774.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 774

Reference: erdosproblems.com/774

Imports (1)

FormalConjecturesUtil

def Set.IsDissociated We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$ line 34 source
Source documentation
We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.InjOn, Finset, Finset.sum, Set.ofPred, LE.le
4 instances, coercions and auxiliary constants left out.
```
def Set.IsDissociated (A : Set ℕ) : Prop :=
…
```
def Set.IsProportionatelyDissociated We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$ line 42 source
Source documentation
We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$.
In other words, there is a (global) $c > 0$ such that every finite $B \subset A$ contains a dissociated set of size $\geq c|B|$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Exists, Real, And, GT.gt, OfNat.ofNat, Finset, LE.le, GE.ge, Finset.card, HMul.hMul, Set.IsDissociated
14 instances, coercions and auxiliary constants left out.
```
def Set.IsProportionatelyDissociated (A : Set ℕ) : Prop :=
…
```
theorem Erdos774.erdos_774 Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets? sorry line 51 source
Source documentation
Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Set.Infinite, Set.IsProportionatelyDissociated, Exists, And, Membership.mem, Set.IsDissociated, Set.Finite, Eq, Set.sUnion
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_774 : answer(sorry) ↔ ∀ (A : Set ℕ), A.Infinite → A.IsProportionatelyDissociated →
    ∃ (T : Set (Set ℕ)), (∀ S ∈ T, S.IsDissociated) ∧ T.Finite ∧ A = ⋃₀ T := by
…
```