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 ≝
21 rel: arrows1 ? concr form
24 interpretation "basic pair relation" 'Vdash2 x y c = (fun21 ??? (rel c) x y).
25 interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
27 alias symbol "eq" = "setoid1 eq".
28 alias symbol "compose" = "category1 composition".
29 record relation_pair (BP1,BP2: basic_pair): Type1 ≝
30 { concr_rel: arrows1 ? (concr BP1) (concr BP2);
31 form_rel: arrows1 ? (form BP1) (form BP2);
32 commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
36 interpretation "concrete relation" 'concr_rel r = (concr_rel ?? r).
37 interpretation "formal relation" 'form_rel r = (form_rel ?? r).
39 definition relation_pair_equality:
40 ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
43 [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
56 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
59 [ apply (relation_pair b b1)
60 | apply relation_pair_equality
64 definition relation_pair_of_relation_pair_setoid :
65 ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
66 coercion relation_pair_of_relation_pair_setoid.
69 ∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
70 intros 7 (o1 o2 r r' H c1 f2);
72 [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
73 lapply (if ?? (H c1 f2) H2) as H3;
74 apply (if ?? (commute ?? r' c1 f2) H3);
75 | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
76 lapply (fi ?? (H c1 f2) H2) as H3;
77 apply (if ?? (commute ?? r c1 f2) H3);
81 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
85 | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
86 lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
91 lemma relation_pair_composition:
92 ∀o1,o2,o3: basic_pair.
93 relation_pair_setoid o1 o2 → relation_pair_setoid o2 o3 → relation_pair_setoid o1 o3.
97 [ apply (r1 \sub\c ∘ r \sub\c)
98 | apply (r1 \sub\f ∘ r \sub\f)
99 | lapply (commute ?? r) as H;
100 lapply (commute ?? r1) as H1;
101 alias symbol "trans" = "trans1".
102 alias symbol "assoc" = "category1 assoc".
105 alias symbol "invert" = "setoid1 symmetry".
106 apply (.= ASSOC ^ -1);
111 lemma relation_pair_composition_is_morphism:
112 ∀o1,o2,o3: basic_pair.
113 ∀a,a':relation_pair_setoid o1 o2.
114 ∀b,b':relation_pair_setoid o2 o3.
116 relation_pair_composition o1 o2 o3 a b
117 = relation_pair_composition o1 o2 o3 a' b'.
120 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
121 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
122 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
125 apply (.= #‡(commute ?? b'));
126 apply (.= ASSOC ^ -1);
129 apply (.= #‡(commute ?? b')\sup -1);
133 definition relation_pair_composition_morphism:
134 ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
137 [ apply relation_pair_composition;
138 | apply relation_pair_composition_is_morphism;]
141 lemma relation_pair_composition_morphism_assoc:
146 .Πa12:relation_pair_setoid o1 o2
147 .Πa23:relation_pair_setoid o2 o3
148 .Πa34:relation_pair_setoid o3 o4
149 .relation_pair_composition_morphism o1 o3 o4
150 (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
151 =relation_pair_composition_morphism o1 o2 o4 a12
152 (relation_pair_composition_morphism o2 o3 o4 a23 a34).
154 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
155 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
156 alias symbol "refl" = "refl1".
157 alias symbol "prop2" = "prop21".
161 lemma relation_pair_composition_morphism_respects_id:
162 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
163 relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
165 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
166 apply ((id_neutral_right1 ????)‡#);
169 lemma relation_pair_composition_morphism_respects_id_r:
170 ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
171 relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
173 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
174 apply ((id_neutral_left1 ????)‡#);
177 definition BP: category1.
180 | apply relation_pair_setoid
181 | apply id_relation_pair
182 | apply relation_pair_composition_morphism
183 | apply relation_pair_composition_morphism_assoc;
184 | apply relation_pair_composition_morphism_respects_id;
185 | apply relation_pair_composition_morphism_respects_id_r;]
188 definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
189 coercion basic_pair_of_BP.
191 definition relation_pair_setoid_of_arrows1_BP :
192 ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
193 coercion relation_pair_setoid_of_arrows1_BP.
195 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
196 intros; constructor 1;
197 [ apply (ext ? ? (rel o));
203 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
204 intros; constructor 1;
205 [ apply (minus_image ?? (rel o));
206 | intros; apply (#‡e); ]
209 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
210 intros (o); constructor 1;
211 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
212 intros; simplify; apply (.= (†e)‡#); apply refl1
213 | intros; split; simplify; intros;
214 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
215 | apply (. #‡((†e)‡(†e1))); assumption]]
218 interpretation "fintersects" 'fintersects U V = (fun21 ??? (fintersects ?) U V).
220 definition fintersectsS:
221 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
222 intros (o); constructor 1;
223 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
224 intros; simplify; apply (.= (†e)‡#); apply refl1
225 | intros; split; simplify; intros;
226 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
227 | apply (. #‡((†e)‡(†e1))); assumption]]
230 interpretation "fintersectsS" 'fintersects U V = (fun21 ??? (fintersectsS ?) U V).
232 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
233 intros (o); constructor 1;
234 [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩⎽o y);
235 | intros; split; intros; cases e2; exists [1,3: apply w]
236 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
237 | apply (. (#‡e1)‡(e‡#)); assumption]]
240 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
241 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).