MovingSofa.lean`

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

28 declarations, 5 contain sorry, 0 declared as axioms

Moving Sofa Problem

References:

Wikipedia
[Ge92] Gerver, J. L., On moving a sofa around a corner. Geometriae Dedicata 42.3 (1992): 267-283.
[Ro18] Romik, D. Differential equations and exact solutions in the moving sofa problem. Experimental mathematics 27.3 (2018): 316-330.
[Ba24] Baek, J. Optimality of Gerver's Sofa. arXiv preprint arXiv:2411.19826 (2024).

Imports (1)

FormalConjecturesUtil

def MovingSofa.horizontalHallway The horizontal side of the hallway is $(-\infty, 1] \times [0, 1]$ line 40 source
Source documentation
The horizontal side of the hallway is $(-\infty, 1] \times [0, 1]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Set.ofPred, Exists, And, LE.le, Eq, WithLp, ENNReal, WithLp.toLp, Matrix.vecCons, Matrix.vecEmpty
12 instances, coercions and auxiliary constants left out.
```
def horizontalHallway : Set ℝ² := {!₂[x, y] | (x) (y) (_ : x ≤ 1 ∧ 0 ≤ y ∧ y ≤ 1)}
```
def MovingSofa.verticalHallway The vertical side of the hallway is $[0, 1] \times (-\infty, 1]$ line 43 source
Source documentation
The vertical side of the hallway is $[0, 1] \times (-\infty, 1]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Set.ofPred, Exists, And, LE.le, Eq, WithLp, ENNReal, WithLp.toLp, Matrix.vecCons, Matrix.vecEmpty
12 instances, coercions and auxiliary constants left out.
```
def verticalHallway : Set ℝ² := {!₂[x, y] | (x) (y) (_ : 0 ≤ x ∧ x ≤ 1 ∧ y ≤ 1)}
```
def MovingSofa.hallway The hallway is the union of its horizontal and vertical sides line 46 source
Source documentation
The hallway is the union of its horizontal and vertical sides.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: Union.union, MovingSofa.horizontalHallway, MovingSofa.verticalHallway
3 instances, coercions and auxiliary constants left out.
```
def hallway : Set ℝ² := horizontalHallway ∪ verticalHallway
…
```
Mentions: MovingSofa.horizontalHallway, MovingSofa.verticalHallway
instance MovingSofa.(instance at line 50) line 50 source
structure MovingSofa.IsMovingSofa A connected closed set line 59 source
Source documentation
A connected closed set $s$ is a moving sofa according to a rigid motion $m:I\to\mathrm{SE}(2)$, if the sofa is initially in the horizontal side of the hallway and ends up in the vertical side. Here, since $\mathrm{SE}(2)$ is not in Mathlib yet, we use $\mathrm{E}(2)$ and rely on continuity and $m(0) = \mathrm{id}$ to ensure $m$ is in $\mathrm{SE}(2)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, unitInterval, AffineIsometryEquiv, ENNReal
23 instances, coercions and auxiliary constants left out.
```
structure IsMovingSofa (s : Set ℝ²) (m : I → E(2)) : Prop where
…
```
def MovingSofa.unitSquare The unit square line 69 source
Source documentation
The unit square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: parallelepiped, ENNReal, OrthonormalBasis, EuclideanSpace.basisFun
31 instances, coercions and auxiliary constants left out.
```
def unitSquare : Set ℝ² := parallelepiped (EuclideanSpace.basisFun (Fin 2) ℝ)
…
```
lemma MovingSofa.mem_Icc_of_mem_unitSquare Coordinates of points in the unit square lie in [0,1] line 73 source
Source documentation
Coordinates of points in the unit square lie in [0,1].
```
private lemma mem_Icc_of_mem_unitSquare {p : ℝ²} (hp : p ∈ unitSquare) (i : Fin 2) :
    p i ∈ Set.Icc (0:ℝ) 1 := by
…
```
Mentions: MovingSofa.unitSquare
theorem MovingSofa.isMovingSofa_unitSquare The unit square $[0,1]^2$ is a valid moving sofa (with the identity motion) line 87 source
Source documentation
The unit square $[0,1]^2$ is a valid moving sofa (with the identity motion). It sits in the corner where both hallways overlap, so the stationary motion works. This is a sanity check that the IsMovingSofa definition is not vacuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Real, unitInterval, AffineIsometryEquiv, EuclideanSpace, Fin, OfNat.ofNat, Nat, ENNReal, MovingSofa.IsMovingSofa, MovingSofa.unitSquare
Proof uses: Exists.intro, AffineIsometryEquiv.refl, MovingSofa.IsMovingSofa.mk, id, IsConnected, parallelepiped, OrthonormalBasis, EuclideanSpace.basisFun, And.intro, Set.Nonempty, Set.image, Finset.sum, Finset.univ, HSMul.hSMul, Set.Icc, IsPreconnected, Membership.mem, Set, And, Eq and 66 more
196 instances, coercions and auxiliary constants left out.

