Library Coq.Init.Nat


Require Import Notations Logic Datatypes.

Local Open Scope nat_scope.

Peano natural numbers, definitions of operations

This file is meant to be used as a whole module, without importing it, leading to qualified definitions (e.g. Nat.pred)

Definition t := nat.

Constants


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

Basic operations


Definition succ := S.

Definition pred n :=
  match n with
    | 0 => n
    | S u => u
  end.

Fixpoint add n m :=
  match n with
  | 0 => m
  | S p => S (p + m)
  end

where "n + m" := (add n m) : nat_scope.

Definition double n := n + n.

Fixpoint mul n m :=
  match n with
  | 0 => 0
  | S p => m + p * m
  end

where "n * m" := (mul n m) : nat_scope.

Truncated subtraction: n-m is 0 if n<=m

Fixpoint sub n m :=
  match n, m with
  | S k, S l => k - l
  | _, _ => n
  end

where "n - m" := (sub n m) : nat_scope.

Comparisons


Fixpoint eqb n m : bool :=
  match n, m with
    | 0, 0 => true
    | 0, S _ => false
    | S _, 0 => false
    | S n', S m' => eqb n' m'
  end.

Fixpoint leb n m : bool :=
  match n, m with
    | 0, _ => true
    | _, 0 => false
    | S n', S m' => leb n' m'
  end.

Definition ltb n m := leb (S n) m.

Infix "=?" := eqb (at level 70) : nat_scope.
Infix "<=?" := leb (at level 70) : nat_scope.
Infix "<?" := ltb (at level 70) : nat_scope.

Fixpoint compare n m : comparison :=
  match n, m with
   | 0, 0 => Eq
   | 0, S _ => Lt
   | S _, 0 => Gt
   | S n', S m' => compare n' m'
  end.

Infix "?=" := compare (at level 70) : nat_scope.

Minimum, maximum


Fixpoint max n m :=
  match n, m with
    | 0, _ => m
    | S n', 0 => n
    | S n', S m' => S (max n' m')
  end.

Fixpoint min n m :=
  match n, m with
    | 0, _ => 0
    | S n', 0 => 0
    | S n', S m' => S (min n' m')
  end.

Parity tests


Fixpoint even n : bool :=
  match n with
    | 0 => true
    | 1 => false
    | S (S n') => even n'
  end.

Definition odd n := negb (even n).

Power


Fixpoint pow n m :=
  match m with
    | 0 => 1
    | S m => n * (n^m)
  end

where "n ^ m" := (pow n m) : nat_scope.

Euclidean division

This division is linear and tail-recursive. In divmod, y is the predecessor of the actual divisor, and u is y minus the real remainder

Fixpoint divmod x y q u :=
  match x with
    | 0 => (q,u)
    | S x' => match u with
                | 0 => divmod x' y (S q) y
                | S u' => divmod x' y q u'
              end
  end.

Definition div x y :=
  match y with
    | 0 => y
    | S y' => fst (divmod x y' 0 y')
  end.

Definition modulo x y :=
  match y with
    | 0 => y
    | S y' => y' - snd (divmod x y' 0 y')
  end.

Infix "/" := div : nat_scope.
Infix "mod" := modulo (at level 40, no associativity) : nat_scope.

Greatest common divisor

We use Euclid algorithm, which is normally not structural, but Coq is now clever enough to accept this (behind modulo there is a subtraction, which now preserves being a subterm)

Fixpoint gcd a b :=
  match a with
   | O => b
   | S a' => gcd (b mod (S a')) (S a')
  end.

Square


Definition square n := n * n.

Square root

The following square root function is linear (and tail-recursive). With Peano representation, we can't do better. For faster algorithm, see Psqrt/Zsqrt/Nsqrt...
We search the square root of n = k + p^2 + (q - r) with q = 2p and 0<=r<=q. We start with p=q=r=0, hence looking for the square root of n = k. Then we progressively decrease k and r. When k = S k' and r=0, it means we can use (S p) as new sqrt candidate, since (S k')+p^2+2p = k'+(S p)^2. When k reaches 0, we have found the biggest p^2 square contained in n, hence the square root of n is p.

Fixpoint sqrt_iter k p q r :=
  match k with
    | O => p
    | S k' => match r with
                | O => sqrt_iter k' (S p) (S (S q)) (S (S q))
                | S r' => sqrt_iter k' p q r'
              end
  end.

Definition sqrt n := sqrt_iter n 0 0 0.

Log2

This base-2 logarithm is linear and tail-recursive.
In log2_iter, we maintain the logarithm p of the counter q, while r is the distance between q and the next power of 2, more precisely q + S r = 2^(S p) and r<2^p. At each recursive call, q goes up while r goes down. When r is 0, we know that q has almost reached a power of 2, and we increase p at the next call, while resetting r to q.
Graphically (numbers are q, stars are r) :
                    10
                  9
                8
              7   *
            6       *
          5           ...
        4
      3   *
    2       *
  1   *       *
0   *   *       *
We stop when k, the global downward counter reaches 0. At that moment, q is the number we're considering (since k+q is invariant), and p its logarithm.

Fixpoint log2_iter k p q r :=
  match k with
    | O => p
    | S k' => match r with
                | O => log2_iter k' (S p) (S q) q
                | S r' => log2_iter k' p (S q) r'
              end
  end.

Definition log2 n := log2_iter (pred n) 0 1 0.

Iterator on natural numbers

Definition iter (n:nat) {A} (f:A->A) (x:A) : A :=
 nat_rect (fun _ => A) x (fun _ => f) n.

Bitwise operations
We provide here some bitwise operations for unary numbers. Some might be really naive, they are just there for fullfiling the same interface as other for natural representations. As soon as binary representations such as NArith are available, it is clearly better to convert to/from them and use their ops.

Fixpoint div2 n :=
  match n with
  | 0 => 0
  | S 0 => 0
  | S (S n') => S (div2 n')
  end.

Fixpoint testbit a n : bool :=
 match n with
   | 0 => odd a
   | S n => testbit (div2 a) n
 end.

Definition shiftl a := nat_rect _ a (fun _ => double).
Definition shiftr a := nat_rect _ a (fun _ => div2).

Fixpoint bitwise (op:bool->bool->bool) n a b :=
 match n with
  | 0 => 0
  | S n' =>
    (if op (odd a) (odd b) then 1 else 0) +
    2*(bitwise op n' (div2 a) (div2 b))
 end.

Definition land a b := bitwise andb a a b.
Definition lor a b := bitwise orb (max a b) a b.
Definition ldiff a b := bitwise (fun b b' => andb b (negb b')) a a b.
Definition lxor a b := bitwise xorb (max a b) a b.