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 "formal_topology/relations.ma".
16 include "formal_topology/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 [ (** screenshot "screen-pow". *) 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^A ≝ λA,x.x.
52 coercion powerset_of_POW'.
54 definition orelation_of_relation: ∀o1,o2:REL. o1 ⇒_\r1 o2 → (POW' o1) ⇒_\o2 (POW' o2).
61 | intros; split; intro;
62 [ intros 2; intros 2; apply (f y); exists[apply a] split; assumption;
63 | intros 2; change with (a ∈ q); cases f1; cases x; clear f1 x;
64 apply (f w f3); assumption; ]
65 | unfold foo; intros; split; intro;
66 [ intros 2; intros 2; apply (f x); exists [apply a] split; assumption;
67 | intros 2; change with (a ∈ q); cases f1; cases x; apply (f w f3); assumption;]
68 | intros; split; unfold foo; unfold image_coercion; simplify; intro; cases f; clear f;
69 [ cases x; cases x2; clear x x2; exists; [apply w1]
70 [ assumption | exists; [apply w] split; assumption]
71 | cases x1; cases x2; clear x1 x2; exists; [apply w1]
72 [ exists; [apply w] split; assumption;
76 lemma orelation_of_relation_preserves_equality:
77 ∀o1,o2:REL.∀t,t': o1 ⇒_\r1 o2.
78 t = t' → orelation_of_relation ?? t =_2 orelation_of_relation ?? t'.
79 intros; split; unfold orelation_of_relation; unfold foo; simplify;
80 change in e with (t =_2 t'); unfold image_coercion; apply (†e);
83 lemma minus_image_id : ∀o:REL.((id1 REL o))⎻ =_1 (id2 SET1 Ω^o).
84 unfold foo; intro o; intro; unfold minus_image; simplify; split; simplify; intros;
85 [ cases e; cases x; change with (a1 ∈ a); change in f with (a1 =_1 w);
86 apply (. f‡#); assumption;
87 | change in f with (a1 ∈ a); exists [ apply a1] split; try assumption;
88 change with (a1 =_1 a1); apply refl1;]
91 lemma star_image_id : ∀o:REL. ((id1 REL o))* =_1 (id2 SET1 Ω^o).
92 unfold foo; intro o; intro; unfold star_image; simplify; split; simplify; intros;
93 [ change with (a1 ∈ a); apply f; change with (a1 =_1 a1); apply rule refl1;
94 | change in f1 with (a1 =_1 y); apply (. f1^-1‡#); apply f;]
97 lemma orelation_of_relation_preserves_identity:
98 ∀o1:REL. orelation_of_relation ?? (id1 ? o1) =_2 id2 OA (POW' o1).
100 (unfold orelation_of_relation; unfold OA; unfold foo; simplify);
101 [ apply (minus_star_image_id o1);
102 | apply (minus_image_id o1);
103 | apply (image_id o1);
104 | apply (star_image_id o1) ]
109 [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
110 apply (f a1); change with (a1 = a1); apply refl1;
111 | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
112 change in f1 with (x = a1); apply (. f1‡#); apply f;
113 | alias symbol "and" = "and_morphism".
114 change with ((∃y:o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
115 intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
116 apply (. f‡#); apply f1;
117 | change with (a1 ∈ a → ∃y:o1.a1 ♮(id1 REL o1) y ∧ y∈a);
118 intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
119 | change with ((∃x:o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
120 intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
121 apply (. f^-1‡#); apply f1;
122 | change with (a1 ∈ a → ∃x:o1.x ♮(id1 REL o1) a1∧x∈a);
123 intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
124 | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
125 apply (f a1); change with (a1 = a1); apply refl1;
126 | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
127 change in f1 with (a1 = y); apply (. f1^-1‡#); apply f;]
131 (* CSC: ???? forse un uncertain mancato *)
132 alias symbol "eq" = "setoid2 eq".
133 alias symbol "compose" = "category1 composition".
134 lemma orelation_of_relation_preserves_composition:
135 ∀o1,o2,o3:REL.∀F: o1 ⇒_\r1 o2.∀G: o2 ⇒_\r1 o3.
136 orelation_of_relation ?? (G ∘ F) =
137 comp2 OA ??? (orelation_of_relation ?? F) (orelation_of_relation ?? G).
