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 "logic/cprop_connectives.ma".
17 definition Type0 := Type.
18 definition Type1 := Type.
19 definition Type2 := Type.
20 definition Type3 := Type.
21 definition Type0_lt_Type1 := (Type0 : Type1).
22 definition Type1_lt_Type2 := (Type1 : Type2).
23 definition Type2_lt_Type3 := (Type2 : Type3).
25 definition Type_OF_Type0: Type0 → Type := λx.x.
26 definition Type_OF_Type1: Type1 → Type := λx.x.
27 definition Type_OF_Type2: Type2 → Type := λx.x.
28 definition Type_OF_Type3: Type3 → Type := λx.x.
29 coercion Type_OF_Type0.
30 coercion Type_OF_Type1.
31 coercion Type_OF_Type2.
32 coercion Type_OF_Type3.
34 definition CProp0 := CProp.
35 definition CProp1 := CProp.
36 definition CProp2 := CProp.
37 definition CProp0_lt_CProp1 := (CProp0 : CProp1).
38 definition CProp1_lt_CProp2 := (CProp1 : CProp2).
40 definition CProp_OF_CProp0: CProp0 → CProp := λx.x.
41 definition CProp_OF_CProp1: CProp1 → CProp := λx.x.
42 definition CProp_OF_CProp2: CProp2 → CProp := λx.x.
44 record equivalence_relation (A:Type0) : Type1 ≝
45 { eq_rel:2> A → A → CProp0;
46 refl: reflexive ? eq_rel;
47 sym: symmetric ? eq_rel;
48 trans: transitive ? eq_rel
51 record setoid : Type1 ≝
53 eq: equivalence_relation carr
56 definition reflexive1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
57 definition symmetric1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
58 definition transitive1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
60 record equivalence_relation1 (A:Type1) : Type2 ≝
61 { eq_rel1:2> A → A → CProp1;
62 refl1: reflexive1 ? eq_rel1;
63 sym1: symmetric1 ? eq_rel1;
64 trans1: transitive1 ? eq_rel1
67 record setoid1: Type2 ≝
69 eq1: equivalence_relation1 carr1
72 definition setoid1_of_setoid: setoid → setoid1.
84 (* questa coercion e' necessaria per problemi di unificazione *)
85 coercion setoid1_of_setoid.
87 definition reflexive2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
88 definition symmetric2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
89 definition transitive2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
91 record equivalence_relation2 (A:Type2) : Type3 ≝
92 { eq_rel2:2> A → A → CProp2;
93 refl2: reflexive2 ? eq_rel2;
94 sym2: symmetric2 ? eq_rel2;
95 trans2: transitive2 ? eq_rel2
98 record setoid2: Type3 ≝
100 eq2: equivalence_relation2 carr2
103 definition setoid2_of_setoid1: setoid1 → setoid2.
115 (*coercion setoid2_of_setoid1.*)
118 definition Leibniz: Type → setoid.
123 [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
124 | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
126 | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
128 | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
135 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
136 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
137 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
138 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
139 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
140 interpretation "setoid symmetry" 'invert r = (sym ____ r).
141 notation ".= r" with precedence 50 for @{'trans $r}.
142 interpretation "trans2" 'trans r = (trans2 _____ r).
143 interpretation "trans1" 'trans r = (trans1 _____ r).
144 interpretation "trans" 'trans r = (trans _____ r).
146 record unary_morphism (A,B: setoid) : Type0 ≝
148 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
151 record unary_morphism1 (A,B: setoid1) : Type1 ≝
153 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
156 record unary_morphism2 (A,B: setoid2) : Type2 ≝
158 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
161 record binary_morphism (A,B,C:setoid) : Type0 ≝
163 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
166 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
167 { fun21:2> A → B → C;
168 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
171 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
172 { fun22:2> A → B → C;
173 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
176 notation "† c" with precedence 90 for @{'prop1 $c }.
177 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
178 notation "#" with precedence 90 for @{'refl}.
179 interpretation "prop1" 'prop1 c = (prop1 _____ c).
180 interpretation "prop11" 'prop1 c = (prop11 _____ c).
181 interpretation "prop12" 'prop1 c = (prop12 _____ c).
182 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
183 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
184 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
185 interpretation "refl" 'refl = (refl ___).
186 interpretation "refl1" 'refl = (refl1 ___).
187 interpretation "refl2" 'refl = (refl2 ___).
189 definition CPROP: setoid1.
194 | intros 1; split; intro; assumption
195 | intros 3; cases H; split; assumption
196 | intros 5; cases H; cases H1; split; intro;
197 [ apply (H4 (H2 x1)) | apply (H3 (H5 z1))]]]
200 alias symbol "eq" = "setoid1 eq".
201 definition if': ∀A,B:CPROP. A = B → A → B.
202 intros; apply (if ?? e); assumption.
205 notation ". r" with precedence 50 for @{'if $r}.
206 interpretation "if" 'if r = (if' __ r).
