Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 237:776cda2286c8
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 08 Sep 2013 05:55:56 +0900 |
| parents | e20b81102eee |
| children | c8db99cdf72a |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 205 | 6 -- c --------> a ----------> b |
| 208 | 7 -- ^ . ----------> |
| 205 | 8 -- | . g |
| 230 | 9 -- |k . |
| 10 -- | . h | |
| 11 -- d | |
| 205 | 12 -- |
| 13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 14 ---- | |
| 15 | |
| 230 | 16 open import Category -- https://github.com/konn/category-agda |
| 205 | 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 | |
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23 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 |
| 205 | 24 field |
| 221 | 25 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 26 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 27 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 ] |
| 230 | 28 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 | 29 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 30 equalizer : Hom A c a |
| 31 equalizer = e | |
| 206 | 32 |
| 230 | 33 -- |
| 225 | 34 -- Flat Equational Definition of Equalizer |
| 230 | 35 -- |
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36 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 |
| 206 | 37 field |
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38 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 |
| 214 | 39 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 |
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40 δ : {a b c : Obj A } → {e : Hom A c a } → (f : Hom A a b) → Hom A a c |
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41 b1 : A [ A [ f o α {a} {b} {a} f g {id1 A a} ] ≈ A [ g o α {a} {b} {a} f g {id1 A a} ] ] |
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42 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} ] ] |
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43 b3 : A [ A [ α {a} {b} {a} f f {id1 A a} o δ {a} {b} {a} {id1 A a} f ] ≈ id1 A a ] |
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44 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
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45 b4 : {d : Obj A } {k : Hom A d c} → |
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46 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 ] |
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47 -- A [ α f g o β f g h ] ≈ h |
| 214 | 48 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 |
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49 β {d} {e} {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 ]) ] |
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50 |
| 209 | 51 open Equalizer |
| 225 | 52 open Burroni |
| 209 | 53 |
| 225 | 54 -- |
| 55 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 56 -- | |
| 219 | 57 |
| 224 | 58 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 ] ] |
| 59 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 60 | |
| 226 | 61 f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] |
| 230 | 62 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 63 |
| 224 | 64 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 65 (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 ] ] ] | |
| 230 | 66 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 67 begin |
| 68 f o ( e o h ) | |
| 69 ≈⟨ assoc ⟩ | |
| 230 | 70 (f o e ) o h |
| 224 | 71 ≈⟨ car eq ⟩ |
| 230 | 72 (g o e ) o h |
| 224 | 73 ≈↑⟨ assoc ⟩ |
| 74 g o ( e o h ) | |
| 75 ∎ | |
| 219 | 76 |
| 225 | 77 -- |
| 78 -- | |
| 79 -- An isomorphic element c' of c makes another equalizer | |
| 80 -- | |
| 230 | 81 -- e eqa f g f |
| 222 | 82 -- c ----------> a ------->b |
| 230 | 83 -- |^ |
| 84 -- || | |
| 222 | 85 -- h || h-1 |
| 230 | 86 -- v| |
| 87 -- c' | |
| 222 | 88 |
| 234 | 89 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 90 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 91 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 92 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
| 93 ( eqa : Equalizer A e f g ) | |
| 94 → Equalizer A (A [ e o h-1 ] ) f g | |
| 95 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
| 222 | 96 fe=ge = fe=ge1 ; |
| 97 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 98 ek=h = ek=h1 ; |
| 222 | 99 uniqueness = uniqueness1 |
| 100 } where | |
| 234 | 101 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 102 fe=ge1 = fe=ge' | |
| 222 | 103 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 104 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 105 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 106 begin | |
| 234 | 107 ( e o h-1 ) o ( h o k eqa j eq ) |
| 108 ≈↑⟨ assoc ⟩ | |
| 109 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 110 ≈⟨ cdr assoc ⟩ | |
| 111 e o (( h-1 o h) o k eqa j eq ) | |
| 112 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 113 e o (id1 A c o k eqa j eq ) | |
| 114 ≈⟨ cdr idL ⟩ | |
| 115 e o k eqa j eq | |
| 222 | 116 ≈⟨ ek=h eqa ⟩ |
| 117 j | |
| 118 ∎ | |
| 119 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 234 | 120 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 121 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 122 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 123 begin | |
| 124 h o k eqa h' eq | |
| 234 | 125 ≈⟨ cdr (uniqueness eqa ( begin |
| 126 e o ( h-1 o j ) | |
| 127 ≈⟨ assoc ⟩ | |
| 128 (e o h-1 ) o j | |
| 129 ≈⟨ ej=h ⟩ | |
| 130 h' | |
| 131 ∎ )) ⟩ | |
| 132 h o ( h-1 o j ) | |
| 133 ≈⟨ assoc ⟩ | |
| 134 (h o h-1 ) o j | |
| 135 ≈⟨ car hh-1=1 ⟩ | |
| 136 id1 A c' o j | |
| 137 ≈⟨ idL ⟩ | |
| 222 | 138 j |
| 139 ∎ | |
| 140 | |
| 225 | 141 -- |
| 142 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 143 -- | |
