ((f ∙ g) ∙ h)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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79 ∎
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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80 identityL : (a : DataObj) (b : DataObj) (f : DataHom a b) → (DataId ∙ f) ≗ f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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81 identityL a b f = begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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82 continuation (DataId ∙ f)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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83 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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84 rev-code b o code b o continuation f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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85 ≈⟨ assoc ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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86 (rev-code b o code b ) o continuation f
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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87 ≈⟨ car ( id-right b) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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88 id1 A (dom b) o continuation f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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89 ≈⟨ idL ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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90 continuation f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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91 ∎
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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92 identityR : (a : DataObj) (b : DataObj) (f : DataHom a b) → (f ∙ DataId ) ≗ f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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93 identityR a b f = begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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94 continuation (f ∙ DataId)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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95 ≈⟨⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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96 ( continuation f o ( code a o rev-code a ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
97 ≈⟨ cdr (id-left a) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
98 ( continuation f o id1 A (codom a) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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99 ≈⟨ idR ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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100 continuation f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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101 ∎
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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102 isEquivalence : {a : DataObj } {b : DataObj } →
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 IsEquivalence {_} {_} {DataHom a b } _≗_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104 isEquivalence {C} {D} = -- this is the same function as A's equivalence but has different types
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105 record { refl = refl-hom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
106 ; sym = sym
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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107 ; trans = trans-hom
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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108 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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109 DataCategory : Category (c₁ ⊔ c₂ ⊔ ℓ) (c₁ ⊔ c₂ ⊔ ℓ) ℓ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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110 DataCategory =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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111 record { Obj = DataObj
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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112 ; Hom = DataHom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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113 ; _o_ = _∙_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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114 ; _≈_ = _≗_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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115 ; Id = DataId
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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116 ; isCategory = isDataCategory
355
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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117 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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118
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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119
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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120
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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121 open Functor
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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122 open NTrans
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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123
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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124 F : Obj A -> Obj DataCategory
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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125 F d = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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126 dom = d
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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127 ; codom = d
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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128 ; code = id1 A d
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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129 ; rev-code = id1 A d
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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130 ; id-left = idL
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
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131 ; id-right = idR
355
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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132 } where open ≈-Reasoning (A)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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133
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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134 U : Functor DataCategory A
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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135 U = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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136 FObj = \d -> codom d
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
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137 ; FMap = \f -> A [ code ( data-codom f ) o continuation f ]
355
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
138 ; isFunctor = record {
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
139 ≈-cong = \{a} {b} {f} {g} -> ≈-cong-1 {a} {b} {f} {g}
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
140 ; identity = \{a} -> identity-1 {a}
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
141 ; distr = \{a b c f g} -> distr-1 {a} {b} {c} {f} {g}
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
142 }
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
143 } where
