2 (**************************************************************************)
5 (* ||A|| A project by Andrea Asperti *)
7 (* ||I|| Developers: *)
8 (* ||T|| The HELM team. *)
9 (* ||A|| http://helm.cs.unibo.it *)
11 (* \ / This file is distributed under the terms of the *)
12 (* v GNU General Public License Version 2 *)
14 (**************************************************************************)
16 (******************* SETS OVER TYPES *****************)
18 include "logic/connectives.ma".
20 nrecord powerclass (A: Type[0]) : Type[1] ≝ { mem: A → CProp[0] }.
22 interpretation "mem" 'mem a S = (mem ? S a).
23 interpretation "powerclass" 'powerset A = (powerclass A).
24 interpretation "subset construction" 'subset \eta.x = (mk_powerclass ? x).
26 ndefinition subseteq ≝ λA.λU,V.∀a:A. a ∈ U → a ∈ V.
27 interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
29 ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
30 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
32 ndefinition intersect ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∧ x ∈ V }.
33 interpretation "intersect" 'intersects U V = (intersect ? U V).
35 ndefinition union ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∨ x ∈ V }.
36 interpretation "union" 'union U V = (union ? U V).
38 ndefinition big_union ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
40 ndefinition big_intersection ≝ λA,B.λT:Ω \sup A.λf:A → Ω \sup B.{ x | ∀i. i ∈ T → x ∈ f i }.
42 ndefinition full_set: ∀A. Ω \sup A ≝ λA.{ x | True }.
43 ncoercion full_set : ∀A:Type[0]. Ω \sup A ≝ full_set on A: Type[0] to (Ω \sup ?).
45 nlemma subseteq_refl: ∀A.∀S: Ω \sup A. S ⊆ S.
46 #A; #S; #x; #H; nassumption.
49 nlemma subseteq_trans: ∀A.∀S,T,U: Ω \sup A. S ⊆ T → T ⊆ U → S ⊆ U.
50 #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
53 include "properties/relations1.ma".
55 ndefinition seteq: ∀A. equivalence_relation1 (Ω \sup A).
56 #A; napply mk_equivalence_relation1
57 [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
58 | #S; napply conj; napply subseteq_refl
59 | #S; #S'; *; #H1; #H2; napply conj; nassumption
60 | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; napply conj; napply subseteq_trans;
61 ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
64 include "sets/setoids1.ma".
66 ndefinition powerclass_setoid: Type[0] → setoid1.
72 include "hints_declaration.ma".
74 alias symbol "hint_decl" = "hint_decl_Type2".
75 unification hint 0 ≔ A : ? ⊢ carr1 (powerclass_setoid A) ≡ Ω^A.
77 (************ SETS OVER SETOIDS ********************)
79 include "logic/cprop.ma".
81 nrecord qpowerclass (A: setoid) : Type[1] ≝
83 mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc)
86 ndefinition Full_set: ∀A. qpowerclass A.
87 #A; napply mk_qpowerclass
89 | #x; #x'; #H; napply refl1; ##]
92 ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
93 #A; napply mk_equivalence_relation1
94 [ napply (λS,S':qpowerclass A. S = S')
95 | #S; napply (refl1 ? (seteq A))
96 | #S; #S'; napply (sym1 ? (seteq A))
97 | #S; #T; #U; napply (trans1 ? (seteq A))]
100 ndefinition qpowerclass_setoid: setoid → setoid1.
101 #A; napply mk_setoid1
102 [ napply (qpowerclass A)
103 | napply (qseteq A) ]
106 unification hint 0 ≔ A : ? ⊢ carr1 (qpowerclass_setoid A) ≡ qpowerclass A.
