Why3 Standard Library index

# Theory of reals

This file provides the basic theory of real numbers, and several additional theories for classical real functions.

## Real numbers and the basic unary and binary operators

```module Real

constant zero : real = 0.0
constant one  : real = 1.0

val (=) (x y : real) : bool ensures { result <-> x = y }

val function  (-_) real : real
val function  (+)  real real : real
val function  (*)  real real : real
val predicate (<)  real real : bool

let predicate (>)  (x y : real) = y < x
let predicate (<=) (x y : real) = x < y || x = y
let predicate (>=) (x y : real) = y <= x

clone export algebra.OrderedField with type t = real,
constant zero = zero, constant one = one,
function (-_) = (-_), function (+) = (+),
function (*) = (*), predicate (<=) = (<=)

let (-) (x y : real)
ensures { result = x - y }
= x + -y

val (/) (x y:real) : real
requires { y <> 0.0 }
ensures  { result = x / y }

end

```

## Alternative Infix Operators

This theory should be used instead of Real when one wants to use both integer and real binary operators.

```module RealInfix

use Real

let function (+.) (x:real) (y:real) : real = x + y
let function (-.) (x:real) (y:real) : real = x - y
let function ( *.) (x:real) (y:real) : real = x * y
function (/.) (x:real) (y:real) : real = x / y
let function (-._) (x:real) : real = - x
function inv (x:real) : real = Real.inv x

let (=.) (x:real) (y:real) = x = y
let predicate (<=.) (x:real) (y:real) = x <= y
let predicate (>=.) (x:real) (y:real) = x >= y
let predicate ( <.) (x:real) (y:real) = x < y
let predicate ( >.) (x:real) (y:real) = x > y

val (/.) (x y:real) : real
requires { y <> 0.0 }
ensures  { result = x /. y }

end

```

## Absolute Value

```module Abs

use Real

function abs(x : real) : real = if x >= 0.0 then x else -x

lemma Abs_le: forall x y:real. abs x <= y <-> -y <= x <= y

lemma Abs_pos: forall x:real. abs x >= 0.0

lemma Abs_sum: forall x y:real. abs(x+y) <= abs x + abs y

lemma Abs_prod: forall x y:real. abs(x*y) = abs x * abs y

lemma triangular_inequality:
forall x y z:real. abs(x-z) <= abs(x-y) + abs(y-z)

end

```

## Minimum and Maximum

```module MinMax

use Real
clone export relations.MinMax with type t = real, predicate le = (<=), goal .

end

```

## Injection of integers into reals

```module FromInt

use int.Int as Int
use Real

function from_int int : real

axiom Zero: from_int 0 = 0.0
axiom One: from_int 1 = 1.0

forall x y:int. from_int (Int.(+) x y) = from_int x + from_int y
axiom Sub:
forall x y:int. from_int (Int.(-) x y) = from_int x - from_int y
axiom Mul:
forall x y:int. from_int (Int.(*) x y) = from_int x * from_int y
axiom Neg:
forall x:int. from_int (Int.(-_) (x)) = - from_int x

lemma Injective:
forall x y: int. from_int x = from_int y -> x = y
axiom Monotonic:
forall x y:int. Int.(<=) x y -> from_int x <= from_int y

end

```

## Various truncation functions

```module Truncate

use Real
use FromInt

function truncate real : int
```

rounds towards zero

```  axiom Truncate_int :
forall i:int. truncate (from_int i) = i

axiom Truncate_down_pos:
forall x:real. x >= 0.0 ->
from_int (truncate x) <= x < from_int (Int.(+) (truncate x) 1)

axiom Truncate_up_neg:
forall x:real. x <= 0.0 ->
from_int (Int.(-) (truncate x) 1) < x <= from_int (truncate x)

axiom Real_of_truncate:
forall x:real. x - 1.0 <= from_int (truncate x) <= x + 1.0

axiom Truncate_monotonic:
forall x y:real. x <= y -> Int.(<=) (truncate x) (truncate y)

axiom Truncate_monotonic_int1:
forall x:real, i:int. x <= from_int i -> Int.(<=) (truncate x) i

axiom Truncate_monotonic_int2:
forall x:real, i:int. from_int i <= x -> Int.(<=) i (truncate x)

function floor real : int
function ceil real : int
```

roundings up and down

```  axiom Floor_int :
forall i:int. floor (from_int i) = i

axiom Ceil_int :
forall i:int. ceil (from_int i) = i

axiom Floor_down:
forall x:real. from_int (floor x) <= x < from_int (Int.(+) (floor x) 1)

axiom Ceil_up:
forall x:real. from_int (Int.(-) (ceil x) 1) < x <= from_int (ceil x)

axiom Floor_monotonic:
forall x y:real. x <= y -> Int.(<=) (floor x) (floor y)

axiom Ceil_monotonic:
forall x y:real. x <= y -> Int.(<=) (ceil x) (ceil y)

end

```

## Square and Square Root

```module Square

use Real

function sqr (x : real) : real = x * x

val function sqrt real : real

axiom Sqrt_positive:
forall x:real. x >= 0.0 -> sqrt x >= 0.0

axiom Sqrt_square:
forall x:real. x >= 0.0 -> sqr (sqrt x) = x

axiom Square_sqrt:
forall x:real. x >= 0.0 -> sqrt (x*x) = x

axiom Sqrt_mul:
forall x y:real. x >= 0.0 /\ y >= 0.0 ->
sqrt (x*y) = sqrt x * sqrt y

axiom Sqrt_le :
forall x y:real. 0.0 <= x <= y -> sqrt x <= sqrt y

end

