InscribedSquare.lean`

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

6 declarations, 5 contain sorry, 0 declared as axioms

Inscribed square problem

The inscribed square problem or Toeplitz conjecture asks whether every Jordan curve (i.e. simple close curve in ℝ²) admits an inscribed square, i.e. a square whose vertices all lie on the curve. There are several open and solved variants of this conjecture.

References:

Wikipedia
A Survey on the Square Peg Problem by Benjamin Matschke
arxiv/2005.09193

Imports (1)

FormalConjecturesUtil

structure InscribedSquare.IsRectangle Documented declaration: IsRectangle line 45 source
Source documentation
Four points a b c d in the plane form a rectangle with a opposite to c iff the line segments from a to c and from b to d have both the same length and the same midpoint, acting as the diagonals of the rectangle. We also require the rectangle to be nondegenerate and have a given aspect ratio ratio : ℝ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
```
structure IsRectangle (a b c d : ℝ²) (ratio : ℝ) : Prop where
…
```
Mentions: c
theorem InscribedSquare.inscribed_square_problem Inscribed square problem Does every Jordan curve admit an inscribed square? sorry line 57 source
Source documentation
Inscribed square problem Does every Jordan curve admit an inscribed square?

depends on sorry what this means
Statement uses: Iff, True, Circle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Topology.IsEmbedding, ENNReal, Exists, InscribedSquare.IsRectangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inscribed_square_problem :
    answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ),
      ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1 := by
…
```
Mentions: InscribedSquare.IsRectangle
theorem InscribedSquare.inscribed_rectangle_problem Inscribed rectangle problem Does every Jordan curve admit inscribed rectangles of any given aspect ratio? sorry line 67 source
Source documentation
Inscribed rectangle problem Does every Jordan curve admit inscribed rectangles of any given aspect ratio?

depends on sorry what this means
Statement uses: Iff, True, Circle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Topology.IsEmbedding, ENNReal, GT.gt, Exists, InscribedSquare.IsRectangle
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 inscribed_rectangle_problem :
    answer(sorry) ↔ ∀ (γ : Circle → ℝ²) (hγ : IsEmbedding γ) (r : ℝ) (hr : r > 0),
      ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r := by
…
```
Mentions: InscribedSquare.IsRectangle
theorem InscribedSquare.exists_inscribed_rectangle It is known that every Jordan curve admits at least one inscribed rectangle sorry line 76 source
Source documentation
It is known that every Jordan curve admits at least one inscribed rectangle.

depends on sorry what this means
Statement uses: Circle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Topology.IsEmbedding, ENNReal, Exists, InscribedSquare.IsRectangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_inscribed_rectangle (γ : Circle → ℝ²) (hγ : IsEmbedding γ) :
    ∃ t₁ t₂ t₃ t₄ r, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r := by
…
```
Mentions: InscribedSquare.IsRectangle
theorem InscribedSquare.exists_inscribed_rectangle_of_smooth It is known that every smooth Jordan curve admits inscribed rectangles of all aspect ratios sorry line 86 source
Source documentation
It is known that every smooth Jordan curve admits inscribed rectangles of all aspect ratios. Here a smooth Jordan curve is a smooth embedding of the circle into the plane, i.e. a smooth simple closed curve with nowhere vanishing velocity.

depends on sorry what this means
Statement uses: Circle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Manifold.IsSmoothEmbedding, ENNReal, modelWithCornersSelf, WithLp, ENat, Top.top, GT.gt, Exists, InscribedSquare.IsRectangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
74 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_inscribed_rectangle_of_smooth (γ : Circle → ℝ²)
    (hγ : IsSmoothEmbedding (𝓡 1) (𝓡 2) ∞ γ) (r : ℝ) (hr : r > 0) :
    ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) r := by
…
```
Mentions: InscribedSquare.IsRectangle
theorem InscribedSquare.exists_inscribed_square_of_C2 It is also known that every $C^2$ Jordan curve admits an inscribed square sorry line 97 source
Source documentation
It is also known that every $C^2$ Jordan curve admits an inscribed square. Here a $C^2$ Jordan curve is a $C^2$ embedding of the circle into the plane, i.e. a $C^2$ simple closed curve with nowhere vanishing velocity.

depends on sorry what this means
Statement uses: Circle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Manifold.IsSmoothEmbedding, ENNReal, modelWithCornersSelf, WithLp, WithTop, ENat, Exists, InscribedSquare.IsRectangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
76 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_inscribed_square_of_C2 (γ : Circle → ℝ²)
    (hγ : IsSmoothEmbedding (𝓡 1) (𝓡 2) 2 γ) :
    ∃ t₁ t₂ t₃ t₄, IsRectangle (γ t₁) (γ t₂) (γ t₃) (γ t₄) 1 := by
…
```
Mentions: InscribedSquare.IsRectangle