real.mlw

(** {1 Theory of reals}

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

*)

(** {2 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 (<=) = (<=)

  meta "remove_unused:keep" function (+)
  meta "remove_unused:keep" function (-)
(* do not necessarily keep, to allow for linear arithmetic only
  meta "remove_unused:keep" function (*)
  meta "remove_unused:keep" function (/)
*)
  meta "remove_unused:keep" function (-_)
  meta "remove_unused:keep" predicate (<)
  meta "remove_unused:keep" predicate (<=)
  meta "remove_unused:keep" predicate (>)
  meta "remove_unused:keep" predicate (>=)

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

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

(***
  lemma sub_zero: forall x y:real. x - y = 0.0 -> x = y
*)

end

(** {2 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

(** {2 Absolute Value} *)

module Abs

  use Real

  let 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_zero: forall x:real. abs x = 0.0 -> 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

(** {2 Minimum and Maximum} *)

module MinMax

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

end

(** {2 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

  axiom Add:
    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

(** {2 Various truncation functions} *)

module Truncate

  use Real
  use FromInt

  (** rounds towards zero *)
  function truncate real : int

  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

(** {2 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

(** {2 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
  axiom exp_increasing : forall x y. x <= y -> exp(x) <= exp(y)
  axiom exp_positive : forall x. exp(x) > 0.0
  axiom exp_inv : forall x. exp (-x) = inv (exp x)
  lemma exp_sum_opposite: forall x : real. exp(x) + exp(-x) >= 2.0

  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_increasing : forall x y. 0.0 < 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

  lemma log2_increasing : forall x y. 0. < x <= y -> log2(x) <= log2(y)
  lemma log10_increasing : forall x y. 0. < x <= y -> log10(x) <= log10(y)

end

(** {2 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

(** {2 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

(** {2 Trigonometric Functions}

See the {h <a href="http://en.wikipedia.org/wiki/Trigonometric_function">wikipedia page</a>}.

*)

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

(** {2 Hyperbolic Functions}

See the {h <a href="http://en.wikipedia.org/wiki/Hyperbolic_function">wikipedia page</a>}.

*)

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

(** {2 Polar Coordinates}

See the {h <a href="http://en.wikipedia.org/wiki/Atan2">wikipedia page</a>}.

*)

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

module Sum
  use int.Int
  use RealInfix

  let rec ghost function sum (f: int -> real) (a b: int) : real
    variant { b - a }
  = if (b <= a) then 0. else sum f a (b - 1) +. f (b - 1)

end