FormalConjectures/OEIS/111114.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

Integer part of $\mathrm{prime}(n)/\pi(n)$

Here $\mathrm{prime}(n)$ is the $n$-th prime number, and $\pi(n)$ is the prime-counting function.

References:

Imports (1)
  1. def OeisA111114.a a n is the integer part of $\mathrm{prime}(n)/\pi(n)$ line 40 source

    Source documentation

    a n is the integer part of $\mathrm{prime}(n)/\pi(n)$. Here $\mathrm{prime}(n)$ is the $n$-th prime number, and $\pi(n)$ is the prime-counting function. The sequence is defined for $n \ge 2$.

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

    Statement uses: Nat

    Proof uses: HDiv.hDiv, Nat.nth, Nat.Prime, HSub.hSub, OfNat.ofNat, Nat.primeCounting

    5 instances, coercions and auxiliary constants left out.

    noncomputable def a (n : ℕ) : ℕ :=
…
  2. theorem OeisA111114.a_2 line 44 source

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

    Statement uses: Eq, Nat, OeisA111114.a, OfNat.ofNat

    Proof uses: Nat.nth, Nat.Prime, Nat.nth_prime_one_eq_three, id, Eq.mpr, HDiv.hDiv, HSub.hSub, Nat.primeCounting, congrArg, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  3. theorem OeisA111114.a_3 line 51 source

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

    Statement uses: Eq, Nat, OeisA111114.a, OfNat.ofNat

    Proof uses: Nat.nth, Nat.Prime, Nat.nth_prime_two_eq_five, id, Eq.mpr, HDiv.hDiv, HSub.hSub, Nat.primeCounting, congrArg, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  4. theorem OeisA111114.a_4 line 58 source

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

    Statement uses: Eq, Nat, OeisA111114.a, OfNat.ofNat

    Proof uses: Nat.nth, Nat.Prime, Nat.nth_prime_three_eq_seven, id, Eq.mpr, HDiv.hDiv, HSub.hSub, Nat.primeCounting, congrArg, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  5. theorem OeisA111114.a_5 line 65 source

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

    Statement uses: Eq, Nat, OeisA111114.a, OfNat.ofNat

    Proof uses: Nat.nth, Nat.Prime, Nat.nth_prime_four_eq_eleven, id, Eq.mpr, HDiv.hDiv, HSub.hSub, Nat.primeCounting, congrArg, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl

    7 instances, coercions and auxiliary constants left out.

    Tactic trace: 4 steps

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

  6. theorem OeisA111114.conjecture Conjecture: As $n \rightarrow \infty$, there are infinitely many n's such that $a(n)$ is greater than $a(n+1)$ sorry line 78 source

    Source documentation

    Conjecture: As $n \rightarrow \infty$, there are infinitely many n's such that $a(n)$ is greater than $a(n+1)$.

    depends on sorry what this means

    Statement uses: Filter.Frequently, Nat, GT.gt, OeisA111114.a, HAdd.hAdd, OfNat.ofNat, Filter.atTop

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

    10 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

    theorem conjecture : ∃ᶠ n in atTop, a n > a (n + 1) := by
…

    Mentions: OeisA111114.a