| 144 -- h : c → c' h' : c' → c | |
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145 -- e' = h o e |
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146 -- e = h' o e' |
| 225 | 147 |
| 148 | |
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149 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 } |
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150 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 151 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 230 | 152 → Hom A c c' |
| 234 | 153 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 154 |
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155 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 : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 156 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 230 | 157 → Hom A c' c |
| 158 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 223 | 159 |
| 234 | 160 -- e' f |
| 230 | 161 -- c'----------> a ------->b f e j = g e j |
| 162 -- ^ g | |
| 163 -- |k h | |
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164 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 165 -- | |
| 166 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 167 -- | |
| 168 -- h j e f = h j e g → h = 'j e f | |
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169 -- h = j e f -> j = j' |
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170 -- |
| 228 | 171 |
| 234 | 172 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 ) |
| 173 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
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174 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 175 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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176 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 177 ≈⟨⟩ |
| 178 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 179 ≈⟨ ek=h keqa ⟩ | |
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180 id1 A c' |
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181 ∎ |
| 226 | 182 |
| 234 | 183 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 ) |
| 184 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 185 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
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186 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- refl |
| 234 | 187 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] |
| 188 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin | |
| 189 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa | |
| 190 ≈⟨⟩ | |
| 191 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
| 192 ≈⟨⟩ | |
| 193 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 194 ≈⟨ cdr ( begin | |
| 195 k eqa' e (fe=ge eqa ) | |
| 196 ≈⟨ uniqueness eqa' ( begin | |
| 197 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
| 198 ≈⟨ car e'->e ⟩ | |
| 199 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
| 200 ≈↑⟨ assoc ⟩ | |
| 201 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
| 202 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
| 203 e o id1 A c | |
| 204 ≈⟨ idR ⟩ | |
| 205 e | |
| 206 ∎ )⟩ | |
| 207 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
| 208 ∎ )⟩ | |
| 209 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
| 210 ≈⟨ ek=h keqa' ⟩ | |
| 211 id1 A c | |
| 212 ∎ | |
| 213 | |
| 214 | |
| 230 | 215 |
| 225 | 216 ---- |
| 217 -- | |
| 218 -- An equalizer satisfies Burroni equations | |
| 219 -- | |
| 230 | 220 -- b4 is not yet done |
| 225 | 221 ---- |
| 222 | |
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223 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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224 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
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225 → Burroni A {c} {a} {b} f g e |
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226 lemma-equ1 {a} {b} {c} f g e eqa = record { |
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227 α = λ {a} {b} {c} f g {e} → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
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228 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
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229 δ = λ {a} {b} {c} {e} f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f); -- Hom A a c |
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230 b1 = fe=ge (eqa {a} {b} {a} f g {id1 A a}) ; |
| 226 | 231 b2 = lemma-b2 ; |
| 232 b3 = lemma-b3 ; | |
| 230 | 233 b4 = lemma-b4 |
| 211 | 234 } where |
| 216 | 235 -- |
| 236 -- e eqa f g f | |
| 237 -- c ----------> a ------->b | |
| 230 | 238 -- ^ g |
| 239 -- | | |
| 216 | 240 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 241 -- | |
| 216 | 242 -- d |
| 230 | 243 -- |
| 244 -- | |
| 216 | 245 -- e o id1 ≈ e → k e ≈ id |
| 246 | |
| 211 | 247 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 248 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
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249 lemma-b3 : A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 226 | 250 lemma-b3 = let open ≈-Reasoning (A) in |
| 230 | 251 begin |
|
233
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252 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
| 215 | 253 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 254 id1 A a |
| 255 ∎ | |
| 230 | 256 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
|
233
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diff
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257 A [ A [ f o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 258 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 259 begin |
|
233
