Metric.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

9 declarations, 0 contain sorry, 0 declared as axioms

Imports (8)

Imported by 2 files

def unitDistNum The number of pairs of points of a finite set s in a metric space that are distance 1 apart line 35 source
Source documentation
The number of pairs of points of a finite set s in a metric space that are distance 1 apart.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Nat
Proof uses: Finset.card, Sym2, Finset.filter, Eq, Real, Dist.dist, Prod.fst, Quot.out, Prod, Sym2.Rel, Prod.snd, OfNat.ofNat, Finset.sym2
5 instances, coercions and auxiliary constants left out.
```
noncomputable def unitDistNum (s : Finset X) : ℕ := #{p ∈ s.sym2 | dist p.out.1 p.out.2 = 1}
```
def distanceSet The set of distances determined by a finite set of points in a metric space line 38 source
Source documentation
The set of distances determined by a finite set of points in a metric space.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Real
Proof uses: Finset.image, Prod, Dist.dist, Prod.fst, Prod.snd, Finset.offDiag
3 instances, coercions and auxiliary constants left out.
```
noncomputable def distanceSet (points : Finset X) : Finset ℝ :=
…
```
def distinctDistances Given a finite set of points in a metric space, we define the number of distinct distances between pairs of points line 45 source
Source documentation
Given a finite set of points in a metric space, we define the number of distinct distances between pairs of points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Nat
Proof uses: Finset.card, Real, distanceSet
```
noncomputable def distinctDistances (points : Finset X) : ℕ :=
…
```
def minimalDistinctDistances The minimum number of distinct distances determined by a set of n points in X line 50 source
Source documentation
The minimum number of distinct distances determined by a set of n points in X.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, And, Eq, Finset.card, distinctDistances
1 instances, coercions and auxiliary constants left out.
```
noncomputable def minimalDistinctDistances (n : ℕ) : ℕ :=
…
```
def distanceMultiplicity The multiplicity of the distance d determined by points, that is, the number of unordered pairs of distinct points at distance d apart line 55 source
Source documentation
The multiplicity of the distance d determined by points, that is, the number of unordered pairs of distinct points at distance d apart.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Real, Nat
Proof uses: HDiv.hDiv, Finset.card, Prod, Finset.filter, Eq, Dist.dist, Prod.fst, Prod.snd, Finset.offDiag, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.
```
noncomputable def distanceMultiplicity (points : Finset X) (d : ℝ) : ℕ :=
…
```
def distinctDistancesFrom Given a finite set of points in a metric space, we define the number of distinct distances between a given point and all other points line 61 source
Source documentation
Given a finite set of points in a metric space, we define the number of distinct distances between a given point and all other points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Nat
Proof uses: Finset.card, Real, Finset.image, Dist.dist, Finset.erase, Classical.propDecidable, Eq
3 instances, coercions and auxiliary constants left out.
```
noncomputable def distinctDistancesFrom (points : Finset X) (pt : X) : ℕ :=
…
```
theorem unitDistNum_le_choose_two The number of unit-distance pairs of a finite set of n points is at most $\binom{n}{2}$, the total number of unordered pairs of distinct points line 67 source
Source documentation
The number of unit-distance pairs of a finite set of n points is at most $\binom{n}{2}$, the total number of unordered pairs of distinct points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, LE.le, Nat, unitDistNum, Nat.choose, Finset.card, OfNat.ofNat
Proof uses: Eq.mpr, Sym2, Finset.filter, Eq, Real, Dist.dist, Prod.fst, Quot.out, Prod, Sym2.Rel, Prod.snd, Finset.sym2, id, congrArg, unitDistNum.eq_1, Finset.image, Classical.propDecidable, Function.uncurry, Sym2.mk, Finset.offDiag and 26 more
31 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem unitDistNum_le_choose_two (s : Finset X) : unitDistNum s ≤ (#s).choose 2 := by
…
```
Mentions: unitDistNum
theorem even_card_offDiag_filter_dist_eq The ordered pairs of distinct points of points at distance d come in swapped pairs, so there are evenly many of them line 80 source
Source documentation
The ordered pairs of distinct points of points at distance d come in swapped pairs, so there are evenly many of them.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Real, Even, Nat, Finset.card, Prod, Finset.filter, Eq, Dist.dist, Prod.fst, Prod.snd, Finset.offDiag
Proof uses: Eq.mpr, id, congrArg, rfl, ZMod, OfNat.ofNat, Eq.symm, propext, ZMod.natCast_eq_zero_iff_even, Finset.sum, Finset.sum_involution, Membership.mem, Prod.swap, of_decide_eq_true, HAdd.hAdd, Bool, Decidable.decide, Bool.true, Eq.refl, Ne and 20 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
theorem even_card_offDiag_filter_dist_eq (points : Finset X) (d : ℝ) :
    Even #(points.offDiag.filter fun (pair : X × X) => dist pair.1 pair.2 = d) := by
…
```
theorem sum_distanceMultiplicity The multiplicities of the distances determined by points add up to the number of unordered pairs of distinct points line 99 source
Source documentation
The multiplicities of the distances determined by points add up to the number of unordered pairs of distinct points.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MetricSpace, Finset, Eq, Nat, Finset.sum, Real, distanceSet, distanceMultiplicity, Nat.choose, Finset.card, OfNat.ofNat
Proof uses: id, Finset.image, Prod, Dist.dist, Prod.fst, Prod.snd, Finset.offDiag, Eq.mpr, HDiv.hDiv, Finset.filter, congrArg, Eq.symm, Nat.sum_div, Membership.mem, Iff.mp, Even, Dvd.dvd, even_iff_two_dvd, even_card_offDiag_filter_dist_eq, Finset.card_eq_sum_card_image and 19 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem sum_distanceMultiplicity (points : Finset X) :
    ∑ d ∈ distanceSet points, distanceMultiplicity points d = (#points).choose 2 := by
…
```
Mentions: distanceSet, distanceMultiplicity