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 big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
39 ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
41 ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
43 nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
46 nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
49 include "properties/relations1.ma".
51 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
52 #A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
53 #S T U; *; #H1 H2; *; /4/;
56 include "sets/setoids1.ma".
58 (* this has to be declared here, so that it is combined with carr *)
59 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
61 ndefinition powerclass_setoid: Type[0] → setoid1.
65 include "hints_declaration.ma".
67 alias symbol "hint_decl" = "hint_decl_Type2".
68 unification hint 0 ≔ A;
69 R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
70 (*--------------------------------------------------*)⊢
73 (************ SETS OVER SETOIDS ********************)
75 include "logic/cprop.ma".
77 nrecord ext_powerclass (A: setoid) : Type[1] ≝ {
78 ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
79 forse lo si vorrebbe dichiarato con un target più lasco
80 ma la sintassi :> non lo supporta *)
81 ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
84 notation > "𝛀 ^ term 90 A" non associative with precedence 70
85 for @{ 'ext_powerclass $A }.
87 notation < "Ω term 90 A \atop ≈" non associative with precedence 90
88 for @{ 'ext_powerclass $A }.
90 interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
92 ndefinition Full_set: ∀A. 𝛀^A.
93 #A; @[ napply A | #x; #x'; #H; napply refl1]
95 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
97 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
98 #A; @ [ napply (λS,S'. S = S') ] /2/.
101 ndefinition ext_powerclass_setoid: setoid → setoid1.
105 unification hint 0 ≔ A;
106 R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
107 (* ----------------------------------------------------- *) ⊢
108 carr1 R ≡ ext_powerclass A.
110 nlemma mem_ext_powerclass_setoid_is_morph:
111 ∀A. (setoid1_of_setoid A) ⇒_1 (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
112 #A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
113 #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
114 [ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
117 unification hint 0 ≔ AA, x, S;
120 TT ≟ (mk_unary_morphism1 ??
121 (λx:setoid1_of_setoid ?.
122 mk_unary_morphism1 ??
123 (λS:ext_powerclass_setoid ?. x ∈ S)
124 (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA x)))
125 (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
126 XX ≟ (ext_powerclass_setoid AA)
127 (*-------------------------------------*) ⊢
128 fun11 (setoid1_of_setoid AA)
129 (unary_morphism1_setoid1 XX CPROP) TT x S
132 nlemma set_ext : ∀S.∀A,B:Ω^S.A =_1 B → ∀x:S.(x ∈ A) =_1 (x ∈ B).
133 #S A B; *; #H1 H2 x; @; ##[ napply H1 | napply H2] nqed.
135 nlemma ext_set : ∀S.∀A,B:Ω^S.(∀x:S. (x ∈ A) = (x ∈ B)) → A = B.
136 #S A B H; @; #x; ncases (H x); #H1 H2; ##[ napply H1 | napply H2] nqed.
138 nlemma subseteq_is_morph: ∀A.
139 (ext_powerclass_setoid A) ⇒_1
140 (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
141 #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
142 #a; #a'; #b; #b'; *; #H1; #H2; *; /5/ by mk_iff, sym1, subseteq_trans;
145 alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
146 unification hint 0 ≔ A,x,y
147 (*-----------------------------------------------*) ⊢
148 eq_rel ? (eq0 A) x y ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) x y.
150 (* XXX capire come mai questa hint non funziona se porto su (setoid1_of_setoid A) *)
152 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
153 #S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
154 ##[##1,2: napply (. Exy^-1‡#); nassumption;
155 ##|##3,4: napply (. Exy‡#); nassumption]
158 alias symbol "hint_decl" = "hint_decl_Type1".
160 A : setoid, B,C : ext_powerclass A;
161 R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
162 (* ------------------------------------------*) ⊢
163 ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
165 nlemma intersect_is_morph:
166 ∀A. (powerclass_setoid A) ⇒_1 (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
167 #A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
168 #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
171 alias symbol "hint_decl" = "hint_decl_Type1".
172 unification hint 0 ≔ A : Type[0], B,C : Ω^A;
173 R ≟ mk_unary_morphism1 ??
174 (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S)))
175 (prop11 ?? (intersect_is_morph A))
176 (*------------------------------------------------------------------------*) ⊢
177 fun11 ?? (fun11 ?? R B) C ≡ intersect A B C.
179 interpretation "prop21 ext" 'prop2 l r =
180 (prop11 (ext_powerclass_setoid ?)
181 (unary_morphism1_setoid1 (ext_powerclass_setoid ?) ?) ? ?? l ?? r).
183 nlemma intersect_is_ext_morph:
185 (ext_powerclass_setoid A) ⇒_1
186 (unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
187 #A; napply (mk_binary_morphism1 … (intersect_is_ext …));
188 #a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
192 AA : setoid, B,C : 𝛀^AA;
194 R ≟ (mk_unary_morphism1 ??
196 mk_unary_morphism1 ??
198 mk_ext_powerclass AA (S∩S') (ext_prop AA (intersect_is_ext ? S S')))
199 (prop11 ?? (intersect_is_ext_morph AA S)))
200 (prop11 ?? (intersect_is_ext_morph AA))) ,
203 (* ------------------------------------------------------*) ⊢
204 ext_carr AA (R B C) ≡ intersect A BB CC.
