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.
44 #A; #S; #x; #H; nassumption.
47 nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
48 #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
51 include "properties/relations1.ma".
53 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
55 [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
56 | #S; @; napply subseteq_refl
57 | #S; #S'; *; #H1; #H2; @; nassumption
58 | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
59 ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
62 include "sets/setoids1.ma".
64 (* this has to be declared here, so that it is combined with carr *)
65 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
67 ndefinition powerclass_setoid: Type[0] → setoid1.
68 #A; @[ napply (Ω^A)| napply seteq ]
71 include "hints_declaration.ma".
73 alias symbol "hint_decl" = "hint_decl_Type2".
74 unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
76 (************ SETS OVER SETOIDS ********************)
78 include "logic/cprop.ma".
80 nrecord ext_powerclass (A: setoid) : Type[1] ≝
81 { ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso...
82 forse lo si vorrebbe dichiarato con un target più lasco
83 ma la sintassi :> non lo supporta *)
84 ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr)
87 notation > "𝛀 ^ term 90 A" non associative with precedence 70
88 for @{ 'ext_powerclass $A }.
90 notation "Ω term 90 A \atop ≈" non associative with precedence 70
91 for @{ 'ext_powerclass $A }.
93 interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
95 ndefinition Full_set: ∀A. 𝛀^A.
96 #A; @[ napply A | #x; #x'; #H; napply refl1]
98 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
100 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
102 [ napply (λS,S'. S = S')
103 | #S; napply (refl1 ? (seteq A))
104 | #S; #S'; napply (sym1 ? (seteq A))
105 | #S; #T; #U; napply (trans1 ? (seteq A))]
108 ndefinition ext_powerclass_setoid: setoid → setoid1.
110 [ napply (ext_powerclass A)
111 | napply (ext_seteq A) ]
114 unification hint 0 ≔ A;
115 R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
116 (* ----------------------------------------------------- *) ⊢
117 carr1 R ≡ ext_powerclass A.
119 interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
122 ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
123 on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
126 nlemma mem_ext_powerclass_setoid_is_morph:
127 ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
129 [ napply (λx,S. x ∈ S)
130 | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
131 ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
132 ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
137 unification hint 0 ≔ A:setoid, x, S;
139 TT ≟ (mk_binary_morphism1 ???
140 (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
141 (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
142 XX ≟ (ext_powerclass_setoid A)
143 (*-------------------------------------*) ⊢
144 fun21 (setoid1_of_setoid A) XX CPROP TT x S
147 nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
149 [ napply (λS,S'. S ⊆ S')
150 | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
151 [ napply (subseteq_trans … a)
152 [ nassumption | napply (subseteq_trans … b); nassumption ]
153 ##| napply (subseteq_trans … a')
154 [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
157 unification hint 0 ≔ A,a,a'
158 (*-----------------------------------------------------------------*) ⊢
159 eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
161 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
162 #A; #S; #S'; @ (S ∩ S');
163 #a; #a'; #Ha; @; *; #H1; #H2; @
164 [##1,2: napply (. Ha^-1‡#); nassumption;
165 ##|##3,4: napply (. Ha‡#); nassumption]
168 alias symbol "hint_decl" = "hint_decl_Type1".
170 A : setoid, B,C : ext_powerclass A;
171 R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
173 (* ------------------------------------------*) ⊢
174 ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
176 nlemma intersect_is_morph:
177 ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
178 #A; @ (λS,S'. S ∩ S');
179 #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
180 [ napply Ha1; nassumption
181 | napply Hb1; nassumption
182 | napply Ha2; nassumption
183 | napply Hb2; nassumption]
186 alias symbol "hint_decl" = "hint_decl_Type1".
188 A : Type[0], B,C : Ω^A;
189 R ≟ (mk_binary_morphism1 …
191 (prop21 … (intersect_is_morph A)))
193 fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
196 interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
198 nlemma intersect_is_ext_morph:
199 ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
200 #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
201 #H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
206 R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
207 (λS,S':carr1 (ext_powerclass_setoid A).
208 mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
209 (prop21 … (intersect_is_ext_morph A))) ,
212 (* ------------------------------------------------------*) ⊢
215 (ext_powerclass_setoid A)
216 (ext_powerclass_setoid A)
217 (ext_powerclass_setoid A) R B C) ≡
218 intersect (carr A) BB CC.
221 alias symbol "hint_decl" = "hint_decl_Type2".
223 A : setoid, B,C : 𝛀^A ;
226 C1 ≟ (carr1 (powerclass_setoid (carr A))),
227 C2 ≟ (carr1 (ext_powerclass_setoid A))
229 eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡
230 eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
233 A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.
235 nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
236 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
237 #U; #A; #B; #H; #x; #y; #K; #K2;
238 alias symbol "prop2" = "prop21 mem".
239 alias symbol "invert" = "setoid1 symmetry".
245 nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
250 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
251 [##1,2: napply (. Ha^-1‡#); nassumption;
252 ##|##3,4: napply (. Ha‡#); nassumption]##]
253 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
254 [ alias symbol "invert" = "setoid1 symmetry".
255 alias symbol "refl" = "refl".
256 alias symbol "prop2" = "prop21".
257 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
258 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
261 (* unfold if intersect, exposing fun21 *)
262 alias symbol "hint_decl" = "hint_decl_Type1".
264 A : setoid, B,C : ext_powerclass A ⊢
266 (mk_binary_morphism1 …
267 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
268 (prop21 … (intersect_ok A)))
271 ≡ intersect ? (pc ? B) (pc ? C).
273 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
274 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
278 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
279 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
280 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
282 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
283 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
285 (******************* compatible equivalence relations **********************)
287 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
288 { rel:> equivalence_relation A;
289 compatibility: ∀x,x':A. x=x' → rel x x'
292 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
296 (******************* first omomorphism theorem for sets **********************)
298 ndefinition eqrel_of_morphism:
299 ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
302 [ napply (λx,y. f x = f y)
303 | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
304 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
305 napply (.= (†H)); napply refl ]
308 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
310 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
313 ndefinition quotiented_mor:
314 ∀A,B.∀f:unary_morphism A B.
315 unary_morphism (quotient … (eqrel_of_morphism … f)) B.
317 [ napply f | #a; #a'; #H; nassumption]
320 nlemma first_omomorphism_theorem_functions1:
321 ∀A,B.∀f: unary_morphism A B.
322 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
323 #A; #B; #f; #x; napply refl;
326 alias symbol "eq" = "setoid eq".
327 ndefinition surjective ≝
328 λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
329 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
331 ndefinition injective ≝
332 λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
333 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
335 nlemma first_omomorphism_theorem_functions2:
336 ∀A,B.∀f: unary_morphism A B.
337 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
338 #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl;
339 (* bug, prova @ I refl *)
342 nlemma first_omomorphism_theorem_functions3:
343 ∀A,B.∀f: unary_morphism A B.
344 injective … (Full_set ?) (quotiented_mor … f).
345 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
348 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
349 { iso_f:> unary_morphism A B;
350 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
351 f_sur: surjective … S T iso_f;
352 f_inj: injective … S iso_f
355 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
356 #A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
359 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
360 #A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
363 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
364 #A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
368 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
369 { iso_f:> unary_morphism A B;
370 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
377 λxxx:isomorphism A B S T.
379 return λxxx:isomorphism A B S T.
381 ∀x_72: mem (carr A) (pc A S) x.
382 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
383 with [ mk_isomorphism _ yyy ⇒ yyy ] ).