The Coq Proof Assistant

# Library Coq.ZArith.BinIntDef

Require Export BinNums.
Require Import BinPos BinNat.
Local Open Scope Z_scope.

# Binary Integers, Definitions of Operations

Initial author: Pierre Crégut, CNET, Lannion, France
Module Z.
Definition t := Z.

## Nicer names Z.pos and Z.neg for constructors

Notation pos := Zpos.
Notation neg := Zneg.

## Constants

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

## Doubling and variants

Definition double x :=
match x with
| 0 => 0
| pos p => pos p~0
| neg p => neg p~0
end.
Definition succ_double x :=
match x with
| 0 => 1
| pos p => pos p~1
| neg p => neg (Pos.pred_double p)
end.
Definition pred_double x :=
match x with
| 0 => neg 1
| neg p => neg p~1
| pos p => pos (Pos.pred_double p)
end.

## Subtraction of positive into Z

Fixpoint pos_sub (x y:positive) {struct y} : Z :=
match x, y with
| p~1, q~1 => double (pos_sub p q)
| p~1, q~0 => succ_double (pos_sub p q)
| p~1, 1 => pos p~0
| p~0, q~1 => pred_double (pos_sub p q)
| p~0, q~0 => double (pos_sub p q)
| p~0, 1 => pos (Pos.pred_double p)
| 1, q~1 => neg q~0
| 1, q~0 => neg (Pos.pred_double q)
| 1, 1 => Z0
end%positive.

## Addition

Definition add x y :=
match x, y with
| 0, y => y
| x, 0 => x
| pos x', pos y' => pos (x' + y')
| pos x', neg y' => pos_sub x' y'
| neg x', pos y' => pos_sub y' x'
| neg x', neg y' => neg (x' + y')
end.
Infix "+" := add : Z_scope.

## Opposite

Definition opp x :=
match x with
| 0 => 0
| pos x => neg x
| neg x => pos x
end.
Notation "- x" := (opp x) : Z_scope.

## Successor

Definition succ x := x + 1.

## Predecessor

Definition pred x := x + neg 1.

## Subtraction

Definition sub m n := m + -n.
Infix "-" := sub : Z_scope.

## Multiplication

Definition mul x y :=
match x, y with
| 0, _ => 0
| _, 0 => 0
| pos x', pos y' => pos (x' * y')
| pos x', neg y' => neg (x' * y')
| neg x', pos y' => neg (x' * y')
| neg x', neg y' => pos (x' * y')
end.
Infix "*" := mul : Z_scope.

## Power function

Definition pow_pos (z:Z) := Pos.iter (mul z) 1.
Definition pow x y :=
match y with
| pos p => pow_pos x p
| 0 => 1
| neg _ => 0
end.
Infix "^" := pow : Z_scope.

## Square

Definition square x :=
match x with
| 0 => 0
| pos p => pos (Pos.square p)
| neg p => pos (Pos.square p)
end.

## Comparison

Definition compare x y :=
match x, y with
| 0, 0 => Eq
| 0, pos y' => Lt
| 0, neg y' => Gt
| pos x', 0 => Gt
| pos x', pos y' => (x' ?= y')%positive
| pos x', neg y' => Gt
| neg x', 0 => Lt
| neg x', pos y' => Lt
| neg x', neg y' => CompOpp ((x' ?= y')%positive)
end.
Infix "?=" := compare (at level 70, no associativity) : Z_scope.

## Sign function

Definition sgn z :=
match z with
| 0 => 0
| pos p => 1
| neg p => neg 1
end.
Boolean equality and comparisons
Definition leb x y :=
match x ?= y with
| Gt => false
| _ => true
end.
Definition ltb x y :=
match x ?= y with
| Lt => true
| _ => false
end.
Nota: geb and gtb are provided for compatibility, but leb and ltb should rather be used instead, since more results will be available on them.
Definition geb x y :=
match x ?= y with
| Lt => false
| _ => true
end.
Definition gtb x y :=
match x ?= y with
| Gt => true
| _ => false
end.
Definition eqb x y :=
match x, y with
| 0, 0 => true
| pos p, pos q => Pos.eqb p q
| neg p, neg q => Pos.eqb p q
| _, _ => false
end.
Infix "=?" := eqb (at level 70, no associativity) : Z_scope.
Infix "<=?" := leb (at level 70, no associativity) : Z_scope.
Infix "<?" := ltb (at level 70, no associativity) : Z_scope.
Infix ">=?" := geb (at level 70, no associativity) : Z_scope.
Infix ">?" := gtb (at level 70, no associativity) : Z_scope.

