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comparison equalizer.agda @ 235:8835015a3e1a

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passed let;s remove yellow
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 08 Sep 2013 04:55:01 +0900
parents c02287d3d2dc
children e20b81102eee
comparison
equal deleted inserted replaced
234:c02287d3d2dc 235:8835015a3e1a
180 ∎ 180 ∎
181 181
182 c-iso← : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) 182 c-iso← : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g )
183 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) 183 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) )
184 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) 184 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) )
185 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } 185 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- refl
186 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] 186 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ]
187 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin 187 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin
188 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa 188 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa
189 ≈⟨⟩ 189 ≈⟨⟩
190 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) 190 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa )
280 ------ fα = gα 280 ------ fα = gα
281 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j 281 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j
282 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j 282 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j
283 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j 283 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j
284 284
285 eefg : {a b c : Obj A} (f g : Hom A a b) {e : Hom A c a} → Equalizer A e ( A [ f o equalizer (eqa f g) ] ) (A [ g o equalizer (eqa f g) ] )
286 eefg f g {e} = eqa ( A [ f o equalizer (eqa f g) ] ) (A [ g o equalizer (eqa f g) ] )
285 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ 287 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [
286 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o equalizer (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ g o A [ equalizer (eqa f g) o j ] ])) ]) 288 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o equalizer (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ g o A [ equalizer (eqa f g) o j ] ])) ])
287 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o 289 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o
288 k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] 290 k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ equalizer (eqa f g) o j ] ])) ]
289 ≈ j ] 291 ≈ j ]
290 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in 292 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in
291 begin 293 begin
292 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g) o j ) )) (( g o ( equalizer (eqa f g) o j ) ))) )) 294 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g) o j ) )) (( g o ( equalizer (eqa f g) o j ) ))) ))
293 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o 295 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o
294 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) ))) ) 296 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) ))) )
295 ≈⟨ {!!} ⟩ 297 ≈⟨ car ((uniqueness (eqa f g) ( begin
298 equalizer (eqa f g) o j
299 ≈↑⟨ idR ⟩
300 (equalizer (eqa f g) o j ) o id1 A d
301 ≈⟨⟩
302 ((equalizer (eqa f g) o j) o equalizer (eqa (f o equalizer (eqa f g) o j) (g o equalizer (eqa f g) o j)))
303 ∎ ))) ⟩
304 j o k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) )))
305 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin
306 equalizer (eqa (f o equalizer (eqa f g) o j) (f o equalizer (eqa f g) o j))
307 ≈⟨ {!!} ⟩
308 id1 A d
309 ∎ ))) ⟩
310 j o id1 A d
311 ≈⟨ idR ⟩
296 j 312 j
297 ∎ 313 ∎
298 314
299 315
300 -- end 316 -- end
301 317
302 318