Tactic trace: 40 steps
Open to load the checked before-and-after goal states.
```
theorem isMovingSofa_unitSquare : ∃ m, IsMovingSofa unitSquare m := by
…
```
Mentions: MovingSofa.IsMovingSofa, MovingSofa.unitSquare
def MovingSofa.rotateTranslate The rigid motion that translates by $p$ and then rotates counterclockwise by $\alpha$ line 120 source
Source documentation
The rigid motion that translates by $p$ and then rotates counterclockwise by $\alpha$. Note that [Ge92] used this definition while [Ro18] used rotation first and then translation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, AffineIsometryEquiv, ENNReal
Proof uses: AffineIsometryEquiv.trans, AffineIsometryEquiv.vaddConst, LinearIsometryEquiv.toAffineIsometryEquiv, Orientation.rotation, EuclideanGeometry.o
57 instances, coercions and auxiliary constants left out.
```
def rotateTranslate (α : Real.Angle) (p : ℝ²) : E(2) :=
…
```
theorem MovingSofa.rotateTranslate_apply rotateTranslate α p sends $q$ to $R_\alpha(q + p)$: the translation is applied first line 126 source
Source documentation
rotateTranslate α p sends $q$ to $R_\alpha(q + p)$: the translation is applied first.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real.Angle, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Eq, AffineIsometryEquiv, ENNReal, MovingSofa.rotateTranslate, LinearIsometryEquiv, RingHom.id, Orientation.rotation, EuclideanGeometry.o, HAdd.hAdd
Proof uses: rfl
73 instances, coercions and auxiliary constants left out.
```
theorem rotateTranslate_apply (α : Real.Angle) (p q : ℝ²) :
    rotateTranslate α p q = EuclideanGeometry.o.rotation α (q + p) := rfl
```
Mentions: MovingSofa.rotateTranslate
def MovingSofa.sofaOfRotateTranslatePath The sofa according to a rotation path line 135 source
Source documentation
The sofa according to a rotation path $p : [0, \pi/2] \to \mathbb{R}^2$ as in [Ge92] is the intersection over $\alpha \in [0, \pi/2]$ of hallways each translated by $p(\alpha)$ and then rotated by $\alpha$, with the special cases that the hallway at $0$ is the horizontal side and the hallway at $\pi/2$ is the vertical side.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, Set
Proof uses: Inter.inter, Set.image, AffineIsometryEquiv, ENNReal, MovingSofa.rotateTranslate, Real.Angle, MovingSofa.horizontalHallway, HDiv.hDiv, Real.pi, MovingSofa.verticalHallway, Set.iInter, Membership.mem, Set.Icc, MovingSofa.hallway
44 instances, coercions and auxiliary constants left out.
```
def sofaOfRotateTranslatePath (p : ℝ → ℝ²) : Set ℝ² :=
…
```
def MovingSofa.GerversSofa.ABφθSpec Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\varphi$, and $\theta$ of [Ge92] line 151 source
Source documentation
Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\varphi$, and $\theta$ of [Ge92].

