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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope Z_scope.
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(** About parity *)
Notation two_or_two_plus_one := Z_modulo_2 (only parsing).
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(** The biggest power of 2 that is stricly less than [a]  Easy to compute: replace all "1" of the binary representation by "0", except the first "1" (or the first one :-) *)
Fixpoint floor_pos (a:positive) : positive :=
match a with
| xH => 1%positive
| xO a' => xO (floor_pos a')
| xI b' => xO (floor_pos b')
end.
Definition floor (a:positive) := Zpos (floor_pos a).

(* Specifications: *)
Lemma floor_gt0 : forall p:positive, floor p > 0.
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(** Two more induction principles over [Z]. *)
Theorem Z_lt_abs_rec :
forall P:Z -> Set,
(forall n:Z, (forall m:Z, Z.abs m < Z.abs n -> P m) -> P n) ->
forall n:Z, P n.
Theorem Z_lt_abs_induction :
forall P:Z -> Prop,
(forall n:Z, (forall m:Z, Z.abs m < Z.abs n -> P m) -> P n) ->
forall n:Z, P n.