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: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
30 definition symmetric1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
31 definition transitive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ 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 coercion setoid1_of_setoid.
58 prefer coercion Type_OF_setoid.
60 definition reflexive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
61 definition symmetric2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
62 definition transitive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ 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.
89 prefer coercion Type_OF_setoid2.
90 prefer coercion Type_OF_setoid.
91 prefer coercion Type_OF_setoid1.
92 (* we prefer 0 < 1 < 2 *)
94 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
95 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
96 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
97 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
98 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
99 interpretation "setoid symmetry" 'invert r = (sym ____ r).
100 notation ".= r" with precedence 50 for @{'trans $r}.
101 interpretation "trans2" 'trans r = (trans2 _____ r).
102 interpretation "trans1" 'trans r = (trans1 _____ r).
103 interpretation "trans" 'trans r = (trans _____ r).
105 record unary_morphism (A,B: setoid) : Type0 ≝
107 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
110 record unary_morphism1 (A,B: setoid1) : Type1 ≝
112 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
115 record unary_morphism2 (A,B: setoid2) : Type2 ≝
117 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
120 record binary_morphism (A,B,C:setoid) : Type0 ≝
122 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
125 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
126 { fun21:2> A → B → C;
127 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
130 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
131 { fun22:2> A → B → C;
132 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
135 notation "† c" with precedence 90 for @{'prop1 $c }.
136 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
137 notation "#" with precedence 90 for @{'refl}.
138 interpretation "prop1" 'prop1 c = (prop1 _____ c).
139 interpretation "prop11" 'prop1 c = (prop11 _____ c).
140 interpretation "prop12" 'prop1 c = (prop12 _____ c).
141 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
142 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
143 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
144 interpretation "refl" 'refl = (refl ___).
145 interpretation "refl1" 'refl = (refl1 ___).
146 interpretation "refl2" 'refl = (refl2 ___).
148 definition CPROP: setoid1.
153 | intros 1; split; intro; assumption
154 | intros 3; cases i; split; assumption
155 | intros 5; cases i; cases i1; split; intro;
156 [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
159 definition CProp0_of_CPROP: carr1 CPROP → CProp0 ≝ λx.x.
160 coercion CProp0_of_CPROP.
162 alias symbol "eq" = "setoid1 eq".
163 definition fi': ∀A,B:CPROP. A = B → B → A.
164 intros; apply (fi ?? e); assumption.
167 notation ". r" with precedence 50 for @{'fi $r}.
168 interpretation "fi" 'fi r = (fi' __ r).
170 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
173 | intros; split; intro; cases a1; split;
175 | apply (if ?? e1 b1)
177 | apply (fi ?? e1 b1)]]
180 interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
182 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
185 | intros; split; intro; cases o; [1,3:left |2,4: right]
188 | apply (if ?? e1 b1)
189 | apply (fi ?? e1 b1)]]
192 interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
194 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
196 [ apply (λA,B. A → B)
197 | intros; split; intros;
198 [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
199 | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
203 record category : Type1 ≝
205 arrows: objs → objs → setoid;
206 id: ∀o:objs. arrows o o;
207 comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
208 comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
209 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
210 id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
211 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
214 record category1 : Type2 ≝
216 arrows1: objs1 → objs1 → setoid1;
217 id1: ∀o:objs1. arrows1 o o;
218 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
219 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
220 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
221 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
222 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
225 record category2 : Type3 ≝
227 arrows2: objs2 → objs2 → setoid2;
228 id2: ∀o:objs2. arrows2 o o;
229 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
230 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
231 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
232 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
233 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
236 notation "'ASSOC'" with precedence 90 for @{'assoc}.
238 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
239 interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
240 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
241 interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
242 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
243 interpretation "category assoc" 'assoc = (comp_assoc ________).
245 definition unary_morphism_setoid: setoid → setoid → setoid.
248 [ apply (unary_morphism s s1);
250 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
251 | intros 1; simplify; intros; apply refl;
252 | simplify; intros; apply sym; apply f;
253 | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
256 definition SET: category1.
259 | apply rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
260 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
261 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
263 | intros; whd; intros; simplify; whd in H1; whd in H;
264 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
265 [ apply Hletin | apply (e a1); ] | apply e1; ]]
266 | intros; whd; intros; simplify; apply refl;
267 | intros; simplify; whd; intros; simplify; apply refl;
268 | intros; simplify; whd; intros; simplify; apply refl;
272 definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
273 coercion setoid_of_SET.
275 definition unary_morphism_setoid_of_arrows1_SET:
276 ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
277 coercion unary_morphism_setoid_of_arrows1_SET.
279 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
280 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
282 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
285 [ apply (unary_morphism1 s s1);
288 alias symbol "eq" = "setoid1 eq".
289 apply (∀a: carr1 s. f a = g a);
290 | intros 1; simplify; intros; apply refl1;
291 | simplify; intros; apply sym1; apply f;
292 | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
295 definition unary_morphism1_of_unary_morphism1_setoid1 :
296 ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
297 coercion unary_morphism1_of_unary_morphism1_setoid1.
299 definition SET1: category2.
302 | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
303 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
304 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
306 | intros; whd; intros; simplify; whd in H1; whd in H;
307 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
308 [ apply Hletin | apply (e a1); ] | apply e1; ]]
309 | intros; whd; intros; simplify; apply refl1;
310 | intros; simplify; whd; intros; simplify; apply refl1;
311 | intros; simplify; whd; intros; simplify; apply refl1;
315 definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
316 coercion setoid1_of_SET1.
318 definition unary_morphism1_setoid1_of_arrows2_SET1:
319 ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
320 coercion unary_morphism1_setoid1_of_arrows2_SET1.
322 variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
323 coercion objs2_of_category1.
325 prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
326 prefer coercion Type_OF_objs1.
328 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).