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.
121 ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
122 on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
125 nlemma mem_ext_powerclass_setoid_is_morph:
126 ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
128 [ napply (λx,S. x ∈ S)
129 | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
130 ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
131 ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
136 unification hint 0 ≔ A:setoid, x, S;
138 TT ≟ (mk_binary_morphism1 ???
139 (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
140 (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
144 (*-------------------------------------*) ⊢
145 fun21 M1 M2 M3 TT x S ≡ mem A SS x.
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 ndefinition prop21_mem :
197 ∀A,C.∀f:binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) C.
198 ∀a,a':setoid1_of_setoid A.
199 ∀b,b':ext_powerclass_setoid A.a = a' → b = b' → f a b = f a' b'.
200 #A; #C; #f; #a; #a'; #b; #b'; #H1; #H2; napply prop21; nassumption;
203 interpretation "prop21 mem" 'prop2 l r = (prop21_mem ??????? l r).
205 nlemma intersect_is_ext_morph:
206 ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
207 #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
208 #H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
213 R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
214 (λS,S':carr1 (ext_powerclass_setoid A).
215 mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
216 (prop21 … (intersect_is_ext_morph A))) ,
219 (* ------------------------------------------------------*) ⊢
222 (ext_powerclass_setoid A)
223 (ext_powerclass_setoid A)
224 (ext_powerclass_setoid A) R B C) ≡
225 intersect (carr A) BB CC.
230 nlemma test: ∀U.∀A,B:qpowerclass U. A ∩ B = A →
231 ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
232 #U; #A; #B; #H; #x; #y; #K; #K2; napply (. #‡(?));
233 ##[ nchange with (A ∩ B = ?);
234 napply (prop21 ??? (mk_binary_morphism1 … (λS,S'.S ∩ S') (prop21 … (intersect_ok' U))) A A B B ##);
240 nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
245 nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
246 [##1,2: napply (. Ha^-1‡#); nassumption;
247 ##|##3,4: napply (. Ha‡#); nassumption]##]
248 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
249 [ alias symbol "invert" = "setoid1 symmetry".
250 napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
251 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
254 (* unfold if intersect, exposing fun21 *)
255 alias symbol "hint_decl" = "hint_decl_Type1".
257 A : setoid, B,C : qpowerclass A ⊢
259 (mk_binary_morphism1 …
260 (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S')))
261 (prop21 … (intersect_ok A)))
264 ≡ intersect ? (pc ? B) (pc ? C).
266 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
267 #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
271 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
272 λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
273 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
275 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
276 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
278 (******************* compatible equivalence relations **********************)
280 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
281 { rel:> equivalence_relation A;
282 compatibility: ∀x,x':A. x=x' → rel x x'
285 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
289 (******************* first omomorphism theorem for sets **********************)
291 ndefinition eqrel_of_morphism:
292 ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
295 [ napply (λx,y. f x = f y)
296 | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
297 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
298 napply (.= (†H)); napply refl ]
301 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
303 [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
306 ndefinition quotiented_mor:
307 ∀A,B.∀f:unary_morphism A B.
308 unary_morphism (quotient … (eqrel_of_morphism … f)) B.
310 [ napply f | #a; #a'; #H; nassumption]
313 nlemma first_omomorphism_theorem_functions1:
314 ∀A,B.∀f: unary_morphism A B.
315 ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
316 #A; #B; #f; #x; napply refl;
319 ndefinition surjective ≝
320 λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
321 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
323 ndefinition injective ≝
324 λA,B.λS: qpowerclass A.λf:unary_morphism A B.
325 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
327 nlemma first_omomorphism_theorem_functions2:
328 ∀A,B.∀f: unary_morphism A B.
329 surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
330 #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl;
331 (* bug, prova @ I refl *)
334 nlemma first_omomorphism_theorem_functions3:
335 ∀A,B.∀f: unary_morphism A B.
336 injective … (Full_set ?) (quotiented_mor … f).
337 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
340 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : Type[0] ≝
341 { iso_f:> unary_morphism A B;
342 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
343 f_sur: surjective … S T iso_f;
344 f_inj: injective … S iso_f
348 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
349 { iso_f:> unary_morphism A B;
350 f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
357 λxxx:isomorphism A B S T.
359 return λxxx:isomorphism A B S T.
361 ∀x_72: mem (carr A) (pc A S) x.
362 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
363 with [ mk_isomorphism _ yyy ⇒ yyy ] ).