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 := Type0.
35 definition CProp1 := Type1.
36 definition CProp2 := Type2.
38 definition CProp0_lt_CProp1 := (CProp0 : CProp1).
39 definition CProp1_lt_CProp2 := (CProp1 : CProp2).
41 definition CProp_OF_CProp0: CProp0 → CProp := λx.x.
42 definition CProp_OF_CProp1: CProp1 → CProp := λx.x.
43 definition CProp_OF_CProp2: CProp2 → CProp := λx.x.
46 record equivalence_relation (A:Type0) : Type1 ≝
47 { eq_rel:2> A → A → CProp0;
48 refl: reflexive ? eq_rel;
49 sym: symmetric ? eq_rel;
50 trans: transitive ? eq_rel
53 record setoid : Type1 ≝
55 eq: equivalence_relation carr
58 definition reflexive1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
59 definition symmetric1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
60 definition transitive1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
62 record equivalence_relation1 (A:Type1) : Type2 ≝
63 { eq_rel1:2> A → A → CProp1;
64 refl1: reflexive1 ? eq_rel1;
65 sym1: symmetric1 ? eq_rel1;
66 trans1: transitive1 ? eq_rel1
69 record setoid1: Type2 ≝
71 eq1: equivalence_relation1 carr1
74 definition setoid1_of_setoid: setoid → setoid1.
86 (* questa coercion e' necessaria per problemi di unificazione *)
87 coercion setoid1_of_setoid.
89 definition reflexive2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
90 definition symmetric2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
91 definition transitive2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
93 record equivalence_relation2 (A:Type2) : Type3 ≝
94 { eq_rel2:2> A → A → CProp2;
95 refl2: reflexive2 ? eq_rel2;
96 sym2: symmetric2 ? eq_rel2;
97 trans2: transitive2 ? eq_rel2
100 record setoid2: Type3 ≝
102 eq2: equivalence_relation2 carr2
105 definition setoid2_of_setoid1: setoid1 → setoid2.
117 (*coercion setoid2_of_setoid1.*)
120 definition Leibniz: Type → setoid.
125 [ apply (λx,y:T.cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y)
126 | alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
128 | alias id "sym_eq" = "cic:/matita/logic/equality/sym_eq.con".
130 | alias id "trans_eq" = "cic:/matita/logic/equality/trans_eq.con".
137 interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
138 interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
139 interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
140 interpretation "setoid2 symmetry" 'invert r = (sym2 ____ r).
141 interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
142 interpretation "setoid symmetry" 'invert r = (sym ____ r).
143 notation ".= r" with precedence 50 for @{'trans $r}.
144 interpretation "trans2" 'trans r = (trans2 _____ r).
145 interpretation "trans1" 'trans r = (trans1 _____ r).
146 interpretation "trans" 'trans r = (trans _____ r).
148 record unary_morphism (A,B: setoid) : Type0 ≝
150 prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
153 record unary_morphism1 (A,B: setoid1) : Type1 ≝
155 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
158 record unary_morphism2 (A,B: setoid2) : Type2 ≝
160 prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
163 record binary_morphism (A,B,C:setoid) : Type0 ≝
165 prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
168 record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
169 { fun21:2> A → B → C;
170 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
173 record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
174 { fun22:2> A → B → C;
175 prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
178 notation "† c" with precedence 90 for @{'prop1 $c }.
179 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
180 notation "#" with precedence 90 for @{'refl}.
181 interpretation "prop1" 'prop1 c = (prop1 _____ c).
182 interpretation "prop11" 'prop1 c = (prop11 _____ c).
183 interpretation "prop12" 'prop1 c = (prop12 _____ c).
184 interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
185 interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
186 interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
187 interpretation "refl" 'refl = (refl ___).
188 interpretation "refl1" 'refl = (refl1 ___).
189 interpretation "refl2" 'refl = (refl2 ___).
191 definition CPROP: setoid1.
196 | intros 1; split; intro; assumption
197 | intros 3; cases H; split; assumption
198 | intros 5; cases H; cases H1; split; intro;
199 [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
202 alias symbol "eq" = "setoid1 eq".
203 definition if': ∀A,B:CPROP. A = B → A → B.
204 intros; apply (if ?? e); assumption.
207 notation ". r" with precedence 50 for @{'if $r}.
208 interpretation "if" 'if r = (if' __ r).
210 definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
213 | intros; split; intro; cases H; split;
215 | apply (if ?? e1 b1)
217 | apply (fi ?? e1 b1)]]
220 interpretation "and_morphism" 'and a b = (fun21 ___ and_morphism a b).
222 definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
225 | intros; split; intro; cases H; [1,3:left |2,4: right]
228 | apply (if ?? e1 b1)
229 | apply (fi ?? e1 b1)]]
232 interpretation "or_morphism" 'or a b = (fun21 ___ or_morphism a b).
