FormalConjectures/ErdosProblems/1213.lean

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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, 2 contain sorry, 0 declared as axioms

Erdős Problem 1213

References:

Imports (1)
  1. def Erdos1213.HasEqualIntervalSums A finite sequence A 0, …, A (s - 1) has two distinct (nonempty) index intervals [u, v) and [x, y) with the same sum line 36 source

    Source documentation

    A finite sequence A 0, …, A (s - 1) has two distinct (nonempty) index intervals [u, v) and [x, y) with the same sum.

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

    Statement uses: Nat

    Proof uses: Exists, And, LT.lt, LE.le, Ne, Prod, Prod.mk, Eq, Finset.sum, Finset.Ico

    5 instances, coercions and auxiliary constants left out.

    def HasEqualIntervalSums (A : ℕ → ℕ) (s : ℕ) : Prop :=
…
  2. theorem Erdos1213.erdos_1213 sorry line 52 source

    depends on sorry what this means

    Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, LT.lt, Eq, StrictMonoOn, Set.Iio, HAdd.hAdd, HSub.hSub, Erdos1213.HasEqualIntervalSums

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

    16 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

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

  3. theorem Erdos1213.erdos_1213.variants.hegyvari Hegyvári [He86] proved that one can take $f(a,K) \ll ae^{O(K)}$ sorry line 61 source

    Source documentation

    Hegyvári [He86] proved that one can take $f(a,K) \ll ae^{O(K)}$.

    depends on sorry what this means

    Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Eq, StrictMonoOn, Set.Iio, HAdd.hAdd, HSub.hSub, HMul.hMul, Real.exp, Erdos1213.HasEqualIntervalSums

    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.

    theorem erdos_1213.variants.hegyvari : ∃ C : ℝ, 0 < C ∧ ∀ a K : ℕ, 1 ≤ a → 1 ≤ K →
    ∀ (s : ℕ) (A : ℕ → ℕ), 0 < s → A 0 = a → StrictMonoOn A (Set.Iio s) →
      (∀ i, i + 1 < s → A (i + 1) - A i ≤ K) → C * a * Real.exp (C * K) < A (s - 1) →
        HasEqualIntervalSums A s := by
…

    Mentions: Erdos1213.HasEqualIntervalSums