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annotate equalizer.agda @ 209:4e138cc953f3

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equalizer difinition
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 02 Sep 2013 21:59:37 +0900
parents a1e5d2a3d3bd
children 51c57efe89b9
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242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 ---
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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2 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 -- Equalizer
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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5 -- e f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6 -- c --------> a ----------> b
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 -- ^ . ---------->
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242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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8 -- | . g
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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9 -- |k .
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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10 -- | . h
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242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11 -- d
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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14 ----
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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15
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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16 open import Category -- https://github.com/konn/category-agda
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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17 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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18 open import Category.Sets
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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20
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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21 open import HomReasoning
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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22 open import cat-utility
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
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242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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25 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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26 e : Hom A c a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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27 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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28 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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29 ke=h : {d : Obj A} → ∀ {h : Hom A d a} → (eq : A [ A [ f o h ] ≈ A [ g o h ] ] ) → A [ A [ e o k {d} h eq ] ≈ h ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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30 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → (eq : A [ A [ f o h ] ≈ A [ g o h ] ] ) → {k' : Hom A d c } → A [ A [ e o k' ] ≈ h ] →
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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31 A [ k {d} h eq ≈ k' ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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32 equalizer : Hom A c a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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33 equalizer = e
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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34
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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35 record EqEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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36 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 α : {e a : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A e a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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38 γ : {d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A c e
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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39 δ : {a b : Obj A} → (f : Hom A a b) → Hom A a c
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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40 b1 : {e : Obj A } → A [ A [ f o α {e} {a} f g ] ≈ A [ g o α {e} {a} f g ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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41 b2 : {e d : Obj A } → {h : Hom A d a } → A [ A [ α {e} f g o γ f g h ] ≈ A [ h o α {c} (A [ f o h ]) (A [ g o h ]) ] ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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42 b3 : {e : Obj A} → A [ A [ α f f o δ f ] ≈ id1 A a ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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43 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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44 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ f g ( A [ α f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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45 -- A [ α f g o β f g h ] ≈ h
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46 β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d e
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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47 β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ]
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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48
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 open Equalizer
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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50 open EqEqualizer
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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51
4e138cc953f3 equalizer difinition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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52 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c : Obj A} (f g : Hom A a b) → Equalizer A {c} f g → EqEqualizer A {c} f g
4e138cc953f3 equalizer difinition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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53 lemma-equ1 A {a} {b} {c} f g eqa = record {
4e138cc953f3 equalizer difinition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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54 α = λ {e'} {a} f g → ? ; -- e' -> c c -> a, Hom A e' a
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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55 γ = λ {d} {e} {a} {b} f g h → {!!} ; -- Hom A c e
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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56 δ = λ {a} {b} f → {!!} ; -- Hom A a c
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57 b1 = {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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58 b2 = {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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59 b3 = {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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60 b4 = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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61 }