Mercurial > hg > Members > kono > Proof > category

comparison CCChom.agda @ 789:4e1e2f7199c8

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CCC Hom done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Apr 2019 19:20:04 +0900
parents a3e124e36acf
children 1e7319868d77
comparison
equal deleted inserted replaced
788:a3e124e36acf 789:4e1e2f7199c8
56 IsoS A (A × A) c (a * b) (c , c ) (a , b ) 56 IsoS A (A × A) c (a * b) (c , c ) (a , b )
57 ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c 57 ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c
58 IsoS A A a (c ^ b) (a * b) c 58 IsoS A A a (c ^ b) (a * b) c
59 nat-2 : {a b c : Obj A} → {f : Hom A (b * c) (b * c) } → {g : Hom A a (b * c) } 59 nat-2 : {a b c : Obj A} → {f : Hom A (b * c) (b * c) } → {g : Hom A a (b * c) }
60 → (A × A) [ (A × A) [ IsoS.≅→ ccc-2 f o (g , g) ] ≈ IsoS.≅→ ccc-2 ( A [ f o g ] ) ] 60 → (A × A) [ (A × A) [ IsoS.≅→ ccc-2 f o (g , g) ] ≈ IsoS.≅→ ccc-2 ( A [ f o g ] ) ]
61 nat-3 : {a b c : Obj A} → { k : Hom A c (a ^ b ) } → A [ A [ IsoS.≅→ (ccc-3) (id1 A (a ^ b)) o
62 (IsoS.≅← (ccc-2 ) (A [ k o (proj₁ ( IsoS.≅→ ccc-2 (id1 A (c * b)))) ] ,
63 (proj₂ ( IsoS.≅→ ccc-2 (id1 A (c * b) ))))) ] ≈ IsoS.≅→ (ccc-3 ) k ]
61 64
62 open import CCC 65 open import CCC
63 66
64 67
65 record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where 68 record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
78 ; _^_ = CCC._<=_ c 81 ; _^_ = CCC._<=_ c
79 ; isCCChom = record { 82 ; isCCChom = record {
80 ccc-1 = λ {a} {b} {c'} → record { ≅→ = c101 ; ≅← = c102 ; iso→ = c103 {a} {b} {c'} ; iso← = c104 ; cong← = c105 ; cong→ = c106 } 83 ccc-1 = λ {a} {b} {c'} → record { ≅→ = c101 ; ≅← = c102 ; iso→ = c103 {a} {b} {c'} ; iso← = c104 ; cong← = c105 ; cong→ = c106 }
81 ; ccc-2 = record { ≅→ = c201 ; ≅← = c202 ; iso→ = c203 ; iso← = c204 ; cong← = c205; cong→ = c206 } 84 ; ccc-2 = record { ≅→ = c201 ; ≅← = c202 ; iso→ = c203 ; iso← = c204 ; cong← = c205; cong→ = c206 }
82 ; ccc-3 = record { ≅→ = c301 ; ≅← = c302 ; iso→ = c303 ; iso← = c304 ; cong← = c305 ; cong→ = c306 } 85 ; ccc-3 = record { ≅→ = c301 ; ≅← = c302 ; iso→ = c303 ; iso← = c304 ; cong← = c305 ; cong→ = c306 }
83 ; nat-2 = nat-2 86 ; nat-2 = nat-2 ; nat-3 = nat-3
84 } 87 }
85 } where 88 } where
86 c101 : {a : Obj A} → Hom A a (CCC.１ c) → Hom OneCat OneObj OneObj 89 c101 : {a : Obj A} → Hom A a (CCC.１ c) → Hom OneCat OneObj OneObj
87 c101 _ = OneObj 90 c101 _ = OneObj
88 c102 : {a : Obj A} → Hom OneCat OneObj OneObj → Hom A a (CCC.１ c) 91 c102 : {a : Obj A} → Hom OneCat OneObj OneObj → Hom A a (CCC.１ c)
136 nat-2 {a} {b} {c₁} {f} {g} = ( begin 139 nat-2 {a} {b} {c₁} {f} {g} = ( begin
137 ( CCC.π c o f) o g 140 ( CCC.π c o f) o g
138 ≈↑⟨ assoc ⟩ 141 ≈↑⟨ assoc ⟩
139 ( CCC.π c ) o (f o g) 142 ( CCC.π c ) o (f o g)
140 ∎ ) , (sym-hom assoc) where open ≈-Reasoning A 143 ∎ ) , (sym-hom assoc) where open ≈-Reasoning A
144 nat-3 : {a b : Obj A} {c = c₁ : Obj A} {k : Hom A c₁ ((c CCC.<= a) b)} →
145 A [ A [ c301 (id1 A ((c CCC.<= a) b)) o c202 (A [ k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) ] , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b)))) ]
146 ≈ c301 k ]
147 nat-3 {a} {b} { c₁} {k} = begin
148 c301 (id1 A ((c CCC.<= a) b)) o c202 ( k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b))))
149 ≈⟨⟩
150 ( CCC.ε c o CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c))
151 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))
152 ≈↑⟨ assoc ⟩
153 (CCC.ε c) o (( CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c))
