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 nrecord 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 ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
26 interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
28 ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
29 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
31 ndefinition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
32 interpretation "intersect" 'intersects U V = (intersect ? U V).
34 ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
35 interpretation "union" 'union U V = (union ? U V).
37 ndefinition substract ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ ¬ x ∈ V }.
38 interpretation "substract" 'minus U V = (substract ? U V).
41 ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
43 ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
45 ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
47 nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
50 nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
53 include "properties/relations1.ma".
55 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
56 #A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
57 #S T U; *; #H1 H2; *; /4/;
60 include "sets/setoids1.ma".
62 ndefinition singleton ≝ λA:setoid.λa:A.{ x | a = x }.
63 interpretation "singl" 'singl a = (singleton ? a).
65 (* this has to be declared here, so that it is combined with carr *)
66 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
68 ndefinition powerclass_setoid: Type[0] → setoid1.
72 alias symbol "hint_decl" = "hint_decl_Type2".
73 unification hint 0 ≔ A;
74 R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
75 (*--------------------------------------------------*)⊢
78 (************ SETS OVER SETOIDS ********************)
80 include "logic/cprop.ma".
82 nrecord ext_powerclass (A: setoid) : Type[1] ≝ {
83 ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
84 forse lo si vorrebbe dichiarato con un target più lasco
85 ma la sintassi :> non lo supporta *)
86 ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
89 notation > "𝛀 ^ term 90 A" non associative with precedence 70
90 for @{ 'ext_powerclass $A }.
92 notation < "Ω term 90 A \atop ≈" non associative with precedence 90
93 for @{ 'ext_powerclass $A }.
95 interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
97 ndefinition Full_set: ∀A. 𝛀^A.
98 #A; @[ napply A | #x; #x'; #H; napply refl1]
100 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
102 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
103 #A; @ [ napply (λS,S'. S = S') ] /2/.
106 ndefinition ext_powerclass_setoid: setoid → setoid1.
110 unification hint 0 ≔ A;
111 R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
112 (* ----------------------------------------------------- *) ⊢
113 carr1 R ≡ ext_powerclass A.
115 nlemma mem_ext_powerclass_setoid_is_morph:
116 ∀A. (setoid1_of_setoid A) ⇒_1 ((𝛀^A) ⇒_1 CPROP).
117 #A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
118 #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
119 [ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
122 unification hint 0 ≔ AA, x, S;
125 TT ≟ (mk_unary_morphism1 ??
126 (λx:setoid1_of_setoid ?.
127 mk_unary_morphism1 ??
128 (λS:ext_powerclass_setoid ?. x ∈ S)
129 (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA x)))
130 (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
131 XX ≟ (ext_powerclass_setoid AA)
132 (*-------------------------------------*) ⊢
133 fun11 (setoid1_of_setoid AA)
134 (unary_morphism1_setoid1 XX CPROP) TT x S
137 nlemma set_ext : ∀S.∀A,B:Ω^S.A =_1 B → ∀x:S.(x ∈ A) =_1 (x ∈ B).
138 #S A B; *; #H1 H2 x; @; ##[ napply H1 | napply H2] nqed.
140 nlemma ext_set : ∀S.∀A,B:Ω^S.(∀x:S. (x ∈ A) = (x ∈ B)) → A = B.
141 #S A B H; @; #x; ncases (H x); #H1 H2; ##[ napply H1 | napply H2] nqed.
143 nlemma subseteq_is_morph: ∀A. 𝛀^A ⇒_1 𝛀^A ⇒_1 CPROP.
144 #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
145 #a; #a'; #b; #b'; *; #H1; #H2; *; /5/ by mk_iff, sym1, subseteq_trans;
149 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
150 #S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
151 ##[##1,2: napply (. Exy^-1╪_1#); nassumption;
152 ##|##3,4: napply (. Exy‡#); nassumption]
155 alias symbol "hint_decl" = "hint_decl_Type1".
157 A : setoid, B,C : ext_powerclass A;
158 R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
159 (* ------------------------------------------*) ⊢
160 ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
162 nlemma intersect_is_morph: ∀A. Ω^A ⇒_1 Ω^A ⇒_1 Ω^A.
