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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground/lib/subset.ma".
17 (* EXTENSIONS FOR SUBSETS ***************************************************)
19 definition subset_ext_f1 (A1) (A0) (f:A1→A0): 𝒫❨A1❩ → 𝒫❨A0❩ ≝
20 λu1,a0. ∃∃a1. a1 ϵ u1 & f a1 = a0.
22 definition subset_ext_f1_1 (A11) (A21) (A0) (f1:A11→A0) (f2:A21→A0): 𝒫❨A11❩ → 𝒫❨A21❩ → 𝒫❨A0❩ ≝
24 ∨∨ subset_ext_f1 A11 A0 f1 u11 a0
25 | subset_ext_f1 A21 A0 f2 u21 a0.
27 definition subset_ext_p1 (A1) (Q:predicate A1): predicate (𝒫❨A1❩) ≝
28 λu1. ∀a1. a1 ϵ u1 → Q a1.
30 (* Basic constructions ******************************************************)
32 lemma subset_in_ext_f1_dx (A1) (A0) (f) (u1) (a1):
33 a1 ϵ u1 → f a1 ϵ subset_ext_f1 A1 A0 f u1.
34 /2 width=3 by ex2_intro/ qed.
36 lemma subset_in_ext_f1_1_dx_1 (A11) (A21) (A0) (f1) (f2) (u11) (u21) (a11):
37 a11 ϵ u11 → f1 a11 ϵ subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21.
38 /3 width=3 by subset_in_ext_f1_dx, or_introl/ qed.
40 lemma subset_in_ext_f1_1_dx_2 (A11) (A21) (A0) (f1) (f2) (u11) (u21) (a21):
41 a21 ϵ u21 → f2 a21 ϵ subset_ext_f1_1 A11 A21 A0 f1 f2 u11 u21.
42 /3 width=3 by subset_in_ext_f1_dx, or_intror/ qed.
44 (* Basic inversions *********************************************************)
46 lemma subset_in_inv_ext_p1_dx (A1) (Q) (u1) (a1):
47 a1 ϵ u1 → subset_ext_p1 A1 Q u1 → Q a1.