645.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 645

References:

erdosproblems.com/645
[BrLa99] Brown, Tom C. and Landman, Bruce M., Monochromatic arithmetic progressions with large differences. Bull. Austral. Math. Soc. (1999), 21--35.

Imports (1)

FormalConjecturesUtil

theorem Erdos645.erdos_645 If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$ sorry line 44 source
Source documentation
If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$.
This was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible with $d>f(x)$ for any increasing function $f$.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT.

depends on sorry what this means
Statement uses: Nat, Bool, Exists, And, LT.lt, OfNat.ofNat, Eq, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_645 (c : ℕ → Bool) : ∃ x d, 0 < x ∧ x < d ∧
    (∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C) := by
…
```
Mentions: c