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 (**************************************************************************)
17 record binary_relation (A,B: SET) : Type1 ≝
18 { satisfy:> binary_morphism1 A B CPROP }.
20 notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)" with precedence 45 for @{'satisfy $r $x $y}.
21 notation > "hvbox (x \natur term 90 r y)" with precedence 45 for @{'satisfy $r $x $y}.
22 interpretation "relation applied" 'satisfy r x y = (fun21 ___ (satisfy __ r) x y).
24 definition binary_relation_setoid: SET → SET → setoid1.
27 [ apply (binary_relation A B)
29 [ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
30 | simplify; intros 3; split; intro; assumption
31 | simplify; intros 5; split; intro;
32 [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
33 | simplify; intros 7; split; intro;
34 [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
35 [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
39 definition binary_relation_of_binary_relation_setoid :
40 ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
41 coercion binary_relation_of_binary_relation_setoid.
43 definition composition:
45 binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
51 [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
53 split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
54 [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
55 | apply (. (e‡#)‡(#‡e1)); assumption]]
56 | intros 8; split; intro H2; simplify in H2 ⊢ %;
57 cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
58 [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
59 [ lapply (if ?? (e1 w y) H4)| lapply (fi ?? (e1 w y) H4) ]
60 exists; try assumption;
64 definition REL: category1.
67 | intros (T T1); apply (binary_relation_setoid T T1)
68 | intros; constructor 1;
69 constructor 1; unfold setoid1_of_setoid; simplify;
70 [ (* changes required to avoid universe inconsistency *)
71 change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
72 | intros; split; intro; change in a a' b b' with (carr o);
73 change in e with (eq ? a a'); change in e1 with (eq ? b b');
74 [ apply (.= (e ^ -1));
83 cases f (w H); clear f; cases H; clear H;
84 [cases f (w1 H); clear f | cases f1 (w1 H); clear f1]
86 exists; try assumption;
87 split; try assumption;
88 exists; try assumption;
90 |6,7: intros 5; unfold composition; simplify; split; intro;
91 unfold setoid1_of_setoid in x y; simplify in x y;
92 [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
93 [ apply (. (e : eq1 ? x w)‡#); assumption
94 | apply (. #‡(e : eq1 ? w y)^-1); assumption]
95 |2,4: exists; try assumption; split;
96 (* change required to avoid universe inconsistency *)
97 change in x with (carr o1); change in y with (carr o2);
98 first [apply refl | assumption]]]
101 definition setoid_of_REL : objs1 REL → setoid ≝ λx.x.
102 coercion setoid_of_REL.
104 definition binary_relation_setoid_of_arrow1_REL :
105 ∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
106 coercion binary_relation_setoid_of_arrow1_REL.
108 definition full_subset: ∀s:REL. Ω \sup s.
109 apply (λs.{x | True});
110 intros; simplify; split; intro; assumption.
113 coercion full_subset.
115 definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
116 apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
118 alias symbol "trans" = "trans1".
119 alias symbol "prop1" = "prop11".
120 apply (.= †e); apply refl1.
123 interpretation "subset comprehension" 'comprehension s p =
124 (comprehension s (mk_unary_morphism1 __ p _)).
