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 definition reflexive1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
30 definition symmetric1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
31 definition transitive1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
33 record equivalence_relation1 (A:Type1) : Type2 ≝
34 { eq_rel1:2> A → A → CProp1;
35 refl1: reflexive1 ? eq_rel1;
36 sym1: symmetric1 ? eq_rel1;
37 trans1: transitive1 ? eq_rel1
40 record setoid1: Type2 ≝
42 eq1: equivalence_relation1 carr1
45 definition setoid1_of_setoid: setoid → setoid1.
57 (* questa coercion e' necessaria per problemi di unificazione *)
58 coercion setoid1_of_setoid.
60 definition reflexive2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
61 definition symmetric2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
62 definition transitive2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
64 record equivalence_relation2 (A:Type2) : Type3 ≝
65 { eq_rel2:2> A → A → CProp2;
66 refl2: reflexive2 ? eq_rel2;
67 sym2: symmetric2 ? eq_rel2;
68 trans2: transitive2 ? eq_rel2
71 record setoid2: Type3 ≝
73 eq2: equivalence_relation2 carr2
76 definition setoid2_of_setoid1: setoid1 → setoid2.
88 (*coercion setoid2_of_setoid1.*)
91 definition Leibniz: Type → setoid.
96 [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
97 | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
99 | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
101 | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
108 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
109 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
110 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
111 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
112 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
113 interpretation "setoid symmetry" 'invert r = (sym ____ r).
114 notation ".= r" with precedence 50 for @{'trans $r}.
115 interpretation "trans2" 'trans r = (trans2 _____ r).
116 interpretation "trans1" 'trans r = (trans1 _____ r).
117 interpretation "trans" 'trans r = (trans _____ r).
119 record unary_morphism (A,B: setoid) : Type0 ≝
121 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
124 record unary_morphism1 (A,B: setoid1) : Type1 ≝
126 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
129 record unary_morphism2 (A,B: setoid2) : Type2 ≝
131 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
134 record binary_morphism (A,B,C:setoid) : Type0 ≝
136 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
139 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
140 { fun21:2> A → B → C;
141 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
144 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
145 { fun22:2> A → B → C;
146 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
149 notation "† c" with precedence 90 for @{'prop1 $c }.
150 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
151 notation "#" with precedence 90 for @{'refl}.
152 interpretation "prop1" 'prop1 c = (prop1 _____ c).
153 interpretation "prop11" 'prop1 c = (prop11 _____ c).
154 interpretation "prop12" 'prop1 c = (prop12 _____ c).
155 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
156 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
157 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
158 interpretation "refl" 'refl = (refl ___).
159 interpretation "refl1" 'refl = (refl1 ___).
160 interpretation "refl2" 'refl = (refl2 ___).
162 definition CPROP: setoid1.
167 | intros 1; split; intro; assumption
168 | intros 3; cases i; split; assumption
169 | intros 5; cases i; cases i1; split; intro;
170 [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
173 alias symbol "eq" = "setoid1 eq".
174 definition if': ∀A,B:CPROP. A = B → A → B.
175 intros; apply (if ?? e); assumption.
178 notation ". r" with precedence 50 for @{'if $r}.
179 interpretation "if" 'if r = (if' __ r).
181 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
184 | intros; split; intro; cases a1; split;
186 | apply (if ?? e1 b1)
188 | apply (fi ?? e1 b1)]]
191 interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
193 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
196 | intros; split; intro; cases o; [1,3:left |2,4: right]
199 | apply (if ?? e1 b1)
200 | apply (fi ?? e1 b1)]]
203 interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
205 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
207 [ apply (λA,B. A → B)
208 | intros; split; intros;
209 [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
210 | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
214 definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
217 [ apply (eq_rel ? (eq S))
218 | intros; split; intro;
219 [ apply (.= H \sup -1);
228 record category : Type1 ≝
230 arrows: objs → objs → setoid;
231 id: ∀o:objs. arrows o o;
232 comp: ∀o1,o2,o3. binary_morphism1 (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
233 comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
234 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
235 id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
236 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
239 record category1 : Type2 ≝
241 arrows1: objs1 → objs1 → setoid1;
242 id1: ∀o:objs1. arrows1 o o;
243 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
244 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
245 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
246 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
247 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
250 record category2 : Type3 ≝
252 arrows2: objs2 → objs2 → setoid2;
253 id2: ∀o:objs2. arrows2 o o;
254 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
255 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
256 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
257 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
258 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
261 notation "'ASSOC'" with precedence 90 for @{'assoc}.
