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 "o-algebra.ma".
16 include "notation.ma".
18 record Obasic_pair: Type2 ≝
21 Orel: arrows2 ? Oconcr Oform
25 interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
26 interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
28 alias symbol "eq" = "setoid1 eq".
29 alias symbol "compose" = "category1 composition".
32 alias symbol "eq" = "setoid2 eq".
33 alias symbol "compose" = "category2 composition".
34 record Orelation_pair (BP1,BP2: Obasic_pair): Type2 ≝
35 { Oconcr_rel: arrows2 ? (Oconcr BP1) (Oconcr BP2);
36 Oform_rel: arrows2 ? (Oform BP1) (Oform BP2);
37 Ocommute: ⊩ ∘ Oconcr_rel = Oform_rel ∘ ⊩
41 interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel __ r).
42 interpretation "o-formal relation" 'form_rel r = (Oform_rel __ r).
44 definition Orelation_pair_equality:
45 ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
48 [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
61 (* qui setoid1 e' giusto: ma non lo e'!!! *)
62 definition Orelation_pair_setoid: Obasic_pair → Obasic_pair → setoid2.
65 [ apply (Orelation_pair o o1)
66 | apply Orelation_pair_equality
70 definition Orelation_pair_of_Orelation_pair_setoid:
71 ∀P,Q. Orelation_pair_setoid P Q → Orelation_pair P Q ≝ λP,Q,x.x.
72 coercion Orelation_pair_of_Orelation_pair_setoid.
74 lemma eq_to_eq': ∀o1,o2.∀r,r': Orelation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
75 intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
76 apply (.= ((Ocommute ?? r) ^ -1));
78 apply (.= (Ocommute ?? r'));
83 definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
87 | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
88 lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
93 lemma Orelation_pair_composition:
94 ∀o1,o2,o3:Obasic_pair.
95 Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
99 [ apply (r1 \sub\c ∘ r \sub\c)
100 | apply (r1 \sub\f ∘ r \sub\f)
101 | lapply (Ocommute ?? r) as H;
102 lapply (Ocommute ?? r1) as H1;
103 apply rule (.= ASSOC);
105 apply rule (.= ASSOC ^ -1);
111 lemma Orelation_pair_composition_is_morphism:
112 ∀o1,o2,o3:Obasic_pair.
113 Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
115 Orelation_pair_composition o1 o2 o3 a b
116 = Orelation_pair_composition o1 o2 o3 a' b'.
118 change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
119 change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
120 change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
121 apply rule (.= ASSOC);
123 apply (.= #‡(Ocommute ?? b'));
124 apply rule (.= ASSOC \sup -1);
126 apply rule (.= ASSOC);
127 apply (.= #‡(Ocommute ?? b')\sup -1);
128 apply rule (ASSOC \sup -1);
131 definition Orelation_pair_composition_morphism:
132 ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
133 intros; constructor 1;
134 [ apply Orelation_pair_composition;
135 | apply Orelation_pair_composition_is_morphism;]
138 lemma Orelation_pair_composition_morphism_assoc:
139 ∀o1,o2,o3,o4:Obasic_pair
140 .Πa12:Orelation_pair_setoid o1 o2
141 .Πa23:Orelation_pair_setoid o2 o3
142 .Πa34:Orelation_pair_setoid o3 o4
143 .Orelation_pair_composition_morphism o1 o3 o4
144 (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
145 =Orelation_pair_composition_morphism o1 o2 o4 a12
146 (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).
148 change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
149 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
150 apply rule (ASSOC‡#);
153 lemma Orelation_pair_composition_morphism_respects_id:
156 .Πa:Orelation_pair_setoid o1 o2
157 .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
159 change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
160 apply ((id_neutral_right2 ????)‡#);
163 lemma Orelation_pair_composition_morphism_respects_id_r:
166 .Πa:Orelation_pair_setoid o1 o2
167 .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
169 change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
170 apply ((id_neutral_left2 ????)‡#);
173 definition OBP: category2.
176 | apply Orelation_pair_setoid
177 | apply Oid_relation_pair
178 | apply Orelation_pair_composition_morphism
179 | apply Orelation_pair_composition_morphism_assoc;
180 | apply Orelation_pair_composition_morphism_respects_id;
181 | apply Orelation_pair_composition_morphism_respects_id_r;]
184 definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.
185 coercion Obasic_pair_of_objs2_OBP.
187 definition Orelation_pair_setoid_of_arrows2_OBP:
188 ∀P,Q.arrows2 OBP P Q → Orelation_pair_setoid P Q ≝ λP,Q,c.c.
189 coercion Orelation_pair_setoid_of_arrows2_OBP.
192 definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
193 intros; constructor 1;
194 [ apply (ext ? ? (rel o));
200 definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
205 definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
206 intros (o); constructor 1;
207 [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
208 intros; simplify; apply (.= (†H)‡#); apply refl1
209 | intros; split; simplify; intros;
210 [ apply (. #‡((†H)‡(†H1))); assumption
211 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
214 interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
216 definition fintersectsS:
217 ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
218 intros (o); constructor 1;
219 [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
220 intros; simplify; apply (.= (†H)‡#); apply refl1
221 | intros; split; simplify; intros;
222 [ apply (. #‡((†H)‡(†H1))); assumption
223 | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
226 interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
230 definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
231 intros (o); constructor 1;
232 [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
233 | intros; split; intros; cases H2; exists [1,3: apply w]
234 [ apply (. (#‡H1)‡(H‡#)); assumption
235 | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
238 interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
239 interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
242 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
243 notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
244 interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (Orel x)).
246 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
247 notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
248 interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (Orel x)).
250 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
251 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
252 interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (Orel x)).
254 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
255 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
256 interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (Orel x)).