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annotate S.agda @ 790:1e7319868d77

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Sets is CCC
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Apr 2019 23:42:19 +0900
parents 749df4959d19
children
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608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1 --I'd like to write the Limit in Sets Category using Agda. Assuming local smallness, a functor is a pair of map on Set OC and I, like this.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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2 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 -- sobj : OC → Set c₂
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4 -- smap : { i j : OC } → (f : I ) → sobj i → sobj j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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5 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6 --A cone for the functor is a record with two fields. Using the record, commutativity of the cone and the propertiesy of the Limit
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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7 --are easity shown, except uniquness. The uniquness of the Limit turned out that congruence of the record with two fields.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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8 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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9 --In the following agda code, I'd like to prove snat-cong lemma.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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10
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11 open import Level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12 module S where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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13
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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14 open import Relation.Binary.Core
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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15 open import Function
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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16 import Relation.Binary.PropositionalEquality
663
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
17 open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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19 record snat { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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20 ( smap : { i j : OC } → (f : I ) → sobj i → sobj j ) : Set c₂ where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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21 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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22 snmap : ( i : OC ) → sobj i
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23 sncommute : ( i j : OC ) → ( f : I ) → smap f ( snmap i ) ≡ snmap j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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24 smap0 : { i j : OC } → (f : I ) → sobj i → sobj j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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25 smap0 {i} {j} f x = smap f x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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26
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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27 open snat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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28
672
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
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29 -- snat-cong' : { c : Level } { I OC : Set c } { SObj : OC → Set c } { SMap : { i j : OC } → (f : I )→ SObj i → SObj j }
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
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30 -- ( s t : snat SObj SMap )
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
diff changeset
31 -- → ( ( λ i → snmap s i ) ≡ ( λ i → snmap t i ) )
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
diff changeset
32 -- → ( ( λ i j f → smap0 s f ( snmap s i ) ≡ snmap s j ) ≡ ( ( λ i j f → smap0 t f ( snmap t i ) ≡ snmap t j ) ) )
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
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33 -- → s ≡ t
749df4959d19 fix completeness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 663
diff changeset
34 -- snat-cong' s t refl refl = {!!}
663
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
35
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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36 snat-cong : {c : Level}
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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37 {I OC : Set c}
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
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38 {sobj : OC → Set c}
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
39 {smap : {i j : OC} → (f : I) → sobj i → sobj j}
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
40 → (s t : snat sobj smap)
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
41 → (snmap-≡ : snmap s ≡ snmap t)
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
42 → (sncommute-≅ : sncommute s ≅ sncommute t)
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
43 → s ≡ t
855e497a9c8f introducd HeterogeneousEquality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 608
diff changeset
44 snat-cong _ _ refl refl = refl
608
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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45
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46 --This is quite simlar to the answer of
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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47 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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48 -- Equality on dependent record types
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 -- https://stackoverflow.com/questions/37488098/equality-on-dependent-record-types
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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50 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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51 --So it should work something like
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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52 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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53 -- snat-cong s t refl refl = refl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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54 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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55 --but it gives an error like this.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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56 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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57 -- .sncommute i j f != sncommute t i j f of type
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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58 -- .SMap f (snmap t i) ≡ snmap t j
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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60 --Is there any help?