108 nlemma mem_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
109 #A; napply mk_binary_morphism1
110 [ #x; napply (λS: qpowerclass_setoid ?. x ∈ S) (* ERROR CSC: ??? *)
111 | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; napply mk_iff; #H;
112 ##[ napply Hb1; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha^-1;##]
113 ##| napply Hb2; napply (. (mem_ok' …)); ##[##3: napply H| napply Ha;##]
119 A : setoid, x : ?, S : ? ⊢ (mem_ok A) x S ≡ mem A S x.
121 nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
122 #A; napply mk_binary_morphism1
123 [ napply (λS,S': qpowerclass_setoid ?. S ⊆ S')
124 | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
125 [ napply (subseteq_trans … a)
126 [ nassumption | napply (subseteq_trans … b); nassumption ]
127 ##| napply (subseteq_trans … a')
128 [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
131 nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
132 #A; napply mk_binary_morphism1
133 [ #S; #S'; napply mk_qpowerclass
135 | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
136 [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##2,5: nassumption |##*: ##skip]
137 ##|##3,4: napply (. (mem_ok' …)) [##1,3,4,6: nassumption |##*: ##skip]##]##]
138 ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
139 [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
140 | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
145 A : setoid, U : qpowerclass_setoid A, V : ? ⊢ (intersect_ok A) U V ≡ U ∩ V.
148 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
149 #A; #U; #V; #x; #x'; #H; #p;
150 (* CSC: senza la change non funziona! *)
151 nchange with (x' ∈ (fun21 ??? (intersect_ok A) U V));
152 napply (. (H^-1‡#)); nassumption.
156 (* qui non funziona una cippa *)
157 ndefinition image: ∀A,B. (carr A → carr B) → Ω \sup A → Ω \sup B ≝
158 λA,B:setoid.λf:carr A → carr B.λSa:Ω \sup A.
159 {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) ? ?(*(f x) y*)}.
160 ##[##2: napply (f x); ##|##3: napply y]
161 #a; #a'; #H; nwhd; nnormalize; (* per togliere setoid1_of_setoid *) napply (mk_iff ????);
162 *; #x; #Hx; napply (ex_intro … x)
163 [ napply (. (#‡(#‡#)));
165 ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
166 λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
169 (******************* compatible equivalence relations **********************)
171 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
172 { rel:> equivalence_relation A;
173 compatibility: ∀x,x':A. x=x' → eq_rel ? rel x x' (* coercion qui non va *)
176 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
177 #A; #R; napply mk_setoid
182 (******************* first omomorphism theorem for sets **********************)
184 ndefinition eqrel_of_morphism:
185 ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
186 #A; #B; #f; napply mk_compatible_equivalence_relation
187 [ napply mk_equivalence_relation
188 [ napply (λx,y. f x = f y)
189 | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
190 ##| #x; #x'; #H; nwhd; napply (.= (†H)); napply refl ]
193 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
194 #A; #R; napply mk_unary_morphism
195 [ napply (λx.x) | #a; #a'; #H; napply (compatibility ? R … H) ]
198 ndefinition quotiented_mor:
199 ∀A,B.∀f:unary_morphism A B.
200 unary_morphism (quotient ? (eqrel_of_morphism ?? f)) B.
201 #A; #B; #f; napply mk_unary_morphism
202 [ napply f | #a; #a'; #H; nassumption]
205 nlemma first_omomorphism_theorem_functions1:
206 ∀A,B.∀f: unary_morphism A B.
207 ∀x. f x = quotiented_mor ??? (canonical_proj ? (eqrel_of_morphism ?? f) x).
208 #A; #B; #f; #x; napply refl;
211 ndefinition surjective ≝
212 λA,B.λS: qpowerclass A.λT: qpowerclass B.λf:unary_morphism A B.
213 ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
215 ndefinition injective ≝
216 λA,B.λS: qpowerclass A.λf:unary_morphism A B.
217 ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
219 nlemma first_omomorphism_theorem_functions2:
220 ∀A,B.∀f: unary_morphism A B. surjective ?? (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism ?? f)).
221 #A; #B; #f; nwhd; #y; #Hy; napply (ex_intro … y); napply conj
222 [ napply I | napply refl]
225 nlemma first_omomorphism_theorem_functions3:
226 ∀A,B.∀f: unary_morphism A B. injective ?? (Full_set ?) (quotiented_mor ?? f).
227 #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
230 nrecord isomorphism (A) (B) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
231 { iso_f:> unary_morphism A B;
232 f_closed: ∀x. x ∈ S → iso_f x ∈ T;
233 f_sur: surjective ?? S T iso_f;
234 f_inj: injective ?? S iso_f