Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 253:24e83b8b81be
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 11 Sep 2013 20:26:48 +0900 |
| parents | e0835b8dd51b |
| children | 45b81fcb8a64 |
| 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 | |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
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 |
| 253 | 33 |
| 230 | 34 -- |
| 251 | 35 -- Burroni's Flat Equational Definition of Equalizer |
| 230 | 36 -- |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
37 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 | 38 field |
| 245 | 39 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 | 40 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 |
| 245 | 41 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c |
| 242 | 42 cong-α : {a b c : Obj A } → { e : Hom A c a } |
| 245 | 43 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
| 242 | 44 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
| 243 | 45 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
| 245 | 46 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' ] |
| 47 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
| 48 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) ] ] | |
| 49 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 ] | |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
50 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
51 b4 : {d : Obj A } {k : Hom A d c} → |
| 245 | 52 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 ] |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
53 -- A [ α f g o β f g h ] ≈ h |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
54 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 |
| 245 | 55 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 ]) ] |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
56 |
| 209 | 57 open Equalizer |
| 225 | 58 open Burroni |
| 209 | 59 |
| 225 | 60 -- |
| 61 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 62 -- | |
| 219 | 63 |
| 224 | 64 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 ] ] |
| 65 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 66 | |
| 226 | 67 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 | 68 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 69 |
| 224 | 70 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 71 (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 | 72 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 73 begin |
| 74 f o ( e o h ) | |
| 75 ≈⟨ assoc ⟩ | |
| 230 | 76 (f o e ) o h |
| 224 | 77 ≈⟨ car eq ⟩ |
| 230 | 78 (g o e ) o h |
| 224 | 79 ≈↑⟨ assoc ⟩ |
| 80 g o ( e o h ) | |
| 81 ∎ | |
| 219 | 82 |
| 251 | 83 -------------------------------- |
| 225 | 84 -- |
| 85 -- | |
| 249 | 86 -- An isomorphic arrow c' to c makes another equalizer |
| 225 | 87 -- |
| 230 | 88 -- e eqa f g f |
| 222 | 89 -- c ----------> a ------->b |
| 230 | 90 -- |^ |
| 91 -- || | |
| 222 | 92 -- h || h-1 |
| 230 | 93 -- v| |
| 94 -- c' | |
| 222 | 95 |
| 234 | 96 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 97 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 98 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 99 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
| 100 ( eqa : Equalizer A e f g ) | |
| 101 → Equalizer A (A [ e o h-1 ] ) f g | |
| 102 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
| 222 | 103 fe=ge = fe=ge1 ; |
| 104 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 105 ek=h = ek=h1 ; |
| 222 | 106 uniqueness = uniqueness1 |
| 107 } where | |
| 234 | 108 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 109 fe=ge1 = fe=ge' | |
| 222 | 110 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 111 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 112 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 113 begin | |
| 234 | 114 ( e o h-1 ) o ( h o k eqa j eq ) |
| 115 ≈↑⟨ assoc ⟩ | |
| 116 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 117 ≈⟨ cdr assoc ⟩ | |
| 118 e o (( h-1 o h) o k eqa j eq ) | |
| 119 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 253 | 120 e o (id c o k eqa j eq ) |
| 234 | 121 ≈⟨ cdr idL ⟩ |
| 122 e o k eqa j eq | |
| 222 | 123 ≈⟨ ek=h eqa ⟩ |
| 124 j | |
| 125 ∎ | |
| 126 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 | 127 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 128 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 129 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 130 begin | |
| 131 h o k eqa h' eq | |
| 234 | 132 ≈⟨ cdr (uniqueness eqa ( begin |
| 133 e o ( h-1 o j ) | |
| 134 ≈⟨ assoc ⟩ | |
| 135 (e o h-1 ) o j | |
| 136 ≈⟨ ej=h ⟩ | |
| 137 h' | |
| 138 ∎ )) ⟩ | |
| 139 h o ( h-1 o j ) | |
| 140 ≈⟨ assoc ⟩ | |
| 141 (h o h-1 ) o j | |
| 142 ≈⟨ car hh-1=1 ⟩ | |
| 253 | 143 id c' o j |
| 234 | 144 ≈⟨ idL ⟩ |
| 222 | 145 j |
| 146 ∎ | |
| 147 | |
| 251 | 148 -------------------------------- |
| 225 | 149 -- |
| 150 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 151 -- | |
| 152 -- h : c → c' h' : c' → c | |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
153 -- e' = h o e |