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
144 open ≈-Reasoning (A)
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
145 ≈-cong-1 : {a : Obj DataCategory} {b : Obj DataCategory} {f g : Hom DataCategory a b} → DataCategory [ f ≈ g ] →
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
146 A [ A [ code (data-codom f) o continuation f ] ≈ A [ code (data-codom g) o continuation g ] ]
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
147 ≈-cong-1 {a} {b} {f} {g} f≈g = begin
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
148 code (data-codom f) o continuation f
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
149 ≈⟨⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
150 code b o continuation f
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
151 ≈⟨ cdr f≈g ⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
152 code b o continuation g
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
153 ≈⟨⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
154 code (data-codom g) o continuation g
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155 ∎
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
156 identity-1 : {a : Obj DataCategory} → A [ A [ code (data-codom (DataId {a})) o continuation (DataId {a}) ] ≈ id1 A (codom a) ]
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
157 identity-1 {a} = begin
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
158 code (data-codom (DataId {a} )) o continuation (DataId {a} )
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
159 ≈⟨⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
160 code a o rev-code a
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
161 ≈⟨ id-left a ⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
162 id1 A (codom a)
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
163 ∎
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
164 distr-1 : {a b c : Obj DataCategory} {f : Hom DataCategory a b} {g : Hom DataCategory b c} →
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
165 A [ A [ code (data-codom ( g ∙ f )) o continuation ( g ∙ f ) ] ≈
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
166 A [ A [ code (data-codom g) o continuation g ] o A [ code (data-codom f) o continuation f ] ] ]
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
167 distr-1 {a} {b} {c} {f} {g} = begin
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
168 code (data-codom (g ∙ f )) o continuation ( g ∙ f )
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
169 ≈⟨⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
170 code c o continuation g o code b o continuation f
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
171 ≈⟨ assoc ⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
172 (code c o continuation g ) o code b o continuation f
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
173 ≈⟨⟩
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
174 ( code (data-codom g) o continuation g ) o ( code (data-codom f) o continuation f )
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
175 ∎
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
176
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
177 eta-map : (a : Obj A) → Hom A a ( FObj U (F a) )
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
178 eta-map a = id1 A a
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
179
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
180
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
181 Lemma1 : UniversalMapping A DataCategory U F eta-map
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
182 Lemma1 = record {
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
183 _* = solution ;
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
184 isUniversalMapping = record {
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
185 universalMapping = \{a} {b} {f} -> universalMapping {a} {b} {f} ;
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
186 uniquness = \{a} {b} {f} {g} -> uniqueness {a} {b} {f} {g}
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
187 }
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
188 } where
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
189 open ≈-Reasoning (A)
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
190 solution : {a : Obj A} {b : Obj DataCategory} → Hom A a (FObj U b) → Hom DataCategory (F a) b
356
500c813af3c9 history to continuation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 355
diff changeset
191 solution {a} {b} f = record { continuation = A [ rev-code b o f ] }
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
192 universalMapping : {a : Obj A} {b : Obj DataCategory} {f : Hom A a (FObj U b)} → A [ A [ FMap U (solution {a} {b} f) o eta-map a ] ≈ f ]
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
193 universalMapping {a} {b} {f} = begin
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
194 FMap U (solution {a} {b} f) o eta-map a
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
195 ≈⟨⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
196 (code b o ( rev-code b o f)) o id1 A a
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
197 ≈⟨ idR ⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
198 code b o ( rev-code b o f)
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
199 ≈⟨ assoc ⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
200 (code b o rev-code b ) o f
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
201 ≈⟨ car (id-left b) ⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
202 id1 A (codom b) o f
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
203 ≈⟨ idL ⟩
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
204 f
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
205 ∎
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
206 uniqueness : {a : Obj A} {b : Obj DataCategory} {f : Hom A a (FObj U b)}
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
207 {g : Hom DataCategory (F a) b} →
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
208 A [ A [ FMap U g o eta-map a ] ≈ f ] →
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
209 DataCategory [ solution f ≈ g ]
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
210 uniqueness {a} {b} {f} {g} Uη≈f = begin
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
211 continuation (solution {a} {b} f)
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
212 ≈⟨⟩
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
213 rev-code b o f
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
214 ≈⟨ sym ( cdr Uη≈f ) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
215 rev-code b o ( code b o continuation g ) o id1 A (codom (F a))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
216 ≈⟨ sym ( cdr assoc) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
217 rev-code b o code b o continuation g o id1 A (codom (F a))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
218 ≈⟨ assoc ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
219 (rev-code b o code b ) o continuation g o id1 A (codom (F a))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
220 ≈⟨ car ( id-right b ) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
221 id (dom b) o continuation g o id1 A (codom (F a))
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
222 ≈⟨ idL ⟩
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
223 ( continuation g ) o id1 A (codom (F a))
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
224 ≈⟨ idR ⟩
357
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 356
diff changeset
225 continuation g
355
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
226 ∎
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
227
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
228
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
229
8fd085379380 add DataCategory Universal mapping
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
230