1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "cprop_connectives.ma".
17 record equivalence_relation (A:Type0) : Type1 ≝
18 { eq_rel:2> A → A → CProp0;
19 refl: reflexive ? eq_rel;
20 sym: symmetric ? eq_rel;
21 trans: transitive ? eq_rel
24 record setoid : Type1 ≝
26 eq: equivalence_relation carr
29 record equivalence_relation1 (A:Type1) : Type2 ≝
30 { eq_rel1:2> A → A → CProp1;
31 refl1: reflexive1 ? eq_rel1;
32 sym1: symmetric1 ? eq_rel1;
33 trans1: transitive1 ? eq_rel1
36 record setoid1: Type2 ≝
38 eq1: equivalence_relation1 carr1
41 definition setoid1_of_setoid: setoid → setoid1.
53 coercion setoid1_of_setoid.
54 prefer coercion Type_OF_setoid.
56 record equivalence_relation2 (A:Type2) : Type3 ≝
57 { eq_rel2:2> A → A → CProp2;
58 refl2: reflexive2 ? eq_rel2;
59 sym2: symmetric2 ? eq_rel2;
60 trans2: transitive2 ? eq_rel2
63 record setoid2: Type3 ≝
65 eq2: equivalence_relation2 carr2
68 definition setoid2_of_setoid1: setoid1 → setoid2.
80 coercion setoid2_of_setoid1.
81 prefer coercion Type_OF_setoid2.
82 prefer coercion Type_OF_setoid.
83 prefer coercion Type_OF_setoid1.
84 (* we prefer 0 < 1 < 2 *)
86 record equivalence_relation3 (A:Type3) : Type4 ≝
87 { eq_rel3:2> A → A → CProp3;
88 refl3: reflexive3 ? eq_rel3;
89 sym3: symmetric3 ? eq_rel3;
90 trans3: transitive3 ? eq_rel3
93 record setoid3: Type4 ≝
95 eq3: equivalence_relation3 carr3
99 interpretation "setoid3 eq" 'eq x y = (eq_rel3 _ (eq3 _) x y).
100 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
101 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
102 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
103 interpretation "setoid3 symmetry" 'invert r = (sym3 ____ r).
104 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
105 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
106 interpretation "setoid symmetry" 'invert r = (sym ____ r).
107 notation ".= r" with precedence 50 for @{'trans $r}.
108 interpretation "trans3" 'trans r = (trans3 _____ r).
109 interpretation "trans2" 'trans r = (trans2 _____ r).
110 interpretation "trans1" 'trans r = (trans1 _____ r).
111 interpretation "trans" 'trans r = (trans _____ r).
113 record unary_morphism (A,B: setoid) : Type0 ≝
115 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
118 record unary_morphism1 (A,B: setoid1) : Type1 ≝
120 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
123 record unary_morphism2 (A,B: setoid2) : Type2 ≝
125 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
128 record unary_morphism3 (A,B: setoid3) : Type3 ≝
130 prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
133 record binary_morphism (A,B,C:setoid) : Type0 ≝
135 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
138 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
139 { fun21:2> A → B → C;
140 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
143 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
144 { fun22:2> A → B → C;
145 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
148 record binary_morphism3 (A,B,C:setoid3) : Type3 ≝
149 { fun23:2> A → B → C;
150 prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 a' b')
153 notation "† c" with precedence 90 for @{'prop1 $c }.
154 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
155 notation "#" with precedence 90 for @{'refl}.
156 interpretation "prop1" 'prop1 c = (prop1 _____ c).
157 interpretation "prop11" 'prop1 c = (prop11 _____ c).
158 interpretation "prop12" 'prop1 c = (prop12 _____ c).
159 interpretation "prop13" 'prop1 c = (prop13 _____ c).
160 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
161 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
162 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
163 interpretation "prop23" 'prop2 l r = (prop23 ________ l r).
164 interpretation "refl" 'refl = (refl ___).
165 interpretation "refl1" 'refl = (refl1 ___).
166 interpretation "refl2" 'refl = (refl2 ___).
167 interpretation "refl3" 'refl = (refl3 ___).
169 definition unary_morphism2_of_unary_morphism1: ∀S,T.unary_morphism1 S T → unary_morphism2 S T.
172 [ apply (fun11 ?? u);
173 | apply (prop11 ?? u); ]
176 definition CPROP: setoid1.
181 | intros 1; split; intro; assumption
182 | intros 3; cases i; split; assumption
183 | intros 5; cases i; cases i1; split; intro;
184 [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
187 definition CProp0_of_CPROP: carr1 CPROP → CProp0 ≝ λx.x.
188 coercion CProp0_of_CPROP.
190 alias symbol "eq" = "setoid1 eq".
191 definition fi': ∀A,B:CPROP. A = B → B → A.
192 intros; apply (fi ?? e); assumption.
195 notation ". r" with precedence 50 for @{'fi $r}.
196 interpretation "fi" 'fi r = (fi' __ r).