163 #A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
164 #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
167 alias symbol "hint_decl" = "hint_decl_Type1".
168 unification hint 0 ≔ A : Type[0], B,C : Ω^A;
169 T ≟ powerclass_setoid A,
170 R ≟ mk_unary_morphism1 ??
171 (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
172 (prop11 ?? (intersect_is_morph A))
173 (*------------------------------------------------------------------------*) ⊢
174 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C ≡ intersect A B C.
176 interpretation "prop21 ext" 'prop2 l r =
177 (prop11 (ext_powerclass_setoid ?)
178 (unary_morphism1_setoid1 (ext_powerclass_setoid ?) ?) ? ?? l ?? r).
180 nlemma intersect_is_ext_morph: ∀A. 𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
181 #A; napply (mk_binary_morphism1 … (intersect_is_ext …));
182 #a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
186 AA : setoid, B,C : 𝛀^AA;
188 T ≟ ext_powerclass_setoid AA,
189 R ≟ (mk_unary_morphism1 ??
191 mk_unary_morphism1 ??
193 mk_ext_powerclass AA (S∩S') (ext_prop AA (intersect_is_ext ? S S')))
194 (prop11 ?? (intersect_is_ext_morph AA S)))
195 (prop11 ?? (intersect_is_ext_morph AA))) ,
198 (* ---------------------------------------------------------------------------------------*) ⊢
199 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ intersect A BB CC.
203 nlemma union_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
204 #X; napply (mk_binary_morphism1 … (λA,B.A ∪ B));
205 #A1 A2 B1 B2 EA EB; napply ext_set; #x;
206 nchange in match (x ∈ (A1 ∪ B1)) with (?∨?);
207 napply (.= (set_ext ??? EA x)‡#);
208 napply (.= #‡(set_ext ??? EB x)); //;
211 nlemma union_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
212 #S A B; @ (A ∪ B); #x y Exy; @; *; #H1;
213 ##[##1,3: @; ##|##*: @2 ]
214 ##[##1,3: napply (. (Exy^-1)╪_1#)
215 ##|##2,4: napply (. Exy╪_1#)]
219 alias symbol "hint_decl" = "hint_decl_Type1".
221 A : setoid, B,C : 𝛀^A;
222 R ≟ (mk_ext_powerclass ? (B ∪ C) (ext_prop ? (union_is_ext ? B C)))
223 (*-------------------------------------------------------------------------*) ⊢
224 ext_carr A R ≡ union ? (ext_carr ? B) (ext_carr ? C).
226 unification hint 0 ≔ S:Type[0], A,B:Ω^S;
227 T ≟ powerclass_setoid S,
228 MM ≟ mk_unary_morphism1 ??
229 (λA.mk_unary_morphism1 ?? (λB.A ∪ B) (prop11 ?? (union_is_morph S A)))
230 (prop11 ?? (union_is_morph S))
231 (*--------------------------------------------------------------------------*) ⊢
232 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A ∪ B.
234 nlemma union_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
235 #A; napply (mk_binary_morphism1 … (union_is_ext …));
236 #x1 x2 y1 y2 Ex Ey; napply (prop11 … (union_is_morph A)); nassumption.
240 AA : setoid, B,C : 𝛀^AA;
242 T ≟ ext_powerclass_setoid AA,
243 R ≟ (mk_unary_morphism1 ??
245 mk_unary_morphism1 ??
247 mk_ext_powerclass AA (S ∪ S') (ext_prop AA (union_is_ext ? S S')))
248 (prop11 ?? (union_is_ext_morph AA S)))
249 (prop11 ?? (union_is_ext_morph AA))) ,
252 (*------------------------------------------------------*) ⊢
253 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ union A BB CC.
