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).
53 [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
56 | #S; #T; #U; *; #H1; #H2; *; /3/]
59 include "sets/setoids1.ma".
61 (* this has to be declared here, so that it is combined with carr *)
62 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
64 ndefinition powerclass_setoid: Type[0] → setoid1.
68 include "hints_declaration.ma".
70 alias symbol "hint_decl" = "hint_decl_Type2".
71 unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
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 70
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 interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
113 ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
114 on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
117 nlemma mem_ext_powerclass_setoid_is_morph:
118 ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
120 [ napply (λx,S. x ∈ S)
121 | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
122 ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
123 ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
128 unification hint 0 ≔ A:setoid, x, S;
130 TT ≟ (mk_binary_morphism1 ???
131 (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
132 (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
133 XX ≟ (ext_powerclass_setoid A)
134 (*-------------------------------------*) ⊢
135 fun21 (setoid1_of_setoid A) XX CPROP TT x S
138 nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
140 [ napply (λS,S'. S ⊆ S')
141 | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *;/4/]
144 unification hint 0 ≔ A,a,a'
145 (*-----------------------------------------------------------------*) ⊢
146 eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
148 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
149 #A; #S; #S'; @ (S ∩ S');
150 #a; #a'; #Ha; @; *; #H1; #H2; @
151 [##1,2: napply (. Ha^-1‡#); nassumption;
152 ##|##3,4: napply (. Ha‡#); 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)))
160 (* ------------------------------------------*) ⊢
161 ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
163 nlemma intersect_is_morph:
164 ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
165 #A; @ (λS,S'. S ∩ S');
166 #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
169 alias symbol "hint_decl" = "hint_decl_Type1".
171 A : Type[0], B,C : Ω^A;
172 R ≟ (mk_binary_morphism1 …
174 (prop21 … (intersect_is_morph A)))
176 fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
179 interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
181 nlemma intersect_is_ext_morph:
182 ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
183 #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
184 #H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
189 R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
190 (λS,S':carr1 (ext_powerclass_setoid A).
191 mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
192 (prop21 … (intersect_is_ext_morph A))) ,
195 (* ------------------------------------------------------*) ⊢
198 (ext_powerclass_setoid A)
199 (ext_powerclass_setoid A)
200 (ext_powerclass_setoid A) R B C) ≡
201 intersect (carr A) BB CC.
204 alias symbol "hint_decl" = "hint_decl_Type2".
206 A : setoid, B,C : 𝛀^A ;
209 C1 ≟ (carr1 (powerclass_setoid (carr A))),
210 C2 ≟ (carr1 (ext_powerclass_setoid A))
212 eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
213 eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
216 A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
218 nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
219 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
220 #U; #A; #B; #H; #x; #y; #K; #K2;
221 alias symbol "prop2" = "prop21 mem".
222 alias symbol "invert" = "setoid1 symmetry".
228 nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
233 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
234 [##1,2: napply (. Ha^-1‡#); nassumption;
235 ##|##3,4: napply (. Ha‡#); nassumption]##]
236 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
237 [ alias symbol "invert" = "setoid1 symmetry".
238 alias symbol "refl" = "refl".
239 alias symbol "prop2" = "prop21".
240 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
241 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
244 (* unfold if intersect, exposing fun21 *)
245 alias symbol "hint_decl" = "hint_decl_Type1".
247 A : setoid, B,C : ext_powerclass A ⊢
249 (mk_binary_morphism1 …
250 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
251 (prop21 … (intersect_ok A)))
254 ≡ intersect ? (pc ? B) (pc ? C).
256 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
257 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
261 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
262 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
263 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
265 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
266 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
268 (******************* compatible equivalence relations **********************)
270 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
271 { rel:> equivalence_relation A;
272 compatibility: ∀x,x':A. x=x' → rel x x'
275 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
279 (******************* first omomorphism theorem for sets **********************)
281 ndefinition eqrel_of_morphism:
282 ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
284 [ @ [ napply (λx,y. f x = f y) ] /2/;
285 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
286 napply (.= (†H)); // ]
289 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
291 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
294 ndefinition quotiented_mor:
295 ∀A,B.∀f:unary_morphism A B.
296 unary_morphism (quotient … (eqrel_of_morphism … f)) B.
297 #A; #B; #f; @ [ napply f ] //.
300 nlemma first_omomorphism_theorem_functions1:
301 ∀A,B.∀f: unary_morphism A B.
302 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
305 alias symbol "eq" = "setoid eq".
306 ndefinition surjective ≝
307 λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
308 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
310 ndefinition injective ≝
311 λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
312 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
314 nlemma first_omomorphism_theorem_functions2:
315 ∀A,B.∀f: unary_morphism A B.
316 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
319 nlemma first_omomorphism_theorem_functions3:
320 ∀A,B.∀f: unary_morphism A B.
321 injective … (Full_set ?) (quotiented_mor … f).
322 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
325 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
326 { iso_f:> unary_morphism A B;
327 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
328 f_sur: surjective … S T iso_f;
329 f_inj: injective … S iso_f
332 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
333 #A; #U; #V; #W; *; #H; #x; *; /2/.
336 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
337 #A; #U; #V; #W; #H; #H1; #x; *; /2/.
340 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
344 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
345 { iso_f:> unary_morphism A B;
346 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
353 λxxx:isomorphism A B S T.
355 return λxxx:isomorphism A B S T.
357 ∀x_72: mem (carr A) (pc A S) x.
358 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
359 with [ mk_isomorphism _ yyy ⇒ yyy ] ).