1. FormalConjectures
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Erdős Problem 42: Maximal Sidon Sets and Disjoint Difference Sets

Reference: erdosproblems.com/42

This problem asks whether maximal Sidon sets can coexist with other Sidon sets that have disjoint difference sets (apart from 0).

open Function Set Filteropen scoped Pointwise namespace Erdos42

Erdős Problem 42: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every maximal Sidon set A ⊆ {1,…,N} there is another Sidon set B ⊆ {1,…,N} of size M such that (A - A) ∩ (B - B) = {0}?

This was proved for all $M$ by GPT 5.5 Pro (prompted by Sandhu), see discussion thread for more details.

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P42/CompactCayley/Proof.lean"] theorem declaration uses 'sorry'erdos_42 : answer(True) ↔ ∀ M ≥ 1, ∀ᶠ N in atTop, ∀ (A : Set ℕ) (_ : IsMaximalSidonSetIn A N), ∃ᵉ (B : Set ℕ), B ⊆ Set.Icc 1 N ∧ IsSidon B ∧ B.ncard = M ∧ ((A - A) ∩ (B - B)) = {0} := ⊢ True ↔ ∀ M ≥ 1, ∀ᶠ (N : ℕ) in atTop, ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0} All goals completed! 🐙

A variant asking for explicit bounds on how large N needs to be in terms of M.

This version provides a constructive function f such that for all M ≥ 1 and N ≥ f(M), every maximal Sidon set A ⊆ {1,…,N} has another Sidon set B ⊆ {1,…,N} of size M with disjoint difference sets (apart from 0).

@[category research open, AMS 5 11] theorem declaration uses 'sorry'erdos_42.variants.constructive : answer(sorry) ↔ ∃ (f : ℕ → ℕ), ∀ (M N : ℕ) (_ : 1 ≤ M) (_ : f M ≤ N), ∀ (A : Set ℕ) (_ : IsMaximalSidonSetIn A N), ∃ᵉ (B : Set ℕ), B ⊆ Set.Icc 1 N ∧ IsSidon B ∧ B.ncard = M ∧ ((A - A) ∩ (B - B)) = {0} := ⊢ True ↔ ∃ f, ∀ (M N : ℕ), 1 ≤ M → f M ≤ N → ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0} All goals completed! 🐙

The set {1, 2, 4} is a maximal Sidon set in {1, ..., 4}.

