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 (******************* SETS OVER TYPES *****************)
17 include "logic/connectives.ma".
19 record powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
21 interpretation "mem" 'mem a S = (mem ? S a).
22 interpretation "powerclass" 'powerset A = (powerclass A).
23 interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
25 definition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
26 interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
28 definition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
29 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
31 definition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
32 interpretation "intersect" 'intersects U V = (intersect ? U V).
34 definition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
35 interpretation "union" 'union U V = (union ? U V).
37 definition substract ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ ¬ x ∈ V }.
38 interpretation "substract" 'minus U V = (substract ? U V).
41 definition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
43 definition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
45 definition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
47 lemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
50 lemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
53 include "properties/relations1.ma".
55 definition seteq: ∀A. equivalence_relation1 (Ω^A).
56 #A % [@(λS,S'. S ⊆ S' ∧ S' ⊆ S)]
58 #S #T #U * #H1 #H2 * /4/
61 include "sets/setoids1.ma".
63 ndefinition singleton ≝ λA:setoid.λa:A.{ x | a = x }.
64 interpretation "singl" 'singl a = (singleton ? a).
66 (* this has to be declared here, so that it is combined with carr *)
67 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
69 ndefinition powerclass_setoid: Type[0] → setoid1.
73 alias symbol "hint_decl" = "hint_decl_Type2".
74 unification hint 0 ≔ A;
75 R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
76 (*--------------------------------------------------*)⊢
79 (************ SETS OVER SETOIDS ********************)
81 include "logic/cprop.ma".
83 nrecord ext_powerclass (A: setoid) : Type[1] ≝ {
84 ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
85 forse lo si vorrebbe dichiarato con un target più lasco
86 ma la sintassi :> non lo supporta *)
87 ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
90 notation > "𝛀 ^ term 90 A" non associative with precedence 70
91 for @{ 'ext_powerclass $A }.
93 notation < "Ω term 90 A \atop ≈" non associative with precedence 90
94 for @{ 'ext_powerclass $A }.
96 interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
98 ndefinition Full_set: ∀A. 𝛀^A.
99 #A; @[ napply A | #x; #x'; #H; napply refl1]
101 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
103 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
104 #A; @ [ napply (λS,S'. S = S') ] /2/.
107 ndefinition ext_powerclass_setoid: setoid → setoid1.
111 unification hint 0 ≔ A;
112 R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
113 (* ----------------------------------------------------- *) ⊢
114 carr1 R ≡ ext_powerclass A.
116 nlemma mem_ext_powerclass_setoid_is_morph:
117 ∀A. (setoid1_of_setoid A) ⇒_1 ((𝛀^A) ⇒_1 CPROP).
118 #A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
119 #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
120 [ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
123 unification hint 0 ≔ AA : setoid, S : 𝛀^AA, x : carr AA;
126 TT ≟ (mk_unary_morphism1 ??
127 (λx:carr1 (setoid1_of_setoid ?).
128 mk_unary_morphism1 ??
129 (λS:carr1 (ext_powerclass_setoid ?). x ∈ (ext_carr ? S))
130 (prop11 ?? (fun11 ?? (mem_ext_powerclass_setoid_is_morph AA) x)))
131 (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
132 T2 ≟ (ext_powerclass_setoid AA)
133 (*---------------------------------------------------------------------------*) ⊢
134 fun11 T2 CPROP (fun11 (setoid1_of_setoid AA) (unary_morphism1_setoid1 T2 CPROP) TT x) S ≡ mem A SS x.
136 nlemma set_ext : ∀S.∀A,B:Ω^S.A =_1 B → ∀x:S.(x ∈ A) =_1 (x ∈ B).
137 #S A B; *; #H1 H2 x; @; ##[ napply H1 | napply H2] nqed.
139 nlemma ext_set : ∀S.∀A,B:Ω^S.(∀x:S. (x ∈ A) = (x ∈ B)) → A = B.
140 #S A B H; @; #x; ncases (H x); #H1 H2; ##[ napply H1 | napply H2] nqed.
142 nlemma subseteq_is_morph: ∀A. 𝛀^A ⇒_1 𝛀^A ⇒_1 CPROP.