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: And, LE.le, OfNat.ofNat, HDiv.hDiv, Real.pi, Eq, HAdd.hAdd, HSub.hSub, HMul.hMul, Real.cos, Real.sin, HPow.hPow, Nat
24 instances, coercions and auxiliary constants left out.
```
def ABφθSpec (A B φ θ : ℝ) : Prop :=
…
```
Mentions: MovingSofa.GerversSofa.A, MovingSofa.GerversSofa.B, MovingSofa.GerversSofa.φ, MovingSofa.GerversSofa.θ
theorem MovingSofa.GerversSofa.ABφθSpec.existsUnique sorry line 163 source
depends on sorry what this means
Statement uses: ExistsUnique, Prod, Real, MovingSofa.GerversSofa.ABφθSpec, Prod.fst, Prod.snd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
def MovingSofa.GerversSofa.A line 167 source
depends on sorry what this means
Statement uses: Real
Proof uses: Prod.fst, Prod, Exists.choose, And, MovingSofa.GerversSofa.ABφθSpec, Prod.snd, Eq, MovingSofa.GerversSofa.ABφθSpec.existsUnique
def MovingSofa.GerversSofa.B line 168 source
depends on sorry what this means
Statement uses: Real
Proof uses: Prod.fst, Prod, Prod.snd, Exists.choose, And, MovingSofa.GerversSofa.ABφθSpec, Eq, MovingSofa.GerversSofa.ABφθSpec.existsUnique
def MovingSofa.GerversSofa.φ line 169 source
depends on sorry what this means
Statement uses: Real
Proof uses: Prod.fst, Prod.snd, Prod, Exists.choose, And, MovingSofa.GerversSofa.ABφθSpec, Eq, MovingSofa.GerversSofa.ABφθSpec.existsUnique
def MovingSofa.GerversSofa.θ line 170 source
depends on sorry what this means
Statement uses: Real
Proof uses: Prod.snd, Prod, Exists.choose, And, MovingSofa.GerversSofa.ABφθSpec, Prod.fst, Eq, MovingSofa.GerversSofa.ABφθSpec.existsUnique
def MovingSofa.GerversSofa.r line 172 source
depends on sorry what this means
Statement uses: Real
Proof uses: ite, LE.le, MovingSofa.GerversSofa.φ, HDiv.hDiv, OfNat.ofNat, MovingSofa.GerversSofa.θ, HSub.hSub, HAdd.hAdd, MovingSofa.GerversSofa.A, Real.pi, MovingSofa.GerversSofa.B, HMul.hMul, HPow.hPow, Nat
24 instances, coercions and auxiliary constants left out.
def MovingSofa.GerversSofa.y line 184 source
depends on sorry what this means
Statement uses: Real
Proof uses: intervalIntegral, HMul.hMul, MovingSofa.GerversSofa.r, Real.sin, HSub.hSub, HDiv.hDiv, Real.pi, OfNat.ofNat, MovingSofa.GerversSofa.φ, MeasureTheory.MeasureSpace.volume
16 instances, coercions and auxiliary constants left out.
def MovingSofa.GerversSofa.x line 187 source
depends on sorry what this means
Statement uses: Real
Proof uses: HSub.hSub, OfNat.ofNat, intervalIntegral, HMul.hMul, MovingSofa.GerversSofa.r, Real.cos, HDiv.hDiv, Real.pi, MovingSofa.GerversSofa.φ, MeasureTheory.MeasureSpace.volume
18 instances, coercions and auxiliary constants left out.
def MovingSofa.GerversSofa.p line 190 source
depends on sorry what this means
Statement uses: Real, EuclideanSpace, Fin, OfNat.ofNat, Nat
Proof uses: WithLp.toLp, ENNReal, Matrix.vecCons, ite, LE.le, MovingSofa.GerversSofa.φ, HSub.hSub, Real.cos, HAdd.hAdd, HMul.hMul, MovingSofa.GerversSofa.x, HDiv.hDiv, Real.pi, MovingSofa.GerversSofa.y, Real.sin, Neg.neg, Matrix.vecEmpty
26 instances, coercions and auxiliary constants left out.
def MovingSofa.gerversSofa Gerver's sofa is the sofa according to the rotation path GerversSofa.p line 201 source
Source documentation
Gerver's sofa is the sofa according to the rotation path GerversSofa.p.

