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".
17 record basic_pair: Type1 ≝
20 rel: arrows1 ? concr form
23 notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
24 notation "⊩" with precedence 60 for @{'Vdash}.
26 interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
27 interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
29 alias symbol "eq" = "setoid1 eq".
30 alias symbol "compose" = "category1 composition".
31 record relation_pair (BP1,BP2: basic_pair): Type1 ≝
32 { concr_rel: arrows1 ? (concr BP1) (concr BP2);
33 form_rel: arrows1 ? (form BP1) (form BP2);
34 commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
37 notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
38 notation "hvbox (r \sub \f)" with precedence 90 for @{'form_rel $r}.
40 interpretation "concrete relation" 'concr_rel r = (concr_rel __ r).
41 interpretation "formal relation" 'form_rel r = (form_rel __ r).
43 definition relation_pair_equality:
44 ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
47 [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
60 definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
63 [ apply (relation_pair b b1)
64 | apply relation_pair_equality
68 lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
69 intros 7 (o1 o2 r r' H c1 f2);
71 [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
72 lapply (if ?? (H c1 f2) H2) as H3;
73 apply (if ?? (commute ?? r' c1 f2) H3);
74 | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
75 lapply (fi ?? (H c1 f2) H2) as H3;
76 apply (if ?? (commute ?? r c1 f2) H3);
80 definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
84 | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
85 lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
90 definition relation_pair_composition:
91 ∀o1,o2,o3. binary_morphism1 (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
96 [ apply (r1 \sub\c ∘ r \sub\c)
97 | apply (r1 \sub\f ∘ r \sub\f)
98 | lapply (commute ?? r) as H;
99 lapply (commute ?? r1) as H1;
100 alias symbol "trans" = "trans1".
101 alias symbol "assoc" = "category1 assoc".
104 alias symbol "invert" = "setoid1 symmetry".
105 apply (.= ASSOC ^ -1);
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);
114 apply (.= #‡(commute ?? b'));
115 apply (.= ASSOC ^ -1);
118 apply (.= #‡(commute ?? b')\sup -1);
122 definition BP: category1.
125 | apply relation_pair_setoid
126 | apply id_relation_pair
127 | apply relation_pair_composition
129 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
130 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
131 alias symbol "refl" = "refl1".
132 alias symbol "prop2" = "prop21".
135 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
136 apply ((id_neutral_right1 ????)‡#);
138 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
139 apply ((id_neutral_left1 ????)‡#);]
142 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
143 intros; constructor 1;
144 [ apply (ext ? ? (rel o));
150 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
151 intros; constructor 1;
152 [ apply (minus_image ?? (rel o));
153 | intros; apply (#‡e); ]
156 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
157 intros (o); constructor 1;
158 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
159 intros; simplify; apply (.= (†e)‡#); apply refl1
160 | intros; split; simplify; intros;
161 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
162 | apply (. #‡((†e)‡(†e1))); assumption]]
165 interpretation "fintersects" 'fintersects U V = (fun21 ___ (fintersects _) U V).
167 definition fintersectsS:
168 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
169 intros (o); constructor 1;
170 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
171 intros; simplify; apply (.= (†e)‡#); apply refl1
172 | intros; split; simplify; intros;
173 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
174 | apply (. #‡((†e)‡(†e1))); assumption]]
177 interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (fintersectsS _) U V).
179 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
180 intros (o); constructor 1;
181 [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩ y);
182 | intros; split; intros; cases e2; exists [1,3: apply w]
183 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
184 | apply (. (#‡e1)‡(e‡#)); assumption]]
187 interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun21 (concr _) __ (relS _) x y).
188 interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun21 ___ (relS _)).