FormalConjectures/GreensOpenProblems/44.lean

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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

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

2 declarations, 2 contain sorry, 0 declared as axioms

Green's Open Problem 44

References:

Imports (1)
  1. theorem Green44.green_44 Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$ sorry line 43 source

    Source documentation

    Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\frac{1}{10} N$?

    We interpret "half the residue classes" as $\lfloor p_i / 2 \rfloor$.

    depends on sorry what this means

    Statement uses: Iff, True, Nat, Fin, OfNat.ofNat, Finset, ZMod, Finset.filter, Not, Membership.mem, Finset.Icc, Nat.Prime, StrictMono, LT.lt, HPow.hPow, Eq, Finset.card, HDiv.hDiv, LE.le, HMul.hMul

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    58 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_44 :
    answer(sorry) ↔ ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),
      let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)
…
  2. theorem Green44.green_44.variants.less_than_sqrt The answer is affirmative if the primes are all less than $N^{1/2}$, by the large sieve sorry line 55 source

    Source documentation

    The answer is affirmative if the primes are all less than $N^{1/2}$, by the large sieve. [Gr24]

    depends on sorry what this means

    Statement uses: Nat, Fin, OfNat.ofNat, Finset, ZMod, Finset.filter, Not, Membership.mem, Finset.Icc, Nat.Prime, StrictMono, LT.lt, HPow.hPow, Eq, Finset.card, HDiv.hDiv, LE.le, HMul.hMul

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    58 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem green_44.variants.less_than_sqrt :
    ∀ (N : ℕ) (p : Fin 1000 → ℕ) (A : (i : Fin 1000) → Finset (ZMod (p i))),
      let remaining := (Finset.Icc 1 N).filter (fun x => ∀ i, (x : ZMod (p i)) ∉ A i)
…