FormalConjectures/OEIS/114831.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

6 declarations, 1 contain sorry, 0 declared as axioms

Each term is previous term plus floor of harmonic mean of two previous terms.

$a(1) = 1, a(2) = 2$ and $a(n) = a(n-1) + \lfloor \frac{2 a(n-1) a(n-2)}{a(n-1) + a(n-2)} \rfloor$ for $n \ge 3$.

References:

Imports (1)
  1. def OeisA114831.a The primary defining sequence a line 40 source

    Source documentation

    The primary defining sequence a. Each term is previous term plus floor of harmonic mean of two previous terms.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Nat

    2 instances, coercions and auxiliary constants left out.

    noncomputable def a : ℕ → ℕ
| 0 => 0 -- Dummy value, sequence starts at index 1
| 1 => 1
| 2 => 2
| n + 3 =>
  let an1 : ℕ := a (n + 2)
…
  2. theorem OeisA114831.a_1 line 53 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA114831.a, OfNat.ofNat

    Proof uses: of_eq_true, Eq.trans, True, eq_self, eq_true, True.intro

    2 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem OeisA114831.a_2 line 56 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA114831.a, OfNat.ofNat

    Proof uses: Eq.mpr, id, congrArg, OeisA114831.a.eq_3, Eq.refl

    2 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  4. theorem OeisA114831.a_3 line 59 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA114831.a, OfNat.ofNat

    Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HAdd.hAdd, Int.toNat, Rat.floor, HDiv.hDiv, Rat, HMul.hMul, Int, ite, Rat.den, Rat.num, False, Int.toNat.match_1, ite_congr, congr, Mathlib.Meta.NormNum.IsNat.to_eq and 33 more

    47 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  5. theorem OeisA114831.a_4 line 63 source

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Eq, Nat, OeisA114831.a, OfNat.ofNat

    Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HAdd.hAdd, Int.toNat, Rat.floor, HDiv.hDiv, Rat, HMul.hMul, congr, Int, ite, Rat.den, Rat.num, False, Int.toNat.match_1, ite_congr, Mathlib.Meta.NormNum.IsNat.to_eq and 34 more

    47 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  6. theorem OeisA114831.conjecture3 sorry line 79 source

    depends on sorry what this means

    Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, OeisA114831.a, HAdd.hAdd, OfNat.ofNat, Filter.atTop, nhds, Real.sqrt

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

    30 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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