143 #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
144 #a; #a'; #b; #b'; *; #H1; #H2; *; /5/ by mk_iff, sym1, subseteq_trans;
148 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
149 #S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
150 ##[##1,2: napply (. Exy^-1╪_1#); nassumption;
151 ##|##3,4: napply (. Exy‡#); nassumption]
154 alias symbol "hint_decl" = "hint_decl_Type1".
155 unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
159 R ≟ (mk_ext_powerclass ?
160 (ext_carr ? B ∩ ext_carr ? C)
161 (ext_prop ? (intersect_is_ext ? B C)))
162 (* ------------------------------------------*) ⊢
163 ext_carr A R ≡ intersect AA BB CC.
165 nlemma intersect_is_morph: ∀A. Ω^A ⇒_1 Ω^A ⇒_1 Ω^A.
166 #A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
167 #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
170 alias symbol "hint_decl" = "hint_decl_Type1".
171 unification hint 0 ≔ A : Type[0], B,C : Ω^A;
172 T ≟ powerclass_setoid A,
173 R ≟ mk_unary_morphism1 ??
174 (λX. mk_unary_morphism1 ??
175 (λY.X ∩ Y) (prop11 ?? (fun11 ?? (intersect_is_morph A) X)))
176 (prop11 ?? (intersect_is_morph A))
177 (*------------------------------------------------------------------------*) ⊢
178 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C ≡ intersect A B C.
180 interpretation "prop21 ext" 'prop2 l r =
181 (prop11 (ext_powerclass_setoid ?)
182 (unary_morphism1_setoid1 (ext_powerclass_setoid ?) ?) ? ?? l ?? r).
184 nlemma intersect_is_ext_morph: ∀A. 𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
185 #A; napply (mk_binary_morphism1 … (intersect_is_ext …));
186 #a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
190 AA : setoid, B,C : 𝛀^AA;
192 T ≟ ext_powerclass_setoid AA,
193 R ≟ (mk_unary_morphism1 ?? (λX:𝛀^AA.
194 mk_unary_morphism1 ?? (λY:𝛀^AA.
196 (ext_carr ? X ∩ ext_carr ? Y)
197 (ext_prop AA (intersect_is_ext ? X Y)))
198 (prop11 ?? (fun11 ?? (intersect_is_ext_morph AA) X)))
199 (prop11 ?? (intersect_is_ext_morph AA))) ,
202 (* ---------------------------------------------------------------------------------------*) ⊢
203 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ intersect A BB CC.
207 nlemma union_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
208 #X; napply (mk_binary_morphism1 … (λA,B.A ∪ B));
209 #A1 A2 B1 B2 EA EB; napply ext_set; #x;
210 nchange in match (x ∈ (A1 ∪ B1)) with (?∨?);
211 napply (.= (set_ext ??? EA x)‡#);
212 napply (.= #‡(set_ext ??? EB x)); //;
215 nlemma union_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
216 #S A B; @ (A ∪ B); #x y Exy; @; *; #H1;
217 ##[##1,3: @; ##|##*: @2 ]
218 ##[##1,3: napply (. (Exy^-1)╪_1#)
219 ##|##2,4: napply (. Exy╪_1#)]
223 alias symbol "hint_decl" = "hint_decl_Type1".
224 unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
228 R ≟ mk_ext_powerclass ?
229 (ext_carr ? B ∪ ext_carr ? C) (ext_prop ? (union_is_ext ? B C))
230 (*-------------------------------------------------------------------------*) ⊢
231 ext_carr A R ≡ union AA BB CC.
233 unification hint 0 ≔ S:Type[0], A,B:Ω^S;
234 T ≟ powerclass_setoid S,
235 MM ≟ mk_unary_morphism1 ??
236 (λA.mk_unary_morphism1 ??
237 (λB.A ∪ B) (prop11 ?? (fun11 ?? (union_is_morph S) A)))
238 (prop11 ?? (union_is_morph S))
239 (*--------------------------------------------------------------------------*) ⊢
240 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A ∪ B.
242 nlemma union_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
243 #A; napply (mk_binary_morphism1 … (union_is_ext …));
244 #x1 x2 y1 y2 Ex Ey; napply (prop11 … (union_is_morph A)); nassumption.
248 AA : setoid, B,C : 𝛀^AA;
250 T ≟ ext_powerclass_setoid AA,
251 R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
252 mk_unary_morphism1 ?? (λY:𝛀^AA.
