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annotate equalizer.agda @ 221:ea0407fb8f02
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 04 Sep 2013 20:35:43 +0900 |
parents | 5d96be63053f |
children | 0bc85361b7d0 |
rev | line source |
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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 |
221 | 26 fe=ge : 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 ] |
207
22811f7a04e1
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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 ] |
207
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parents:
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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 ]) ] | |
207
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47 |
209 | 48 open Equalizer |
49 open EqEqualizer | |
50 | |
219 | 51 |
52 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 ] ] | |
53 f1=g1 eq = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
54 | |
55 | |
56 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
57 A [ e eqa ≈ id1 A a ] → | |
58 A [ k eqa (id1 A a) (f1=g1 eq) ≈ id1 A a ] | |
59 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in | |
60 begin | |
61 k eqa (id1 A a) (f1=g1 eq) | |
62 ≈⟨ uniqueness eqa ( begin | |
63 e eqa o id1 A a | |
64 ≈⟨ idR ⟩ | |
65 e eqa | |
66 ≈⟨ e=1 ⟩ | |
67 id1 A a | |
68 ∎ )⟩ | |
69 id1 A a | |
70 ∎ | |
71 | |
217 | 72 -- e eqa f g f |
73 -- c ----------> a ------->b | |
218 | 74 -- ^ ---> d ---> |
75 -- | i h | |
76 -- j| k' (d' → d) | |
77 -- | k (d' → a) | |
78 -- d' | |
217 | 79 |
218 | 80 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 ) |
81 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
217 | 82 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
218 | 83 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
84 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
221 | 85 fe=ge = fe=ge1 ; |
217 | 86 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
87 ek=h = ek=h1 ; | |
88 uniqueness = uniqueness1 | |
89 } where | |
90 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
91 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
92 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
93 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
94 begin | |
95 f o ( h o j ) | |
96 ≈⟨ assoc ⟩ | |
97 (f o h ) o j | |
98 ≈⟨ eq' ⟩ | |
99 (g o h ) o j | |
100 ≈↑⟨ assoc ⟩ | |
101 g o ( h o j ) | |
102 ∎ | |
221 | 103 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
104 fe=ge1 = let open ≈-Reasoning (A) in | |
217 | 105 begin |
106 ( f o h ) o i | |
107 ≈↑⟨ assoc ⟩ | |
108 f o (h o i ) | |
109 ≈⟨ cdr eq ⟩ | |
110 f o (e eqa) | |
221 | 111 ≈⟨ fe=ge eqa ⟩ |
217 | 112 g o (e eqa) |
113 ≈↑⟨ cdr eq ⟩ | |
114 g o (h o i ) | |
115 ≈⟨ assoc ⟩ | |
116 ( g o h ) o i | |
117 ∎ | |
218 | 118 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' ] ]} → |
119 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
120 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
217 | 121 begin |
218 | 122 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') |
123 ≈↑⟨ idL ⟩ | |
124 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
125 ≈↑⟨ car h-1-id ⟩ | |
126 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
127 ≈↑⟨ assoc ⟩ | |
128 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
129 ≈⟨ cdr assoc ⟩ | |
130 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
131 ≈⟨ cdr (car eq ) ⟩ | |
132 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
133 ≈⟨ cdr (ek=h eqa) ⟩ | |
134 h-1 o ( h o k' ) | |
135 ≈⟨ assoc ⟩ | |
136 ( h-1 o h ) o k' | |
137 ≈⟨ car h-1-id ⟩ | |
138 id1 A d o k' | |
139 ≈⟨ idL ⟩ | |
140 k' | |
217 | 141 ∎ |
142 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} → | |
143 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
144 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
145 begin | |