depends on sorry what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: MovingSofa.sofaOfRotateTranslatePath, MovingSofa.GerversSofa.p
1 instances, coercions and auxiliary constants left out.
```
def gerversSofa : Set ℝ² :=
…
```
theorem MovingSofa.isMovingSofa_gerversSofa sorry line 208 source
depends on sorry what this means
Statement uses: Exists, Real, unitInterval, AffineIsometryEquiv, EuclideanSpace, Fin, OfNat.ofNat, Nat, ENNReal, MovingSofa.IsMovingSofa, MovingSofa.gerversSofa
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def MovingSofa.sofaConstant The sofa constant is the maximal area of a moving sofa line 215 source
Source documentation
The sofa constant is the maximal area of a moving sofa.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ENNReal
Proof uses: iSup, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Exists, unitInterval, AffineIsometryEquiv, MovingSofa.IsMovingSofa, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume
40 instances, coercions and auxiliary constants left out.
```
def sofaConstant : ℝ≥0∞ := ⨆ (s : Set ℝ²) (_ : ∃ m, IsMovingSofa s m), volume s
…
```
Mentions: MovingSofa.IsMovingSofa
theorem MovingSofa.one_le_sofaConstant The sofa constant is at least 1, as witnessed by the unit square line 219 source
Source documentation
The sofa constant is at least 1, as witnessed by the unit square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, ENNReal, OfNat.ofNat, MovingSofa.sofaConstant
Proof uses: Trans.trans, Eq, MeasureTheory.Measure, EuclideanSpace, Real, Fin, Nat, Set, MeasureTheory.MeasureSpace.volume, MovingSofa.unitSquare, Eq.symm, parallelepiped, OrthonormalBasis, EuclideanSpace.basisFun, OrthonormalBasis.volume_parallelepiped, le_iSup₂, Exists, unitInterval, AffineIsometryEquiv, MovingSofa.IsMovingSofa and 1 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem one_le_sofaConstant : 1 ≤ sofaConstant := by
…
```
Mentions: MovingSofa.sofaConstant
theorem MovingSofa.sofaConstant_eq sorry line 228 source
depends on sorry what this means
Statement uses: Eq, ENNReal, MovingSofa.sofaConstant, MeasureTheory.Measure, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Set, MeasureTheory.MeasureSpace.volume, MovingSofa.gerversSofa
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
94 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem MovingSofa.sofaConstant_eq_volume_gerversSofa sorry line 235 source
depends on sorry what this means
Statement uses: Eq, ENNReal, MovingSofa.sofaConstant, MeasureTheory.Measure, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Set, MeasureTheory.MeasureSpace.volume, MovingSofa.gerversSofa
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
94 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem MovingSofa.volume_eq_sofaConstant_iff_congruent_gerversSofa Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion sorry line 246 source
Source documentation
Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion.
The motion is needed: horizontalHallway is $(-\infty, 1] \times [0, 1]$, so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion. It has the same area, so uniqueness cannot hold on the nose.

depends on sorry what this means
Statement uses: Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Exists, unitInterval, AffineIsometryEquiv, ENNReal, MovingSofa.IsMovingSofa, Iff, Eq, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, MovingSofa.sofaConstant, Set.image, MovingSofa.gerversSofa
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
122 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem volume_eq_sofaConstant_iff_congruent_gerversSofa (s : Set ℝ²)
    (hs : ∃ m, IsMovingSofa s m) :
    volume s = sofaConstant ↔ ∃ g : E(2), s = g '' gerversSofa := by
…
```
Mentions: MovingSofa.IsMovingSofa, MovingSofa.sofaConstant, MovingSofa.gerversSofa