257 nlemma substract_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
258 #X; napply (mk_binary_morphism1 … (λA,B.A - B));
259 #A1 A2 B1 B2 EA EB; napply ext_set; #x;
260 nchange in match (x ∈ (A1 - B1)) with (?∧?);
261 napply (.= (set_ext ??? EA x)‡#); @; *; #H H1; @; //; #H2; napply H1;
262 ##[ napply (. (set_ext ??? EB x)); ##| napply (. (set_ext ??? EB^-1 x)); ##] //;
265 nlemma substract_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
266 #S A B; @ (A - B); #x y Exy; @; *; #H1 H2; @; ##[##2,4: #H3; napply H2]
267 ##[##1,4: napply (. Exy╪_1#); // ##|##2,3: napply (. Exy^-1╪_1#); //]
270 alias symbol "hint_decl" = "hint_decl_Type1".
272 A : setoid, B,C : 𝛀^A;
273 R ≟ (mk_ext_powerclass ? (B - C) (ext_prop ? (substract_is_ext ? B C)))
274 (*-------------------------------------------------------------------------*) ⊢
275 ext_carr A R ≡ substract ? (ext_carr ? B) (ext_carr ? C).
277 unification hint 0 ≔ S:Type[0], A,B:Ω^S;
278 T ≟ powerclass_setoid S,
279 MM ≟ mk_unary_morphism1 ??
280 (λA.mk_unary_morphism1 ?? (λB.A - B) (prop11 ?? (substract_is_morph S A)))
281 (prop11 ?? (substract_is_morph S))
282 (*--------------------------------------------------------------------------*) ⊢
283 fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A - B.
285 nlemma substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
286 #A; napply (mk_binary_morphism1 … (substract_is_ext …));
287 #x1 x2 y1 y2 Ex Ey; napply (prop11 … (substract_is_morph A)); nassumption.
291 AA : setoid, B,C : 𝛀^AA;
293 T ≟ ext_powerclass_setoid AA,
294 R ≟ (mk_unary_morphism1 ??
296 mk_unary_morphism1 ??
298 mk_ext_powerclass AA (S - S') (ext_prop AA (substract_is_ext ? S S')))
299 (prop11 ?? (substract_is_ext_morph AA S)))
300 (prop11 ?? (substract_is_ext_morph AA))) ,
303 (*------------------------------------------------------*) ⊢
304 ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ substract A BB CC.
307 nlemma single_is_morph : ∀A:setoid. (setoid1_of_setoid A) ⇒_1 Ω^A.
308 #X; @; ##[ napply (λx.{(x)}); ##]
309 #a b E; napply ext_set; #x; @; #H; /3/; nqed.
311 nlemma single_is_ext: ∀A:setoid. A → 𝛀^A.
312 #X a; @; ##[ napply ({(a)}); ##] #x y E; @; #H; /3/; nqed.
314 alias symbol "hint_decl" = "hint_decl_Type1".
315 unification hint 0 ≔ A : setoid, a:A;
316 R ≟ (mk_ext_powerclass ? {(a)} (ext_prop ? (single_is_ext ? a)))
317 (*-------------------------------------------------------------------------*) ⊢
318 ext_carr A R ≡ singleton A a.
320 unification hint 0 ≔ A:setoid, a:A;
321 T ≟ setoid1_of_setoid A,
323 MM ≟ mk_unary_morphism1 ??
324 (λa:setoid1_of_setoid A.{(a)}) (prop11 ?? (single_is_morph A))
325 (*--------------------------------------------------------------------------*) ⊢
326 fun11 T (powerclass_setoid AA) MM a ≡ {(a)}.
328 nlemma single_is_ext_morph:∀A:setoid.(setoid1_of_setoid A) ⇒_1 𝛀^A.
329 #A; @; ##[ #a; napply (single_is_ext ? a); ##] #a b E; @; #x; /3/; nqed.
333 T ≟ ext_powerclass_setoid AA,
334 R ≟ mk_unary_morphism1 ??
335 (λa:setoid1_of_setoid AA.
336 mk_ext_powerclass AA {(a)} (ext_prop ? (single_is_ext AA a)))
337 (prop11 ?? (single_is_ext_morph AA))
338 (*------------------------------------------------------*) ⊢
339 ext_carr AA (fun11 (setoid1_of_setoid AA) T R a) ≡ singleton AA a.
