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/saturations.ma".
18 record basic_topology: Type1 ≝
20 A: Ω^carrbt ⇒_1 Ω^carrbt;
21 J: Ω^carrbt ⇒_1 Ω^carrbt;
22 A_is_saturation: is_saturation ? A;
23 J_is_reduction: is_reduction ? J;
24 compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
27 definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
29 record continuous_relation (S,T: basic_topology) : Type1 ≝
30 { cont_rel:> S ⇒_\r1 T;
31 reduced: ∀U. U =_1 J ? U → image_coercion ?? cont_rel U =_1 J ? (image_coercion ?? cont_rel U);
32 saturated: ∀U. U =_1 A ? U → (foo ?? cont_rel)⎻* U = _1A ? ((foo ?? cont_rel)⎻* U)
35 definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
36 intros (S T); constructor 1;
37 [ apply (continuous_relation S T)
39 [ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
40 | simplify; intros; apply refl1;
41 | simplify; intros (x y H); apply sym1; apply H
42 | simplify; intros; apply trans1; [2: apply f |3: apply f1; |1: skip]]]
45 definition continuos_relation_of_continuous_relation_setoid :
46 ∀P,Q. continuous_relation_setoid P Q → continuous_relation P Q ≝ λP,Q,x.x.
47 coercion continuos_relation_of_continuous_relation_setoid.
49 axiom continuous_relation_eq':
50 ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
51 a = a' → ∀X.(foo ?? a)⎻* (A o1 X) = (foo ?? a')⎻* (A o1 X).
53 intros; split; intro; unfold minus_star_image; simplify; intros;
54 [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
55 lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
56 cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption]
57 lapply (fi ?? (A_is_saturation ???) Hcut);
58 apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify;
59 [ apply I | assumption ]
60 | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
61 lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
62 cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption]
63 lapply (fi ?? (A_is_saturation ???) Hcut);
64 apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
65 [ apply I | assumption ]]
68 axiom continuous_relation_eq_inv':
69 ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
70 (∀X.(foo ?? a)⎻* (A o1 X) = (foo ?? a')⎻* (A o1 X)) → a=a'.
72 cut (∀a,a': continuous_relation_setoid o1 o2.
73 (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) →
74 ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
75 [2: clear b H a' a; intros;
76 lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
77 (* fundamental adjunction here! to be taken out *)
78 cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
79 [2: intro; intros 2; unfold minus_star_image; simplify; intros;
80 apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
82 cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
83 [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
84 (* second half of the fundamental adjunction here! to be taken out too *)
85 intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
86 unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
87 whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
88 apply (if ?? (A_is_saturation ???));
89 intros 2 (x H); lapply (Hletin V ? x ?);
90 [ apply refl | cases H; assumption; ]
91 change with (x ∈ A ? (ext ?? a V));
92 apply (. #‡(†(extS_singleton ????)));
94 split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
98 definition continuous_relation_comp:
100 continuous_relation_setoid o1 o2 →
101 continuous_relation_setoid o2 o3 →
102 continuous_relation_setoid o1 o3.
103 intros (o1 o2 o3 r s); constructor 1;
104 [ alias symbol "compose" (instance 1) = "category1 composition".
108 (*change in ⊢ (? ? ? (? ? ? ? %) ?) with (image_coercion ?? (s ∘ r) U);*)
109 apply (.= †(image_comp ??????));
110 apply (.= (reduced ?? s (image_coercion ?? r U) ?)^-1);
111 [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
112 | change in ⊢ (? ? ? % ?) with ((image_coercion ?? s ∘ image_coercion ?? r) U);
113 apply (.= (image_comp ??????)^-1);
116 apply sym1; unfold foo;
117 apply (.= †(minus_star_image_comp ??????));
118 apply (.= (saturated ?? s ((foo ?? r)⎻* U) ?)^-1);
119 [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
120 | change in ⊢ (? ? ? % ?) with (((foo ?? s)⎻* ∘ (foo ?? r)⎻* ) U);
121 apply (.= (minus_star_image_comp ??????)^-1);
125 definition BTop: category1.
127 [ apply basic_topology
128 | apply continuous_relation_setoid
129 | intro; constructor 1;
132 apply (.= (image_id ??));
134 apply (.= †(image_id ??));
138 apply (.= (minus_star_image_id ??));
140 apply (.= †(minus_star_image_id ??));
143 | intros; constructor 1;
144 [ apply continuous_relation_comp;
145 | intros; simplify; intro x; simplify;
146 lapply depth=0 (continuous_relation_eq' ???? e) as H';
147 lapply depth=0 (continuous_relation_eq' ???? e1) as H1';
148 letin K ≝ (λX.H1' (minus_star_image ?? (foo ?? a) (A ? X))); clearbody K;
150 minus_star_image o2 o3 (foo ?? b) (A o2 (minus_star_image o1 o2 (foo ?? a) (A o1 X)))
151 =_1 minus_star_image o2 o3 (foo ?? b') (A o2 (minus_star_image o1 o2 (foo ?? a') (A o1 X))));
152 [2: intro; apply sym1;
153 apply (.= (†(†((H' X)^-1)))); apply sym1; apply (K X);]
155 alias symbol "compose" (instance 1) = "category1 composition".
156 alias symbol "compose" (instance 1) = "category1 composition".
157 alias symbol "compose" (instance 1) = "category1 composition".
159 minus_star_image ?? (foo ?? (b ∘ a)) (A o1 X) =_1 minus_star_image ?? (foo ?? (b'∘a')) (A o1 X));
160 [2: intro; unfold foo;
161 apply (.= (minus_star_image_comp ??????));
162 change in ⊢ (? ? ? % ?) with ((foo ?? b)⎻* ((foo ?? a)⎻* (A o1 X)));
163 apply (.= †(saturated ?????));
164 [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
166 apply (.= (minus_star_image_comp ??????));
167 change in ⊢ (? ? ? % ?) with ((foo ?? b')⎻* ((foo ?? a')⎻* (A o1 X)));
168 apply (.= †(saturated ?????));
169 [ apply ((saturation_idempotent ????)^-1); apply A_is_saturation ]
171 clear Hcut; generalize in match x; clear x;
172 apply (continuous_relation_eq_inv');
174 | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
175 alias symbol "trans" (instance 1) = "trans1".
176 alias symbol "refl" (instance 5) = "refl1".
177 alias symbol "prop2" (instance 3) = "prop21".
178 alias symbol "prop1" (instance 2) = "prop11".
179 alias symbol "assoc" (instance 4) = "category1 assoc".
180 apply (.= †(ASSOC‡#));
182 | intros; simplify; intro; unfold continuous_relation_comp; simplify;
183 apply (.= †((id_neutral_right1 ????)‡#));
185 | intros; simplify; intro; simplify;
186 apply (.= †((id_neutral_left1 ????)‡#));
192 (* this proof is more logic-oriented than set/lattice oriented *)
193 theorem continuous_relation_eqS:
194 ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
195 a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
197 cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
198 [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
199 try assumption; split; assumption]
200 cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y));
201 [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
203 apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
204 assumption;] clear Hcut;
205 split; apply (if ?? (A_is_saturation ???)); intros 2;
206 [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
207 cases Hletin; clear Hletin; cases x; clear x;
208 cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
209 [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
210 exists [1,3: apply w] split; assumption;]
211 cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
212 [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
213 apply Hcut2; assumption.