|
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
154 -- e = h' o e' |
| 225 | 155 |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
156 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 } |
|
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
157 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 158 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 159 → Hom A c c' -- should be e' = c-sio-l o e |
| 234 | 160 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 161 |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
162 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 | 163 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 164 → Hom A c' c -- e = c-sio-r o e' |
| 230 | 165 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
| 223 | 166 |
| 234 | 167 -- e' f |
| 230 | 168 -- c'----------> a ------->b f e j = g e j |
| 169 -- ^ g | |
| 170 -- |k h | |
|
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
171 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 172 -- | |
| 173 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 174 -- | |
| 175 -- h j e f = h j e g → h = 'j e f | |
|
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
176 -- h = j e f -> j = j' |
|
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
177 -- |
| 228 | 178 |
| 251 | 179 e←e' : { 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 ) |
| 250 | 180 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 181 → A [ A [ e' o c-iso-l eqa eqa' keqa ] ≈ e ] | |
| 251 | 182 e←e' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 183 e' o c-iso-l eqa eqa' keqa |
| 184 ≈⟨⟩ | |
| 185 e' o k eqa' e (fe=ge eqa) | |
| 186 ≈⟨⟩ | |
| 187 equalizer eqa' o k eqa' e (fe=ge eqa) | |
| 188 ≈⟨ ek=h eqa' ⟩ | |
| 189 e | |
| 190 ∎ | |
| 191 | |
| 251 | 192 e'←e : { 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 ) |
| 250 | 193 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 194 → A [ A [ e o c-iso-r eqa eqa' keqa ] ≈ e' ] | |
| 251 | 195 e'←e {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 196 e o c-iso-r eqa eqa' keqa |
| 197 ≈⟨⟩ | |
| 253 | 198 e o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 199 ≈↑⟨ car (ek=h eqa' ) ⟩ |
| 253 | 200 ( equalizer eqa' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 201 ≈⟨⟩ |
| 253 | 202 ( e' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 203 ≈↑⟨ assoc ⟩ |
| 253 | 204 e' o (( k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) ) |
| 250 | 205 ≈⟨⟩ |
| 253 | 206 e' o (equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) ) |
| 250 | 207 ≈⟨ cdr ( ek=h keqa ) ⟩ |
| 253 | 208 e' o id c' |
| 250 | 209 ≈⟨ idR ⟩ |
| 210 e' | |
| 211 ∎ | |
| 212 | |
| 253 | 213 -- e←e' e'←e e = e |
| 214 -- e'←e e←e e' = e' is enough for isomorphism but we can prove l o r = id also. | |
| 252 | 215 |
| 234 | 216 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 ) |
| 217 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
|
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
218 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 219 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
|
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
220 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 221 ≈⟨⟩ |
| 253 | 222 equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 234 | 223 ≈⟨ ek=h keqa ⟩ |
| 253 | 224 id c' |
|
229
68b2681ea9df
c in equalizer is equal up to iso done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
225 ∎ |
| 226 | 226 |
| 234 | 227 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 ) |
| 228 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 229 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
| 230 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] | |
| 250 | 231 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' = let open ≈-Reasoning (A) in begin |
| 234 | 232 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 233 ≈⟨⟩ | |
| 253 | 234 k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) o k eqa' e (fe=ge eqa ) |
| 234 | 235 ≈⟨⟩ |
| 236 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 237 ≈⟨ cdr ( begin | |
| 238 k eqa' e (fe=ge eqa ) | |
| 239 ≈⟨ uniqueness eqa' ( begin | |
| 253 | 240 e' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 251 | 241 ≈↑⟨ car (e'←e eqa eqa' keqa ) ⟩ |
| 253 | 242 ( e o equalizer keqa' ) o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 234 | 243 ≈↑⟨ assoc ⟩ |
| 253 | 244 e o ( equalizer keqa' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c))) |
| 234 | 245 ≈⟨ cdr ( ek=h keqa' ) ⟩ |
| 253 | 246 e o id c |
| 234 | 247 ≈⟨ idR ⟩ |
| 248 e | |
| 249 ∎ )⟩ | |
| 253 | 250 k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) |
| 234 | 251 ∎ )⟩ |
| 253 | 252 equalizer keqa' o k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) ≈⟨ ek=h keqa' ⟩ |
| 253 id c | |
| 234 | 254 ∎ |
| 255 | |
| 256 | |
| 230 | 257 |
| 251 | 258 -------------------------------- |
| 225 | 259 ---- |
| 260 -- | |
| 261 -- An equalizer satisfies Burroni equations | |
| 262 -- | |
| 263 ---- | |
| 264 | |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