126 definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
127 apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
128 [ intros; simplify; apply (.= (e‡#)); apply refl1
129 | intros; simplify; split; intros; simplify;
130 [ change with (∀x. x ♮a b → x ♮a' b'); intros;
131 apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
132 | change with (∀x. x ♮a' b' → x ♮a b); intros;
133 apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
136 definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
137 (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
138 intros (X S r); constructor 1;
139 [ intro F; constructor 1; constructor 1;
140 [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
141 | intros; split; intro; cases f (H1 H2); clear f; split;
142 [ apply (. (H‡#)); assumption
143 |3: apply (. (H\sup -1‡#)); assumption
144 |2,4: cases H2 (w H3); exists; [1,3: apply w]
145 [ apply (. (#‡(H‡#))); assumption
146 | apply (. (#‡(H \sup -1‡#))); assumption]]]
147 | intros; split; simplify; intros; cases f; cases H1; split;
149 |2,4: exists; [1,3: apply w]
150 [ apply (. (#‡H)‡#); assumption
151 | apply (. (#‡H\sup -1)‡#); assumption]]]
154 lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
156 unfold extS; simplify;
158 [ intros 2; change with (a ∈ X);
162 change in f2 with (eq1 ? a w);
163 apply (. (f2\sup -1‡#));
165 | intros 2; change in f with (a ∈ X);
168 | exists; [ apply a ]
171 | change with (a = a); apply refl]]]
174 lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
175 intros; unfold extS; simplify; split; intros; simplify; intros;
176 [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
177 cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
178 exists; [apply w1] split [2: assumption] constructor 1; [assumption]
179 exists; [apply w] split; assumption
180 | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
181 cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
182 cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
187 (* the same as ⋄ for a basic pair *)
188 definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
189 intros; constructor 1;
190 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
191 intros; simplify; split; intro; cases e1; exists [1,3: apply w]
192 [ apply (. (#‡e^-1)‡#); assumption
193 | apply (. (#‡e)‡#); assumption]
194 | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
195 [ apply (. #‡(#‡e1^-1)); cases x; split; try assumption;
196 apply (if ?? (e ??)); assumption
197 | apply (. #‡(#‡e1)); cases x; split; try assumption;
198 apply (if ?? (e ^ -1 ??)); assumption]]
201 (* the same as □ for a basic pair *)
202 definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
203 intros; constructor 1;
204 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
205 intros; simplify; split; intros; apply f;
206 [ apply (. #‡e); assumption
207 | apply (. #‡e ^ -1); assumption]
208 | intros; split; simplify; intros; [ apply (. #‡e1^ -1); | apply (. #‡e1 )]
209 apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
212 (* the same as Rest for a basic pair *)
213 definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
214 intros; constructor 1;
215 [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
216 intros; simplify; split; intros; apply f;
217 [ apply (. e ‡#); assumption
218 | apply (. e^ -1‡#); assumption]
219 | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
220 apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
223 (* the same as Ext for a basic pair *)
224 definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
225 intros; constructor 1;
226 [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
227 exT ? (λy:V.x ♮r y ∧ y ∈ S) });
228 intros; simplify; split; intro; cases e1; exists [1,3: apply w]
229 [ apply (. (e ^ -1‡#)‡#); assumption
230 | apply (. (e‡#)‡#); assumption]
231 | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
232 [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
233 apply (if ?? (e ??)); assumption
234 | apply (. #‡(#‡e1)); cases x; split; try assumption;
235 apply (if ?? (e ^ -1 ??)); assumption]]
239 (* minus_image is the same as ext *)
241 theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
242 intros; unfold image; simplify; split; simplify; intros;
243 [ change with (a ∈ U);
244 cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
245 | change in f with (a ∈ U);
246 exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
249 theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
250 intros; unfold minus_star_image; simplify; split; simplify; intros;
251 [ change with (a ∈ U); apply H; change with (a=a); apply refl
252 | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
255 theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
256 intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
257 clear x; [ cases f; clear f; | cases f1; clear f1 ]
258 exists; try assumption; cases x; clear x; split; try assumption;
259 exists; try assumption; split; assumption.
262 theorem minus_star_image_comp:
264 minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
265 intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
266 [ apply H; exists; try assumption; split; assumption
267 | change with (x ∈ X); cases f; cases x1; apply H; assumption]
275 ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
277 unfold ext; unfold extS; simplify; split; intro; simplify; intros;
278 cases f; clear f; split; try assumption;
279 [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
280 [1: split] assumption;
281 | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
282 [2: cases f] assumption]
285 theorem extS_singleton:
286 ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
287 intros; unfold extS; unfold ext; unfold singleton; simplify;
288 split; intros 2; simplify; cases f; split; try assumption;
289 [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
291 | exists; try assumption; split; try assumption; change with (x = x); apply refl]