Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 220:5d96be63053f
eefg
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 04 Sep 2013 18:45:17 +0900 |
| parents | 2ae029454fb6 |
| children | ea0407fb8f02 |
| 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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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 | |
| 217 | 51 -- Equalizer is unique up to iso |
| 52 | |
| 53 equalizer-iso : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) | |
| 54 → Hom A c c' --- != id1 A c | |
| 55 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (ef=eg eqa) | |
| 56 | |
| 219 | 57 equalizer-iso-eq : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) → {eff : Equalizer A {c} f f} |
| 58 → A [ A [ equalizer-iso eqa eqa' o equalizer-iso eqa' eqa ] ≈ id1 A c' ] | |
| 59 equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' {eff} = let open ≈-Reasoning (A) in | |
| 60 begin | |
| 61 equalizer-iso eqa eqa' o equalizer-iso eqa' eqa | |
| 62 ≈⟨⟩ | |
| 63 k eqa' (e eqa) (ef=eg eqa) o k eqa (e eqa') (ef=eg eqa') | |
| 64 ≈⟨ car (uniqueness eqa' {!!}) ⟩ | |
| 65 {!!} o k eqa (e eqa') (ef=eg eqa') | |
| 66 ≈⟨ {!!} ⟩ | |
| 67 id1 A c' | |
| 68 ∎ | |
| 69 | |
| 70 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 | |
| 71 -- ke = e' | |
| 72 -- k'ke = k'e' = e kk' = 1 | |
| 73 | |
| 74 -- x e = e -> x = id? | |
| 75 | |
| 76 f1=g1 : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → A [ A [ f o id1 A a ] ≈ A [ g o id1 A a ] ] | |
| 77 f1=g1 eq = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 78 | |
| 79 | |
| 80 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
| 81 A [ e eqa ≈ id1 A a ] → | |
| 82 A [ k eqa (id1 A a) (f1=g1 eq) ≈ id1 A a ] | |
| 83 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in | |
| 84 begin | |
| 85 k eqa (id1 A a) (f1=g1 eq) | |
| 86 ≈⟨ uniqueness eqa ( begin | |
| 87 e eqa o id1 A a | |
| 88 ≈⟨ idR ⟩ | |
| 89 e eqa | |
| 90 ≈⟨ e=1 ⟩ | |
| 91 id1 A a | |
| 92 ∎ )⟩ | |
| 93 id1 A a | |
| 94 ∎ | |
| 95 | |
| 217 | 96 -- e eqa f g f |
| 97 -- c ----------> a ------->b | |
| 218 | 98 -- ^ ---> d ---> |
| 99 -- | i h | |
| 100 -- j| k' (d' → d) | |
| 101 -- | k (d' → a) | |
| 102 -- d' | |
| 217 | 103 |
| 218 | 104 equalizer+h : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (i : Hom A c d ) → (h : Hom A d a ) → (h-1 : Hom A a d ) |
| 105 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
| 217 | 106 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 107 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 108 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 217 | 109 ef=eg = ef=eg1 ; |
| 110 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; | |
| 111 ek=h = ek=h1 ; | |
| 112 uniqueness = uniqueness1 | |
| 113 } where | |
| 114 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
| 115 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 116 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 117 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 118 begin | |
| 119 f o ( h o j ) | |
| 120 ≈⟨ assoc ⟩ | |
| 121 (f o h ) o j | |
| 122 ≈⟨ eq' ⟩ | |
| 123 (g o h ) o j | |
| 124 ≈↑⟨ assoc ⟩ | |
| 125 g o ( h o j ) | |
| 126 ∎ | |
| 127 ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] | |
| 128 ef=eg1 = let open ≈-Reasoning (A) in | |
| 129 begin | |
| 130 ( f o h ) o i | |
| 131 ≈↑⟨ assoc ⟩ | |
| 132 f o (h o i ) | |
| 133 ≈⟨ cdr eq ⟩ | |
| 134 f o (e eqa) | |
| 135 ≈⟨ ef=eg eqa ⟩ | |
| 136 g o (e eqa) | |
| 137 ≈↑⟨ cdr eq ⟩ | |
| 138 g o (h o i ) | |
| 139 ≈⟨ assoc ⟩ | |
| 140 ( g o h ) o i | |
| 141 ∎ | |
| 218 | 142 ek=h1 : {d' : Obj A} {k' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → |
| 143 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 144 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 145 begin |
| 218 | 146 i o k eqa (h o k' ) (fhj=ghj k' eq') -- h-1 (h o i ) o k eqa (h o k' ) = h-1 (h o k') |
| 147 ≈↑⟨ idL ⟩ | |
| 148 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 149 ≈↑⟨ car h-1-id ⟩ | |
| 150 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 151 ≈↑⟨ assoc ⟩ | |
