# Library Coq.NArith.BinNatDef

Require Export BinNums.
Require Import BinPos.
Local Open Scope N_scope.

# Binary natural numbers, definitions of operations

Module N.
Definition t := N.

## Nicer name N.pos for contructor Npos

Notation pos := Npos.

## Constants

Definition zero := 0.
Definition one := 1.
Definition two := 2.

## Operation x -> 2*x+1

Definition succ_double x :=
match x with
| 0 => 1
| pos p => pos p ~1
end.

## Operation x -> 2*x

Definition double n :=
match n with
| 0 => 0
| pos p => pos p ~0
end.

## Successor

Definition succ n :=
match n with
| 0 => 1
| pos p => pos (Pos.succ p)
end.

## Predecessor

Definition pred n :=
match n with
| 0 => 0
| pos p => Pos.pred_N p
end.

## The successor of a N can be seen as a positive

Definition succ_pos (n : N) : positive :=
match n with
| 0 => 1%positive
| pos p => Pos.succ p
end.

## Addition

Definition add n m :=
match n, m with
| 0, _ => m
| _, 0 => n
| pos p, pos q => pos (p + q)
end.
Infix "+" := add : N_scope.
Subtraction
Definition sub n m :=
match n, m with
| 0, _ => 0
| n, 0 => n
| pos n', pos m' =>
match Pos.sub_mask n' m' with
| IsPos p => pos p
| _ => 0
end
end.
Infix "-" := sub : N_scope.
Multiplication
Definition mul n m :=
match n, m with
| 0, _ => 0
| _, 0 => 0
| pos p, pos q => pos (p * q)
end.
Infix "*" := mul : N_scope.
Order
Definition compare n m :=
match n, m with
| 0, 0 => Eq
| 0, pos m' => Lt
| pos n', 0 => Gt
| pos n', pos m' => (n' ?= m')%positive
end.
Infix "?=" := compare (at level 70, no associativity) : N_scope.
Boolean equality and comparison
Fixpoint eqb n m :=
match n, m with
| 0, 0 => true
| pos p, pos q => Pos.eqb p q
| _, _ => false
end.
Definition leb x y :=
match x ?= y with Gt => false | _ => true end.
Definition ltb x y :=
match x ?= y with Lt => true | _ => false end.
Infix "=?" := eqb (at level 70, no associativity) : N_scope.
Infix "<=?" := leb (at level 70, no associativity) : N_scope.
Infix "<?" := ltb (at level 70, no associativity) : N_scope.
Min and max
Definition min n n' := match n ?= n' with
| Lt | Eq => n
| Gt => n'
end.
Definition max n n' := match n ?= n' with
| Lt | Eq => n'
| Gt => n
end.
Dividing by 2
Definition div2 n :=
match n with
| 0 => 0
| 1 => 0
| pos (p ~0) => pos p
| pos (p ~1) => pos p
end.
Parity
Definition even n :=
match n with
| 0 => true
| pos (xO _) => true
| _ => false
end.
Definition odd n := negb (even n).
Power
Definition pow n p :=
match p, n with
| 0, _ => 1
| _, 0 => 0
| pos p, pos q => pos (q ^ p)
end.
Infix "^" := pow : N_scope.
Square
Definition square n :=
match n with
| 0 => 0
| pos p => pos (Pos.square p)
end.
Base-2 logarithm
Definition log2 n :=
match n with
| 0 => 0
| 1 => 0
| pos (p ~0) => pos (Pos.size p)
| pos (p ~1) => pos (Pos.size p)
end.
How many digits in a number ? Number 0 is said to have no digits at all.
Definition size n :=
match n with
| 0 => 0
| pos p => pos (Pos.size p)
end.
Definition size_nat n :=
match n with
| 0 => O
| pos p => Pos.size_nat p
end.
Euclidean division
Fixpoint pos_div_eucl (a:positive)(b:N) : N * N :=
match a with
| xH =>
match b with 1 => (1,0) | _ => (0,1) end
| xO a' =>
let (q, r) := pos_div_eucl a' b in
let r' := double r in
if b <=? r' then ( succ_double q , r' - b )
else ( double q , r' )
| xI a' =>
let (q, r) := pos_div_eucl a' b in
let r' := succ_double r in
if b <=? r' then ( succ_double q , r' - b )
else ( double q , r' )
end.
Definition div_eucl (a b:N) : N * N :=
match a, b with
| 0, _ => (0, 0)
| _, 0 => (0, a )
| pos na, _ => pos_div_eucl na b
end.
Definition div a b := fst (div_eucl a b).
Definition modulo a b := snd (div_eucl a b).
Infix "/" := div : N_scope.
Infix "mod" := modulo (at level 40, no associativity) : N_scope.
Greatest common divisor
Definition gcd a b :=
match a, b with
| 0, _ => b
| _, 0 => a
| pos p, pos q => pos (Pos.gcd p q)
end.
Generalized Gcd, also computing rests of a and b after division by gcd.
Definition ggcd a b :=
match a, b with
| 0, _ => ( b ,(0,1))
| _, 0 => ( a ,(1,0))
| pos p, pos q =>
let '( g ,( aa , bb )) := Pos.ggcd p q in
( pos g , ( pos aa , pos bb ))
end.
Square root
Definition sqrtrem n :=
match n with
| 0 => (0, 0)
| pos p =>
match Pos.sqrtrem p with
| ( s , IsPos r ) => ( pos s , pos r )
| ( s , _ ) => ( pos s , 0)
end
end.
Definition sqrt n :=
match n with
| 0 => 0
| pos p => pos (Pos.sqrt p)
end.
Operation over bits of a N number.
Logical or
Definition lor n m :=
match n, m with
| 0, _ => m
| _, 0 => n
| pos p, pos q => pos (Pos.lor p q)
end.
Logical and
Definition land n m :=
match n, m with
| 0, _ => 0
| _, 0 => 0
| pos p, pos q => Pos.land p q
end.
Logical diff
Fixpoint ldiff n m :=
match n, m with
| 0, _ => 0
| _, 0 => n
| pos p, pos q => Pos.ldiff p q
end.
xor
Definition lxor n m :=
match n, m with
| 0, _ => m
| _, 0 => n
| pos p, pos q => Pos.lxor p q
end.
Shifts
Definition shiftl_nat (a:N)(n:nat) := nat_iter n double a.
Definition shiftr_nat (a:N)(n:nat) := nat_iter n div2 a.
Definition shiftl a n :=
match a with
| 0 => 0
| pos a => pos (Pos.shiftl a n)
end.
Definition shiftr a n :=
match n with
| 0 => a
| pos p => Pos.iter p div2 a
end.
Checking whether a particular bit is set or not
Definition testbit_nat (a:N) :=
match a with
| 0 => fun _ => false
| pos p => Pos.testbit_nat p
end.
Same, but with index in N
Definition testbit a n :=
match a with
| 0 => false
| pos p => Pos.testbit p n
end.
Translation from N to nat and back.
Definition to_nat (a:N) :=
match a with
| 0 => O
| pos p => Pos.to_nat p
end.
Definition of_nat (n:nat) :=
match n with
| O => 0
| S n' => pos (Pos.of_succ_nat n')
end.
Iteration of a function
Definition iter (n:N) {A} (f:A->A) (x:A) : A :=
match n with
| 0 => x
| pos p => Pos.iter p f x
end.
End N.