PisotNumber.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

Pisot–Vijayaraghavan numbers

A Pisot–Vijayaraghavan number (or PV number) is a real algebraic integer θ > 1 all of whose Galois conjugates other than θ itself lie strictly inside the complex unit disc. Equivalently, every root in ℂ of the minimal polynomial of θ over ℚ, apart from θ, has modulus < 1.

References:

Wikipedia, Pisot–Vijayaraghavan number

Main definitions

IsPisot: a real number is a Pisot–Vijayaraghavan number.

Main statements

isPisot_goldenRatio: the golden ratio is a Pisot number.

Imports (5)

Imported by 1 file

FormalConjecturesForMathlib.lean

def IsPisot Documented declaration: IsPisot line 52 source
Source documentation
A Pisot–Vijayaraghavan number is a real algebraic integer θ > 1 all of whose other conjugates lie strictly inside the complex unit disc: every root in ℂ of the minimal polynomial of θ over ℚ, except θ itself, has modulus < 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: And, LT.lt, OfNat.ofNat, IsIntegral, Int, Complex, Membership.mem, Multiset, Polynomial.aroots, Rat, minpoly, Ne, Norm.norm
18 instances, coercions and auxiliary constants left out.
```
def IsPisot (θ : ℝ) : Prop :=
…
```
theorem isPisot_goldenRatio Documented declaration: isPisot_goldenRatio line 59 source
Source documentation
The golden ratio φ = (1 + √5) / 2 is a Pisot number: it is an algebraic integer greater than 1, its minimal polynomial over ℚ is X ^ 2 - X - 1, and its only other conjugate ψ = (1 - √5) / 2 has modulus (√5 - 1) / 2 < 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsPisot, Real.goldenRatio
Proof uses: And.intro, LT.lt, Real, OfNat.ofNat, And, IsIntegral, Int, Complex, Membership.mem, Multiset, Polynomial.aroots, Rat, minpoly, Ne, Norm.norm, Real.one_lt_goldenRatio, Exists.intro, Polynomial, Polynomial.Monic, Eq and 195 more
193 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
```
theorem isPisot_goldenRatio : IsPisot goldenRatio := by
…
```
Mentions: IsPisot