138 intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
139 [ whd; intros; apply f; exists; [ apply x] split; assumption;
140 | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
141 | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
142 split; [ assumption | exists; [apply w] split; assumption ]
143 | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
144 split; [ exists; [apply w] split; assumption | assumption ]
145 | unfold arrows1_of_ORelation_setoid;
146 cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
147 split; [ assumption | exists; [apply w] split; assumption ]
148 | unfold arrows1_of_ORelation_setoid in e;
149 cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
150 split; [ exists; [apply w] split; assumption | assumption ]
151 | whd; intros; apply f; exists; [ apply y] split; assumption;
152 | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
155 definition POW: carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
158 | intros; constructor 1;
159 [ apply (orelation_of_relation S T);
160 | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
161 | apply orelation_of_relation_preserves_identity;
162 | apply orelation_of_relation_preserves_composition; ]
165 theorem POW_faithful: faithful2 ?? POW.
166 intros 5; unfold POW in e; simplify in e; cases e;
167 unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
168 intros 2; simplify; unfold image_coercion in e3; cases (e3 {(x)});
169 split; intro; [ lapply (s y); | lapply (s1 y); ]
170 [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
171 |*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
176 lemma currify: ∀A,B,C. (A × B ⇒_1 C) → A → (B ⇒_1 C).
177 intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
181 include "formal_topology/notation.ma".
183 theorem POW_full: full2 ?? POW.
184 intros 3 (S T); exists;
185 [ constructor 1; constructor 1;
186 [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
187 | apply hide; intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
188 [4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
189 lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
190 | (split; intro; split; simplify);
191 [ change with (∀a1.(∀x. a1 ∈ (f {(x):S}) → x ∈ a) → a1 ∈ f⎻* a);
192 | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f {(x):S} → x ∈ a));
193 | alias symbol "and" (instance 4) = "and_morphism".
194 change with (∀a1.(∃y:carr T. y ∈ f {(a1):S} ∧ y ∈ a) → a1 ∈ f⎻ a);
195 | alias symbol "and" (instance 2) = "and_morphism".
196 change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f {(a1):S} ∧ y ∈ a));
197 | alias symbol "and" (instance 3) = "and_morphism".
198 change with (∀a1.(∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a) → a1 ∈ f a);
199 | change with (∀a1.a1 ∈. f a → (∃x:carr S. a1 ∈ f {(x):S} ∧ x ∈ a));
200 | change with (∀a1.(∀y. y ∈ f {(a1):S} → y ∈ a) → a1 ∈ f* a);
201 | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f {(a1):S} → y ∈ a)); ]
202 [ intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)^-1) ? a1);
203 [ intros 2; apply (f1 a2); change in f2 with (a2 ∈ f⎻ (singleton ? a1));
204 lapply (. (or_prop3 ?? f (singleton ? a2) (singleton ? a1)));
205 [ cases Hletin; change in x1 with (eq1 ? a1 w);
206 apply (. x1‡#); assumption;
207 | exists; [apply a2] [change with (a2=a2); apply rule #; | assumption]]
208 | change with (a1 = a1); apply rule #; ]
209 | intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)) ? x);
210 [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f⎻* a); apply (. f3^-1‡#);
212 | lapply (. (or_prop3 ?? f (singleton ? x) (singleton ? a1))^-1);
213 [ cases Hletin; change in x1 with (eq1 ? x w);
214 change with (x ∈ f⎻ (singleton ? a1)); apply (. x1‡#); assumption;
215 | exists; [apply a1] [assumption | change with (a1=a1); apply rule #; ]]]
216 | intros; cases e; cases x; clear e x;
217 lapply (. (or_prop3 ?? f (singleton ? a1) a)^-1);
218 [ cases Hletin; change in x with (eq1 ? a1 w1); apply (. x‡#); assumption;
219 | exists; [apply w] assumption ]
220 | intros; lapply (. (or_prop3 ?? f (singleton ? a1) a));
221 [ cases Hletin; exists; [apply w] split; assumption;
222 | exists; [apply a1] [change with (a1=a1); apply rule #; | assumption ]]
223 | intros; cases e; cases x; clear e x;
224 apply (f_image_monotone ?? f (singleton ? w) a ? a1);
225 [ intros 2; change in f3 with (eq1 ? w a2); change with (a2 ∈ a);
226 apply (. f3^-1‡#); assumption;
228 | intros; lapply (. (or_prop3 ?? f a (singleton ? a1))^-1);
229 [ cases Hletin; exists; [apply w] split;
230 [ lapply (. (or_prop3 ?? f (singleton ? w) (singleton ? a1)));
231 [ cases Hletin1; change in x3 with (eq1 ? a1 w1); apply (. x3‡#); assumption;
232 | exists; [apply w] [change with (w=w); apply rule #; | assumption ]]
234 | exists; [apply a1] [ assumption; | change with (a1=a1); apply rule #;]]
235 | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)^-1) ? a1);
236 [ apply f1; | change with (a1=a1); apply rule #; ]
237 | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)) ? y);
238 [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f* a);
239 apply (. f3^-1‡#); assumption;