## Minimum and maximum

Definition max n m :=
match n ?= m with
| Eq | Gt => n
| Lt => m
end.
Definition min n m :=
match n ?= m with
| Eq | Lt => n
| Gt => m
end.

## Absolute value

Definition abs z :=
match z with
| 0 => 0
| pos p => pos p
| neg p => pos p
end.

## Conversions

From Z to nat via absolute value
Definition abs_nat (z:Z) : nat :=
match z with
| 0 => 0%nat
| pos p => Pos.to_nat p
| neg p => Pos.to_nat p
end.
From Z to N via absolute value
Definition abs_N (z:Z) : N :=
match z with
| 0 => 0%N
| pos p => N.pos p
| neg p => N.pos p
end.
From Z to nat by rounding negative numbers to 0
Definition to_nat (z:Z) : nat :=
match z with
| pos p => Pos.to_nat p
| _ => O
end.
From Z to N by rounding negative numbers to 0
Definition to_N (z:Z) : N :=
match z with
| pos p => N.pos p
| _ => 0%N
end.
From nat to Z
Definition of_nat (n:nat) : Z :=
match n with
| O => 0
| S n => pos (Pos.of_succ_nat n)
end.
From N to Z
Definition of_N (n:N) : Z :=
match n with
| 0%N => 0
| N.pos p => pos p
end.
From Z to positive by rounding nonpositive numbers to 1
Definition to_pos (z:Z) : positive :=
match z with
| pos p => p
| _ => 1%positive
end.
Conversion with a decimal representation for printing/parsing
Definition of_uint (d:Decimal.uint) := of_N (Pos.of_uint d).
Definition of_hex_uint (d:Hexadecimal.uint) := of_N (Pos.of_hex_uint d).
Definition of_num_uint (d:Number.uint) :=
match d with
| Number.UIntDecimal d => of_uint d
| Number.UIntHexadecimal d => of_hex_uint d
end.
Definition of_int (d:Decimal.int) :=
match d with
| Decimal.Pos d => of_uint d
| Decimal.Neg d => opp (of_uint d)
end.
Definition of_hex_int (d:Hexadecimal.int) :=
match d with
| Hexadecimal.Pos d => of_hex_uint d
| Hexadecimal.Neg d => opp (of_hex_uint d)
end.
Definition of_num_int (d:Number.int) :=
match d with
| Number.IntDecimal d => of_int d
| Number.IntHexadecimal d => of_hex_int d
end.
Definition to_int n :=
match n with
| 0 => Decimal.Pos Decimal.zero
| pos p => Decimal.Pos (Pos.to_uint p)
| neg p => Decimal.Neg (Pos.to_uint p)
end.
Definition to_hex_int n :=
match n with
| 0 => Hexadecimal.Pos Hexadecimal.zero
| pos p => Hexadecimal.Pos (Pos.to_hex_uint p)
| neg p => Hexadecimal.Neg (Pos.to_hex_uint p)
end.
Definition to_num_int n := Number.IntDecimal (to_int n).
Definition to_num_hex_int n := Number.IntHexadecimal (to_hex_int n).

## Iteration of a function

By convention, iterating a negative number of times is identity.
Definition iter (n:Z) {A} (f:A -> A) (x:A) :=
match n with
| pos p => Pos.iter f x p
| _ => x
end.

## Euclidean divisions for binary integers

Concerning the many possible variants of integer divisions, see the headers of the generic files ZDivFloor, ZDivTrunc, ZDivEucl, and the article by R. Boute mentioned there. We provide here two flavours, Floor and Trunc, while the Euclid convention can be found in file Zeuclid.v For non-zero b, they all satisfy a = b*(a/b) + (a mod b) and |a mod b| < |b| , but the sign of the modulo will differ when a<0 and/or b<0.

## Floor division

div_eucl provides a Truncated-Toward-Bottom (a.k.a Floor) Euclidean division. Its projections are named div (noted "/") and modulo (noted with an infix "mod"). These functions correspond to the `div` and `mod` of Haskell. This is the historical convention of Coq.
The main properties of this convention are :

- we have sgn (a mod b) = sgn (b)
- div a b is the greatest integer smaller or equal to the exact fraction a/b.
- there is no easy sign rule.

