FormalConjectures/ErdosProblems/812.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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 812

References:

Imports (1)
  1. theorem Erdos812.erdos_812.parts.i Is it true that $\frac{R(n+1)}{R(n)}\geq 1+c$ for some constant $c>0$, for all large $n$? sorry line 43 source

    Source documentation

    Is it true that $\frac{R(n+1)}{R(n)}\geq 1+c$ for some constant $c>0$, for all large $n$?

    depends on sorry what this means

    Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, GE.ge, HDiv.hDiv, Combinatorics.hypergraphRamsey, HAdd.hAdd, Filter.atTop

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

    32 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_812.parts.i :
    answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in atTop, (R (n + 1) : ℝ) / (R n : ℝ) ≥ 1 + c:= by
…

    Mentions: c

  2. theorem Erdos812.erdos_812.parts.ii Is it true that $R(n+1)-R(n) \gg n^2$? sorry line 51 source

    Source documentation

    Is it true that $R(n+1)-R(n) \gg n^2$?

    depends on sorry what this means

    Statement uses: Iff, True, Asymptotics.IsBigO, Nat, Real, Filter.atTop, HPow.hPow, OfNat.ofNat, HSub.hSub, Combinatorics.hypergraphRamsey, HAdd.hAdd

    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 erdos_812.parts.ii :
    answer(sorry) ↔
      (fun n : ℕ ↦ (R (n + 1) : ℝ) - (R n : ℝ)) ≫ (fun n : ℕ ↦ (n : ℝ) ^ 2) := by
…
  3. theorem Erdos812.erdos_812.variants.lower_bound Burr, Erdős, Faudree, and Schelp [BEFS89] proved that $R(n+1)-R(n) \geq 4n-8$ for all $n\geq 2$ sorry line 60 source

    Source documentation

    Burr, Erdős, Faudree, and Schelp [BEFS89] proved that $R(n+1)-R(n) \geq 4n-8$ for all $n\geq 2$.

    depends on sorry what this means

    Statement uses: Nat, GE.ge, OfNat.ofNat, Int, HSub.hSub, Combinatorics.hypergraphRamsey, HAdd.hAdd, HMul.hMul

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

    24 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem erdos_812.variants.lower_bound :
    ∀ n : ℕ, n ≥ 2 → (R (n + 1) : ℤ) - (R n : ℤ) ≥ 4 * (n : ℤ) - 8 := by
…