```

## Exponential and Logarithm

```module ExpLog

use Real

val function exp real : real
axiom Exp_zero : exp(0.0) = 1.0
axiom Exp_sum : forall x y:real. exp (x+y) = exp x * exp y

constant e : real = exp 1.0

val function log real : real
axiom Log_one : log 1.0 = 0.0
axiom Log_mul :
forall x y:real. x > 0.0 /\ y > 0.0 -> log (x*y) = log x + log y

axiom Log_exp: forall x:real. log (exp x) = x

axiom Exp_log: forall x:real. x > 0.0 -> exp (log x) = x

function log2 (x : real) : real = log x / log 2.0
function log10 (x : real) : real = log x / log 10.0

end

```

## Power of a real to an integer

```module PowerInt

use int.Int
use RealInfix

clone export int.Exponentiation with
type t = real, constant one = Real.one, function (*) = Real.(*),
goal Assoc, goal Unit_def_l, goal Unit_def_r, axiom Power_0, axiom Power_s

lemma Pow_ge_one:
forall x:real, n:int. 0 <= n /\ 1.0 <=. x -> 1.0 <=. power x n

end

```

## Power of a real to a real exponent

```module PowerReal

use Real
use ExpLog

function pow real real : real

axiom Pow_def:
forall x y:real. x > 0.0 -> pow x y = exp (y * log x)

lemma Pow_pos:
forall x y. x > 0.0 -> pow x y > 0.0

lemma Pow_plus :
forall x y z. z > 0.0 -> pow z (x + y) = pow z x * pow z y

lemma Pow_mult :
forall x y z. x > 0.0 -> pow (pow x y) z = pow x (y * z)

lemma Pow_x_zero:
forall x:real. x > 0.0 -> pow x 0.0 = 1.0

lemma Pow_x_one:
forall x:real. x > 0.0 -> pow x 1.0 = x

lemma Pow_one_y:
forall y:real. pow 1.0 y = 1.0

use Square

lemma Pow_x_two:
forall x:real. x > 0.0 -> pow x 2.0 = sqr x

lemma Pow_half:
forall x:real. x > 0.0 -> pow x 0.5 = sqrt x

use FromInt
use int.Power

lemma pow_from_int: forall x y: int. Int.(<) 0 x -> Int.(<=) 0 y ->
pow (from_int x) (from_int y) = from_int (power x y)

end

```

## Trigonometric Functions

```module Trigonometry

use Real
use Square
use Abs

function cos real : real
function sin real : real

axiom Pythagorean_identity:
forall x:real. sqr (cos x) + sqr (sin x) = 1.0

lemma Cos_le_one: forall x:real. abs (cos x) <= 1.0
lemma Sin_le_one: forall x:real. abs (sin x) <= 1.0

axiom Cos_0: cos 0.0 = 1.0
axiom Sin_0: sin 0.0 = 0.0

val constant pi : real

axiom Pi_double_precision_bounds:
0x1.921fb54442d18p+1 < pi < 0x1.921fb54442d19p+1
(*
axiom Pi_interval:
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196
< pi <
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038197
*)

axiom Cos_pi: cos pi = -1.0
axiom Sin_pi: sin pi = 0.0

axiom Cos_pi2: cos (0.5 * pi) = 0.0
axiom Sin_pi2: sin (0.5 * pi) = 1.0

axiom Cos_plus_pi: forall x:real. cos (x + pi) = - cos x
axiom Sin_plus_pi: forall x:real. sin (x + pi) = - sin x

axiom Cos_plus_pi2: forall x:real. cos (x + 0.5*pi) = - sin x
axiom Sin_plus_pi2: forall x:real. sin (x + 0.5*pi) = cos x

axiom Cos_neg:
forall x:real. cos (-x) = cos x
axiom Sin_neg:
forall x:real. sin (-x) = - sin x

axiom Cos_sum:
forall x y:real. cos (x+y) = cos x * cos y - sin x * sin y
axiom Sin_sum:
forall x y:real. sin (x+y) = sin x * cos y + cos x * sin y

function tan (x : real) : real = sin x / cos x
function atan real : real
axiom Tan_atan:
forall x:real. tan (atan x) = x

end

```

## Hyperbolic Functions

```module Hyperbolic

use Real
use Square
use ExpLog

function sinh (x : real) : real = 0.5 * (exp x - exp (-x))
function cosh (x : real) : real = 0.5 * (exp x + exp (-x))
function tanh (x : real) : real = sinh x / cosh x

function asinh (x : real) : real = log (x + sqrt (sqr x + 1.0))
function acosh (x : real) : real
axiom Acosh_def:
forall x:real. x >= 1.0 -> acosh x = log (x + sqrt (sqr x - 1.0))
function atanh (x : real) : real
axiom Atanh_def:
forall x:real. -1.0 < x < 1.0 -> atanh x = 0.5 * log ((1.0+x)/(1.0-x))

end

```

## Polar Coordinates

```module Polar

use Real
use Square
use Trigonometry

function hypot (x y : real) : real = sqrt (sqr x + sqr y)
function atan2 real real : real

axiom X_from_polar:
forall x y:real. x = hypot x y * cos (atan2 y x)

axiom Y_from_polar:
forall x y:real. y = hypot x y * sin (atan2 y x)

end
```

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