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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/relations.ma".
16 include "formal_topology/notation.ma".
18 record basic_pair: Type[1] ≝ {
19 concr: REL; form: REL; rel: concr ⇒_\r1 form
22 interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ??? (rel c) x y).
23 interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
25 record relation_pair (BP1,BP2: basic_pair): Type[1] ≝ {
26 concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
27 commute: comp1 REL ??? concr_rel (rel ?) =_1 form_rel ∘ ⊩
30 interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
31 interpretation "formal relation" 'form_rel r = (form_rel ?? r).
33 definition relation_pair_equality: ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
34 intros; constructor 1; [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
35 | simplify; intros; apply refl1;
36 | simplify; intros 2; apply sym1;
37 | simplify; intros 3; apply trans1; ]
40 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
43 [ apply (relation_pair b b1)
44 | apply relation_pair_equality
48 definition relation_pair_of_relation_pair_setoid :
49 ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
50 coercion relation_pair_of_relation_pair_setoid.
52 alias symbol "compose" (instance 1) = "category1 composition".
54 ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ =_1 r'\sub\f ∘ ⊩.
55 intros 5 (o1 o2 r r' H);
56 apply (.= (commute ?? r)^-1);
57 change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
59 apply (commute ?? r').
62 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
66 | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
67 lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
72 lemma relation_pair_composition:
73 ∀o1,o2,o3: basic_pair.
74 relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
78 [ apply (r1 \sub\c ∘ r \sub\c)
79 | apply (r1 \sub\f ∘ r \sub\f)
80 | lapply (commute ?? r) as H;
81 lapply (commute ?? r1) as H1;
82 alias symbol "trans" = "trans1".
83 alias symbol "assoc" = "category1 assoc".
86 alias symbol "invert" = "setoid1 symmetry".
87 apply (.= ASSOC ^ -1);
92 lemma relation_pair_composition_is_morphism:
93 ∀o1,o2,o3: basic_pair.
94 ∀a,a':relation_pair_setoid o1 o2.
95 ∀b,b':relation_pair_setoid o2 o3.
97 relation_pair_composition o1 o2 o3 a b
98 = relation_pair_composition o1 o2 o3 a' b'.
101 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
102 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
103 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
106 apply (.= #‡(commute ?? b'));
107 apply (.= ASSOC ^ -1);
110 apply (.= #‡(commute ?? b')^-1);
114 definition relation_pair_composition_morphism:
115 ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
118 [ apply relation_pair_composition;
119 | apply relation_pair_composition_is_morphism;]
122 lemma relation_pair_composition_morphism_assoc:
127 .Πa12:relation_pair_setoid o1 o2
128 .Πa23:relation_pair_setoid o2 o3
129 .Πa34:relation_pair_setoid o3 o4
130 .relation_pair_composition_morphism o1 o3 o4
131 (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
132 =relation_pair_composition_morphism o1 o2 o4 a12
133 (relation_pair_composition_morphism o2 o3 o4 a23 a34).
135 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
136 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
137 alias symbol "refl" = "refl1".
138 alias symbol "prop2" = "prop21".
142 lemma relation_pair_composition_morphism_respects_id:
143 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
144 relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
146 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
147 apply ((id_neutral_right1 ????)‡#);
150 lemma relation_pair_composition_morphism_respects_id_r:
151 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
152 relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
154 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
155 apply ((id_neutral_left1 ????)‡#);
158 definition BP: category1.
161 | apply relation_pair_setoid
162 | apply id_relation_pair
163 | apply relation_pair_composition_morphism
164 | apply relation_pair_composition_morphism_assoc;
165 | apply relation_pair_composition_morphism_respects_id;
166 | apply relation_pair_composition_morphism_respects_id_r;]
169 definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
170 coercion basic_pair_of_BP.
172 definition relation_pair_setoid_of_arrows1_BP :
173 ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
174 coercion relation_pair_setoid_of_arrows1_BP.
177 definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
178 intros; constructor 1;
179 [ apply (ext ? ? (rel o));
185 definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(concr o).
186 intros; constructor 1;
187 [ apply (minus_image ?? (rel o));
188 | intros; apply (#‡e); ]
191 definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
192 intros (o); constructor 1;
193 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
194 intros; simplify; apply (.= (†e)‡#); apply refl1
195 | intros; split; simplify; intros;
196 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
197 | apply (. #‡((†e)‡(†e1))); assumption]]
200 interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
202 definition fintersectsS:
203 ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(form o).
204 intros (o); constructor 1;
205 [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
206 intros; simplify; apply (.= (†e)‡#); apply refl1
207 | intros; split; simplify; intros;
208 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
209 | apply (. #‡((†e)‡(†e1))); assumption]]
212 interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
214 definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
215 intros (o); constructor 1;
216 [ apply (λx:concr o.λS: Ω^(form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
217 | intros; split; intros; cases e2; exists [1,3: apply w]
218 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
219 | apply (. (#‡e1)‡(e‡#)); assumption]]
222 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
223 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).