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 "o-algebra.ma".
18 definition POW': objs1 SET → OAlgebra.
19 intro A; constructor 1;
23 | apply big_intersects;
26 simplify; intros; apply (refl1 ? (eq1 CPROP));
27 | apply ({x | False});
28 simplify; intros; apply (refl1 ? (eq1 CPROP));
29 | intros; whd; intros; assumption
30 | intros; whd; split; assumption
31 | intros; apply transitive_subseteq_operator; [2: apply f; | skip | assumption]
32 | intros; cases f; exists [apply w] assumption
33 | intros; split; [ intros 4; apply (f ? f1 i); | intros 3; intro; apply (f i ? f1); ]
35 [ intros 4; apply f; exists; [apply i] assumption;
36 | intros 3; intros; cases f1; apply (f w a x); ]
38 | intros 3; constructor 1;
39 | intros; cases f; exists; [apply w]
41 | whd; intros; cases i; simplify; assumption]
42 | intros; split; intro;
43 [ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
44 | cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
45 | intros; intros 2; cases (f {(a)} ?);
46 [ exists; [apply a] [assumption | change with (a = a); apply refl1;]
47 | change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
51 definition powerset_of_POW': ∀A.oa_P (POW' A) → Ω \sup A ≝ λA,x.x.
52 coercion powerset_of_POW'.
54 definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (POW' o1) (POW' o2).
58 [ apply (λU.image ?? c U);
59 | intros; apply (#‡e); ]
61 [ apply (λU.minus_star_image ?? c U);
62 | intros; apply (#‡e); ]
64 [ apply (λU.star_image ?? c U);
65 | intros; apply (#‡e); ]
67 [ apply (λU.minus_image ?? c U);
68 | intros; apply (#‡e); ]
69 | intros; split; intro;
70 [ change in f with (∀a. a ∈ image ?? c p → a ∈ q);
71 change with (∀a:o1. a ∈ p → a ∈ star_image ?? c q);
72 intros 4; apply f; exists; [apply a] split; assumption;
73 | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? c q);
74 change with (∀a. a ∈ image ?? c p → a ∈ q);
75 intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
76 | intros; split; intro;
77 [ change in f with (∀a. a ∈ minus_image ?? c p → a ∈ q);
78 change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c q);
79 intros 4; apply f; exists; [apply a] split; assumption;
80 | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c q);
81 change with (∀a. a ∈ minus_image ?? c p → a ∈ q);
82 intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
83 | intros; split; intro; cases f; clear f;
84 [ cases x; cases x2; clear x x2; exists; [apply w1]
86 | exists; [apply w] split; assumption]
87 | cases x1; cases x2; clear x1 x2; exists; [apply w1]
88 [ exists; [apply w] split; assumption;
92 lemma orelation_of_relation_preserves_equality:
93 ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. t = t' → eq2 ? (orelation_of_relation ?? t) (orelation_of_relation ?? t').
94 intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
95 simplify; whd in o1 o2;
96 [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
98 | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
100 | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
101 apply (. #‡(e ^ -1‡#));
102 | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
104 | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
105 apply (. #‡(e ^ -1‡#));
106 | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
108 | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
109 apply (. #‡(e ^ -1‡#));
110 | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
114 lemma orelation_of_relation_preserves_identity:
115 ∀o1:REL. eq2 ? (orelation_of_relation ?? (id1 ? o1)) (id2 OA (POW' o1)).
116 intros; split; intro; split; whd; intro;
117 [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
118 apply (f a1); change with (a1 = a1); apply refl1;
119 | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
120 change in f1 with (x = a1); apply (. f1‡#); apply f;
121 | alias symbol "and" = "and_morphism".
122 change with ((∃y:o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
123 intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
124 apply (. f‡#); apply f1;
125 | change with (a1 ∈ a → ∃y:o1.a1 ♮(id1 REL o1) y ∧ y∈a);
126 intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
127 | change with ((∃x:o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
128 intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
129 apply (. f^-1‡#); apply f1;
130 | change with (a1 ∈ a → ∃x:o1.x ♮(id1 REL o1) a1∧x∈a);
131 intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
132 | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
133 apply (f a1); change with (a1 = a1); apply refl1;
134 | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
135 change in f1 with (a1 = y); apply (. f1^-1‡#); apply f;]
138 (* CSC: ???? forse un uncertain mancato *)
139 alias symbol "eq" = "setoid2 eq".
140 alias symbol "compose" = "category1 composition".
