Mercurial > hg > Members > kono > Proof > category
changeset 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 |
files | equalizer.agda |
diffstat | 1 files changed, 46 insertions(+), 8 deletions(-) [+] |
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--- a/equalizer.agda Tue Sep 03 02:38:23 2013 +0900 +++ b/equalizer.agda Tue Sep 03 04:29:07 2013 +0900 @@ -23,10 +23,10 @@ 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 field e : Hom A c a - ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] + ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c - 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 ] - uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → (eq : A [ A [ f o h ] ≈ A [ g o h ] ] ) → {k' : Hom A d c } → + 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 ] + 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 ] → A [ k {d} h eq ≈ k' ] equalizer : Hom A c a equalizer = e @@ -40,7 +40,7 @@ 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 ]) ] ] b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] - 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 ] -- k + 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 ] -- A [ α f g o β f g h ] ≈ h 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 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] @@ -48,6 +48,17 @@ open Equalizer open EqEqualizer +ff-equal : {a b : Obj A} (f : Hom A a b) → (eqa : Equalizer A f f ) → A [ e eqa ≈ id1 A a ] +ff-equal {a} {b} f eqa = let open ≈-Reasoning (A) in + begin + e eqa + ≈↑⟨ ek=h eqa ⟩ + e eqa o k eqa (e eqa) refl-hom + ≈⟨ {!!} ⟩ + id1 A a + ∎ + + lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g lemma-equ1 A {a} {b} {c} f g eqa = record { @@ -57,7 +68,7 @@ b1 = ef=eg (eqa f g) ; b2 = lemma-equ5 ; b3 = lemma-equ3 ; - b4 = {!!} + b4 = lemma-equ6 } where lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom @@ -65,7 +76,7 @@ lemma-equ3 = let open ≈-Reasoning (A) in begin e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) - ≈⟨ ke=h (eqa f f ) (lemma-equ2 f) ⟩ + ≈⟨ ek=h (eqa f f ) ⟩ id1 A a ∎ lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → @@ -85,12 +96,39 @@ ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in begin - e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 f g h) - ≈⟨ ke=h (eqa f g) (lemma-equ4 f g h) ⟩ + 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) + ≈⟨ ek=h (eqa f g) ⟩ h o e (eqa (f o h ) ( g o h )) ∎ + lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ + 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 ] ])) ]) + (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o + 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 ] ])) ] + ≈ k₁ ] + lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in + begin + ( 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 ) ))) )) + (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o 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 ) ))) ) + ≈⟨ car ( uniqueness (eqa f g) ( begin + e (eqa f g) o k₁ + ≈⟨ {!!} ⟩ + (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)) + ∎ )) ⟩ + 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 ) ))) + ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin + e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d + ≈⟨ {!!} ⟩ + id1 A d + ∎ )) ⟩ + k₁ o id1 A d + ≈⟨ idR ⟩ + k₁ + ∎ + +