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/relations.ma".
17 include "formal_topology/notation.ma".
19 record basic_pair: Type[1] ≝ {
20 concr: REL; form: REL; rel: concr ⇒_\r1 form
23 interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ??? (rel c) x y).
24 interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
26 record relation_pair (BP1,BP2: basic_pair): Type[1] ≝ {
27 concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
28 commute: comp1 REL ??? concr_rel (rel ?) =_1 form_rel ∘ ⊩
31 interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
32 interpretation "formal relation" 'form_rel r = (form_rel ?? r).
34 definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
35 intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
36 | simplify; intros; apply refl1;
37 | simplify; intros 2; apply sym1;
38 | simplify; intros 3; apply trans1; ]
41 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
44 [ apply (relation_pair b b1)
45 | apply relation_pair_equality
49 definition relation_pair_of_relation_pair_setoid :
50 ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
51 coercion relation_pair_of_relation_pair_setoid.
53 alias symbol "compose" (instance 1) = "category1 composition".
55 ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ =_1 r'\sub\f ∘ ⊩.
56 intros 5 (o1 o2 r r' H);
57 apply (.= (commute ?? r)^-1);
58 change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
60 apply (commute ?? r').
63 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
67 | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
68 lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
73 lemma relation_pair_composition:
74 ∀o1,o2,o3: basic_pair.
75 relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
79 [ apply (r1 \sub\c ∘ r \sub\c)
80 | apply (r1 \sub\f ∘ r \sub\f)
81 | lapply (commute ?? r) as H;
82 lapply (commute ?? r1) as H1;
83 alias symbol "trans" = "trans1".
84 alias symbol "assoc" = "category1 assoc".
87 alias symbol "invert" = "setoid1 symmetry".
88 apply (.= ASSOC ^ -1);
93 lemma relation_pair_composition_is_morphism:
94 ∀o1,o2,o3: basic_pair.
95 ∀a,a':relation_pair_setoid o1 o2.
96 ∀b,b':relation_pair_setoid o2 o3.
98 relation_pair_composition o1 o2 o3 a b
99 = relation_pair_composition o1 o2 o3 a' b'.
102 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
103 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
104 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
107 apply (.= #‡(commute ?? b'));
108 apply (.= ASSOC ^ -1);
111 apply (.= #‡(commute ?? b')^-1);
115 definition relation_pair_composition_morphism:
116 ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
119 [ apply relation_pair_composition;
120 | apply relation_pair_composition_is_morphism;]
123 lemma relation_pair_composition_morphism_assoc:
128 .Πa12:relation_pair_setoid o1 o2
129 .Πa23:relation_pair_setoid o2 o3
130 .Πa34:relation_pair_setoid o3 o4
131 .relation_pair_composition_morphism o1 o3 o4
132 (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
133 =relation_pair_composition_morphism o1 o2 o4 a12
134 (relation_pair_composition_morphism o2 o3 o4 a23 a34).
136 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
137 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
138 alias symbol "refl" = "refl1".
139 alias symbol "prop2" = "prop21".
143 lemma relation_pair_composition_morphism_respects_id:
144 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
145 relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
147 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
148 apply ((id_neutral_right1 ????)‡#);
151 lemma relation_pair_composition_morphism_respects_id_r:
152 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
153 relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
155 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
156 apply ((id_neutral_left1 ????)‡#);
159 definition BP: category1.
162 | apply relation_pair_setoid
163 | apply id_relation_pair
164 | apply relation_pair_composition_morphism
165 | apply relation_pair_composition_morphism_assoc;
166 | apply relation_pair_composition_morphism_respects_id;
167 | apply relation_pair_composition_morphism_respects_id_r;]
170 definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
171 coercion basic_pair_of_BP.
173 definition relation_pair_setoid_of_arrows1_BP :
174 ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
175 coercion relation_pair_setoid_of_arrows1_BP.
178 definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
179 intros; constructor 1;
180 [ apply (ext ? ? (rel o));
186 definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(concr o).
187 intros; constructor 1;
188 [ apply (minus_image ?? (rel o));
189 | intros; apply (#‡e); ]
192 definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
193 intros (o); constructor 1;
194 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
195 intros; simplify; apply (.= (†e)‡#); apply refl1
196 | intros; split; simplify; intros;
197 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
198 | apply (. #‡((†e)‡(†e1))); assumption]]
201 interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
203 definition fintersectsS:
204 ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(form o).
205 intros (o); constructor 1;
206 [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
207 intros; simplify; apply (.= (†e)‡#); apply refl1
208 | intros; split; simplify; intros;
209 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
210 | apply (. #‡((†e)‡(†e1))); assumption]]
213 interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
215 definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
216 intros (o); constructor 1;
217 [ apply (λx:concr o.λS: Ω^(form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
218 | intros; split; intros; cases e2; exists [1,3: apply w]
219 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
220 | apply (. (#‡e1)‡(e‡#)); assumption]]
223 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
224 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).