@[category textbook, AMS 5 11] theorem example_maximal_sidon : IsMaximalSidonSetIn {1, 2, 4} 4 := ⊢ {1, 2, 4}.IsMaximalSidonSetIn 4 ⊢ {1, 2, 4} ⊆ Icc 1 4⊢ IsSidon {1, 2, 4}⊢ ∀ ⦃x : ℕ⦄, x ∈ Icc 1 4 → x ∉ {1, 2, 4} → ¬IsSidon ({1, 2, 4} ∪ {x}) ⊢ {1, 2, 4} ⊆ Icc 1 4 intro x x:ℕhx:x ∈ {1, 2, 4}⊢ x ∈ Icc 1 4 x:ℕhx:x = 1 ∨ x = 2 ∨ x = 4⊢ x ∈ Icc 1 4 ⊢ 1 ∈ Icc 1 4⊢ 2 ∈ Icc 1 4⊢ 4 ∈ Icc 1 4 ⊢ 1 ∈ Icc 1 4⊢ 2 ∈ Icc 1 4⊢ 4 ∈ Icc 1 4 All goals completed! 🐙 ⊢ IsSidon {1, 2, 4} intro i₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}⊢ ∀ j₁ ∈ {1, 2, 4}, ∀ i₂ ∈ {1, 2, 4}, ∀ j₂ ∈ {1, 2, 4}, i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕ⊢ j₁ ∈ {1, 2, 4} → ∀ i₂ ∈ {1, 2, 4}, ∀ j₂ ∈ {1, 2, 4}, i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}⊢ ∀ i₂ ∈ {1, 2, 4}, ∀ j₂ ∈ {1, 2, 4}, i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}i₂:ℕ⊢ i₂ ∈ {1, 2, 4} → ∀ j₂ ∈ {1, 2, 4}, i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}i₂:ℕhi₂:i₂ ∈ {1, 2, 4}⊢ ∀ j₂ ∈ {1, 2, 4}, i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}i₂:ℕhi₂:i₂ ∈ {1, 2, 4}j₂:ℕ⊢ j₂ ∈ {1, 2, 4} → i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}i₂:ℕhi₂:i₂ ∈ {1, 2, 4}j₂:ℕhj₂:j₂ ∈ {1, 2, 4}⊢ i₁ + i₂ = j₁ + j₂ → i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕhi₁:i₁ ∈ {1, 2, 4}j₁:ℕhj₁:j₁ ∈ {1, 2, 4}i₂:ℕhi₂:i₂ ∈ {1, 2, 4}j₂:ℕhj₂:j₂ ∈ {1, 2, 4}hsum:i₁ + i₂ = j₁ + j₂⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ i₁:ℕj₁:ℕi₂:ℕj₂:ℕhsum:i₁ + i₂ = j₁ + j₂hi₁:i₁ = 1 ∨ i₁ = 2 ∨ i₁ = 4hj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4⊢ i₁ = j₁ ∧ i₂ = j₂ ∨ i₁ = j₂ ∧ i₂ = j₁ j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + i₂ = j₁ + j₂⊢ 1 = j₁ ∧ i₂ = j₂ ∨ 1 = j₂ ∧ i₂ = j₁j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + i₂ = j₁ + j₂⊢ 2 = j₁ ∧ i₂ = j₂ ∨ 2 = j₂ ∧ i₂ = j₁j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = j₁ + j₂⊢ 4 = j₁ ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = j₁ j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + i₂ = j₁ + j₂⊢ 1 = j₁ ∧ i₂ = j₂ ∨ 1 = j₂ ∧ i₂ = j₁j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + i₂ = j₁ + j₂⊢ 2 = j₁ ∧ i₂ = j₂ ∨ 2 = j₂ ∧ i₂ = j₁j₁:ℕi₂:ℕj₂:ℕhj₁:j₁ = 1 ∨ j₁ = 2 ∨ j₁ = 4hi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = j₁ + j₂⊢ 4 = j₁ ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = j₁ i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 1 + j₂⊢ 4 = 1 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 1i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 2 + j₂⊢ 4 = 2 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 2i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 4 + j₂⊢ 4 = 4 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 4 i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + i₂ = 1 + j₂⊢ 1 = 1 ∧ i₂ = j₂ ∨ 1 = j₂ ∧ i₂ = 1i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + i₂ = 2 + j₂⊢ 1 = 2 ∧ i₂ = j₂ ∨ 1 = j₂ ∧ i₂ = 2i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + i₂ = 4 + j₂⊢ 1 = 4 ∧ i₂ = j₂ ∨ 1 = j₂ ∧ i₂ = 4i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + i₂ = 1 + j₂⊢ 2 = 1 ∧ i₂ = j₂ ∨ 2 = j₂ ∧ i₂ = 1i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + i₂ = 2 + j₂⊢ 2 = 2 ∧ i₂ = j₂ ∨ 2 = j₂ ∧ i₂ = 2i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + i₂ = 4 + j₂⊢ 2 = 4 ∧ i₂ = j₂ ∨ 2 = j₂ ∧ i₂ = 4i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 1 + j₂⊢ 4 = 1 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 1i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 2 + j₂⊢ 4 = 2 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 2i₂:ℕj₂:ℕhi₂:i₂ = 1 ∨ i₂ = 2 ∨ i₂ = 4hj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + i₂ = 4 + j₂⊢ 4 = 4 ∧ i₂ = j₂ ∨ 4 = j₂ ∧ i₂ = 4 