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 *)
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15 include "formal_topology/relations.ma".
17 definition is_saturation: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp[1] ≝
18 λC:REL.λA:Ω^C ⇒_1 Ω^C. ∀U,V. (U ⊆ A V) =_1 (A U ⊆ A V).
20 definition is_reduction: ∀C:REL. Ω^C ⇒_1 Ω^C → CProp[1] ≝
21 λC:REL.λJ: Ω^C ⇒_1 Ω^C. ∀U,V. (J U ⊆ V) =_1 (J U ⊆ J V).
23 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
24 intros; apply (fi ?? (i ??)); apply subseteq_refl.
27 theorem saturation_monotone:
28 ∀C,A. is_saturation C A →
29 ∀U,V. U ⊆ V → A U ⊆ A V.
30 intros; apply (if ?? (i ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
34 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
36 [ apply (if ?? (i ??)); apply subseteq_refl
37 | apply saturation_expansive; assumption]