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Axiomatization of real numbers (work in progress)
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14
15 include "datatypes/subsets.ma".
16 include "formal_topology/relations.ma".
17
18 definition is_saturation ≝
19  λC:REL.λA:unary_morphism (Ω \sup C) (Ω \sup C).
20   ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
21
22 definition is_reduction ≝
23  λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C).
24   ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
25
26 theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U.
27  intros; apply (fi ?? (H ??)); apply subseteq_refl.
28 qed.
29
30 theorem saturation_monotone:
31  ∀C,A. is_saturation C A →
32   ∀U,V. U ⊆ V → A U ⊆ A V.
33  intros; apply (if ?? (H ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ]
34  assumption.
35 qed.
36
37 theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U.
38  intros; split;
39   [ apply (if ?? (H ??)); apply subseteq_refl
40   | apply saturation_expansive; assumption]
41 qed.