347 alias symbol "hint_decl" = "hint_decl_Type2".
349 A : setoid, B,C : 𝛀^A ;
352 C1 ≟ (carr1 (powerclass_setoid (carr A))),
353 C2 ≟ (carr1 (ext_powerclass_setoid A))
355 eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
356 eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
359 A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
361 nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
362 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
363 #U; #A; #B; #H; #x; #y; #K; #K2;
364 alias symbol "prop2" = "prop21 mem".
365 alias symbol "invert" = "setoid1 symmetry".
371 nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
376 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
377 [##1,2: napply (. Ha^-1‡#); nassumption;
378 ##|##3,4: napply (. Ha‡#); nassumption]##]
379 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
380 [ alias symbol "invert" = "setoid1 symmetry".
381 alias symbol "refl" = "refl".
382 alias symbol "prop2" = "prop21".
383 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
384 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
387 (* unfold if intersect, exposing fun21 *)
388 alias symbol "hint_decl" = "hint_decl_Type1".
390 A : setoid, B,C : ext_powerclass A ⊢
392 (mk_binary_morphism1 …
393 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
394 (prop21 … (intersect_ok A)))
397 ≡ intersect ? (pc ? B) (pc ? C).
399 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
400 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
404 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
405 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
406 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq0 B) (f x) y}.
408 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
409 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
411 (******************* compatible equivalence relations **********************)
413 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
414 { rel:> equivalence_relation A;
415 compatibility: ∀x,x':A. x=x' → rel x x'
418 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
422 (******************* first omomorphism theorem for sets **********************)
424 ndefinition eqrel_of_morphism:
425 ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
427 [ @ [ napply (λx,y. f x = f y) ] /2/;
428 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
429 napply (.= (†H)); // ]
432 ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
434 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
437 ndefinition quotiented_mor:
438 ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
439 #A; #B; #f; @ [ napply f ] //.
442 nlemma first_omomorphism_theorem_functions1:
443 ∀A,B.∀f: unary_morphism A B.
444 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
447 alias symbol "eq" = "setoid eq".
448 ndefinition surjective ≝
449 λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
450 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
452 ndefinition injective ≝
453 λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
454 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
456 nlemma first_omomorphism_theorem_functions2:
458 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
461 nlemma first_omomorphism_theorem_functions3:
463 injective … (Full_set ?) (quotiented_mor … f).
464 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
467 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
469 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
470 f_sur: surjective … S T iso_f;
471 f_inj: injective … S iso_f
476 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
477 { iso_f:> unary_morphism A B;
478 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
485 λxxx:isomorphism A B S T.
487 return λxxx:isomorphism A B S T.
489 ∀x_72: mem (carr A) (pc A S) x.
490 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
491 with [ mk_isomorphism _ yyy ⇒ yyy ] ).
499 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
500 #A; #U; #V; #W; *; #H; #x; *; /2/.
503 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
504 #A; #U; #V; #W; #H; #H1; #x; *; /2/.
507 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
510 nlemma cupC : ∀S. ∀a,b:Ω^S.a ∪ b = b ∪ a.
511 #S a b; @; #w; *; nnormalize; /2/; nqed.
513 nlemma cupID : ∀S. ∀a:Ω^S.a ∪ a = a.
514 #S a; @; #w; ##[*; //] /2/; nqed.
516 (* XXX Bug notazione \cup, niente parentesi *)
517 nlemma cupA : ∀S.∀a,b,c:Ω^S.a ∪ b ∪ c = a ∪ (b ∪ c).
518 #S a b c; @; #w; *; /3/; *; /3/; nqed.
520 ndefinition Empty_set : ∀A.Ω^A ≝ λA.{ x | False }.
522 notation "∅" non associative with precedence 90 for @{ 'empty }.
523 interpretation "empty set" 'empty = (Empty_set ?).
525 nlemma cup0 :∀S.∀A:Ω^S.A ∪ ∅ = A.
526 #S p; @; #w; ##[*; //| #; @1; //] *; nqed.