263 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
264 interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
265 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
266 interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
267 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
268 interpretation "category assoc" 'assoc = (comp_assoc ________).
270 (* bug grande come una casa?
271 Ma come fa a passare la quantificazione larga??? *)
272 definition unary_morphism_setoid: setoid → setoid → setoid1.
275 [ apply (unary_morphism s s1);
277 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
278 | intros 1; simplify; intros; apply refl;
279 | simplify; intros; apply sym; apply f;
280 | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
283 definition SET: category1.
286 | apply rule (λS,T:setoid.unary_morphism_setoid S T);
287 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
288 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
290 | intros; whd; intros; simplify; whd in H1; whd in H;
291 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
292 [ apply Hletin | apply (e a1); ] | apply e1; ]]
293 | intros; whd; intros; simplify; apply refl;
294 | intros; simplify; whd; intros; simplify; apply refl;
295 | intros; simplify; whd; intros; simplify; apply refl;
299 definition setoid_of_SET: objs1 SET → setoid.
300 intros; apply o; qed.
301 coercion setoid_of_SET.
303 definition setoid1_of_SET: SET → setoid1.
304 intro; whd in t; apply setoid1_of_setoid; apply t.
306 coercion setoid1_of_SET.
308 definition eq': ∀w:SET.equivalence_relation ? := λw.eq w.
310 definition prop1_SET :
311 ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:Type_OF_objs1 A.eq' ? a b→eq' ? (w a) (w b).
312 intros; apply (prop1 A B w a b e);
316 interpretation "SET dagger" 'prop1 h = (prop1_SET _ _ _ _ _ h).
317 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
318 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
319 interpretation "SET eq" 'eq x y = (eq_rel _ (eq' _) x y).
321 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
324 [ apply (unary_morphism1 s s1);
327 alias symbol "eq" = "setoid1 eq".
328 apply (∀a: carr1 s. f a = g a);
329 | intros 1; simplify; intros; apply refl1;
330 | simplify; intros; apply sym1; apply f;
331 | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
334 definition SET1: category2.
337 | apply rule (λS,T.unary_morphism1_setoid1 S T);
338 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
339 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
341 | intros; whd; intros; simplify; whd in H1; whd in H;
342 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
343 [ apply Hletin | apply (e a1); ] | apply e1; ]]
344 | intros; whd; intros; simplify; apply refl1;
345 | intros; simplify; whd; intros; simplify; apply refl1;
346 | intros; simplify; whd; intros; simplify; apply refl1;
350 definition setoid1_OF_SET1: objs2 SET1 → setoid1.
351 intros; apply o; qed.
353 coercion setoid1_OF_SET1.
355 definition eq'': ∀w:SET1.equivalence_relation1 ? := λw.eq1 w.
357 definition prop11_SET1 :
358 ∀A,B:SET1.∀w:arrows2 SET1 A B.∀a,b:Type_OF_objs2 A.eq'' ? a b→eq'' ? (w a) (w b).
359 intros; apply (prop11 A B w a b e);
362 definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2.
363 intro; apply (setoid2_of_setoid1 t); qed.
364 coercion setoid2_OF_category2.
366 definition objs2_OF_category1: Type_OF_category1 SET → objs2 SET1.
367 intro; apply (setoid1_of_setoid t); qed.
368 coercion objs2_OF_category1.
370 definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
371 intro; whd in t; apply (carr1 t);
373 coercion Type1_OF_SET1.
375 definition Type_OF_setoid1_of_carr: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
376 [ apply setoid1_of_SET; apply U
379 coercion Type_OF_setoid1_of_carr.
381 interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
382 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
383 interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
385 lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
388 coercion unary_morphism1_of_arrows1_SET1.