File: Mathlib/Data/Real/GoldenRatio.lean

full_name: goldenRatio
code: /-- The golden ratio `φ := (1 + √5)/2`. -/
abbrev goldenRatio : ℝ 
goal: ℝ

full_name: goldenConj
code: /-- The conjugate of the golden ratio `ψ := (1 - √5)/2`. -/
abbrev goldenConj : ℝ 
goal: ℝ

full_name: goldenRatio._aux_Mathlib_Data_Real_GoldenRatio___unexpand_goldenRatio_1
code: @[inherit_doc goldenRatio] scoped[goldenRatio] notation "φ" => goldenRatio
goal: Lean.PrettyPrinter.Unexpander

full_name: goldenRatio._aux_Mathlib_Data_Real_GoldenRatio___macroRules_goldenRatio_termφ_1
code: @[inherit_doc goldenRatio] scoped[goldenRatio] notation "φ" => goldenRatio
goal: Lean.Macro

full_name: goldenRatio._aux_Mathlib_Data_Real_GoldenRatio___unexpand_goldenConj_1
code: @[inherit_doc goldenConj] scoped[goldenRatio] notation "ψ" => goldenConj
goal: Lean.PrettyPrinter.Unexpander

full_name: goldenRatio._aux_Mathlib_Data_Real_GoldenRatio___macroRules_goldenRatio_termψ_1
code: @[inherit_doc goldenConj] scoped[goldenRatio] notation "ψ" => goldenConj
goal: Lean.Macro

full_name: inv_gold
code: /-- The inverse of the golden ratio is the opposite of its conjugate. -/
theorem inv_gold : φ⁻¹ = -ψ 
goal: φ⁻¹ = -ψ

full_name: inv_goldConj
code: /-- The opposite of the golden ratio is the inverse of its conjugate. -/
theorem inv_goldConj : ψ⁻¹ = -φ 
goal: ψ⁻¹ = -φ

full_name: gold_mul_goldConj
code: @[simp]
theorem gold_mul_goldConj : φ * ψ = -1 
goal: φ * ψ = -1

full_name: goldConj_mul_gold
code: @[simp]
theorem goldConj_mul_gold : ψ * φ = -1 
goal: ψ * φ = -1

full_name: gold_add_goldConj
code: @[simp]
theorem gold_add_goldConj : φ + ψ = 1 
goal: φ + ψ = 1

full_name: one_sub_goldConj
code: theorem one_sub_goldConj : 1 - φ = ψ 
goal: 1 - φ = ψ

full_name: one_sub_gold
code: theorem one_sub_gold : 1 - ψ = φ 
goal: 1 - ψ = φ

full_name: gold_sub_goldConj
code: @[simp]
theorem gold_sub_goldConj : φ - ψ = √5 
goal: φ - ψ = √5

full_name: gold_pow_sub_gold_pow
code: theorem gold_pow_sub_gold_pow (n : ℕ) : φ ^ (n + 2) - φ ^ (n + 1) = φ ^ n 
goal: φ ^ (n + 2) - φ ^ (n + 1) = φ ^ n

full_name: gold_sq
code: @[simp 1200]
theorem gold_sq : φ ^ 2 = φ + 1 
goal: φ ^ 2 = φ + 1

full_name: goldConj_sq
code: @[simp 1200]
theorem goldConj_sq : ψ ^ 2 = ψ + 1 
goal: ψ ^ 2 = ψ + 1

full_name: gold_pos
code: theorem gold_pos : 0 < φ 
goal: 0 < φ

full_name: gold_ne_zero
code: theorem gold_ne_zero : φ ≠ 0 
goal: φ ≠ 0

full_name: one_lt_gold
code: theorem one_lt_gold : 1 < φ 
goal: 1 < φ

full_name: gold_lt_two
code: theorem gold_lt_two : φ < 2 
goal: φ < 2

full_name: goldConj_neg
code: theorem goldConj_neg : ψ < 0 
goal: ψ < 0

full_name: goldConj_ne_zero
code: theorem goldConj_ne_zero : ψ ≠ 0 
goal: ψ ≠ 0

full_name: neg_one_lt_goldConj
code: theorem neg_one_lt_goldConj : -1 < ψ 
goal: -1 < ψ

full_name: gold_irrational
code: /-- The golden ratio is irrational. -/
theorem gold_irrational : Irrational φ 
goal: Irrational φ

full_name: goldConj_irrational
code: /-- The conjugate of the golden ratio is irrational. -/
theorem goldConj_irrational : Irrational ψ 
goal: Irrational ψ

full_name: fibRec
code: /-- The recurrence relation satisfied by the Fibonacci sequence. -/
def fibRec : LinearRecurrence α 
goal: LinearRecurrence α

full_name: fibRec_charPoly_eq
code: /-- The characteristic polynomial of `fibRec` is `X² - (X + 1)`. -/
theorem fibRec_charPoly_eq {β : Type*} [CommRing β] :
    fibRec.charPoly = X ^ 2 - (X + (1 : β[X])) 
goal: fibRec.charPoly = Polynomial.X ^ 2 - (Polynomial.X + 1)

full_name: fib_isSol_fibRec
code: /-- As expected, the Fibonacci sequence is a solution of `fibRec`. -/
theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) 
goal: fibRec.IsSolution fun x => ↑(Nat.fib x)

full_name: geom_gold_isSol_fibRec
code: /-- The geometric sequence `fun n ↦ φ^n` is a solution of `fibRec`. -/
theorem geom_gold_isSol_fibRec : fibRec.IsSolution (φ ^ ·) 
goal: fibRec.IsSolution fun x => φ ^ x

full_name: geom_goldConj_isSol_fibRec
code: /-- The geometric sequence `fun n ↦ ψ^n` is a solution of `fibRec`. -/
theorem geom_goldConj_isSol_fibRec : fibRec.IsSolution (ψ ^ ·) 
goal: fibRec.IsSolution fun x => ψ ^ x

full_name: Real.coe_fib_eq'
code: /-- Binet's formula as a function equality. -/
theorem Real.coe_fib_eq' :
    (fun n => Nat.fib n : ℕ → ℝ) = fun n => (φ ^ n - ψ ^ n) / √5 
goal: (fun n => ↑(Nat.fib n)) = fun n => (φ ^ n - ψ ^ n) / √5

full_name: Real.coe_fib_eq
code: /-- Binet's formula as a dependent equality. -/
theorem Real.coe_fib_eq : ∀ n, (Nat.fib n : ℝ) = (φ ^ n - ψ ^ n) / √5 
goal: ↑(Nat.fib n) = (φ ^ n - ψ ^ n) / √5

full_name: fib_golden_conj_exp
code: /-- Relationship between the Fibonacci Sequence, Golden Ratio and its conjugate's exponents --/
theorem fib_golden_conj_exp (n : ℕ) : Nat.fib (n + 1) - φ * Nat.fib n = ψ ^ n 
goal: ↑(Nat.fib (n + 1)) - φ * ↑(Nat.fib n) = ψ ^ n

full_name: fib_golden_exp'
code: /-- Relationship between the Fibonacci Sequence, Golden Ratio and its exponents --/
theorem fib_golden_exp' (n : ℕ) : φ * Nat.fib (n + 1) + Nat.fib n = φ ^ (n + 1) 
goal: φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)