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Nota : this file is OBSOLETE, and left only for compatibility.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 Even.
Local Open Scope nat_scope.
Implicit Type n :
nat.
Here we define n/2 and prove some of its properties
Since div2 is recursively defined on 0, 1 and (S (S n)), it is
useful to prove the corresponding induction principle
0 <n => n/2 < n
Properties related to the parity
Properties related to the double (2n)
Specializations
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)
Doubling before dividing by two brings back to the initial number.