# Library Coq.Arith.Div2

Nota : this file is OBSOLETE, and left only for compatibility. Please consider using Nat.div2 directly, and results about it (see file PeanoNat).

Require Import PeanoNat.

Local Open Scope nat_scope.

Implicit Type n : nat.

Here we define n/2 and prove some of its properties

#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.div2 instead.")]
Notation div2 := Nat.div2 (only parsing).

Since div2 is recursively defined on 0, 1 and (S (S n)), it is useful to prove the corresponding induction principle

#[local]
Definition ind_0_1_SS_stt :
forall P:nat -> Prop,
P 0 -> P 1 -> (forall n, P n -> P (S (S n))) -> forall n, P n.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete.")]
Notation ind_0_1_SS := ind_0_1_SS_stt.

0 <n => n/2 < n

#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.lt_div2 instead.")]
Notation lt_div2 := Nat.lt_div2 (only parsing).

#[global]
Hint Resolve Nat.lt_div2: arith.

Properties related to the parity

#[local]
Definition even_div2_stt n : Nat.Even_alt n -> Nat.div2 n = Nat.div2 (S n).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Even_div2 (together with Nat.Even_alt_Even) instead.")]
Notation even_div2 := even_div2_stt.

#[local]
Definition odd_div2_stt n : Nat.Odd_alt n -> S (Nat.div2 n) = Nat.div2 (S n).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Odd_div2 (together with Nat.Odd_alt_Odd) instead.")]
Notation odd_div2 := odd_div2_stt.

#[local]
Definition div2_even_stt n : Nat.div2 n = Nat.div2 (S n) -> Nat.Even_alt n.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.div2_Even (together with Nat.Even_alt_Even) instead.")]
Notation div2_even := div2_even_stt.

#[local]
Definition div2_odd_stt n : S (Nat.div2 n) = Nat.div2 (S n) -> Nat.Odd_alt n.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.div2_Odd (together with Nat.Odd_alt_Odd) instead.")]
Notation div2_odd := div2_odd_stt.

#[global]
Hint Resolve even_div2_stt div2_even_stt odd_div2_stt div2_odd_stt: arith.

#[local]
Definition even_odd_div2_stt n :
(Nat.Even_alt n <-> Nat.div2 n = Nat.div2 (S n)) /\
(Nat.Odd_alt n <-> S (Nat.div2 n) = Nat.div2 (S n)).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Even_Odd_div2 (together with Nat.Even_alt_Even and Nat.Odd_alt_Odd) instead.")]
Notation even_odd_div2 := even_odd_div2_stt.

Properties related to the double (2n)

#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.double instead.")]
Notation double := Nat.double (only parsing).

#[global]
Hint Unfold Nat.double: arith.

#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.double_S instead.")]
Notation double_S := Nat.double_S.

#[global]
Hint Resolve Nat.double_S: arith.

#[local]
Definition even_odd_double_stt n :
(Nat.Even_alt n <-> n = Nat.double (Nat.div2 n)) /\ (Nat.Odd_alt n <-> n = S (Nat.double (Nat.div2 n))).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Even_Odd_double (together with Nat.Even_alt_Even and Nat.Odd_alt_Odd) instead.")]
Notation even_odd_double := even_odd_double_stt.

Specializations

#[local]
Definition even_double_stt n : Nat.Even_alt n -> n = Nat.double (Nat.div2 n).
Proof proj1 (proj1 (even_odd_double_stt n)).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Even_double (together with Nat.Even_alt_Even) instead.")]
Notation even_double := even_double_stt.

#[local]
Definition double_even_stt n : n = Nat.double (Nat.div2 n) -> Nat.Even_alt n.
Proof proj2 (proj1 (even_odd_double_stt n)).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.double_Even (together with Nat.Even_alt_Even) instead.")]
Notation double_even := double_even_stt.

#[local]
Definition odd_double_stt n : Nat.Odd_alt n -> n = S (Nat.double (Nat.div2 n)).
Proof proj1 (proj2 (even_odd_double_stt n)).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Odd_double (together with Nat.Odd_alt_Odd) instead.")]
Notation odd_double := odd_double_stt.

#[local]
Definition double_odd_stt n : n = S (Nat.double (Nat.div2 n)) -> Nat.Odd_alt n.
Proof proj2 (proj2 (even_odd_double_stt n)).
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.double_Odd (together with Nat.Odd_alt_Odd) instead.")]
Notation double_odd := double_odd_stt.

#[global]
Hint Resolve even_double_stt double_even_stt odd_double_stt double_odd_stt: arith.

Application:
• if n is even then there is a p such that n = 2p
• if n is odd then there is a p such that n = 2p+1
(Immediate: it is n/2)

#[local]
Definition even_2n_stt : forall n, Nat.Even_alt n -> {p : nat | n = Nat.double p}.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Even_alt_Even instead.")]
Notation even_2n := even_2n_stt.

#[local]
Definition odd_S2n_stt : forall n, Nat.Odd_alt n -> {p : nat | n = S (Nat.double p)}.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.Odd_alt_Odd instead.")]
Notation odd_S2n := odd_S2n_stt.

Doubling before dividing by two brings back to the initial number.

#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.div2_double instead.")]
Notation div2_double := Nat.div2_double.
#[deprecated(since="8.16",note="The Arith.Div2 file is obsolete. Use Nat.div2_succ_double instead.")]
Notation div2_double_plus_one := Nat.div2_succ_double.

Require Import Even.