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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground/notation/functions/downarrow_1.ma".
16 include "ground/arith/pnat_split.ma".
18 (* PREDECESSOR FOR POSITIVE INTEGERS ****************************************)
20 definition ppred (p): pnat ≝ psplit … (𝟏) (λp.p) p.
23 "predecessor (positive integers)"
24 'DownArrow p = (ppred p).
26 (* Basic constructions ******************************************************)
28 lemma ppred_unit: 𝟏 = ↓𝟏.
31 lemma ppred_succ (p): p = ↓↑p.
34 (* Basic inversions *********************************************************)
36 lemma ppred_inv_refl (p): p = ↓p → 𝟏 = p.
38 #p #IH #H /2 width=1 by/
41 lemma pnat_split_unit_pos (p): ∨∨ 𝟏 = p | p = ↑↓p.
43 /2 width=1 by or_introl, or_intror/