146 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
147 ≈⟨ uniqueness eqa ( begin | |
148 e eqa o k' | |
149 ≈↑⟨ car eq ⟩ | |
150 (h o i ) o k' | |
151 ≈↑⟨ assoc ⟩ | |
152 h o (i o k') | |
153 ≈⟨ cdr ik=h ⟩ | |
154 h o h' | |
155 ∎ ) ⟩ | |
156 k' | |
157 ∎ | |
215 | 158 |
218 | 159 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 ) |
160 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
161 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
162 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
163 e = e eqa ; | |
221 | 164 fe=ge = fe=ge1 ; |
218 | 165 k = λ j eq' → k eqa j (fj=gj j eq') ; |
166 ek=h = ek=h1 ; | |
167 uniqueness = uniqueness1 | |
168 } where | |
169 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 ] ] | |
170 fj=gj j eq = let open ≈-Reasoning (A) in | |
171 begin | |
172 f o j | |
173 ≈↑⟨ idL ⟩ | |
174 id1 A b o ( f o j ) | |
175 ≈↑⟨ car h-1-id ⟩ | |
176 (h-1 o h ) o ( f o j ) | |
177 ≈↑⟨ assoc ⟩ | |
178 h-1 o (h o ( f o j )) | |
179 ≈⟨ cdr assoc ⟩ | |
180 h-1 o ((h o f) o j ) | |
181 ≈⟨ cdr eq ⟩ | |
182 h-1 o ((h o g) o j ) | |
183 ≈↑⟨ cdr assoc ⟩ | |
184 h-1 o (h o ( g o j )) | |
185 ≈⟨ assoc ⟩ | |
186 (h-1 o h) o ( g o j ) | |
187 ≈⟨ car h-1-id ⟩ | |
188 id1 A b o ( g o j ) | |
189 ≈⟨ idL ⟩ | |
190 g o j | |
191 ∎ | |
221 | 192 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
193 fe=ge1 = let open ≈-Reasoning (A) in | |
218 | 194 begin |
195 ( h o f ) o e eqa | |
196 ≈↑⟨ assoc ⟩ | |
197 h o (f o e eqa ) | |
221 | 198 ≈⟨ cdr (fe=ge eqa) ⟩ |
218 | 199 h o (g o e eqa ) |
200 ≈⟨ assoc ⟩ | |
201 ( h o g ) o e eqa | |
202 ∎ | |
203 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 ] ]} → | |
204 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
205 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
206 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} → | |
207 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
208 uniqueness1 = uniqueness eqa | |
209 | |
220 | 210 eefeg : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
211 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
212 eefeg {a} {b} {c} {d} {f} {g} eqa = record { | |
213 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
221 | 214 fe=ge = fe=ge1 ; |
220 | 215 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
216 ek=h = ek=h1 ; | |
217 uniqueness = uniqueness1 | |
218 } where | |
219 i = id1 A c | |
220 h = e eqa | |
221 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → | |
222 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
223 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
224 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
225 begin | |
226 f o ( h o j ) | |
227 ≈⟨ assoc ⟩ | |
228 (f o h ) o j | |
229 ≈⟨ eq' ⟩ | |
230 (g o h ) o j | |
231 ≈↑⟨ assoc ⟩ | |
232 g o ( h o j ) | |
233 ∎ | |
221 | 234 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
235 fe=ge1 = let open ≈-Reasoning (A) in | |
220 | 236 begin |
237 ( f o h ) o i | |
221 | 238 ≈⟨ car ( fe=ge eqa ) ⟩ |
220 | 239 ( g o h ) o i |
240 ∎ | |
241 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' ] ]} → | |
242 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
243 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
244 begin | |
245 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') | |
246 ≈⟨ idL ⟩ | |
247 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
248 ≈⟨ uniqueness eqa refl-hom ⟩ | |
249 k' | |
250 ∎ | |
251 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} → | |
252 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
253 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
254 begin | |
255 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
256 ≈⟨ uniqueness eqa ( begin | |
257 e eqa o k' | |
258 ≈↑⟨ cdr idL ⟩ | |
259 e eqa o (id1 A c o k' ) | |
260 ≈⟨ cdr ik=h ⟩ | |
261 e eqa o h' | |
262 ∎ ) ⟩ | |
263 k' | |
264 ∎ | |
265 | |
221 | 266 -- Equalizer is unique up to iso |
267 | |
