PierceBirkhoff.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 3 contain sorry, 0 declared as axioms

Pierce–Birkhoff conjecture

Reference: Wikipedia

The Pierce-Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated in 1956 by Garrett Birkhoff and Richard S. Pierce, though the modern rigorous formulation is due to Melvin Henriksen and John R. Isbell.

The conjecture has been proved for n = 1 and n = 2 by Louis Mahé.

Imports (1)

FormalConjecturesUtil

def PierceBirkhoff.IsSemiAlgebraic Documented declaration: IsSemiAlgebraic line 41 source
Source documentation
A set is semi-algebraic in ℝⁿ if it can be described by a finite union of sets, each defined by finitely many simultaneous multivariate polynomial equations and strict inequalities.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real
Proof uses: Exists, Finset, MvPolynomial, And, Finite, Eq, Set.iUnion, Set.ofPred, Membership.mem, RingHom, Finsupp, MvPolynomial.eval, OfNat.ofNat, GT.gt
16 instances, coercions and auxiliary constants left out.
```
def IsSemiAlgebraic {n : ℕ} (S : Set (Fin n → ℝ)) : Prop :=
…
```
def PierceBirkhoff.IsSemiAlgebraic₁ A set is semi-algebraic in ℝ if it can be described by a finite boolean combination of polynomial equations and inequalities line 50 source
Source documentation
A set is semi-algebraic in ℝ if it can be described by a finite boolean combination of polynomial equations and inequalities.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Exists, Polynomial, And, Finite, Eq, Union.union, Set.iUnion, Set.ofPred, Polynomial.eval, OfNat.ofNat, GT.gt
5 instances, coercions and auxiliary constants left out.
```
def IsSemiAlgebraic₁ (S : Set ℝ) : Prop :=
…
```
def PierceBirkhoff.IsPiecewiseMvPolynomial Documented declaration: IsPiecewiseMvPolynomial line 61 source
Source documentation
A function f : ℝⁿ → ℝ is piecewise polynomial if there exists a finite covering of ℝⁿ by closed semi-algebraic sets such that the restriction of f to each set in the covering is polynomial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Real
Proof uses: Exists, Set, MvPolynomial, And, Finite, IsClosed, PierceBirkhoff.IsSemiAlgebraic, Eq, Set.iUnion, Set.univ, Membership.mem, RingHom, Finsupp, MvPolynomial.eval
16 instances, coercions and auxiliary constants left out.
```
def IsPiecewiseMvPolynomial {n : ℕ} (f : (Fin n → ℝ) → ℝ) : Prop :=
…
```
def PierceBirkhoff.IsPiecewisePolynomial Documented declaration: IsPiecewisePolynomial line 75 source
Source documentation
A function f : ℝ → ℝ is piecewise polynomial if there exists a finite covering of ℝ by closed semi-algebraic sets such that the restriction of f to each set in the covering is polynomial.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: Exists, Set, Polynomial, And, Finite, IsClosed, PierceBirkhoff.IsSemiAlgebraic₁, Eq, Set.iUnion, Set.univ, Membership.mem, Polynomial.eval
5 instances, coercions and auxiliary constants left out.
```
def IsPiecewisePolynomial (f : ℝ → ℝ) : Prop :=
…
```
theorem PierceBirkhoff.pierce_birkhoff_conjecture Documented declaration: pierce_birkhoff_conjecture sorry line 90 source
Source documentation
The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function f : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that f = supᵢ infⱼ(gᵢⱼ).

depends on sorry what this means
Statement uses: Nat, Fin, Real, PierceBirkhoff.IsPiecewiseMvPolynomial, Exists, MvPolynomial, And, Finite, Eq, iSup, iInf, RingHom, Finsupp, MvPolynomial.eval
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pierce_birkhoff_conjecture {n : ℕ} (f : (Fin n → ℝ) → ℝ)
    (hf : IsPiecewiseMvPolynomial f) :
    ∃ (ι κ : Type) (g : ι → κ → MvPolynomial (Fin n) ℝ), Finite ι ∧ Finite κ ∧
      ∀ x, f x = ⨆ i, ⨅ j, MvPolynomial.eval x (g i j) := by
…
```
Mentions: PierceBirkhoff.IsPiecewiseMvPolynomial
theorem PierceBirkhoff.pierce_birkhoff_conjecture_dim_one The Pierce-Birkhoff conjecture holds for n = 1 sorry line 101 source
Source documentation
The Pierce-Birkhoff conjecture holds for n = 1. This was proved by Louis Mahé.

depends on sorry what this means
Statement uses: Real, PierceBirkhoff.IsPiecewisePolynomial, Exists, Polynomial, And, Finite, Eq, iSup, iInf, Polynomial.eval
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pierce_birkhoff_conjecture_dim_one (f : ℝ → ℝ)
    (hf : IsPiecewisePolynomial f) :
    ∃ (ι κ : Type) (g : ι → κ → Polynomial ℝ), Finite ι ∧ Finite κ ∧
      ∀ x, f x = ⨆ i, ⨅ j, Polynomial.eval x (g i j) := by
…
```
Mentions: PierceBirkhoff.IsPiecewisePolynomial
theorem PierceBirkhoff.pierce_birkhoff_conjecture_dim_two The Pierce-Birkhoff conjecture holds for n = 2 sorry line 112 source
Source documentation
The Pierce-Birkhoff conjecture holds for n = 2. This was proved by Louis Mahé.

depends on sorry what this means
Statement uses: Fin, OfNat.ofNat, Nat, Real, PierceBirkhoff.IsPiecewiseMvPolynomial, Exists, MvPolynomial, And, Finite, Eq, iSup, iInf, RingHom, Finsupp, MvPolynomial.eval
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem pierce_birkhoff_conjecture_dim_two
    (f : (Fin 2 → ℝ) → ℝ) (hf : IsPiecewiseMvPolynomial f) :
    ∃ (ι κ : Type) (g : ι → κ → MvPolynomial (Fin 2) ℝ), Finite ι ∧ Finite κ ∧
      ∀ x, f x = ⨆ i, ⨅ j, MvPolynomial.eval x (g i j) := by
…
```
Mentions: PierceBirkhoff.IsPiecewiseMvPolynomial