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freyd2
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 12 Jun 2017 10:50:02 +0900 |
| parents | f526f4b68565 |
| children |
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--- -- -- Equalizer -- -- e f -- c -------→ a ---------→ b -- ^ . ---------→ -- | . g -- |k . -- | . h -- d -- -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> ---- open import Category -- https://github.com/konn/category-agda open import Level module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where open import HomReasoning open import cat-utility -- in cat-utility -- record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where -- field -- fe=ge : 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 -- 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 -- -- Burroni's Flat Equational Definition of Equalizer -- record Burroni { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) (e : Hom A c a) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (e : Hom A c a ) → Hom A c a 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 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c cong-α : {a b c : Obj A } → { e : Hom A c a } → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α {a} {b} {c} f g e ) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) (id1 A d) ] ] b3 : {a b d : Obj A} → (f : Hom A a b ) → {h : Hom A d a } → A [ A [ α {a} {b} {d} f f h o δ {a} {b} {d} h 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 e o k ] ) o ( δ {d} {b} {d} (id1 A d) (A [ f o A [ α {a} {b} {c} f g e o k ] ] ) )] ≈ k ] -- A [ α f g o β f g h ] ≈ h β : { d 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} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ {d} {b} {d} (id1 A d) (A [ f o h ]) ] open Equalizer open IsEqualizer open Burroni -- -- Some obvious conditions for k (fe = ge) → ( fh = gh ) -- f1=g1 : { a b c : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → (h : Hom A c a) → A [ A [ f o h ] ≈ A [ g o h ] ] f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] f1=f1 f = let open ≈-Reasoning (A) in refl-hom f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → (eq : A [ A [ f o e ] ≈ A [ g o e ] ] ) → A [ A [ f o A [ e o h ] ] ≈ A [ g o A [ e o h ] ] ] f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in begin f o ( e o h ) ≈⟨ assoc ⟩ (f o e ) o h ≈⟨ car eq ⟩ (g o e ) o h ≈↑⟨ assoc ⟩ g o ( e o h ) ∎ ------------------------------- -- -- Every equalizer is monic -- -- e i = e j → i = j -- -- e eqa f g f -- c ---------→ a ------→b -- ^^ -- || -- i||j -- || -- d monoic : { c a b d : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A f g) → { i j : Hom A d (equalizer-c eqa) } → A [ A [ equalizer eqa o i ] ≈ A [ equalizer eqa o j ] ] → A [ i ≈ j ] monoic {c} {a} {b} {d} {f} {g} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin i ≈↑⟨ uniqueness (isEqualizer eqa) ( begin equalizer eqa o i ≈⟨ ei=ej ⟩ equalizer eqa o j ∎ )⟩ k (isEqualizer eqa) (equalizer eqa o j) ( f1=gh (fe=ge (isEqualizer eqa) ) ) ≈⟨ uniqueness (isEqualizer eqa) ( begin equalizer eqa o j ≈⟨⟩ equalizer eqa o j ∎ )⟩ j ∎ -------------------------------- -- -- -- Isomorphic arrows from c' to c makes another equalizer -- -- e eqa f g f -- c ---------→ a ------→b -- |^ -- || -- h || h-1 -- v| -- c' equalizer+iso : {a b c' : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A f g ) → (h-1 : Hom A c' (equalizer-c eqa) ) → (h : Hom A (equalizer-c eqa) c' ) → A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A (equalizer-c eqa) ] → IsEqualizer A (A [ equalizer eqa o h-1 ] ) f g equalizer+iso {a} {b} {c'} {f} {g} eqa h-1 h hh-1=1 h-1h=1 = record { fe=ge = fe=ge1 ; k = λ j eq → A [ h o k (isEqualizer eqa) j eq ] ; ek=h = ek=h1 ; uniqueness = uniqueness1 } where e = equalizer eqa fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] fe=ge1 = f1=gh ( fe=ge (isEqualizer eqa) ) ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → A [ A [ A [ e o h-1 ] o A [ h o k (isEqualizer eqa) j eq ] ] ≈ j ] ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in begin ( e o h-1 ) o ( h o k (isEqualizer eqa) j eq ) ≈↑⟨ assoc ⟩ e o ( h-1 o ( h o k (isEqualizer eqa) j eq ) ) ≈⟨ cdr assoc ⟩ e o (( h-1 o h) o k (isEqualizer eqa) j eq ) ≈⟨ cdr (car h-1h=1 ) ⟩ e o (id1 A (equalizer-c eqa) o k (isEqualizer eqa) j eq ) ≈⟨ cdr idL ⟩ e o k (isEqualizer eqa) j eq ≈⟨ ek=h (isEqualizer eqa) ⟩ j ∎ uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → A [ A [ h o k (isEqualizer eqa) h' eq ] ≈ j ] uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in begin h o k (isEqualizer eqa) h' eq ≈⟨ cdr (uniqueness (isEqualizer eqa) ( begin e o ( h-1 o j ) ≈⟨ assoc ⟩ (e o h-1 ) o j ≈⟨ ej=h ⟩ h' ∎ )) ⟩ h o ( h-1 o j ) ≈⟨ assoc ⟩ (h o h-1 ) o j ≈⟨ car hh-1=1 ⟩ id c' o j ≈⟨ idL ⟩ j ∎ -------------------------------- -- -- If we have two equalizers on c and c', there are isomorphic pair h, h' -- -- h : c → c' h' : c' → c -- e' = h o e -- e = h' o e' -- -- -- -- e eqa f g f -- c ---------→a ------→b -- ^ ^ g -- | | -- |k = id c' | -- v | -- c'-----------+ -- e eqa' f g c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → Hom A c c' c-iso-l {c} {c'} {a} {b} {f} {g} {e} eqa eqa' = k eqa' e ( fe=ge eqa ) c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → Hom A c' c c-iso-r {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = k eqa e' ( fe=ge eqa' ) c-iso-lr : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → A [ A [ c-iso-l eqa eqa' o c-iso-r eqa eqa' ] ≈ id1 A c' ] c-iso-lr {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = let open ≈-Reasoning (A) in begin c-iso-l eqa eqa' o c-iso-r eqa eqa' ≈⟨⟩ k eqa' e ( fe=ge eqa ) o k eqa e' ( fe=ge eqa' ) ≈↑⟨ uniqueness eqa' ( begin e' o ( k eqa' e (fe=ge eqa) o k eqa e' (fe=ge eqa') ) ≈⟨ assoc ⟩ ( e' o k eqa' e (fe=ge eqa) ) o k eqa e' (fe=ge eqa') ≈⟨ car (ek=h eqa') ⟩ e o k eqa e' (fe=ge eqa') ≈⟨ ek=h eqa ⟩ e' ∎ )⟩ k eqa' e' ( fe=ge eqa' ) ≈⟨ uniqueness eqa' ( begin e' o id c' ≈⟨ idR ⟩ e' ∎ )⟩ id c' ∎ -- c-iso-rl is obvious from the symmetry -------------------------------- ---- -- -- Existence of equalizer satisfies Burroni equations -- ---- lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → IsEqualizer A e f g ) → Burroni A {c} {a} {b} f g e lemma-equ1 {a} {b} {c} f g e eqa = record { α = λ {a} {b} {c} f g e → equalizer1 (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a γ = λ {a} {b} {c} {d} f g h → k (eqa {a} {b} {c} f g ) {d} ( A [ h o (equalizer1 ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) {a} (id1 A a) (f1=f1 f); -- Hom A a c cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; b2 = λ {d} {h} → lemma-b2 {d} {h}; b3 = lemma-b3 ; b4 = lemma-b4 } where -- -- e eqa f g f -- c ---------→ a ------→b -- ^ g -- | -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) -- | -- d -- -- -- e o id1 ≈ e → k e ≈ id lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer1 (eqa f f ) o k (eqa f f) (id1 A a) (f1=f1 f) ] ≈ id1 A a ] lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in begin equalizer1 (eqa f f) o k (eqa f f) (id a) (f1=f1 f) ≈⟨ ek=h (eqa f f ) ⟩ id a ∎ lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → A [ A [ f o A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in begin f o ( h o equalizer1 (eqa (f o h) ( g o h ))) ≈⟨ assoc ⟩ (f o h) o equalizer1 (eqa (f o h) ( g o h )) ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ (g o h) o equalizer1 (eqa (f o h) ( g o h )) ≈↑⟨ assoc ⟩ g o ( h o equalizer1 (eqa (f o h) ( g o h ))) ∎ cong-α1 : {a b c : Obj A } → { e : Hom A c a } → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer1 (eqa {a} {b} {c} f g {e} )≈ equalizer1 (eqa {a} {b} {c} f g' {e} ) ] cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom cong-γ1 : {a c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → { e : Hom A c a} → A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer1 ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin k (eqa f g ) {d} ( A [ h o (equalizer1 ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ≈⟨ uniqueness (eqa f g) ( begin e o k (eqa f g ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ≈⟨ ek=h (eqa f g ) ⟩ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ≈↑⟨ car h=h' ⟩ h o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ∎ )⟩ k (eqa f g ) {d} ( A [ h' o (equalizer1 ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ∎ cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) ≈ k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in begin k (eqa {a} {b} {c} f f {e} ) (id a) (f1=f1 f) ≈⟨ uniqueness (eqa f f) ( begin e o k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ id a ∎ )⟩ k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') ∎ lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ A [ equalizer1 (eqa f g) o k (eqa f g) (A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] ≈ A [ h o equalizer1 (eqa (A [ f o h ]) (A [ g o h ])) ] ] lemma-b2 {d} {h} = let open ≈-Reasoning (A) in begin equalizer1 (eqa f g) o k (eqa f g) (h o equalizer1 (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) ≈⟨ ek=h (eqa f g) ⟩ h o equalizer1 (eqa (f o h ) ( g o h )) ∎ lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ A [ k (eqa f g) (A [ A [ equalizer1 (eqa f g) o j ] o equalizer1 (eqa (A [ f o A [ equalizer1 (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer1 (eqa f g {e} ) o j ] ])) ]) (lemma-equ4 {a} {b} {c} f g (A [ equalizer1 (eqa f g) o j ])) o k (eqa (A [ f o A [ equalizer1 (eqa f g) o j ] ]) (A [ f o A [ equalizer1 (eqa f g) o j ] ])) (id1 A d) (f1=f1 (A [ f o A [ equalizer1 (eqa f g) o j ] ])) ] ≈ j ] lemma-b4 {d} {j} = let open ≈-Reasoning (A) in begin ( k (eqa f g) (( ( equalizer1 (eqa f g) o j ) o equalizer1 (eqa (( f o ( equalizer1 (eqa f g {e}) o j ) )) (( g o ( equalizer1 (eqa f g {e}) o j ) ))) )) (lemma-equ4 {a} {b} {c} f g (( equalizer1 (eqa f g) o j ))) o k (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer1 (eqa f g) o j ) ))) ) ≈⟨ car ((uniqueness (eqa f g) ( begin equalizer1 (eqa f g) o j ≈↑⟨ idR ⟩ (equalizer1 (eqa f g) o j ) o id d ≈⟨⟩ -- here we decide e (fej) (gej)' is id d ((equalizer1 (eqa f g) o j) o equalizer1 (eqa (f o equalizer1 (eqa f g {e}) o j) (g o equalizer1 (eqa f g {e}) o j))) ∎ ))) ⟩ j o k (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer1 (eqa f g) o j ) ))) ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer1 (eqa f g) o j ) )) (( f o ( equalizer1 (eqa f g) o j ) ))) ( begin equalizer1 (eqa (f o equalizer1 (eqa f g {e} ) o j) (f o equalizer1 (eqa f g {e}) o j)) o id d ≈⟨ idR ⟩ equalizer1 (eqa (f o equalizer1 (eqa f g {e}) o j) (f o equalizer1 (eqa f g {e}) o j)) ≈⟨⟩ -- here we decide e (fej) (fej)' is id d id d ∎ ))) ⟩ j o id d ≈⟨ idR ⟩ j ∎ -------------------------------- -- -- Bourroni equations gives an Equalizer -- lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) → ( bur : Burroni A {c} {a} {b} f g e ) → IsEqualizer A {c} {a} {b} (α bur f g e) f g lemma-equ2 {a} {b} {c} f g e bur = record { fe=ge = fe=ge1 ; k = k1 ; ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h } where k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c k1 {d} h fh=gh = β bur {d} {a} {b} f g h fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] fe=ge1 = b1 bur ek=h1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ (α bur f g e) o k1 {d} h eq ] ≈ h ] ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in begin α bur f g e o k1 h eq ≈⟨⟩ α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) ) ≈⟨ assoc ⟩ ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id d) (f o h) ≈⟨ car (b2 bur) ⟩ ( h o ( α bur ( f o h ) ( g o h ) (id d))) o δ bur {d} {b} {d} (id d) (f o h) ≈↑⟨ assoc ⟩ h o ((( α bur ( f o h ) ( g o h ) (id d) )) o δ bur {d} {b} {d} (id d) (f o h) ) ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ h o ((( α bur ( f o h ) ( f o h ) (id d)))o δ bur {d} {b} {d} (id d) (f o h) ) ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id d} ) ⟩ h o id d ≈⟨ idR ⟩ h ∎ uniqueness1 : {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 [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in begin k1 {d} h eq ≈⟨⟩ γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) (f o h) ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) ( f o ( α bur f g e o k') ) ≈⟨ b4 bur ⟩ k' ∎ -- end