254 (ext_carr ? X ∪ ext_carr ? Y) (ext_prop AA (union_is_ext ? X Y)))
255 (prop11 ?? (fun11 ?? (union_is_ext_morph AA) X)))
256 (prop11 ?? (union_is_ext_morph AA)),
259 (*------------------------------------------------------*) ⊢
260 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ union A BB CC.
264 nlemma substract_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
265 #X; napply (mk_binary_morphism1 … (λA,B.A - B));
266 #A1 A2 B1 B2 EA EB; napply ext_set; #x;
267 nchange in match (x ∈ (A1 - B1)) with (?∧?);
268 napply (.= (set_ext ??? EA x)‡#); @; *; #H H1; @; //; #H2; napply H1;
269 ##[ napply (. (set_ext ??? EB x)); ##| napply (. (set_ext ??? EB^-1 x)); ##] //;
272 nlemma substract_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
273 #S A B; @ (A - B); #x y Exy; @; *; #H1 H2; @; ##[##2,4: #H3; napply H2]
274 ##[##1,4: napply (. Exy╪_1#); // ##|##2,3: napply (. Exy^-1╪_1#); //]
277 alias symbol "hint_decl" = "hint_decl_Type1".
278 unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
282 R ≟ mk_ext_powerclass ?
283 (ext_carr ? B - ext_carr ? C)
284 (ext_prop ? (substract_is_ext ? B C))
285 (*---------------------------------------------------*) ⊢
286 ext_carr A R ≡ substract AA BB CC.
288 unification hint 0 ≔ S:Type[0], A,B:Ω^S;
289 T ≟ powerclass_setoid S,
290 MM ≟ mk_unary_morphism1 ??
291 (λA.mk_unary_morphism1 ??
292 (λB.A - B) (prop11 ?? (fun11 ?? (substract_is_morph S) A)))
293 (prop11 ?? (substract_is_morph S))
294 (*--------------------------------------------------------------------------*) ⊢
295 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A - B.
297 nlemma substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
298 #A; napply (mk_binary_morphism1 … (substract_is_ext …));
299 #x1 x2 y1 y2 Ex Ey; napply (prop11 … (substract_is_morph A)); nassumption.
303 AA : setoid, B,C : 𝛀^AA;
305 T ≟ ext_powerclass_setoid AA,
306 R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
307 mk_unary_morphism1 ?? (λY:𝛀^AA.
309 (ext_carr ? X - ext_carr ? Y)
310 (ext_prop AA (substract_is_ext ? X Y)))
311 (prop11 ?? (fun11 ?? (substract_is_ext_morph AA) X)))
312 (prop11 ?? (substract_is_ext_morph AA)),
315 (*------------------------------------------------------*) ⊢
316 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ substract A BB CC.
319 nlemma single_is_morph : ∀A:setoid. (setoid1_of_setoid A) ⇒_1 Ω^A.
320 #X; @; ##[ napply (λx.{(x)}); ##]
321 #a b E; napply ext_set; #x; @; #H; /3/; nqed.
323 nlemma single_is_ext: ∀A:setoid. A → 𝛀^A.
324 #X a; @; ##[ napply ({(a)}); ##] #x y E; @; #H; /3/; nqed.
326 alias symbol "hint_decl" = "hint_decl_Type1".
327 unification hint 0 ≔ A : setoid, a : carr A;
328 R ≟ (mk_ext_powerclass ? {(a)} (ext_prop ? (single_is_ext ? a)))
329 (*-------------------------------------------------------------------------*) ⊢
330 ext_carr A R ≡ singleton A a.
332 unification hint 0 ≔ A:setoid, a : carr A;
333 T ≟ setoid1_of_setoid A,
335 MM ≟ mk_unary_morphism1 ??
336 (λa:carr1 (setoid1_of_setoid A).{(a)}) (prop11 ?? (single_is_morph A))
337 (*--------------------------------------------------------------------------*) ⊢
338 fun11 T (powerclass_setoid AA) MM a ≡ {(a)}.
340 nlemma single_is_ext_morph:∀A:setoid.(setoid1_of_setoid A) ⇒_1 𝛀^A.
341 #A; @; ##[ #a; napply (single_is_ext ? a); ##] #a b E; @; #x; /3/; nqed.
343 unification hint 1 ≔ AA : setoid, a: carr AA;
344 T ≟ ext_powerclass_setoid AA,
345 R ≟ mk_unary_morphism1 ??
346 (λa:carr1 (setoid1_of_setoid AA).
