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 definition relation_pair_composition:
92 ∀o1,o2,o3. binary_morphism1 (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);
110 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
111 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
112 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
115 apply (.= #‡(commute ?? b'));
116 apply (.= ASSOC ^ -1);
119 apply (.= #‡(commute ?? b')\sup -1);
123 definition BP: category1.
126 | apply relation_pair_setoid
127 | apply id_relation_pair
128 | apply relation_pair_composition
130 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
131 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
132 alias symbol "refl" = "refl1".
133 alias symbol "prop2" = "prop21".
136 change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
137 apply ((id_neutral_right1 ????)‡#);
139 change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
140 apply ((id_neutral_left1 ????)‡#);]
143 definition basic_pair_of_BP : objs1 BP → basic_pair ≝ λx.x.
144 coercion basic_pair_of_BP.
146 definition relation_pair_setoid_of_arrows1_BP :
147 ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
148 coercion relation_pair_setoid_of_arrows1_BP.
150 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
151 intros; constructor 1;
152 [ apply (ext ? ? (rel o));
158 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o).
159 intros; constructor 1;
160 [ apply (minus_image ?? (rel o));
161 | intros; apply (#‡e); ]
164 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
165 intros (o); constructor 1;
166 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
167 intros; simplify; apply (.= (†e)‡#); apply refl1
168 | intros; split; simplify; intros;
169 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
170 | apply (. #‡((†e)‡(†e1))); assumption]]
173 interpretation "fintersects" 'fintersects U V = (fun21 ___ (fintersects _) U V).
175 definition fintersectsS:
176 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
177 intros (o); constructor 1;
178 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
179 intros; simplify; apply (.= (†e)‡#); apply refl1
180 | intros; split; simplify; intros;
181 [ apply (. #‡((†e^-1)‡(†e1^-1))); assumption
182 | apply (. #‡((†e)‡(†e1))); assumption]]
185 interpretation "fintersectsS" 'fintersects U V = (fun21 ___ (fintersectsS _) U V).
187 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
188 intros (o); constructor 1;
189 [ apply (λx:concr o.λS: Ω \sup (form o).∃y:form o.y ∈ S ∧ x ⊩_o y);
190 | intros; split; intros; cases e2; exists [1,3: apply w]
191 [ apply (. (#‡e1^-1)‡(e^-1‡#)); assumption
192 | apply (. (#‡e1)‡(e‡#)); assumption]]
195 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr _) __ (relS c) x y).
196 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ___ (relS c)).