268 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 ) | |
269 → Hom A c c' --- != id1 A c | |
270 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (fe=ge eqa) | |
220 | 271 |
221 | 272 -- |
273 -- | |
274 -- e eqa f g f | |
275 -- c ----------> a ------->b | |
276 -- | |
277 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 ) | |
278 → A [ A [ equalizer-iso eqa eqa' o equalizer-iso eqa' eqa ] ≈ id1 A c' ] | |
279 equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' = let open ≈-Reasoning (A) in | |
280 begin | |
281 k eqa' (e eqa) (fe=ge eqa) o k eqa (e eqa' ) (fe=ge eqa' ) | |
282 ≈⟨ {!!} ⟩ | |
283 id1 A c' | |
284 ∎ | |
285 | |
286 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 | |
287 -- ke = e' | |
288 -- k'ke = k'e' = e kk' = 1 | |
289 | |
290 -- x e = e -> x = id? | |
218 | 291 |
211 | 292 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
293 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
209 | 294 lemma-equ1 A {a} {b} {c} f g eqa = record { |
216 | 295 α = λ f g → e (eqa f g ) ; -- Hom A c a |
214 | 296 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 | 297 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
221 | 298 b1 = fe=ge (eqa f g) ; |
212 | 299 b2 = lemma-equ5 ; |
300 b3 = lemma-equ3 ; | |
215 | 301 b4 = lemma-equ6 |
211 | 302 } where |
216 | 303 -- |
304 -- e eqa f g f | |
305 -- c ----------> a ------->b | |
306 -- ^ g | |
307 -- | | |
308 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
309 -- | | |
310 -- d | |
311 -- | |
312 -- | |
313 -- e o id1 ≈ e → k e ≈ id | |
314 | |
211 | 315 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
316 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
213 | 317 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
318 lemma-equ3 = let open ≈-Reasoning (A) in | |
211 | 319 begin |
320 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
215 | 321 ≈⟨ ek=h (eqa f f ) ⟩ |
211 | 322 id1 A a |
323 ∎ | |
214 | 324 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
212 | 325 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 | 326 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
212 | 327 begin |
328 f o ( h o e (eqa (f o h) ( g o h ))) | |
329 ≈⟨ assoc ⟩ | |
330 (f o h) o e (eqa (f o h) ( g o h )) | |
221 | 331 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
212 | 332 (g o h) o e (eqa (f o h) ( g o h )) |
333 ≈↑⟨ assoc ⟩ | |
334 g o ( h o e (eqa (f o h) ( g o h ))) | |
335 ∎ | |
336 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
214 | 337 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 | 338 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
339 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
340 begin | |
215 | 341 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) |
342 ≈⟨ ek=h (eqa f g) ⟩ | |
212 | 343 h o e (eqa (f o h ) ( g o h )) |
344 ∎ | |
215 | 345 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
346 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 ] ])) ]) | |
347 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
348 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 ] ])) ] | |
349 ≈ k₁ ] | |
350 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
351 begin | |
352 ( 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 ) ))) )) | |
353 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
354 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 ) ))) ) | |
355 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
356 e (eqa f g) o k₁ | |
357 ≈⟨ {!!} ⟩ | |
358 (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)) | |
359 ∎ )) ⟩ | |
360 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 ) ))) | |
361 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
362 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
363 ≈⟨ {!!} ⟩ | |
364 id1 A d | |
365 ∎ )) ⟩ | |
366 k₁ o id1 A d | |
367 ≈⟨ idR ⟩ | |
368 k₁ | |
369 ∎ | |
211 | 370 |
371 | |
212 | 372 |
373 | |
374 | |
215 | 375 |
376 |