234 definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
236 [ apply (λA,B. A → B)
237 | intros; split; intros;
238 [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
239 | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
243 definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
246 [ apply (eq_rel ? (eq S))
247 | intros; split; intro;
248 [ apply (.= H \sup -1);
257 record category : Type1 ≝
259 arrows: objs → objs → setoid;
260 id: ∀o:objs. arrows o o;
261 comp: ∀o1,o2,o3. binary_morphism1 (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
262 comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
263 comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
264 id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
265 id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
268 record category1 : Type2 ≝
270 arrows1: objs1 → objs1 → setoid1;
271 id1: ∀o:objs1. arrows1 o o;
272 comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
273 comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
274 comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
275 id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
276 id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
279 record category2 : Type3 ≝
281 arrows2: objs2 → objs2 → setoid2;
282 id2: ∀o:objs2. arrows2 o o;
283 comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
284 comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
285 comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 = comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
286 id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a = a;
287 id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
290 notation "'ASSOC'" with precedence 90 for @{'assoc}.
292 interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
293 interpretation "category2 assoc" 'assoc = (comp_assoc2 ________).
294 interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
295 interpretation "category1 assoc" 'assoc = (comp_assoc1 ________).
296 interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
297 interpretation "category assoc" 'assoc = (comp_assoc ________).
299 (* bug grande come una casa?
300 Ma come fa a passare la quantificazione larga??? *)
301 definition unary_morphism_setoid: setoid → setoid → setoid1.
304 [ apply (unary_morphism s s1);
306 [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
307 | intros 1; simplify; intros; apply refl;
308 | simplify; intros; apply sym; apply f;
309 | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
312 definition SET: category1.
315 | apply rule (λS,T:setoid.unary_morphism_setoid S T);
316 | intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
317 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
319 | intros; whd; intros; simplify; whd in H1; whd in H;
320 apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
321 [ apply Hletin | apply (e a1); ] | apply e1; ]]
322 | intros; whd; intros; simplify; apply refl;
323 | intros; simplify; whd; intros; simplify; apply refl;
324 | intros; simplify; whd; intros; simplify; apply refl;
328 definition setoid_of_SET: objs1 SET → setoid.
329 intros; apply o; qed.
330 coercion setoid_of_SET.
332 definition setoid1_of_SET: SET → setoid1.
333 intro; whd in t; apply setoid1_of_setoid; apply t.
335 coercion setoid1_of_SET.
337 definition eq': ∀w:SET.equivalence_relation ? := λw.eq w.
339 definition prop1_SET :
340 ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:Type_OF_objs1 A.eq' ? a b→eq' ? (w a) (w b).
341 intros; apply (prop1 A B w a b e);
345 interpretation "SET dagger" 'prop1 h = (prop1_SET _ _ _ _ _ h).
346 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
347 interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
348 interpretation "SET eq" 'eq x y = (eq_rel _ (eq' _) x y).
350 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid2.
353 [ apply (unary_morphism1 s s1);
356 alias symbol "eq" = "setoid1 eq".
357 apply (∀a: carr1 s. f a = g a);
358 | intros 1; simplify; intros; apply refl1;
359 | simplify; intros; apply sym1; apply f;
360 | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
363 definition SET1: category2.
366 | apply rule (λS,T.unary_morphism1_setoid1 S T);
367 | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
368 | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
370 | intros; whd; intros; simplify; whd in H1; whd in H;
371 apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
372 [ apply Hletin | apply (e a1); ] | apply e1; ]]
373 | intros; whd; intros; simplify; apply refl1;
374 | intros; simplify; whd; intros; simplify; apply refl1;
375 | intros; simplify; whd; intros; simplify; apply refl1;
379 definition setoid1_OF_SET1: objs2 SET1 → setoid1.
380 intros; apply o; qed.
382 coercion setoid1_OF_SET1.
384 definition eq'': ∀w:SET1.equivalence_relation1 ? := λw.eq1 w.
386 definition prop11_SET1 :
387 ∀A,B:SET1.∀w:arrows2 SET1 A B.∀a,b:Type_OF_objs2 A.eq'' ? a b→eq'' ? (w a) (w b).
388 intros; apply (prop11 A B w a b e);
391 definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2.
392 intro; apply (setoid2_of_setoid1 t); qed.
393 coercion setoid2_OF_category2.
395 definition objs2_OF_category1: Type_OF_category1 SET → objs2 SET1.
396 intro; apply (setoid1_of_setoid t); qed.
397 coercion objs2_OF_category1.
399 definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
400 intro; whd in t; apply (carr1 t);
402 coercion Type1_OF_SET1.
404 interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
405 interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
406 interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).
408 lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
411 coercion unary_morphism1_of_arrows1_SET1.