Documentation

Std.Data.Rat.Basic

Basics for the Rational Numbers #

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

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    • One or more equations did not get rendered due to their size.
    instance instReprRat :
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    • One or more equations did not get rendered due to their size.
    theorem Rat.den_pos (self : Rat) :
    0 < self.den
    @[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)) :

    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
      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)
      @[inline]
      def Rat.normalize (num : Int) (den : optParam Nat 1) (den_nz : autoParam (den 0) _auto✝) :

      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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      • One or more equations did not get rendered due to their size.
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        def mkRat (num : Int) (den : Nat) :

        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) :

          Embedding of Int in the rational numbers.

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            instance Rat.instOfNatRat {n : Nat} :
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            • Rat.instOfNatRat = { ofNat := n }
            @[inline]
            def Rat.isInt (a : Rat) :

            Is this rational number integral?

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              def Rat.divInt :
              IntIntRat

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

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

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                  @[irreducible]
                  def Rat.ofScientific (m : Nat) (s : Bool) (e : Nat) :

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

                    Rational number strictly less than relation, as a Bool.

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                    • One or more equations did not get rendered due to their size.
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                      instance Rat.instLTRat :
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                      instance Rat.instLERat :
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                      @[irreducible]
                      def Rat.mul (a : Rat) (b : Rat) :

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

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                        @[irreducible]
                        def Rat.inv (a : 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 :
                          RatRatRat

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

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                            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
                            @[irreducible]
                            def Rat.add (a : Rat) (b : 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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                            • One or more equations did not get rendered due to their size.
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                              def Rat.neg (a : Rat) :

                              Negation of rational numbers.

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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
                                @[irreducible]
                                def Rat.sub (a : Rat) (b : 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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                                • One or more equations did not get rendered due to their size.
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                                  def Rat.floor (a : Rat) :

                                  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) :

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

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                                    • Rat.ceil a = if a.den = 1 then a.num else a.num / a.den + 1
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