265 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
|
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
266 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
|
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
267 → Burroni A {c} {a} {b} f g e |
|
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
268 lemma-equ1 {a} {b} {c} f g e eqa = record { |
| 245 | 269 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
| 242 | 270 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 ] ))) ] ) |
| 271 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
| 249 | 272 δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f); -- Hom A a c |
|
246
80d9ef47566b
cong-γ1 done, cong-δ1 remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
245
diff
changeset
|
273 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
| 247 | 274 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
| 245 | 275 cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
276 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
| 226 | 277 b2 = lemma-b2 ; |
| 278 b3 = lemma-b3 ; | |
| 230 | 279 b4 = lemma-b4 |
| 211 | 280 } where |
| 216 | 281 -- |
| 282 -- e eqa f g f | |
| 283 -- c ----------> a ------->b | |
| 230 | 284 -- ^ g |
| 285 -- | | |
| 216 | 286 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 287 -- | |
| 216 | 288 -- d |
| 230 | 289 -- |
| 290 -- | |
| 216 | 291 -- e o id1 ≈ e → k e ≈ id |
| 292 | |
| 249 | 293 lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (f1=f1 f) ] ≈ id1 A a ] |
| 240 | 294 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
| 230 | 295 begin |
| 253 | 296 equalizer (eqa f f) o k (eqa f f) (id a) (f1=f1 f) |
| 215 | 297 ≈⟨ ek=h (eqa f f ) ⟩ |
| 253 | 298 id a |
| 211 | 299 ∎ |
| 230 | 300 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
301 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 | 302 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 303 begin |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
304 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 305 ≈⟨ assoc ⟩ |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
306 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 307 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
308 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 309 ≈↑⟨ assoc ⟩ |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
310 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 311 ∎ |
| 245 | 312 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
| 313 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer (eqa {a} {b} {c} f g {e} )≈ equalizer (eqa {a} {b} {c} f g' {e} ) ] | |
| 314 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
| 247 | 315 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} → |
|
246
80d9ef47566b
cong-γ1 done, cong-δ1 remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
245
diff
changeset
|
316 A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) |
|
80d9ef47566b
cong-γ1 done, cong-δ1 remains
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
245
diff
changeset
|
317 ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] |
| 247 | 318 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
| 245 | 319 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 ) |
| 320 ≈⟨ uniqueness (eqa f g) ( begin | |
| 248 | 321 e o 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' ) |
| 322 ≈⟨ ek=h (eqa f g ) ⟩ | |
| 323 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 324 ≈↑⟨ car h=h' ⟩ | |
| 325 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 245 | 326 ∎ )⟩ |
| 327 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' ) | |
| 328 ∎ | |
| 249 | 329 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) ≈ |
| 330 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
| 247 | 331 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
| 332 begin | |
| 253 | 333 k (eqa {a} {b} {c} f f {e} ) (id a) (f1=f1 f) |
| 247 | 334 ≈⟨ uniqueness (eqa f f) ( begin |
| 253 | 335 e o k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 336 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
| 253 | 337 id a |
| 247 | 338 ∎ )⟩ |
| 253 | 339 k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 340 ∎ |
| 341 | |
| 230 | 342 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
343 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) ] |
|
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
344 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 345 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 346 begin |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
347 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 | 348 ≈⟨ ek=h (eqa f g) ⟩ |
|
233
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
349 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 350 ∎ |
| 230 | 351 |
| 352 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
353 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
|
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
354 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
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
355 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
