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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 ?? (H ??)) | apply (if ?? (H ??))] assumption
33 | simplify; intros 7; split; intro;
34 [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
35 [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
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 H (w H3); clear H; 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 H (w H1); clear H; 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.
107 definition setoid1_of_REL: REL → setoid ≝ λS. S.
109 coercion setoid1_of_REL.
111 definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
112 apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
113 intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
116 interpretation "subset comprehension" 'comprehension s p =
117 (comprehension s (mk_unary_morphism __ p _)).
119 definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
120 apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
121 [ intros; simplify; apply (.= (H‡#)); apply refl1
122 | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
123 [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
124 | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
127 definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
128 (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
129 intros (X S r); constructor 1;
130 [ intro F; constructor 1; constructor 1;
131 [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
132 | intros; split; intro; cases f (H1 H2); clear f; split;
133 [ apply (. (H‡#)); assumption
134 |3: apply (. (H\sup -1‡#)); assumption
135 |2,4: cases H2 (w H3); exists; [1,3: apply w]
136 [ apply (. (#‡(H‡#))); assumption
137 | apply (. (#‡(H \sup -1‡#))); assumption]]]
138 | intros; split; simplify; intros; cases f; cases H1; split;
140 |2,4: exists; [1,3: apply w]
141 [ apply (. (#‡H)‡#); assumption
142 | apply (. (#‡H\sup -1)‡#); assumption]]]
145 lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
147 unfold extS; simplify;
149 [ intros 2; change with (a ∈ X);
153 change in f2 with (eq1 ? a w);
154 apply (. (f2\sup -1‡#));
156 | intros 2; change in f with (a ∈ X);
159 | exists; [ apply a ]
162 | change with (a = a); apply refl]]]
165 lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
166 intros; unfold extS; simplify; split; intros; simplify; intros;
167 [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
168 cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
169 exists; [apply w1] split [2: assumption] constructor 1; [assumption]
170 exists; [apply w] split; assumption
171 | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
172 cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
173 cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
178 (* the same as ⋄ for a basic pair *)
179 definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) ?(*(Ω \sup V)*).
181 intros; constructor 1;
182 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
183 intros; simplify; split; intro; cases H; exists [1,3: apply w]
184 [ apply (. (#‡e)‡#); assumption
185 | apply (. (#‡e ^ -1)‡#); assumption]
186 | intros; split; simplify; intros; cases H; exists [1,3: apply w]
187 [ apply (. #‡(#‡e1)); cases x; split; try assumption;
188 apply (if ?? (e ??)); assumption
189 | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
190 apply (if ?? (e ^ -1 ??)); assumption]]
194 (* the same as □ for a basic pair *)
195 definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
196 intros; constructor 1;
197 [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
198 intros; simplify; split; intros; apply H1;
199 [ apply (. #‡H \sup -1); assumption
200 | apply (. #‡H); assumption]
201 | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
202 apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
205 (* minus_image is the same as ext *)
207 theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
208 intros; unfold image; simplify; split; simplify; intros;
209 [ change with (a ∈ U);
210 cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
211 | change in f with (a ∈ U);
212 exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
215 theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
216 intros; unfold minus_star_image; simplify; split; simplify; intros;
217 [ change with (a ∈ U); apply H; change with (a=a); apply refl
218 | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
221 theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
222 intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
223 clear x; [ cases f; clear f; | cases f1; clear f1 ]
224 exists; try assumption; cases x; clear x; split; try assumption;
225 exists; try assumption; split; assumption.
228 theorem minus_star_image_comp:
230 minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
231 intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
232 [ apply H; exists; try assumption; split; assumption
233 | change with (x ∈ X); cases f; cases x1; apply H; assumption]
241 ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
243 unfold ext; unfold extS; simplify; split; intro; simplify; intros;
244 cases f; clear f; split; try assumption;
245 [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
246 [1: split] assumption;
247 | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
248 [2: cases f] assumption]
251 theorem extS_singleton:
252 ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
253 intros; unfold extS; unfold ext; unfold singleton; simplify;
254 split; intros 2; simplify; cases f; split; try assumption;
255 [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
257 | exists; try assumption; split; try assumption; change with (x = x); apply refl]
261 include "o-algebra.ma".
263 definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).