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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 *)
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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 → SET1.
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 composition:
41 binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
47 [ alias symbol "and" = "and_morphism".
48 (* carr to avoid universe inconsistency *)
49 apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
51 split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
52 [ apply (. (e‡#)‡(#‡e1)); assumption
53 | apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
54 | intros 8; split; intro H2; simplify in H2 ⊢ %;
55 cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
56 [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
57 [ lapply (if ?? (e1 w y) H4)| lapply (fi ?? (e1 w y) H4) ]
58 exists; try assumption;
62 definition REL: category1.
65 | intros (T T1); apply (binary_relation_setoid T T1)
66 | intros; constructor 1;
67 constructor 1; unfold setoid1_of_setoid; simplify;
68 [ (* changes required to avoid universe inconsistency *)
69 change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
70 | intros; split; intro; change in a a' b b' with (carr o);
71 change in e with (eq ? a a'); change in e1 with (eq ? b b');
72 [ apply (.= (e ^ -1));
81 cases f (w H); clear f; cases H; clear H;
82 [cases f (w1 H); clear f | cases f1 (w1 H); clear f1]
84 exists; try assumption;
85 split; try assumption;
86 exists; try assumption;
88 |6,7: intros 5; unfold composition; simplify; split; intro;
89 unfold setoid1_of_setoid in x y; simplify in x y;
90 [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
91 [ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
92 | apply (. #‡(e : eq1 ? w y)); assumption]
93 |2,4: exists; try assumption; split;
94 (* change required to avoid universe inconsistency *)
95 change in x with (carr o1); change in y with (carr o2);
96 first [apply refl | assumption]]]
100 definition full_subset: ∀s:REL. Ω \sup s.
101 apply (λs.{x | True});
102 intros; simplify; split; intro; assumption.
105 coercion full_subset.
108 definition setoid1_of_REL: REL → setoid ≝ λS. S.
109 coercion setoid1_of_REL.
111 lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
112 [ apply (setoid1_of_SET o1);
115 coercion Type_OF_setoid1_of_REL.
118 definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
119 apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
120 intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
123 interpretation "subset comprehension" 'comprehension s p =
124 (comprehension s (mk_unary_morphism __ 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 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
128 [ intros; simplify; apply (.= (H‡#)); apply refl1
129 | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
130 [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
131 | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
134 definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
135 (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
136 intros (X S r); constructor 1;
137 [ intro F; constructor 1; constructor 1;
138 [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
139 | intros; split; intro; cases f (H1 H2); clear f; split;
140 [ apply (. (H‡#)); assumption
141 |3: apply (. (H\sup -1‡#)); assumption
142 |2,4: cases H2 (w H3); exists; [1,3: apply w]
143 [ apply (. (#‡(H‡#))); assumption
144 | apply (. (#‡(H \sup -1‡#))); assumption]]]
145 | intros; split; simplify; intros; cases f; cases H1; split;
147 |2,4: exists; [1,3: apply w]
148 [ apply (. (#‡H)‡#); assumption
149 | apply (. (#‡H\sup -1)‡#); assumption]]]
152 lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
154 unfold extS; simplify;
156 [ intros 2; change with (a ∈ X);
160 change in f2 with (eq1 ? a w);
161 apply (. (f2\sup -1‡#));
163 | intros 2; change in f with (a ∈ X);
166 | exists; [ apply a ]
169 | change with (a = a); apply refl]]]
172 lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
173 intros; unfold extS; simplify; split; intros; simplify; intros;
174 [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
175 cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
176 exists; [apply w1] split [2: assumption] constructor 1; [assumption]
177 exists; [apply w] split; assumption
178 | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
179 cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
180 cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
185 (* the same as ⋄ for a basic pair *)
186 definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
187 intros; constructor 1;
188 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:carr U. x ♮r y ∧ x ∈ S });
189 intros; simplify; split; intro; cases e1; exists [1,3: apply w]
190 [ apply (. (#‡e)‡#); assumption
191 | apply (. (#‡e ^ -1)‡#); assumption]
192 | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
193 [ apply (. #‡(#‡e1)); cases x; split; try assumption;
194 apply (if ?? (e ??)); assumption
195 | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
196 apply (if ?? (e ^ -1 ??)); assumption]]
199 (* the same as □ for a basic pair *)
200 definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
201 intros; constructor 1;
202 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
203 intros; simplify; split; intros; apply f;
204 [ apply (. #‡e ^ -1); assumption
205 | apply (. #‡e); assumption]
206 | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
207 apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
210 (* the same as Rest for a basic pair *)
211 definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
212 intros; constructor 1;
213 [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr V. x ♮r y → y ∈ S});
214 intros; simplify; split; intros; apply f;
215 [ apply (. e ^ -1‡#); assumption
216 | apply (. e‡#); assumption]
217 | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
218 apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
221 (* the same as Ext for a basic pair *)
222 definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
223 intros; constructor 1;
224 [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
225 exT ? (λy:carr V.x ♮r y ∧ y ∈ S) });
226 intros; simplify; split; intro; cases e1; exists [1,3: apply w]
227 [ apply (. (e‡#)‡#); assumption
228 | apply (. (e ^ -1‡#)‡#); assumption]
229 | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
230 [ apply (. #‡(#‡e1)); cases x; split; try assumption;
231 apply (if ?? (e ??)); assumption
232 | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
233 apply (if ?? (e ^ -1 ??)); assumption]]
237 (* minus_image is the same as ext *)
239 theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
240 intros; unfold image; simplify; split; simplify; intros;
241 [ change with (a ∈ U);
242 cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
243 | change in f with (a ∈ U);
244 exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
247 theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
248 intros; unfold minus_star_image; simplify; split; simplify; intros;
249 [ change with (a ∈ U); apply H; change with (a=a); apply refl
250 | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
253 theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
254 intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
255 clear x; [ cases f; clear f; | cases f1; clear f1 ]
256 exists; try assumption; cases x; clear x; split; try assumption;
257 exists; try assumption; split; assumption.
260 theorem minus_star_image_comp:
262 minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
263 intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
264 [ apply H; exists; try assumption; split; assumption
265 | change with (x ∈ X); cases f; cases x1; apply H; assumption]
273 ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
275 unfold ext; unfold extS; simplify; split; intro; simplify; intros;
276 cases f; clear f; split; try assumption;
277 [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
278 [1: split] assumption;
279 | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
280 [2: cases f] assumption]
283 theorem extS_singleton:
284 ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
285 intros; unfold extS; unfold ext; unfold singleton; simplify;
286 split; intros 2; simplify; cases f; split; try assumption;
287 [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
289 | exists; try assumption; split; try assumption; change with (x = x); apply refl]