347 mk_ext_powerclass AA {(a)} (ext_prop ? (single_is_ext AA a)))
348 (prop11 ?? (single_is_ext_morph AA))
349 (*------------------------------------------------------*) ⊢
350 ext_carr AA (fun11 (setoid1_of_setoid AA) T R a) ≡ singleton AA a.
354 alias symbol "hint_decl" = "hint_decl_Type2".
356 A : setoid, B,C : 𝛀^A ;
359 C1 ≟ (carr1 (powerclass_setoid (carr A))),
360 C2 ≟ (carr1 (ext_powerclass_setoid A))
362 eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
363 eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
366 A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
368 nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
369 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
370 #U; #A; #B; #H; #x; #y; #K; #K2;
371 alias symbol "prop2" = "prop21 mem".
372 alias symbol "invert" = "setoid1 symmetry".
378 nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
383 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
384 [##1,2: napply (. Ha^-1‡#); nassumption;
385 ##|##3,4: napply (. Ha‡#); nassumption]##]
386 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
387 [ alias symbol "invert" = "setoid1 symmetry".
388 alias symbol "refl" = "refl".
389 alias symbol "prop2" = "prop21".
390 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
391 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
394 (* unfold if intersect, exposing fun21 *)
395 alias symbol "hint_decl" = "hint_decl_Type1".
397 A : setoid, B,C : ext_powerclass A ⊢
399 (mk_binary_morphism1 …
400 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
401 (prop21 … (intersect_ok A)))
404 ≡ intersect ? (pc ? B) (pc ? C).
406 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
407 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
411 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
412 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
413 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq0 B) (f x) y}.
415 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
416 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
418 (******************* compatible equivalence relations **********************)
420 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
421 { rel:> equivalence_relation A;
422 compatibility: ∀x,x':A. x=x' → rel x x'
425 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
429 (******************* first omomorphism theorem for sets **********************)
431 ndefinition eqrel_of_morphism:
432 ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
434 [ @ [ napply (λx,y. f x = f y) ] /2/;
435 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
436 napply (.= (†H)); // ]
439 ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
441 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
444 ndefinition quotiented_mor:
445 ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
446 #A; #B; #f; @ [ napply f ] //.
449 nlemma first_omomorphism_theorem_functions1:
450 ∀A,B.∀f: unary_morphism A B.
451 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
454 alias symbol "eq" = "setoid eq".
455 ndefinition surjective ≝
456 λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
457 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
459 ndefinition injective ≝
460 λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
461 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
463 nlemma first_omomorphism_theorem_functions2:
465 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
468 nlemma first_omomorphism_theorem_functions3:
470 injective … (Full_set ?) (quotiented_mor … f).
471 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
474 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
476 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
477 f_sur: surjective … S T iso_f;
478 f_inj: injective … S iso_f
483 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
484 { iso_f:> unary_morphism A B;
485 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
492 λxxx:isomorphism A B S T.
494 return λxxx:isomorphism A B S T.
496 ∀x_72: mem (carr A) (pc A S) x.
497 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
498 with [ mk_isomorphism _ yyy ⇒ yyy ] ).
506 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
507 #A; #U; #V; #W; *; #H; #x; *; /2/.
510 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
511 #A; #U; #V; #W; #H; #H1; #x; *; /2/.
514 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
517 nlemma cupC : ∀S. ∀a,b:Ω^S.a ∪ b = b ∪ a.
518 #S a b; @; #w; *; nnormalize; /2/; nqed.
520 nlemma cupID : ∀S. ∀a:Ω^S.a ∪ a = a.
521 #S a; @; #w; ##[*; //] /2/; nqed.
523 (* XXX Bug notazione \cup, niente parentesi *)
524 nlemma cupA : ∀S.∀a,b,c:Ω^S.a ∪ b ∪ c = a ∪ (b ∪ c).
525 #S a b c; @; #w; *; /3/; *; /3/; nqed.
527 ndefinition Empty_set : ∀A.Ω^A ≝ λA.{ x | False }.
529 notation "∅" non associative with precedence 90 for @{ 'empty }.
530 interpretation "empty set" 'empty = (Empty_set ?).
532 nlemma cup0 :∀S.∀A:Ω^S.A ∪ ∅ = A.
533 #S p; @; #w; ##[*; //| #; @1; //] *; nqed.