Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 215:637b5f58ed28
equ6...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 03 Sep 2013 04:29:07 +0900 |
| parents | f8afdb9ed99a |
| children | 0135419f375c |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 205 | 6 -- c --------> a ----------> b |
| 208 | 7 -- ^ . ----------> |
| 205 | 8 -- | . g |
| 208 | 9 -- |k . |
| 10 -- | . h | |
| 205 | 11 -- d |
| 12 -- | |
| 13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 14 ---- | |
| 15 | |
| 16 open import Category -- https://github.com/konn/category-agda | |
| 17 open import Level | |
| 18 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where | |
| 19 | |
| 20 open import HomReasoning | |
| 21 open import cat-utility | |
| 22 | |
| 209 | 23 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 |
| 205 | 24 field |
| 209 | 25 e : Hom A c a |
| 215 | 26 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 28 ek=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 ] |
| 29 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → | |
| 214 | 30 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 31 equalizer : Hom A c a |
| 32 equalizer = e | |
| 206 | 33 |
| 209 | 34 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 |
| 206 | 35 field |
| 212 | 36 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
| 214 | 37 γ : {a b c d : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 212 | 38 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
| 213 | 39 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
| 214 | 40 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
| 213 | 41 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
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Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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42 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 215 | 43 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] |
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44 -- A [ α f g o β f g h ] ≈ h |
| 214 | 45 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 c |
| 46 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] | |
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47 |
| 209 | 48 open Equalizer |
| 49 open EqEqualizer | |
| 50 | |
| 215 | 51 ff-equal : {a b : Obj A} (f : Hom A a b) → (eqa : Equalizer A f f ) → A [ e eqa ≈ id1 A a ] |
| 52 ff-equal {a} {b} f eqa = let open ≈-Reasoning (A) in | |
| 53 begin | |
| 54 e eqa | |
| 55 ≈↑⟨ ek=h eqa ⟩ | |
| 56 e eqa o k eqa (e eqa) refl-hom | |
| 57 ≈⟨ {!!} ⟩ | |
| 58 id1 A a | |
| 59 ∎ | |
| 60 | |
| 61 | |
| 211 | 62 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 63 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 64 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 211 | 65 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 66 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (e ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
| 213 | 67 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 211 | 68 b1 = ef=eg (eqa f g) ; |
| 212 | 69 b2 = lemma-equ5 ; |
| 70 b3 = lemma-equ3 ; | |
| 215 | 71 b4 = lemma-equ6 |
| 211 | 72 } where |
| 73 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] | |
| 74 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 75 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 76 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 77 begin |
| 78 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 79 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 80 id1 A a |
| 81 ∎ | |
| 214 | 82 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 83 A [ A [ f o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 84 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 85 begin |
| 86 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 87 ≈⟨ assoc ⟩ | |
| 88 (f o h) o e (eqa (f o h) ( g o h )) | |
| 89 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 90 (g o h) o e (eqa (f o h) ( g o h )) | |
| 91 ≈↑⟨ assoc ⟩ | |
| 92 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 93 ∎ | |
| 94 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 95 A [ e (eqa f g) o k (eqa f g) (A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
| 212 | 96 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 97 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 98 begin | |
| 215 | 99 e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 100 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 101 h o e (eqa (f o h ) ( g o h )) |
| 102 ∎ | |
| 215 | 103 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
| 104 A [ k (eqa f g) (A [ A [ e (eqa f g) o k₁ ] o e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ g o A [ e (eqa f g) o k₁ ] ])) ]) | |
| 105 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
| 106 k (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ f o A [ e (eqa f g) o k₁ ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o k₁ ] ])) ] | |
| 107 ≈ k₁ ] | |
| 108 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
| 109 begin | |
| 110 ( k (eqa f g) (( ( e (eqa f g) o k₁ ) o e (eqa (( f o ( e (eqa f g) o k₁ ) )) (( g o ( e (eqa f g) o k₁ ) ))) )) | |
| 111 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
| 112 k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ) | |
| 113 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
| 114 e (eqa f g) o k₁ | |
| 115 ≈⟨ {!!} ⟩ | |
| 116 (e (eqa f g) o k₁) o e (eqa (f o e (eqa f g) o k₁) (g o e (eqa f g) o k₁)) | |
| 117 ∎ )) ⟩ | |
| 118 k₁ o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) | |
| 119 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
| 120 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
| 121 ≈⟨ {!!} ⟩ | |
| 122 id1 A d | |
| 123 ∎ )) ⟩ | |
| 124 k₁ o id1 A d | |
| 125 ≈⟨ idR ⟩ | |
| 126 k₁ | |
| 127 ∎ | |
| 211 | 128 |
| 129 | |
| 212 | 130 |
| 131 | |
| 132 | |
| 215 | 133 |
| 134 |