Basics for the Rational Numbers #

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theorem Rat.ext_iff (x : Rat) (y : Rat) :

x = y ↔ x.num = y.num ∧ x.den = y.den

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theorem Rat.ext (x : Rat) (y : Rat) (num : x.num = y.num) (den : x.den = y.den) :

x = y

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structure Rat :

Type

mk' :: (

num : Int
The numerator of the rational number is an integer.
den : Nat
The denominator of the rational number is a natural number.
den_nz : s.den ≠ 0
The denominator is nonzero.
reduced : Nat.Coprime (Int.natAbs s.num) s.den
The numerator and denominator are coprime: it is in "reduced form".

)

Rational numbers, implemented as a pair of integers num / den such that the denominator is positive and the numerator and denominator are coprime.

Instances For

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instance instDecidableEqRat :

DecidableEq Rat

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instance instInhabitedRat :

Inhabited Rat

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instance instToStringRat :

ToString Rat

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instance instReprRat :

Repr Rat

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theorem Rat.den_pos (self : Rat) :

0 < self.den

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@[inline]

def Rat.maybeNormalize (num : Int) (den : Nat) (g : Nat) (den_nz : den / g ≠ 0) (reduced : Nat.Coprime (Int.natAbs (Int.div num ↑g)) (den / g)) :

Rat

Auxiliary definition for Rat.normalize. Constructs num / den as a rational number, dividing both num and den by g (which is the gcd of the two) if it is not 1.

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theorem Rat.normalize.den_nz {num : Int} {den : Nat} {g : Nat} (den_nz : den ≠ 0) (e : g = Nat.gcd (Int.natAbs num) den) :

den / g ≠ 0

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theorem Rat.normalize.reduced {num : Int} {den : Nat} {g : Nat} (den_nz : den ≠ 0) (e : g = Nat.gcd (Int.natAbs num) den) :

Nat.Coprime (Int.natAbs (Int.div num ↑g)) (den / g)

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@[inline]

def Rat.normalize (num : Int) (den : optParam Nat 1) (den_nz : autoParam (den ≠ 0) _auto✝) :

Rat

Construct a normalized Rat from a numerator and nonzero denominator. This is a "smart constructor" that divides the numerator and denominator by the gcd to ensure that the resulting rational number is normalized.

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def mkRat (num : Int) (den : Nat) :

Rat

Construct a rational number from a numerator and denominator. This is a "smart constructor" that divides the numerator and denominator by the gcd to ensure that the resulting rational number is normalized, and returns zero if den is zero.

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def Rat.ofInt (num : Int) :

Rat

Embedding of Int in the rational numbers.

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instance Rat.instIntCastRat :

IntCast Rat

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instance Rat.instOfNatRat {n : Nat} :

OfNat Rat n

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@[inline]

def Rat.isInt (a : Rat) :

Bool

Is this rational number integral?

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def Rat.divInt :

Int → Int → Rat

Form the quotient n / d where n d : Int.

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def Rat.«term_/._» :

Lean.TrailingParserDescr

Form the quotient n / d where n d : Int.

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@[irreducible]

def Rat.ofScientific (m : Nat) (s : Bool) (e : Nat) :

Rat

Implements "scientific notation" 123.4e-5 for rational numbers. (This definition is @[irreducible] because you don't want to unfold it. Use Rat.ofScientific_def, Rat.ofScientific_true_def, or Rat.ofScientific_false_def instead.)

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instance Rat.instOfScientificRat :

OfScientific Rat

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def Rat.blt (a : Rat) (b : Rat) :

Bool

Rational number strictly less than relation, as a Bool.

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instance Rat.instLTRat :

LT Rat

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instance Rat.instDecidableLtRatInstLTRat (a : Rat) (b : Rat) :

Decidable (a < b)

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instance Rat.instLERat :

LE Rat

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instance Rat.instDecidableLeRatInstLERat (a : Rat) (b : Rat) :

Decidable (a ≤ b)

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@[irreducible]

def Rat.mul (a : Rat) (b : Rat) :

Rat

Multiplication of rational numbers. (This definition is @[irreducible] because you don't want to unfold it. Use Rat.mul_def instead.)

Instances For

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instance Rat.instMulRat :

Mul Rat

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@[irreducible]

def Rat.inv (a : Rat) :

Rat

The inverse of a rational number. Note: inv 0 = 0. (This definition is @[irreducible] because you don't want to unfold it. Use Rat.inv_def instead.)

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def Rat.div :

Rat → Rat → Rat

Division of rational numbers. Note: div a 0 = 0.

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instance Rat.instDivRat :

Div Rat

Division of rational numbers. Note: div a 0 = 0. Written with a separate function Rat.div as a wrapper so that the definition is not unfolded at .instance transparency.

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theorem Rat.add.aux (a : Rat) (b : Rat) {g : Nat} {ad : Nat} {bd : Nat} (hg : g = Nat.gcd a.den b.den) (had : ad = a.den / g) (hbd : bd = b.den / g) :

let den := ad * b.den; let num := a.num * ↑bd + b.num * ↑ad; Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den

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@[irreducible]

def Rat.add (a : Rat) (b : Rat) :

Rat

Addition of rational numbers. (This definition is @[irreducible] because you don't want to unfold it. Use Rat.add_def instead.)

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instance Rat.instAddRat :

Add Rat

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def Rat.neg (a : Rat) :

Rat

Negation of rational numbers.

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instance Rat.instNegRat :

Neg Rat

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theorem Rat.sub.aux (a : Rat) (b : Rat) {g : Nat} {ad : Nat} {bd : Nat} (hg : g = Nat.gcd a.den b.den) (had : ad = a.den / g) (hbd : bd = b.den / g) :

let den := ad * b.den; let num := a.num * ↑bd - b.num * ↑ad; Nat.gcd (Int.natAbs num) g = Nat.gcd (Int.natAbs num) den

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@[irreducible]

def Rat.sub (a : Rat) (b : Rat) :

Rat

Subtraction of rational numbers. (This definition is @[irreducible] because you don't want to unfold it. Use Rat.sub_def instead.)

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instance Rat.instSubRat :

Sub Rat

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def Rat.floor (a : Rat) :

Int

The floor of a rational number a is the largest integer less than or equal to a.

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def Rat.ceil (a : Rat) :

Int

The ceiling of a rational number a is the smallest integer greater than or equal to a.

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