154 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b)))))
155 ≈⟨ cdr (car (IsCCC.π-cong (CCC.isCCC c ) idL refl-hom ) ) ⟩
156 (CCC.ε c) o ( CCC.<_,_> c (CCC.π c) (CCC.π' c)
157 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b)))))
158 ≈⟨ cdr (car (IsCCC.π-id (CCC.isCCC c))) ⟩
159 (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b )
160 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b)))))
161 ≈⟨ cdr ( cdr ( IsCCC.π-cong (CCC.isCCC c) (cdr idR) idR )) ⟩
162 (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b ) o (CCC.<_,_> c (k o ( CCC.π c )) ( CCC.π' c )))
163 ≈⟨ cdr idL ⟩
164 (CCC.ε c) o (CCC.<_,_> c ( k o (CCC.π c) ) (CCC.π' c))
165 ≈⟨⟩
166 c301 k
167 ∎ where open ≈-Reasoning A
141 168
142 169
143 170
144 open CCChom 171 open CCChom
145 open IsCCChom 172 open IsCCChom
262 ≈⟨ cong← (ccc-2 (isCCChom h)) ( eq1 , eq2 ) ⟩ 289 ≈⟨ cong← (ccc-2 (isCCChom h)) ( eq1 , eq2 ) ⟩
263 ≅← (ccc-2 (isCCChom h)) (f' , g') 290 ≅← (ccc-2 (isCCChom h)) (f' , g')
264 ≈⟨⟩ 291 ≈⟨⟩
265 <,> f' g' 292 <,> f' g'
266 ∎ where open ≈-Reasoning A 293 ∎ where open ≈-Reasoning A
267 e40 : {a c : Obj A} → { f : Hom A (_*_ h a c ) a } → A [ ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) f) ≈ f ]
268 e40 = iso→ (ccc-3 (isCCChom h))
269 e41 : {a c : Obj A} → { f : Hom A a (_^_ h c a )} → A [ ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) f) ≈ f ]
270 e41 = iso← (ccc-3 (isCCChom h))
271 e4a : {a b c : Obj A} → { k : Hom A (c /\ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ] 294 e4a : {a b c : Obj A} → { k : Hom A (c /\ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ]
272 e4a {a} {b} {c} {k} = begin 295 e4a {a} {b} {c} {k} = begin
273 ε o ( <,> ((k *) o π ) π' ) 296 ε o ( <,> ((k *) o π ) π' )
274 ≈⟨⟩ 297 ≈⟨⟩
275 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h)) ((( ≅← (ccc-3 (isCCChom h)) k) o π ) , π')) 298 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h)) ((( ≅← (ccc-3 (isCCChom h)) k) o π ) , π'))
276 ≈⟨ {!!} ⟩ 299 ≈⟨ nat-3 (isCCChom h) ⟩
277 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h))
278 (_[_o_] (A × A) ( ≅← (ccc-3 (isCCChom h)) k , id1 A b ) ( π , π')))
279 ≈⟨ {!!} ⟩
280 ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) k) 300 ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) k)
281 ≈⟨ iso→ (ccc-3 (isCCChom h)) ⟩ 301 ≈⟨ iso→ (ccc-3 (isCCChom h)) ⟩
282 k 302 k
283 ∎ where open ≈-Reasoning A 303 ∎ where open ≈-Reasoning A
284 e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ] 304 e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ]
285 e4b {a} {b} {c} {k} = begin 305 e4b {a} {b} {c} {k} = begin
286 ( ε o ( <,> ( k o π ) π' ) ) * 306 ( ε o ( <,> ( k o π ) π' ) ) *
287 ≈⟨⟩ 307 ≈⟨⟩
288 ≅← (ccc-3 (isCCChom h)) ( ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) o (≅← (ccc-2 (isCCChom h)) ( k o π , π'))) 308 ≅← (ccc-3 (isCCChom h)) ( ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) o (≅← (ccc-2 (isCCChom h)) ( k o π , π')))
289 ≈⟨ {!!} ⟩ 309 ≈⟨ cong← (ccc-3 (isCCChom h)) (nat-3 (isCCChom h)) ⟩
290 ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) k) 310 ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) k)
291 ≈⟨ iso← (ccc-3 (isCCChom h)) ⟩ 311 ≈⟨ iso← (ccc-3 (isCCChom h)) ⟩
292 k 312 k
293 ∎ where open ≈-Reasoning A 313 ∎ where open ≈-Reasoning A
294 *-cong : {a b c : Obj A} {f f' : Hom A (a /\ b) c} → A [ f ≈ f' ] → A [ f * ≈ f' * ] 314 *-cong : {a b c : Obj A} {f f' : Hom A (a /\ b) c} → A [ f ≈ f' ] → A [ f * ≈ f' * ]