198 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
201 | intros; split; intro; cases a1; split;
203 | apply (if ?? e1 b1)
205 | apply (fi ?? e1 b1)]]
208 interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
210 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
213 | intros; split; intro; cases o; [1,3:left |2,4: right]
216 | apply (if ?? e1 b1)
217 | apply (fi ?? e1 b1)]]
220 interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
222 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
224 [ apply (λA,B. A → B)
225 | intros; split; intros;
226 [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
227 | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
231 record category : Type1 ≝
233 arrows: objs → objs → setoid;
234 id: ∀o:objs. arrows o o;
235 comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
236 comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
237 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
238 id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
239 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
242 record category1 : Type2 ≝
244 arrows1: objs1 → objs1 → setoid1;
245 id1: ∀o:objs1. arrows1 o o;
246 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
247 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
248 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
249 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
250 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
253 record category2 : Type3 ≝
255 arrows2: objs2 → objs2 → setoid2;
256 id2: ∀o:objs2. arrows2 o o;
257 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
258 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
259 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
260 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
261 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
264 record category3 : Type4 ≝
266 arrows3: objs3 → objs3 → setoid3;
267 id3: ∀o:objs3. arrows3 o o;
268 comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
269 comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
270 comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 = comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
271 id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a = a;
272 id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) = a
275 notation "'ASSOC'" with precedence 90 for @{'assoc}.
277 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
278 interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
279 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
280 interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
281 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
282 interpretation "category assoc" 'assoc = (comp_assoc ________).
284 definition category2_of_category1: category1 → category2.
288 | intros; apply (setoid2_of_setoid1 (arrows1 c o o1));
292 [ intros; apply (comp1 c o1 o2 o3 c1 c2);
293 | intros; whd in e e1 a a' b b'; change with (eq1 ? (b∘a) (b'∘a')); apply (e‡e1); ]
294 | intros; simplify; whd in a12 a23 a34; whd; apply rule (ASSOC);
295 | intros; simplify; whd in a; whd; apply id_neutral_right1;
296 | intros; simplify; whd in a; whd; apply id_neutral_left1; ]
298 (*coercion category2_of_category1.*)
300 record functor2 (C1: category2) (C2: category2) : Type3 ≝
301 { map_objs2:1> C1 → C2;
302 map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
303 respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
305 ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
306 map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
308 definition functor2_setoid: category2 → category2 → setoid3.
311 [ apply (functor2 C1 C2);
314 apply (∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
315 | simplify; intros; apply cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
316 | simplify; intros; apply cic:/matita/logic/equality/sym_eq.con; apply H;
317 | simplify; intros; apply cic:/matita/logic/equality/trans_eq.con;
318 [2: apply H; | skip | apply H1;]]]
321 definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
322 coercion functor2_of_functor2_setoid.
324 definition CAT2: category3.
327 | apply functor2_setoid;
328 | intros; constructor 1;
330 | intros; constructor 1;
332 | intros; assumption;]
333 | intros; apply rule #;
334 | intros; apply rule #; ]
335 | intros; constructor 1;
336 [ intros; constructor 1;
337 [ intros; apply (c1 (c o));
338 | intros; constructor 1;
339 [ intro; apply (map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
340 | intros; apply (††e); ]
342 apply (.= †(respects_id2 : ?));
343 apply (respects_id2 : ?);
345 apply (.= †(respects_comp2 : ?));
346 apply (respects_comp2 : ?); ]
347 | intros; intro; simplify;
348 apply (cic:/matita/logic/equality/eq_ind.con ????? (e ?));
349 apply (cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
351 | intros; intro; simplify; constructor 1;
352 | intros; intro; simplify; constructor 1;
353 | intros; intro; simplify; constructor 1; ]
356 definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
357 coercion category2_of_objs3_CAT2.
359 definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
360 coercion functor2_setoid_of_arrows3_CAT2.
362 definition unary_morphism_setoid: setoid → setoid → setoid.
365 [ apply (unary_morphism s s1);
367 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
368 | intros 1; simplify; intros; apply refl;
369 | simplify; intros; apply sym; apply f;
370 | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
373 definition SET: category1.
376 | apply rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
377 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
378 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
380 | intros; whd; intros; simplify; whd in H1; whd in H;
381 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
382 [ apply Hletin | apply (e a1); ] | apply e1; ]]
383 | intros; whd; intros; simplify; apply refl;
384 | intros; simplify; whd; intros; simplify; apply refl;
385 | intros; simplify; whd; intros; simplify; apply refl;
389 definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
390 coercion setoid_of_SET.
392 definition unary_morphism_setoid_of_arrows1_SET:
393 ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
394 coercion unary_morphism_setoid_of_arrows1_SET.
396 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
397 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
399 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
402 [ apply (unary_morphism1 s s1);
405 alias symbol "eq" = "setoid1 eq".
406 apply (∀a: carr1 s. f a = g a);
407 | intros 1; simplify; intros; apply refl1;
408 | simplify; intros; apply sym1; apply f;
409 | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
412 definition unary_morphism1_of_unary_morphism1_setoid1 :
413 ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
414 coercion unary_morphism1_of_unary_morphism1_setoid1.
416 definition SET1: objs3 CAT2.
419 | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
420 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
421 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
423 | intros; whd; intros; simplify; whd in H1; whd in H;
424 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
425 [ apply Hletin | apply (e a1); ] | apply e1; ]]
426 | intros; whd; intros; simplify; apply refl1;
427 | intros; simplify; whd; intros; simplify; apply refl1;
428 | intros; simplify; whd; intros; simplify; apply refl1;
432 definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
433 coercion setoid1_of_SET1.
435 definition unary_morphism1_setoid1_of_arrows2_SET1:
436 ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
437 coercion unary_morphism1_setoid1_of_arrows2_SET1.
439 variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
440 coercion objs2_of_category1.
442 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
443 prefer coercion Type_OF_objs1.
445 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).