In addition, note that we take a/0 = 0 and a mod 0 = a. This choice is motivated by the div-mod equation a = (a / b) * b + (a mod b) for b = 0.
First, a division for positive numbers. Even if the second argument is a Z, the answer is arbitrary if it isn't a Zpos.
Fixpoint pos_div_eucl (a:positive) (b:Z) : Z * Z :=
match a with
| xH => if 2 <=? b then (0, 1) else (1, 0)
| xO a' =>
let (q, r) := pos_div_eucl a' b in
let r' := 2 * r in
if r' <? b then (2 * q, r') else (2 * q + 1, r' - b)
| xI a' =>
let (q, r) := pos_div_eucl a' b in
let r' := 2 * r + 1 in
if r' <? b then (2 * q, r') else (2 * q + 1, r' - b)
end.
Then the general euclidean division
Definition div_eucl (a b:Z) : Z * Z :=
match a, b with
| 0, _ => (0, 0)
| _, 0 => (0, a)
| pos a', pos _ => pos_div_eucl a' b
| neg a', pos _ =>
let (q, r) := pos_div_eucl a' b in
match r with
| 0 => (- q, 0)
| _ => (- (q + 1), b - r)
end
| neg a', neg b' =>
let (q, r) := pos_div_eucl a' (pos b') in (q, - r)
| pos a', neg b' =>
let (q, r) := pos_div_eucl a' (pos b') in
match r with
| 0 => (- q, 0)
| _ => (- (q + 1), b + r)
end
end.
Definition div (a b:Z) : Z := let (q, _) := div_eucl a b in q.
Definition modulo (a b:Z) : Z := let (_, r) := div_eucl a b in r.
Infix "/" := div : Z_scope.
Infix "mod" := modulo (at level 40, no associativity) : Z_scope.

## Trunc Division

quotrem provides a Truncated-Toward-Zero Euclidean division. Its projections are named quot (noted "÷") and rem. These functions correspond to the `quot` and `rem` of Haskell. This division convention is used in most programming languages, e.g. Ocaml.
With this convention:

- we have sgn(a rem b) = sgn(a)
- sign rule for division: quot (-a) b = quot a (-b) = -(quot a b)
- and for modulo: a rem (-b) = a rem b and (-a) rem b = -(a rem b)

Note that we take here quot a 0 = 0 and a rem 0 = a. This choice is motivated by the quot-rem equation a = (quot a b) * b + (a rem b) for b = 0.
Definition quotrem (a b:Z) : Z * Z :=
match a, b with
| 0, _ => (0, 0)
| _, 0 => (0, a)
| pos a, pos b =>
let (q, r) := N.pos_div_eucl a (N.pos b) in (of_N q, of_N r)
| neg a, pos b =>
let (q, r) := N.pos_div_eucl a (N.pos b) in (-of_N q, - of_N r)
| pos a, neg b =>
let (q, r) := N.pos_div_eucl a (N.pos b) in (-of_N q, of_N r)
| neg a, neg b =>
let (q, r) := N.pos_div_eucl a (N.pos b) in (of_N q, - of_N r)
end.
Definition quot a b := fst (quotrem a b).
Definition rem a b := snd (quotrem a b).
Infix "÷" := quot (at level 40, left associativity) : Z_scope.
No infix notation for rem, otherwise it becomes a keyword

## Parity functions

Definition even z :=
match z with
| 0 => true
| pos (xO _) => true
| neg (xO _) => true
| _ => false
end.
Definition odd z :=
match z with
| 0 => false
| pos (xO _) => false
| neg (xO _) => false
| _ => true
end.

## Division by two

div2 performs rounding toward bottom, it is hence a particular case of div, and for all relative number n we have: n = 2 * div2 n + if odd n then 1 else 0.
Definition div2 z :=
match z with
| 0 => 0
| pos 1 => 0
| pos p => pos (Pos.div2 p)
| neg p => neg (Pos.div2_up p)
end.
quot2 performs rounding toward zero, it is hence a particular case of quot, and for all relative number n we have: n = 2 * quot2 n + if odd n then sgn n else 0.
Definition quot2 (z:Z) :=
match z with
| 0 => 0
| pos 1 => 0
| pos p => pos (Pos.div2 p)
| neg 1 => 0
| neg p => neg (Pos.div2 p)
end.
NB: Z.quot2 used to be named Z.div2 in Coq <= 8.3

# Base-2 logarithm

Definition log2 z :=
match z with
| pos (p~1) => pos (Pos.size p)
| pos (p~0) => pos (Pos.size p)
| _ => 0
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
| neg _ => (0,0)
end.
Definition sqrt n :=
match n with
| pos p => pos (Pos.sqrt p)
| _ => 0
end.