208 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
211 | intros; split; intro; cases H; split;
213 | apply (if ?? e1 b1)
215 | apply (fi ?? e1 b1)]]
218 interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
220 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
223 | intros; split; intro; cases H; [1,3:left |2,4: right]
226 | apply (if ?? e1 b1)
227 | apply (fi ?? e1 b1)]]
230 interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
232 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
234 [ apply (λA,B. A → B)
235 | intros; split; intros;
236 [ apply (if ?? e1); apply H; apply (fi ?? e); assumption
237 | apply (fi ?? e1); apply H; apply (if ?? e); assumption]]
241 definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
244 [ apply (eq_rel ? (eq S))
245 | intros; split; intro;
246 [ apply (.= H \sup -1);
255 record category : Type1 ≝
257 arrows: objs → objs → setoid;
258 id: ∀o:objs. arrows o o;
259 comp: ∀o1,o2,o3. binary_morphism1 (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
260 comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
261 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
262 id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
263 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
266 record category1 : Type2 ≝
268 arrows1: objs1 → objs1 → setoid1;
269 id1: ∀o:objs1. arrows1 o o;
270 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
271 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
272 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
273 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
274 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
277 record category2 : Type3 ≝
279 arrows2: objs2 → objs2 → setoid2;
280 id2: ∀o:objs2. arrows2 o o;
281 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
282 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
283 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
284 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
285 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
288 notation "'ASSOC'" with precedence 90 for @{'assoc}.
290 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
291 interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
292 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
293 interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
294 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
295 interpretation "category assoc" 'assoc = (comp_assoc ________).
297 (* bug grande come una casa?
298 Ma come fa a passare la quantificazione larga??? *)
299 definition unary_morphism_setoid: setoid → setoid → setoid1.
302 [ apply (unary_morphism s s1);
304 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
305 | intros 1; simplify; intros; apply refl;
306 | simplify; intros; apply sym; apply H;
307 | simplify; intros; apply trans; [2: apply H; | skip | apply H1]]]
310 definition SET: category1.
313 | apply rule (λS,T:setoid.unary_morphism_setoid S T);
314 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
315 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
317 | intros; whd; intros; simplify; whd in H1; whd in H;
318 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
319 [ apply Hletin | apply (e a1); ] | apply e1; ]]
320 | intros; whd; intros; simplify; apply refl;
321 | intros; simplify; whd; intros; simplify; apply refl;
322 | intros; simplify; whd; intros; simplify; apply refl;
326 definition setoid_of_SET: objs1 SET → setoid.
327 intros; apply o; qed.
328 coercion setoid_of_SET.
330 definition setoid1_of_SET: SET → setoid1.
331 intro; whd in t; apply setoid1_of_setoid; apply t.
333 coercion setoid1_of_SET.
335 definition eq': ∀w:SET.equivalence_relation ? := λw.eq w.
337 definition prop1_SET :
338 ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:Type_OF_objs1 A.eq' ? a b→eq' ? (w a) (w b).
339 intros; apply (prop1 A B w a b e);
343 interpretation "SET dagger" 'prop1 h = (prop1_SET _ _ _ _ _ h).
344 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
345 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
346 interpretation "SET eq" 'eq x y = (eq_rel _ (eq' _) x y).
348 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
351 [ apply (unary_morphism1 s s1);
354 alias symbol "eq" = "setoid1 eq".
355 apply (∀a: carr1 s. f a = g a);
356 | intros 1; simplify; intros; apply refl1;
357 | simplify; intros; apply sym1; apply H;
358 | simplify; intros; apply trans1; [2: apply H; | skip | apply H1]]]
361 definition SET1: category2.
364 | apply rule (λS,T.unary_morphism1_setoid1 S T);
365 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
366 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
368 | intros; whd; intros; simplify; whd in H1; whd in H;
369 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
370 [ apply Hletin | apply (e a1); ] | apply e1; ]]
371 | intros; whd; intros; simplify; apply refl1;
372 | intros; simplify; whd; intros; simplify; apply refl1;
373 | intros; simplify; whd; intros; simplify; apply refl1;
377 definition setoid1_OF_SET1: objs2 SET1 → setoid1.
378 intros; apply o; qed.
380 coercion setoid1_OF_SET1.
382 definition eq'': ∀w:SET1.equivalence_relation1 ? := λw.eq1 w.
384 definition prop11_SET1 :
385 ∀A,B:SET1.∀w:arrows2 SET1 A B.∀a,b:Type_OF_objs2 A.eq'' ? a b→eq'' ? (w a) (w b).
386 intros; apply (prop11 A B w a b e);
389 definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2.
390 intro; apply (setoid2_of_setoid1 t); qed.
391 coercion setoid2_OF_category2.
393 definition objs2_OF_category1: Type_OF_category1 SET → objs2 SET1.
394 intro; apply (setoid1_of_setoid t); qed.
395 coercion objs2_OF_category1.
397 definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
398 intro; whd in t; apply (carr1 t);
400 coercion Type1_OF_SET1.
402 interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
403 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
404 interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
406 lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
409 coercion unary_morphism1_of_arrows1_SET1.