| 152 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 153 ≈⟨ cdr assoc ⟩ | |
| 154 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 155 ≈⟨ cdr (car eq ) ⟩ | |
| 156 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 157 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 158 h-1 o ( h o k' ) | |
| 159 ≈⟨ assoc ⟩ | |
| 160 ( h-1 o h ) o k' | |
| 161 ≈⟨ car h-1-id ⟩ | |
| 162 id1 A d o k' | |
| 163 ≈⟨ idL ⟩ | |
| 164 k' | |
| 217 | 165 ∎ |
| 166 uniqueness1 : {d' : Obj A} {h' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
| 167 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 168 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 169 begin | |
| 170 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 171 ≈⟨ uniqueness eqa ( begin | |
| 172 e eqa o k' | |
| 173 ≈↑⟨ car eq ⟩ | |
| 174 (h o i ) o k' | |
| 175 ≈↑⟨ assoc ⟩ | |
| 176 h o (i o k') | |
| 177 ≈⟨ cdr ik=h ⟩ | |
| 178 h o h' | |
| 179 ∎ ) ⟩ | |
| 180 k' | |
| 181 ∎ | |
| 215 | 182 |
| 218 | 183 h+equalizer : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (h : Hom A b d ) |
| 184 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
| 185 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
| 186 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
| 187 e = e eqa ; | |
| 188 ef=eg = ef=eg1 ; | |
| 189 k = λ j eq' → k eqa j (fj=gj j eq') ; | |
| 190 ek=h = ek=h1 ; | |
| 191 uniqueness = uniqueness1 | |
| 192 } where | |
| 193 fj=gj : {e : Obj A} → (j : Hom A e a ) → A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ] → A [ A [ f o j ] ≈ A [ g o j ] ] | |
| 194 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 195 begin | |
| 196 f o j | |
| 197 ≈↑⟨ idL ⟩ | |
| 198 id1 A b o ( f o j ) | |
| 199 ≈↑⟨ car h-1-id ⟩ | |
| 200 (h-1 o h ) o ( f o j ) | |
| 201 ≈↑⟨ assoc ⟩ | |
| 202 h-1 o (h o ( f o j )) | |
| 203 ≈⟨ cdr assoc ⟩ | |
| 204 h-1 o ((h o f) o j ) | |
| 205 ≈⟨ cdr eq ⟩ | |
| 206 h-1 o ((h o g) o j ) | |
| 207 ≈↑⟨ cdr assoc ⟩ | |
| 208 h-1 o (h o ( g o j )) | |
| 209 ≈⟨ assoc ⟩ | |
| 210 (h-1 o h) o ( g o j ) | |
| 211 ≈⟨ car h-1-id ⟩ | |
| 212 id1 A b o ( g o j ) | |
| 213 ≈⟨ idL ⟩ | |
| 214 g o j | |
| 215 ∎ | |
| 216 ef=eg1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] | |
| 217 ef=eg1 = let open ≈-Reasoning (A) in | |
| 218 begin | |
| 219 ( h o f ) o e eqa | |
| 220 ≈↑⟨ assoc ⟩ | |
| 221 h o (f o e eqa ) | |
| 222 ≈⟨ cdr (ef=eg eqa) ⟩ | |
| 223 h o (g o e eqa ) | |
| 224 ≈⟨ assoc ⟩ | |
| 225 ( h o g ) o e eqa | |
| 226 ∎ | |
| 227 ek=h1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} → | |
| 228 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 229 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 230 uniqueness1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} {k' : Hom A d₁ c} → | |
| 231 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 232 uniqueness1 = uniqueness eqa | |
| 233 | |
| 220 | 234 eefeg : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
| 235 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
| 236 eefeg {a} {b} {c} {d} {f} {g} eqa = record { | |
| 237 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 238 ef=eg = ef=eg1 ; | |
| 239 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; | |
| 240 ek=h = ek=h1 ; | |
| 241 uniqueness = uniqueness1 | |
| 242 } where | |
| 243 i = id1 A c | |
| 244 h = e eqa | |
| 245 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → | |
| 246 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 247 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 248 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 249 begin | |
| 250 f o ( h o j ) | |
| 251 ≈⟨ assoc ⟩ | |
| 252 (f o h ) o j | |
| 253 ≈⟨ eq' ⟩ | |
| 254 (g o h ) o j | |
| 255 ≈↑⟨ assoc ⟩ | |
| 256 g o ( h o j ) | |
| 257 ∎ | |
| 258 ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] | |
| 259 ef=eg1 = let open ≈-Reasoning (A) in | |
| 260 begin | |
| 261 ( f o h ) o i | |
| 262 ≈⟨ car ( ef=eg eqa ) ⟩ | |
| 263 ( g o h ) o i | |
| 264 ∎ | |
| 265 ek=h1 : {d' : Obj A} {k' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → | |
| 266 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 267 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 268 begin | |
| 269 i o k eqa (h o k' ) (fhj=ghj k' eq') -- h-1 (h o i ) o k eqa (h o k' ) = h-1 (h o k') | |
| 270 ≈⟨ idL ⟩ | |
| 271 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
| 272 ≈⟨ uniqueness eqa refl-hom ⟩ | |
| 273 k' | |
| 274 ∎ | |
| 275 uniqueness1 : {d' : Obj A} {h' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
| 276 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 277 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 278 begin | |
| 279 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
| 280 ≈⟨ uniqueness eqa ( begin | |
| 281 e eqa o k' | |
| 282 ≈↑⟨ cdr idL ⟩ | |
| 283 e eqa o (id1 A c o k' ) | |
| 284 ≈⟨ cdr ik=h ⟩ | |
| 285 e eqa o h' | |
| 286 ∎ ) ⟩ | |
| 287 k' | |
| 288 ∎ | |
| 289 | |
| 290 | |
| 291 | |
| 218 | 292 |
| 211 | 293 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 294 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 295 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 216 | 296 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 297 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 | 298 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 211 | 299 b1 = ef=eg (eqa f g) ; |
| 212 | 300 b2 = lemma-equ5 ; |
| 301 b3 = lemma-equ3 ; | |
| 215 | 302 b4 = lemma-equ6 |
| 211 | 303 } where |
| 216 | 304 -- |
| 305 -- e eqa f g f | |
| 306 -- c ----------> a ------->b | |
| 307 -- ^ g | |
| 308 -- | | |
| 309 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 310 -- | | |
| 311 -- d | |
| 312 -- | |
| 313 -- | |
| 314 -- e o id1 ≈ e → k e ≈ id | |
| 315 | |
| 211 | 316 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 317 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 318 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 319 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 320 begin |
| 321 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 322 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 323 id1 A a |
| 324 ∎ | |
| 214 | 325 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 326 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 | 327 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 328 begin |
| 329 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 330 ≈⟨ assoc ⟩ | |
| 331 (f o h) o e (eqa (f o h) ( g o h )) | |
| 332 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 333 (g o h) o e (eqa (f o h) ( g o h )) | |
| 334 ≈↑⟨ assoc ⟩ | |
| 335 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 336 ∎ | |
| 337 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 338 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 | 339 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 340 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 341 begin | |
| 215 | 342 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) |
| 343 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 344 h o e (eqa (f o h ) ( g o h )) |
| 345 ∎ | |
| 215 | 346 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
| 347 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 ] ])) ]) | |
| 348 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
| 349 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 ] ])) ] | |
| 350 ≈ k₁ ] | |
| 351 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
| 352 begin | |
| 353 ( 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 ) ))) )) | |
| 354 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
| 355 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 ) ))) ) | |
| 356 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
| 357 e (eqa f g) o k₁ | |
| 358 ≈⟨ {!!} ⟩ | |
| 359 (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)) | |
| 360 ∎ )) ⟩ | |
| 361 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 ) ))) | |
| 362 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
| 363 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
| 364 ≈⟨ {!!} ⟩ | |
| 365 id1 A d | |
| 366 ∎ )) ⟩ | |
| 367 k₁ o id1 A d | |
| 368 ≈⟨ idR ⟩ | |
| 369 k₁ | |
| 370 ∎ | |
| 211 | 371 |
| 372 | |
| 212 | 373 |
| 374 | |
| 375 | |
| 215 | 376 |
| 377 |