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 "relations.ma".
16 include "notation.ma".
18 record basic_pair: Type1 ≝ {
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): Type1 ≝ {
26 concr_rel: (concr BP1) ⇒_\r1 (concr BP2); form_rel: (form BP1) ⇒_\r1 (form BP2);
27 commute: ⊩ ∘ concr_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.
53 ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
54 intros 5 (o1 o2 r r' H);
55 apply (.= (commute ?? r)^-1);
56 change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
58 apply (commute ?? r').
61 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
65 | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
66 lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
71 lemma relation_pair_composition:
72 ∀o1,o2,o3: basic_pair.
73 relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
77 [ apply (r1 \sub\c ∘ r \sub\c)
78 | apply (r1 \sub\f ∘ r \sub\f)
79 | lapply (commute ?? r) as H;
80 lapply (commute ?? r1) as H1;
81 alias symbol "trans" = "trans1".
82 alias symbol "assoc" = "category1 assoc".
85 alias symbol "invert" = "setoid1 symmetry".
86 apply (.= ASSOC ^ -1);
91 lemma relation_pair_composition_is_morphism:
92 ∀o1,o2,o3: basic_pair.
93 ∀a,a':relation_pair_setoid o1 o2.
94 ∀b,b':relation_pair_setoid o2 o3.
96 relation_pair_composition o1 o2 o3 a b
97 = relation_pair_composition o1 o2 o3 a' b'.
100 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
101 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
102 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
105 apply (.= #‡(commute ?? b'));
106 apply (.= ASSOC ^ -1);
109 apply (.= #‡(commute ?? b')\sup -1);
113 definition relation_pair_composition_morphism:
114 ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
117 [ apply relation_pair_composition;
118 | apply relation_pair_composition_is_morphism;]
121 lemma relation_pair_composition_morphism_assoc:
126 .Πa12:relation_pair_setoid o1 o2
127 .Πa23:relation_pair_setoid o2 o3
128 .Πa34:relation_pair_setoid o3 o4
129 .relation_pair_composition_morphism o1 o3 o4
130 (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
131 =relation_pair_composition_morphism o1 o2 o4 a12
132 (relation_pair_composition_morphism o2 o3 o4 a23 a34).
134 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
135 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
136 alias symbol "refl" = "refl1".
137 alias symbol "prop2" = "prop21".
141 lemma relation_pair_composition_morphism_respects_id:
142 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
143 relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
145 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
146 apply ((id_neutral_right1 ????)‡#);
149 lemma relation_pair_composition_morphism_respects_id_r:
150 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
151 relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
153 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
154 apply ((id_neutral_left1 ????)‡#);
157 definition BP: category1.
160 | apply relation_pair_setoid
161 | apply id_relation_pair
162 | apply relation_pair_composition_morphism
163 | apply relation_pair_composition_morphism_assoc;
164 | apply relation_pair_composition_morphism_respects_id;
165 | apply relation_pair_composition_morphism_respects_id_r;]
168 definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
169 coercion basic_pair_of_BP.
171 definition relation_pair_setoid_of_arrows1_BP :
172 ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
173 coercion relation_pair_setoid_of_arrows1_BP.
176 definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
177 intros; constructor 1;
178 [ apply (ext ? ? (rel o));
184 definition BPextS: ∀o: BP. Ω^(form o) ⇒_1 Ω^(concr o).
185 intros; constructor 1;
186 [ apply (minus_image ?? (rel o));
187 | intros; apply (#‡e); ]
190 definition fintersects: ∀o: BP. (form o) × (form o) ⇒_1 Ω^(form o).
191 intros (o); constructor 1;
192 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
193 intros; simplify; apply (.= (†e)‡#); apply refl1
194 | intros; split; simplify; intros;
195 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
196 | apply (. #‡((†e)‡(†e1))); assumption]]
199 interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
201 definition fintersectsS:
202 ∀o:BP. Ω^(form o) × Ω^(form o) ⇒_1 Ω^(form o).
203 intros (o); constructor 1;
204 [ apply (λo: basic_pair.λa,b: Ω^(form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
205 intros; simplify; apply (.= (†e)‡#); apply refl1
206 | intros; split; simplify; intros;
207 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
208 | apply (. #‡((†e)‡(†e1))); assumption]]
211 interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
213 definition relS: ∀o: BP. (concr o) × Ω^(form o) ⇒_1 CPROP.
214 intros (o); constructor 1;
215 [ apply (λx:concr o.λS: Ω^(form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
216 | intros; split; intros; cases e2; exists [1,3: apply w]
217 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
218 | apply (. (#‡e1)‡(e‡#)); assumption]]
221 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
222 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).