## Greatest Common Divisor

Definition gcd a b :=
match a,b with
| 0, _ => abs b
| _, 0 => abs a
| pos a, pos b => pos (Pos.gcd a b)
| pos a, neg b => pos (Pos.gcd a b)
| neg a, pos b => pos (Pos.gcd a b)
| neg a, neg b => pos (Pos.gcd a b)
end.
A generalized gcd, also computing division of a and b by gcd.
Definition ggcd a b : Z*(Z*Z) :=
match a,b with
| 0, _ => (abs b,(0, sgn b))
| _, 0 => (abs a,(sgn a, 0))
| pos a, pos b =>
let '(g,(aa,bb)) := Pos.ggcd a b in (pos g, (pos aa, pos bb))
| pos a, neg b =>
let '(g,(aa,bb)) := Pos.ggcd a b in (pos g, (pos aa, neg bb))
| neg a, pos b =>
let '(g,(aa,bb)) := Pos.ggcd a b in (pos g, (neg aa, pos bb))
| neg a, neg b =>
let '(g,(aa,bb)) := Pos.ggcd a b in (pos g, (neg aa, neg bb))
end.

## Bitwise functions

When accessing the bits of negative numbers, all functions below will use the two's complement representation. For instance, -1 will correspond to an infinite stream of true bits. If this isn't what you're looking for, you can use abs first and then access the bits of the absolute value.
testbit : accessing the n-th bit of a number a. For negative n, we arbitrarily answer false.
Definition testbit a n :=
match n with
| 0 => odd a
| pos p =>
match a with
| 0 => false
| pos a => Pos.testbit a (N.pos p)
| neg a => negb (N.testbit (Pos.pred_N a) (N.pos p))
end
| neg _ => false
end.
Shifts
Nota: a shift to the right by -n will be a shift to the left by n, and vice-versa.
For fulfilling the two's complement convention, shifting to the right a negative number should correspond to a division by 2 with rounding toward bottom, hence the use of div2 instead of quot2.
Definition shiftl a n :=
match n with
| 0 => a
| pos p => Pos.iter (mul 2) a p
| neg p => Pos.iter div2 a p
end.
Definition shiftr a n := shiftl a (-n).
Bitwise operations lor land ldiff lxor
Definition lor a b :=
match a, b with
| 0, _ => b
| _, 0 => a
| pos a, pos b => pos (Pos.lor a b)
| neg a, pos b => neg (N.succ_pos (N.ldiff (Pos.pred_N a) (N.pos b)))
| pos a, neg b => neg (N.succ_pos (N.ldiff (Pos.pred_N b) (N.pos a)))
| neg a, neg b => neg (N.succ_pos (N.land (Pos.pred_N a) (Pos.pred_N b)))
end.
Definition land a b :=
match a, b with
| 0, _ => 0
| _, 0 => 0
| pos a, pos b => of_N (Pos.land a b)
| neg a, pos b => of_N (N.ldiff (N.pos b) (Pos.pred_N a))
| pos a, neg b => of_N (N.ldiff (N.pos a) (Pos.pred_N b))
| neg a, neg b => neg (N.succ_pos (N.lor (Pos.pred_N a) (Pos.pred_N b)))
end.
Definition ldiff a b :=
match a, b with
| 0, _ => 0
| _, 0 => a
| pos a, pos b => of_N (Pos.ldiff a b)
| neg a, pos b => neg (N.succ_pos (N.lor (Pos.pred_N a) (N.pos b)))
| pos a, neg b => of_N (N.land (N.pos a) (Pos.pred_N b))
| neg a, neg b => of_N (N.ldiff (Pos.pred_N b) (Pos.pred_N a))
end.
Definition lxor a b :=
match a, b with
| 0, _ => b
| _, 0 => a
| pos a, pos b => of_N (Pos.lxor a b)
| neg a, pos b => neg (N.succ_pos (N.lxor (Pos.pred_N a) (N.pos b)))
| pos a, neg b => neg (N.succ_pos (N.lxor (N.pos a) (Pos.pred_N b)))
| neg a, neg b => of_N (N.lxor (Pos.pred_N a) (Pos.pred_N b))
end.
End Z.
Re-export the notation for those who just Import BinIntDef