Documentation

Mathlib.Data.Real.GoldenRatio

The golden ratio and its conjugate #

This file defines the golden ratio φ := (1 + √5)/2 and its conjugate ψ := (1 - √5)/2, which are the two real roots of X² - X - 1.

Along with various computational facts about them, we prove their irrationality, and we link them to the Fibonacci sequence by proving Binet's formula.

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@[reducible]
def goldenRatio :
ℝ

The golden ratio φ := (1 + √5)/2.

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    @[reducible]
    def goldenConj :
    ℝ

    The conjugate of the golden ratio ψ := (1 - √5)/2.

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      def goldenRatio.termφ :
      Lean.ParserDescr

      The golden ratio φ := (1 + √5)/2.

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        def goldenRatio.termψ :
        Lean.ParserDescr

        The conjugate of the golden ratio ψ := (1 - √5)/2.

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          theorem inv_gold :
          goldenRatio⁻¹ = -goldenConj

          The inverse of the golden ratio is the opposite of its conjugate.

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          theorem inv_goldConj :
          goldenConj⁻¹ = -goldenRatio

          The opposite of the golden ratio is the inverse of its conjugate.

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          @[simp]
          theorem gold_mul_goldConj :
          goldenRatio * goldenConj = -1
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          @[simp]
          theorem goldConj_mul_gold :
          goldenConj * goldenRatio = -1
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          @[simp]
          theorem gold_add_goldConj :
          goldenRatio + goldenConj = 1
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          theorem one_sub_goldConj :
          1 - goldenRatio = goldenConj
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          theorem one_sub_gold :
          1 - goldenConj = goldenRatio
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          @[simp]
          theorem gold_sub_goldConj :
          goldenRatio - goldenConj = √5
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          theorem gold_pow_sub_gold_pow (n : ℕ) :
          goldenRatio ^ (n + 2) - goldenRatio ^ (n + 1) = goldenRatio ^ n
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          @[simp]
          theorem gold_sq :
          goldenRatio ^ 2 = goldenRatio + 1
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          @[simp]
          theorem goldConj_sq :
          goldenConj ^ 2 = goldenConj + 1
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          theorem gold_pos :
          0 < goldenRatio
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          theorem gold_ne_zero :
          goldenRatio ≠ 0
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          theorem one_lt_gold :
          1 < goldenRatio
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          theorem gold_lt_two :
          goldenRatio < 2
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          theorem goldConj_neg :
          goldenConj < 0
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          theorem goldConj_ne_zero :
          goldenConj ≠ 0
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          theorem neg_one_lt_goldConj :
          -1 < goldenConj

          Irrationality #

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          theorem gold_irrational :
          Irrational goldenRatio

          The golden ratio is irrational.

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          theorem goldConj_irrational :
          Irrational goldenConj

          The conjugate of the golden ratio is irrational.

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          def fibRec {α : Type u_1} [CommSemiring α] :
          LinearRecurrence α

          The recurrence relation satisfied by the Fibonacci sequence.

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          • fibRec = { order := 2, coeffs := ![1, 1] }
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            theorem fibRec_charPoly_eq {β : Type u_2} [CommRing β] :
            LinearRecurrence.charPoly fibRec = Polynomial.X ^ 2 - (Polynomial.X + 1)

            The characteristic polynomial of fibRec is X² - (X + 1).

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            theorem fib_isSol_fibRec {α : Type u_1} [CommSemiring α] :
            LinearRecurrence.IsSolution fibRec fun (x : ℕ) => ↑(Nat.fib x)

            As expected, the Fibonacci sequence is a solution of fibRec.

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            theorem geom_gold_isSol_fibRec :
            LinearRecurrence.IsSolution fibRec fun (x : ℕ) => goldenRatio ^ x

            The geometric sequence fun n ↦ φ^n is a solution of fibRec.

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            theorem geom_goldConj_isSol_fibRec :
            LinearRecurrence.IsSolution fibRec fun (x : ℕ) => goldenConj ^ x

            The geometric sequence fun n ↦ ψ^n is a solution of fibRec.

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            theorem Real.coe_fib_eq' :
            (fun (n : ℕ) => ↑(Nat.fib n)) = fun (n : ℕ) => (goldenRatio ^ n - goldenConj ^ n) / √5

            Binet's formula as a function equality.

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            theorem Real.coe_fib_eq (n : ℕ) :
            ↑(Nat.fib n) = (goldenRatio ^ n - goldenConj ^ n) / √5

            Binet's formula as a dependent equality.

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            theorem fib_golden_conj_exp (n : ℕ) :
            ↑(Nat.fib (n + 1)) - goldenRatio * ↑(Nat.fib n) = goldenConj ^ n

            Relationship between the Fibonacci Sequence, Golden Ratio and its conjugate's exponents -

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            theorem fib_golden_exp' (n : ℕ) :
            goldenRatio * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = goldenRatio ^ (n + 1)

            Relationship between the Fibonacci Sequence, Golden Ratio and its exponents -