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 *)
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15 include "formal_topology/o-algebra.ma".
16 include "formal_topology/notation.ma".
18 record Obasic_pair: Type[2] ≝ {
19 Oconcr: OA; Oform: OA; Orel: arrows2 ? Oconcr Oform
23 interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
24 interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
26 record Orelation_pair (BP1,BP2: Obasic_pair): Type[2] ≝ {
27 Oconcr_rel: (Oconcr BP1) ⇒_\o2 (Oconcr BP2); Oform_rel: (Oform BP1) ⇒_\o2 (Oform BP2);
28 Ocommute: ⊩ ∘ Oconcr_rel =_2 Oform_rel ∘ ⊩
32 interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel ?? r).
33 interpretation "o-formal relation" 'form_rel r = (Oform_rel ?? r).
35 definition Orelation_pair_equality:
36 ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
39 [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
52 (* qui setoid1 e' giusto: ma non lo e'!!! *)
53 definition Orelation_pair_setoid: Obasic_pair → Obasic_pair → setoid2.
56 [ apply (Orelation_pair o o1)
57 | apply Orelation_pair_equality
61 definition Orelation_pair_of_Orelation_pair_setoid:
62 ∀P,Q. Orelation_pair_setoid P Q → Orelation_pair P Q ≝ λP,Q,x.x.
63 coercion Orelation_pair_of_Orelation_pair_setoid.
65 lemma eq_to_eq': ∀o1,o2.∀r,r': Orelation_pair_setoid o1 o2. r =_2 r' → r \sub\f ∘ ⊩ =_2 r'\sub\f ∘ ⊩.
66 intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
67 apply (.= ((Ocommute ?? r) ^ -1));
69 apply (.= (Ocommute ?? r'));
74 definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
78 | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
79 lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
84 lemma Orelation_pair_composition:
85 ∀o1,o2,o3:Obasic_pair.
86 Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
90 [ apply (r1 \sub\c ∘ r \sub\c)
91 | apply (r1 \sub\f ∘ r \sub\f)
92 | lapply (Ocommute ?? r) as H;
93 lapply (Ocommute ?? r1) as H1;
94 apply rule (.= ASSOC);
96 apply rule (.= ASSOC ^ -1);
102 lemma Orelation_pair_composition_is_morphism:
103 ∀o1,o2,o3:Obasic_pair.
104 Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
106 Orelation_pair_composition o1 o2 o3 a b
107 = Orelation_pair_composition o1 o2 o3 a' b'.
109 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
110 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
111 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
112 apply rule (.= ASSOC);
114 apply (.= #‡(Ocommute ?? b'));
115 apply rule (.= ASSOC^-1);
117 apply rule (.= ASSOC);
118 apply (.= #‡(Ocommute ?? b')^-1);
119 apply rule (ASSOC^-1);
122 definition Orelation_pair_composition_morphism:
123 ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
124 intros; constructor 1;
125 [ apply Orelation_pair_composition;
126 | apply Orelation_pair_composition_is_morphism;]
129 lemma Orelation_pair_composition_morphism_assoc:
130 ∀o1,o2,o3,o4:Obasic_pair
131 .Πa12:Orelation_pair_setoid o1 o2
132 .Πa23:Orelation_pair_setoid o2 o3
133 .Πa34:Orelation_pair_setoid o3 o4
134 .Orelation_pair_composition_morphism o1 o3 o4
135 (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
136 =Orelation_pair_composition_morphism o1 o2 o4 a12
137 (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).
139 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
140 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
141 apply rule (ASSOC‡#);
144 lemma Orelation_pair_composition_morphism_respects_id:
147 .Πa:Orelation_pair_setoid o1 o2
148 .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
150 change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
151 apply ((id_neutral_right2 ????)‡#);
154 lemma Orelation_pair_composition_morphism_respects_id_r:
157 .Πa:Orelation_pair_setoid o1 o2
158 .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
160 change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
161 apply ((id_neutral_left2 ????)‡#);
164 definition OBP: category2.
167 | apply Orelation_pair_setoid
168 | apply Oid_relation_pair
169 | apply Orelation_pair_composition_morphism
170 | apply Orelation_pair_composition_morphism_assoc;
171 | apply Orelation_pair_composition_morphism_respects_id;
172 | apply Orelation_pair_composition_morphism_respects_id_r;]
175 definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.
176 coercion Obasic_pair_of_objs2_OBP.
178 definition Orelation_pair_setoid_of_arrows2_OBP:
179 ∀P,Q.arrows2 OBP P Q → Orelation_pair_setoid P Q ≝ λP,Q,c.c.
180 coercion Orelation_pair_setoid_of_arrows2_OBP.
182 notation > "B ⇒_\obp2 C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
183 notation "B ⇒\sub (\obp 2) C" right associative with precedence 72 for @{'arrows2_OBP $B $C}.
184 interpretation "'arrows2_OBP" 'arrows2_OBP A B = (arrows2 OBP A B).
187 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
188 intros; constructor 1;
189 [ apply (ext ? ? (rel o));
195 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
200 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
201 intros (o); constructor 1;
202 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
203 intros; simplify; apply (.= (†H)‡#); apply refl1
204 | intros; split; simplify; intros;
205 [ apply (. #‡((†H)‡(†H1))); assumption
206 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
209 interpretation "fintersects" 'fintersects U V = (fun1 ??? (fintersects ?) U V).
211 definition fintersectsS:
212 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
213 intros (o); constructor 1;
214 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
215 intros; simplify; apply (.= (†H)‡#); apply refl1
216 | intros; split; simplify; intros;
217 [ apply (. #‡((†H)‡(†H1))); assumption
218 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
221 interpretation "fintersectsS" 'fintersects U V = (fun1 ??? (fintersectsS ?) U V).
225 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
226 intros (o); constructor 1;
227 [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
228 | intros; split; intros; cases H2; exists [1,3: apply w]
229 [ apply (. (#‡H1)‡(H‡#)); assumption
230 | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
233 interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr ?) ?? (relS ?) x y).
234 interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ??? (relS ?)).
237 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
238 notation > "□⎽term 90 b" non associative with precedence 90 for @{'box $b}.
239 interpretation "Universal image ⊩⎻*" 'box x = (fun12 ? ? (or_f_minus_star ? ?) (Orel x)).
241 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
242 notation > "◊⎽term 90 b" non associative with precedence 90 for @{'diamond $b}.
243 interpretation "Existential image ⊩" 'diamond x = (fun12 ? ? (or_f ? ?) (Orel x)).
245 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
246 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
247 interpretation "Universal pre-image ⊩*" 'rest x = (fun12 ? ? (or_f_star ? ?) (Orel x)).
249 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
250 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
251 interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 ? ? (or_f_minus ? ?) (Orel x)).