| 249 | 356 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) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] |
| 222 | 357 ≈ j ] |
| 230 | 358 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 359 begin |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
360 ( 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
4bba19bc71be
e is now explict parameter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
232
diff
changeset
|
361 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
| 249 | 362 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) ) |
|
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
363 ≈⟨ car ((uniqueness (eqa f g) ( begin |
|
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
364 equalizer (eqa f g) o j |
|
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
365 ≈↑⟨ idR ⟩ |
| 253 | 366 (equalizer (eqa f g) o j ) o id d |
| 367 ≈⟨⟩ -- here we decide e (fej) (gej)' is id d | |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
368 ((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
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
369 ∎ ))) ⟩ |
| 249 | 370 j o k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) |
|
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
371 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
| 253 | 372 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id d |
|
236
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
373 ≈⟨ idR ⟩ |
|
e20b81102eee
Burroni equational equalizer definition done.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
235
diff
changeset
|
374 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
| 253 | 375 ≈⟨⟩ -- here we decide e (fej) (fej)' is id d |
| 376 id d | |
|
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
377 ∎ ))) ⟩ |
| 253 | 378 j o id d |
|
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
379 ≈⟨ idR ⟩ |
| 222 | 380 j |
|
235
8835015a3e1a
passed let;s remove yellow
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
234
diff
changeset
|
381 ∎ |
| 211 | 382 |
| 251 | 383 -------------------------------- |
| 384 -- | |
| 385 -- Bourroni equations gives an Equalizer | |
| 386 -- | |
| 211 | 387 |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
388 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
| 245 | 389 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g e) f g |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
390 lemma-equ2 {a} {b} {c} f g e bur = record { |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
391 fe=ge = fe=ge1 ; |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
392 k = k1 ; |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
393 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
394 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
395 } where |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
396 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
397 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
| 245 | 398 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
399 fe=ge1 = b1 bur |
| 245 | 400 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 ] |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
401 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
402 begin |
| 245 | 403 α bur f g e o k1 h eq |
| 239 | 404 ≈⟨⟩ |
| 253 | 405 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) ) |
| 239 | 406 ≈⟨ assoc ⟩ |
| 253 | 407 ( 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) |
| 239 | 408 ≈⟨ car (b2 bur) ⟩ |
| 253 | 409 ( h o ( α bur ( f o h ) ( g o h ) (id d))) o δ bur {d} {b} {d} (id d) (f o h) |
| 239 | 410 ≈↑⟨ assoc ⟩ |
| 253 | 411 h o ((( α bur ( f o h ) ( g o h ) (id d) )) o δ bur {d} {b} {d} (id d) (f o h) ) |
| 240 | 412 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
| 253 | 413 h o ((( α bur ( f o h ) ( f o h ) (id d)))o δ bur {d} {b} {d} (id d) (f o h) ) |
| 414 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id d} ) ⟩ | |
| 415 h o id d | |
| 240 | 416 ≈⟨ idR ⟩ |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
417 h |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
418 ∎ |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
419 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
| 245 | 420 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
421 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
422 begin |
| 240 | 423 k1 {d} h eq |
| 239 | 424 ≈⟨⟩ |
| 253 | 425 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) |
| 240 | 426 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
| 253 | 427 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) (f o h) |
| 245 | 428 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ |
| 253 | 429 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') ) |
| 240 | 430 ≈⟨ b4 bur ⟩ |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
431 k' |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
432 ∎ |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
433 |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
434 |
| 225 | 435 -- end |
| 212 | 436 |
| 437 | |
| 438 | |
| 215 | 439 |
| 440 |