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diff
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260 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 261 ≈⟨ assoc ⟩ |
|
233
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diff
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262 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 263 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
|
233
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diff
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264 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 265 ≈↑⟨ assoc ⟩ |
|
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diff
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266 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 267 ∎ |
| 230 | 268 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
|
233
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diff
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269 A [ equalizer (eqa f g) o k (eqa f g) (A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
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diff
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270 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 271 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 272 begin |
|
233
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diff
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273 equalizer (eqa f g) o k (eqa f g) (h o equalizer (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 215 | 274 ≈⟨ ek=h (eqa f g) ⟩ |
|
233
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diff
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275 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 276 ∎ |
| 230 | 277 |
| 278 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
|
236
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diff
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279 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
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diff
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280 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
|
233
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diff
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281 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
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diff
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282 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 ] ])) ] |
| 222 | 283 ≈ j ] |
| 230 | 284 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 285 begin |
|
236
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diff
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286 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g {e}) o j ) )) (( g o ( equalizer (eqa f g {e}) o j ) ))) )) |
|
233
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diff
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287 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
|
236
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diff
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288 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 ) ))) ) |
|
235
8835015a3e1a
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234
diff
changeset
|
289 ≈⟨ car ((uniqueness (eqa f g) ( begin |
|
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234
diff
changeset
|
290 equalizer (eqa f g) o j |
|
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234
diff
changeset
|
291 ≈↑⟨ idR ⟩ |
|
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234
diff
changeset
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292 (equalizer (eqa f g) o j ) o id1 A d |
|
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234
diff
changeset
|
293 ≈⟨⟩ |
|
236
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diff
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294 ((equalizer (eqa f g) o j) o equalizer (eqa (f o equalizer (eqa f g {e}) o j) (g o equalizer (eqa f g {e}) o j))) |
|
235
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234
diff
changeset
|
295 ∎ ))) ⟩ |
|
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234
diff
changeset
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296 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 ) ))) |
|
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234
diff
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297 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
|
236
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diff
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298 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d |
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235
diff
changeset
|
299 ≈⟨ idR ⟩ |
|
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235
diff
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|
300 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
|
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Burroni equational equalizer definition done.
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parents:
235
diff
changeset
|
301 ≈⟨⟩ |
|
235
8835015a3e1a
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234
diff
changeset
|
302 id1 A d |
|
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234
diff
changeset
|
303 ∎ ))) ⟩ |
|
8835015a3e1a
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234
diff
changeset
|
304 j o id1 A d |
|
8835015a3e1a
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234
diff
changeset
|
305 ≈⟨ idR ⟩ |
| 222 | 306 j |
|
235
8835015a3e1a
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234
diff
changeset
|
307 ∎ |
| 211 | 308 |
| 309 | |
| 225 | 310 -- end |
| 212 | 311 |
| 312 | |
| 313 | |
| 215 | 314 |
| 315 |