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 "saturations.ma".
18 record basic_topology: Type1 ≝
20 A: unary_morphism1 (Ω \sup carrbt) (Ω \sup carrbt);
21 J: unary_morphism1 (Ω \sup carrbt) (Ω \sup carrbt);
22 A_is_saturation: is_saturation ? A;
23 J_is_reduction: is_reduction ? J;
24 compatibility: ∀U,V. (A U ≬ J V) = (U ≬ J V)
27 lemma hint: basic_topology → objs1 REL.
28 intro; apply (carrbt b);
32 record continuous_relation (S,T: basic_topology) : Type1 ≝
33 { cont_rel:> arrows1 ? S T;
34 reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
35 saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
38 definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
39 intros (S T); constructor 1;
40 [ apply (continuous_relation S T)
42 [ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
43 | simplify; intros; apply refl1;
44 | simplify; intros; apply sym1; apply H
45 | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
48 definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows1 ? S T ≝ cont_rel.
52 definition cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T → binary_relation S T ≝ cont_rel.
56 theorem continuous_relation_eq':
57 ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
58 a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
59 intros; split; intro; unfold minus_star_image; simplify; intros;
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 ?)‡#); assumption]
63 lapply (fi ?? (A_is_saturation ???) Hcut);
64 apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify;
65 [ apply I | assumption ]
66 | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
67 lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
68 cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption]
69 lapply (fi ?? (A_is_saturation ???) Hcut);
70 apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
71 [ apply I | assumption ]]
74 theorem continuous_relation_eq_inv':
75 ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
76 (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
78 cut (∀a,a': continuous_relation_setoid o1 o2.
79 (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) →
80 ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
81 [2: clear b H a' a; intros;
82 lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
83 (* fundamental adjunction here! to be taken out *)
84 cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
85 [2: intro; intros 2; unfold minus_star_image; simplify; intros;
86 apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
88 cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
89 [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
90 (* second half of the fundamental adjunction here! to be taken out too *)
91 intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
92 unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
93 whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
94 apply (if ?? (A_is_saturation ???));
95 intros 2 (x H); lapply (Hletin V ? x ?);
96 [ apply refl | cases H; assumption; ]
97 change with (x ∈ A ? (ext ?? a V));
98 apply (. #‡(†(extS_singleton ????)));
100 split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
103 definition continuous_relation_comp:
105 continuous_relation_setoid o1 o2 →
106 continuous_relation_setoid o2 o3 →
107 continuous_relation_setoid o1 o3.
108 intros (o1 o2 o3 r s); constructor 1;
112 apply (.= †(image_comp ??????));
113 apply (.= (reduced ?????)\sup -1);
114 [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
115 | apply (.= (image_comp ??????)\sup -1);
119 apply (.= †(minus_star_image_comp ??????));
120 apply (.= (saturated ?????)\sup -1);
121 [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
122 | apply (.= (minus_star_image_comp ??????)\sup -1);
126 definition BTop: category1.
128 [ apply basic_topology
129 | apply continuous_relation_setoid
130 | intro; constructor 1;
133 apply (.= (image_id ??));
135 apply (.= †(image_id ??));
139 apply (.= (minus_star_image_id ??));
141 apply (.= †(minus_star_image_id ??));
144 | intros; constructor 1;
145 [ apply continuous_relation_comp;
146 | intros; simplify; intro x; simplify;
147 lapply depth=0 (continuous_relation_eq' ???? H) as H';
148 lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
149 letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
151 minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
152 = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
153 [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
156 minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
158 apply (.= (minus_star_image_comp ??????));
159 apply (.= #‡(saturated ?????));
160 [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
162 apply (.= (minus_star_image_comp ??????));
163 apply (.= #‡(saturated ?????));
164 [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
165 apply ((Hcut X) \sup -1)]
166 clear Hcut; generalize in match x; clear x;
167 apply (continuous_relation_eq_inv');
169 | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
170 apply (.= †(ASSOC1‡#));
172 | intros; simplify; intro; unfold continuous_relation_comp; simplify;
173 apply (.= †((id_neutral_right1 ????)‡#));
175 | intros; simplify; intro; simplify;
176 apply (.= †((id_neutral_left1 ????)‡#));
182 (* this proof is more logic-oriented than set/lattice oriented *)
183 theorem continuous_relation_eqS:
184 ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
185 a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
187 cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
188 [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
189 try assumption; split; assumption]
190 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));
191 [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
193 apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
194 assumption;] clear Hcut;
195 split; apply (if ?? (A_is_saturation ???)); intros 2;
196 [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
197 cases Hletin; clear Hletin; cases x; clear x;
198 cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
199 [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
200 exists [1,3: apply w] split; assumption;]
201 cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
202 [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
203 apply Hcut2; assumption.