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 include "basics/core_notation/fintersects_2.ma".
16 include "formal_topology/o-algebra.ma".
17 include "formal_topology/notation.ma".
19 record Obasic_pair: Type[2] ≝ {
20 Oconcr: OA; Oform: OA; Orel: arrows2 ? Oconcr Oform
24 interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
25 interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
27 record Orelation_pair (BP1,BP2: Obasic_pair): Type[2] ≝ {
28 Oconcr_rel: (Oconcr BP1) ⇒_\o2 (Oconcr BP2); Oform_rel: (Oform BP1) ⇒_\o2 (Oform BP2);
29 Ocommute: ⊩ ∘ Oconcr_rel =_2 Oform_rel ∘ ⊩
33 interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel ?? r).
34 interpretation "o-formal relation" 'form_rel r = (Oform_rel ?? r).
36 definition Orelation_pair_equality:
37 ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
40 [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
53 (* qui setoid1 e' giusto: ma non lo e'!!! *)
54 definition Orelation_pair_setoid: Obasic_pair → Obasic_pair → setoid2.
57 [ apply (Orelation_pair o o1)
58 | apply Orelation_pair_equality
62 definition Orelation_pair_of_Orelation_pair_setoid:
63 ∀P,Q. Orelation_pair_setoid P Q → Orelation_pair P Q ≝ λP,Q,x.x.
64 coercion Orelation_pair_of_Orelation_pair_setoid.
66 lemma eq_to_eq': ∀o1,o2.∀r,r': Orelation_pair_setoid o1 o2. r =_2 r' → r \sub\f ∘ ⊩ =_2 r'\sub\f ∘ ⊩.
67 intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
68 apply (.= ((Ocommute ?? r) ^ -1));
70 apply (.= (Ocommute ?? r'));
75 definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
79 | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
80 lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
85 lemma Orelation_pair_composition:
86 ∀o1,o2,o3:Obasic_pair.
87 Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
91 [ apply (r1 \sub\c ∘ r \sub\c)
92 | apply (r1 \sub\f ∘ r \sub\f)
93 | lapply (Ocommute ?? r) as H;
94 lapply (Ocommute ?? r1) as H1;
95 apply rule (.= ASSOC);
97 apply rule (.= ASSOC ^ -1);
103 lemma Orelation_pair_composition_is_morphism:
104 ∀o1,o2,o3:Obasic_pair.
105 Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
107 Orelation_pair_composition o1 o2 o3 a b
108 = Orelation_pair_composition o1 o2 o3 a' b'.
110 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
111 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
112 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
113 apply rule (.= ASSOC);
115 apply (.= #‡(Ocommute ?? b'));
116 apply rule (.= ASSOC^-1);
118 apply rule (.= ASSOC);
119 apply (.= #‡(Ocommute ?? b')^-1);
120 apply rule (ASSOC^-1);
123 definition Orelation_pair_composition_morphism:
124 ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
125 intros; constructor 1;
126 [ apply Orelation_pair_composition;
127 | apply Orelation_pair_composition_is_morphism;]
130 lemma Orelation_pair_composition_morphism_assoc:
131 ∀o1,o2,o3,o4:Obasic_pair
132 .Πa12:Orelation_pair_setoid o1 o2
133 .Πa23:Orelation_pair_setoid o2 o3
134 .Πa34:Orelation_pair_setoid o3 o4
135 .Orelation_pair_composition_morphism o1 o3 o4
136 (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
137 =Orelation_pair_composition_morphism o1 o2 o4 a12
138 (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).
140 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
141 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
142 apply rule (ASSOC‡#);
145 lemma Orelation_pair_composition_morphism_respects_id:
148 .Πa:Orelation_pair_setoid o1 o2
149 .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
151 change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
152 apply ((id_neutral_right2 ????)‡#);
155 lemma Orelation_pair_composition_morphism_respects_id_r:
158 .Πa:Orelation_pair_setoid o1 o2
159 .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
161 change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
162 apply ((id_neutral_left2 ????)‡#);
165 definition OBP: category2.
168 | apply Orelation_pair_setoid
169 | apply Oid_relation_pair
170 | apply Orelation_pair_composition_morphism
171 | apply Orelation_pair_composition_morphism_assoc;
172 | apply Orelation_pair_composition_morphism_respects_id;
173 | apply Orelation_pair_composition_morphism_respects_id_r;]
176 definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.
177 coercion Obasic_pair_of_objs2_OBP.
179 definition Orelation_pair_setoid_of_arrows2_OBP:
180 ∀P,Q.arrows2 OBP P Q → Orelation_pair_setoid P Q ≝ λP,Q,c.c.
181 coercion Orelation_pair_setoid_of_arrows2_OBP.
183 notation > "B ⇒_\obp2 C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
184 notation "B ⇒\sub (\obp 2) C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
185 interpretation "'arrows2_OBP" 'arrows2_OBP A B = (arrows2 OBP A B).
188 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
189 intros; constructor 1;
190 [ apply (ext ? ? (rel o));
196 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
201 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
202 intros (o); constructor 1;
203 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
204 intros; simplify; apply (.= (†H)‡#); apply refl1
205 | intros; split; simplify; intros;
206 [ apply (. #‡((†H)‡(†H1))); assumption
207 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
210 interpretation "fintersects" 'fintersects U V = (fun1 ??? (fintersects ?) U V).
212 definition fintersectsS:
213 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
214 intros (o); constructor 1;
215 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
216 intros; simplify; apply (.= (†H)‡#); apply refl1
217 | intros; split; simplify; intros;
218 [ apply (. #‡((†H)‡(†H1))); assumption
219 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
222 interpretation "fintersectsS" 'fintersects U V = (fun1 ??? (fintersectsS ?) U V).
226 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
227 intros (o); constructor 1;
228 [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
229 | intros; split; intros; cases H2; exists [1,3: apply w]
230 [ apply (. (#‡H1)‡(H‡#)); assumption
231 | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
234 interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr ?) ?? (relS ?) x y).
235 interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ??? (relS ?)).
238 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
239 notation > "□⎽term 90 b" non associative with precedence 90 for @{'box $b}.
240 interpretation "Universal image ⊩⎻*" 'box x = (fun12 ? ? (or_f_minus_star ? ?) (Orel x)).
242 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
243 notation > "◊⎽term 90 b" non associative with precedence 90 for @{'diamond $b}.
244 interpretation "Existential image ⊩" 'diamond x = (fun12 ? ? (or_f ? ?) (Orel x)).
246 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
247 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
248 interpretation "Universal pre-image ⊩*" 'rest x = (fun12 ? ? (or_f_star ? ?) (Orel x)).
250 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
251 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
252 interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 ? ? (or_f_minus ? ?) (Orel x)).