206 nlemma union_is_morph :
207 ∀A. (powerclass_setoid A) ⇒_1 (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
208 (*XXX ∀A.Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A). avec non-unif-coerc*)
209 #X; napply (mk_binary_morphism1 … (λA,B.A ∪ B));
210 #A1 A2 B1 B2 EA EB; napply ext_set; #x;
211 nchange in match (x ∈ (A1 ∪ B1)) with (?∨?);
212 napply (.= (set_ext ??? EA x)‡#);
213 napply (.= #‡(set_ext ??? EB x)); //;
216 nlemma union_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
217 #S A B; @ (A ∪ B); #x y Exy; @; *; #H1;
218 ##[##1,3: @; ##|##*: @2 ]
219 ##[##1,3: napply (. (Exy^-1)╪_1#)
220 ##|##2,4: napply (. Exy╪_1#)]
224 alias symbol "hint_decl" = "hint_decl_Type1".
226 A : setoid, B,C : 𝛀^A;
227 R ≟ (mk_ext_powerclass ? (B ∪ C) (ext_prop ? (union_is_ext ? B C)))
228 (*-------------------------------------------------------------------------*) ⊢
229 ext_carr A R ≡ union ? (ext_carr ? B) (ext_carr ? C).
231 unification hint 0 ≔ S:Type[0], A,B:Ω^S;
232 MM ≟ mk_unary_morphism1 ??
233 (λA.mk_unary_morphism1 ?? (λB.A ∪ B) (prop11 ?? (union_is_morph S A)))
234 (prop11 ?? (union_is_morph S))
235 (*--------------------------------------------------------------------------*) ⊢
236 fun11 ?? (fun11 ?? MM A) B ≡ A ∪ B.
238 nlemma union_is_ext_morph:∀A.
239 (ext_powerclass_setoid A) ⇒_1
240 (unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
241 (*XXX ∀A:setoid.𝛀^A ⇒_1 (𝛀^A ⇒_1 𝛀^A). with coercion non uniformi *)
242 #A; napply (mk_binary_morphism1 … (union_is_ext …));
243 #x1 x2 y1 y2 Ex Ey; napply (prop11 … (union_is_morph A)); nassumption.
247 AA : setoid, B,C : 𝛀^AA;
249 R ≟ (mk_unary_morphism1 ??
251 mk_unary_morphism1 ??
253 mk_ext_powerclass AA (S ∪ S') (ext_prop AA (union_is_ext ? S S')))
254 (prop11 ?? (union_is_ext_morph AA S)))
255 (prop11 ?? (union_is_ext_morph AA))) ,
258 (*------------------------------------------------------*) ⊢
259 ext_carr AA (R B C) ≡ union A BB CC.
262 alias symbol "hint_decl" = "hint_decl_Type2".
264 A : setoid, B,C : 𝛀^A ;
267 C1 ≟ (carr1 (powerclass_setoid (carr A))),
268 C2 ≟ (carr1 (ext_powerclass_setoid A))
270 eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
271 eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
274 A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
276 nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
277 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
278 #U; #A; #B; #H; #x; #y; #K; #K2;
279 alias symbol "prop2" = "prop21 mem".
280 alias symbol "invert" = "setoid1 symmetry".
286 nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
291 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
292 [##1,2: napply (. Ha^-1‡#); nassumption;
293 ##|##3,4: napply (. Ha‡#); nassumption]##]
294 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
295 [ alias symbol "invert" = "setoid1 symmetry".
296 alias symbol "refl" = "refl".
297 alias symbol "prop2" = "prop21".
298 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
299 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
302 (* unfold if intersect, exposing fun21 *)
303 alias symbol "hint_decl" = "hint_decl_Type1".
305 A : setoid, B,C : ext_powerclass A ⊢
307 (mk_binary_morphism1 …
308 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
309 (prop21 … (intersect_ok A)))
312 ≡ intersect ? (pc ? B) (pc ? C).
314 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
315 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
319 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
320 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
321 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq0 B) (f x) y}.
323 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
324 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
326 (******************* compatible equivalence relations **********************)
328 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
329 { rel:> equivalence_relation A;
330 compatibility: ∀x,x':A. x=x' → rel x x'
333 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
337 (******************* first omomorphism theorem for sets **********************)
339 ndefinition eqrel_of_morphism:
340 ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
342 [ @ [ napply (λx,y. f x = f y) ] /2/;
343 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
344 napply (.= (†H)); // ]
347 ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
349 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
352 ndefinition quotiented_mor:
353 ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
354 #A; #B; #f; @ [ napply f ] //.
357 nlemma first_omomorphism_theorem_functions1:
358 ∀A,B.∀f: unary_morphism A B.
359 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
362 alias symbol "eq" = "setoid eq".
363 ndefinition surjective ≝
364 λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
365 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
367 ndefinition injective ≝
368 λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
369 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
371 nlemma first_omomorphism_theorem_functions2:
373 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
376 nlemma first_omomorphism_theorem_functions3:
378 injective … (Full_set ?) (quotiented_mor … f).
379 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
382 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
384 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
385 f_sur: surjective … S T iso_f;
386 f_inj: injective … S iso_f
389 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
390 #A; #U; #V; #W; *; #H; #x; *; /2/.
393 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
394 #A; #U; #V; #W; #H; #H1; #x; *; /2/.
397 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
401 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
402 { iso_f:> unary_morphism A B;
403 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
410 λxxx:isomorphism A B S T.
412 return λxxx:isomorphism A B S T.
414 ∀x_72: mem (carr A) (pc A S) x.
415 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
416 with [ mk_isomorphism _ yyy ⇒ yyy ] ).