141 lemma orelation_of_relation_preserves_composition:
142 ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
143 orelation_of_relation ?? (G ∘ F) =
144 comp2 OA (POW' o1) (POW' o2) (POW' o3)
145 ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
146 [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
147 intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
148 [ whd; intros; apply f; exists; [ apply x] split; assumption;
149 | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
150 | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
151 split; [ assumption | exists; [apply w] split; assumption ]
152 | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
153 split; [ exists; [apply w] split; assumption | assumption ]
154 | unfold arrows1_of_ORelation_setoid;
155 cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
156 split; [ assumption | exists; [apply w] split; assumption ]
157 | unfold arrows1_of_ORelation_setoid in e;
158 cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
159 split; [ exists; [apply w] split; assumption | assumption ]
160 | whd; intros; apply f; exists; [ apply y] split; assumption;
161 | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
164 definition POW: carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
167 | intros; constructor 1;
168 [ apply (orelation_of_relation S T);
169 | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
170 | apply orelation_of_relation_preserves_identity;
171 | apply orelation_of_relation_preserves_composition; ]
174 theorem POW_faithful:
175 ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
176 map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f=g.
177 intros; unfold POW in e; simplify in e; cases e;
178 unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
179 intros 2; cases (e3 {(x)});
180 split; intro; [ lapply (s y); | lapply (s1 y); ]
181 [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
182 |*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
186 lemma currify: ∀A,B,C. binary_morphism1 A B C → A → unary_morphism1 B C.
187 intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
191 alias symbol "singl" = "singleton".
192 alias symbol "eq" = "setoid eq".
193 lemma in_singleton_to_eq : ∀A:setoid.∀y,x:A.y ∈ {(x)} → (eq1 A) y x.
194 intros; apply sym1; apply f;
197 lemma eq_to_in_singleton : ∀A:setoid.∀y,x:A.eq1 A y x → y ∈ {(x)}.
198 intros; apply (e^-1);
202 interpretation "lifting singl" 'singl x =
203 (fun11 ? (objs2 (POW ?)) (singleton ?) x).
205 theorem POW_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? POW S T g = f).
207 [ constructor 1; constructor 1;
208 [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
209 | intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
210 [4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
211 lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
212 | whd; split; whd; intro; simplify; unfold map_arrows2; simplify;
214 [ change with (∀a1.(∀x. a1 ∈ f (singleton S x) → x ∈ a) → a1 ∈ f⎻* a);
215 | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f (singleton S x) → x ∈ a)); ]
217 [ change with (∀a1.(∃y:carr T. y ∈ f (singleton S a1) ∧ y ∈ a) → a1 ∈ f⎻ a);
218 | change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f (singleton S a1) ∧ y ∈ a)); ]
220 [ change with (∀a1.(∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a) → a1 ∈ f a);
221 | change with (∀a1.a1 ∈ f a → (∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a)); ]
223 [ change with (∀a1.(∀y. y ∈ f (singleton S a1) → y ∈ a) → a1 ∈ f* a);
224 | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f (singleton S a1) → y ∈ a)); ]]
225 [ intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)^-1) ? a1);
226 [ intros 2; apply (f1 a2); change in f2 with (a2 ∈ f⎻ (singleton ? a1));
227 lapply (. (or_prop3 ?? f (singleton ? a2) (singleton ? a1)));
228 [ cases Hletin; change in x1 with (eq1 ? a1 w);
229 apply (. x1‡#); assumption;
230 | exists; [apply a2] [change with (a2=a2); apply rule #; | assumption]]
231 | change with (a1 = a1); apply rule #; ]
232 | intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)) ? x);
233 [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f⎻* a); apply (. f3^-1‡#);
235 | lapply (. (or_prop3 ?? f (singleton ? x) (singleton ? a1))^-1);
236 [ cases Hletin; change in x1 with (eq1 ? x w);
237 change with (x ∈ f⎻ (singleton ? a1)); apply (. x1‡#); assumption;
238 | exists; [apply a1] [assumption | change with (a1=a1); apply rule #; ]]]
239 | intros; cases e; cases x; clear e x;
240 lapply (. (or_prop3 ?? f (singleton ? a1) a)^-1);
241 [ cases Hletin; change in x with (eq1 ? a1 w1); apply (. x‡#); assumption;
242 | exists; [apply w] assumption ]
243 | intros; lapply (. (or_prop3 ?? f (singleton ? a1) a));
244 [ cases Hletin; exists; [apply w] split; assumption;
245 | exists; [apply a1] [change with (a1=a1); apply rule #; | assumption ]]
246 | intros; cases e; cases x; clear e x;
247 apply (f_image_monotone ?? f (singleton ? w) a ? a1);
248 [ intros 2; change in f3 with (eq1 ? w a2); change with (a2 ∈ a);
249 apply (. f3^-1‡#); assumption;
251 | intros; lapply (. (or_prop3 ?? f a (singleton ? a1))^-1);
252 [ cases Hletin; exists; [apply w] split;
253 [ lapply (. (or_prop3 ?? f (singleton ? w) (singleton ? a1)));
254 [ cases Hletin1; change in x3 with (eq1 ? a1 w1); apply (. x3‡#); assumption;
255 | exists; [apply w] [change with (w=w); apply rule #; | assumption ]]
257 | exists; [apply a1] [ assumption; | change with (a1=a1); apply rule #;]]
258 | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)^-1) ? a1);
259 [ apply f1; | change with (a1=a1); apply rule #; ]
260 | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)) ? y);
261 [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f* a);
262 apply (. f3^-1‡#); assumption;