1071.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1071

References:

erdosproblems.com/1071
[Da85] Danzer, L., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985), 167-177.

Imports (1)

FormalConjecturesUtil

def Erdos1071.SegmentsDisjoint Two segments are disjoint if they only intersect at their endpoints (if at all) line 36 source
Source documentation
Two segments are disjoint if they only intersect at their endpoints (if at all).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Prod, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat
Proof uses: LE.le, Set, Inter.inter, segment, ENNReal, Prod.fst, Prod.snd, Insert.insert, Singleton.singleton
21 instances, coercions and auxiliary constants left out.
```
def SegmentsDisjoint (seg1 seg2 : ℝ² × ℝ²) : Prop :=
…
```
theorem Erdos1071.erdos_1071.parts.i Can a finite set of disjoint unit segments in a unit square be maximal? sorry line 46 source
Source documentation
Can a finite set of disjoint unit segments in a unit square be maximal? Solved affirmatively by [Da85], who gave an explicit construction.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

depends on sorry what this means
Statement uses: Iff, True, Exists, Finset, Prod, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Maximal, And, Membership.mem, Eq, Dist.dist, ENNReal, Prod.fst, Prod.snd, Set, Set.Icc and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1071.parts.i :
    answer(True) ↔ ∃ S : Finset (ℝ² × ℝ²),
      Maximal (fun T : Finset (ℝ² × ℝ²) =>
        (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧
          seg.1 0 ∈ Icc 0 1 ∧ seg.1 1 ∈ Icc 0 1 ∧
          seg.2 0 ∈ Icc 0 1 ∧ seg.2 1 ∈ Icc 0 1) ∧
          (T : Set (ℝ² × ℝ²)).Pairwise SegmentsDisjoint) S := by
…
```
Mentions: Erdos1071.SegmentsDisjoint
theorem Erdos1071.erdos_1071.parts.ii Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite? sorry line 63 source
Source documentation
Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite? Here a unit line segment is a rotated and translated copy of the open interval $(0,1)$, and it is required to be contained in $R$. Solved affirmatively by [Fo99], who gave an explicit construction.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Prod, And, IsOpen, ENNReal, IsConnected, Set.Countable, Set.Infinite, Maximal, Membership.mem, Eq, Dist.dist and 6 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1071.parts.ii :
    answer(True) ↔ ∃ (R : Set ℝ²) (S : Set (ℝ² × ℝ²)),
      IsOpen R ∧ IsConnected R ∧ S.Countable ∧ S.Infinite ∧
      Maximal (fun T : Set (ℝ² × ℝ²) =>
        (∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ openSegment ℝ seg.1 seg.2 ⊆ R) ∧
        T.Pairwise SegmentsDisjoint) S := by
…
```
Mentions: Erdos1071.SegmentsDisjoint