j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 1 = 4 + j₂⊢ 4 = 4 ∧ 1 = j₂ ∨ 4 = j₂ ∧ 1 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 2 = 4 + j₂⊢ 4 = 4 ∧ 2 = j₂ ∨ 4 = j₂ ∧ 2 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 4 = 4 + j₂⊢ 4 = 4 ∧ 4 = j₂ ∨ 4 = j₂ ∧ 4 = 4 j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 1 = 1 + j₂⊢ 1 = 1 ∧ 1 = j₂ ∨ 1 = j₂ ∧ 1 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 2 = 1 + j₂⊢ 1 = 1 ∧ 2 = j₂ ∨ 1 = j₂ ∧ 2 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 4 = 1 + j₂⊢ 1 = 1 ∧ 4 = j₂ ∨ 1 = j₂ ∧ 4 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 1 = 2 + j₂⊢ 1 = 2 ∧ 1 = j₂ ∨ 1 = j₂ ∧ 1 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 2 = 2 + j₂⊢ 1 = 2 ∧ 2 = j₂ ∨ 1 = j₂ ∧ 2 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 4 = 2 + j₂⊢ 1 = 2 ∧ 4 = j₂ ∨ 1 = j₂ ∧ 4 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 1 = 4 + j₂⊢ 1 = 4 ∧ 1 = j₂ ∨ 1 = j₂ ∧ 1 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 2 = 4 + j₂⊢ 1 = 4 ∧ 2 = j₂ ∨ 1 = j₂ ∧ 2 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:1 + 4 = 4 + j₂⊢ 1 = 4 ∧ 4 = j₂ ∨ 1 = j₂ ∧ 4 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 1 = 1 + j₂⊢ 2 = 1 ∧ 1 = j₂ ∨ 2 = j₂ ∧ 1 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 2 = 1 + j₂⊢ 2 = 1 ∧ 2 = j₂ ∨ 2 = j₂ ∧ 2 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 4 = 1 + j₂⊢ 2 = 1 ∧ 4 = j₂ ∨ 2 = j₂ ∧ 4 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 1 = 2 + j₂⊢ 2 = 2 ∧ 1 = j₂ ∨ 2 = j₂ ∧ 1 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 2 = 2 + j₂⊢ 2 = 2 ∧ 2 = j₂ ∨ 2 = j₂ ∧ 2 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 4 = 2 + j₂⊢ 2 = 2 ∧ 4 = j₂ ∨ 2 = j₂ ∧ 4 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 1 = 4 + j₂⊢ 2 = 4 ∧ 1 = j₂ ∨ 2 = j₂ ∧ 1 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 2 = 4 + j₂⊢ 2 = 4 ∧ 2 = j₂ ∨ 2 = j₂ ∧ 2 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:2 + 4 = 4 + j₂⊢ 2 = 4 ∧ 4 = j₂ ∨ 2 = j₂ ∧ 4 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 1 = 1 + j₂⊢ 4 = 1 ∧ 1 = j₂ ∨ 4 = j₂ ∧ 1 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 2 = 1 + j₂⊢ 4 = 1 ∧ 2 = j₂ ∨ 4 = j₂ ∧ 2 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 4 = 1 + j₂⊢ 4 = 1 ∧ 4 = j₂ ∨ 4 = j₂ ∧ 4 = 1j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 1 = 2 + j₂⊢ 4 = 2 ∧ 1 = j₂ ∨ 4 = j₂ ∧ 1 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 2 = 2 + j₂⊢ 4 = 2 ∧ 2 = j₂ ∨ 4 = j₂ ∧ 2 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 4 = 2 + j₂⊢ 4 = 2 ∧ 4 = j₂ ∨ 4 = j₂ ∧ 4 = 2j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 1 = 4 + j₂⊢ 4 = 4 ∧ 1 = j₂ ∨ 4 = j₂ ∧ 1 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 2 = 4 + j₂⊢ 4 = 4 ∧ 2 = j₂ ∨ 4 = j₂ ∧ 2 = 4j₂:ℕhj₂:j₂ = 1 ∨ j₂ = 2 ∨ j₂ = 4hsum:4 + 4 = 4 + j₂⊢ 4 = 4 ∧ 4 = j₂ ∨ 4 = j₂ ∧ 4 = 4 hsum:4 + 4 = 4 + 1⊢ 4 = 4 ∧ 4 = 1 ∨ 4 = 1 ∧ 4 = 4hsum:4 + 4 = 4 + 2⊢ 4 = 4 ∧ 4 = 2 ∨ 4 = 2 ∧ 4 = 4hsum:4 + 4 = 4 + 4⊢ 4 = 4 ∧ 4 = 4 ∨ 4 = 4 ∧ 4 = 4 hsum:1 + 1 = 1 + 1⊢ 1 = 1 ∧ 1 = 1 ∨ 1 = 1 ∧ 1 = 1hsum:1 + 1 = 1 + 2⊢ 1 = 1 ∧ 1 = 2 ∨ 1 = 2 ∧ 1 = 1hsum:1 + 1 = 1 + 4⊢ 1 = 1 ∧ 1 = 4 ∨ 1 = 4 ∧ 1 = 1hsum:1 + 2 = 1 + 1⊢ 1 = 1 ∧ 2 = 1 ∨ 1 = 1 ∧ 2 = 1hsum:1 + 2 = 1 + 2⊢ 1 = 1 ∧ 2 = 2 ∨ 1 = 2 ∧ 2 = 1hsum:1 + 2 = 1 + 4⊢ 1 = 1 ∧ 2 = 4 ∨ 1 = 4 ∧ 2 = 1hsum:1 + 4 = 1 + 1⊢ 1 = 1 ∧ 4 = 1 ∨ 1 = 1 ∧ 4 = 1hsum:1 + 4 = 1 + 2⊢ 1 = 1 ∧ 4 = 2 ∨ 1 = 2 ∧ 4 = 1hsum:1 + 4 = 1 + 4⊢ 1 = 1 ∧ 4 = 4 ∨ 1 = 4 ∧ 4 = 1hsum:1 + 1 = 2 + 1⊢ 1 = 2 ∧ 1 = 1 ∨ 1 = 1 ∧ 1 = 2hsum:1 + 1 = 2 + 2⊢ 1 = 2 ∧ 1 = 2 ∨ 1 = 2 ∧ 1 = 2hsum:1 + 1 = 2 + 4⊢ 1 = 2 ∧ 1 = 4 ∨ 1 = 4 ∧ 1 = 2hsum:1 + 2 = 2 + 1⊢ 1 = 2 ∧ 2 = 1 ∨ 1 = 1 ∧ 2 = 2hsum:1 + 2 = 2 + 2⊢ 1 = 2 ∧ 2 = 2 ∨ 1 = 2 ∧ 2 = 2hsum:1 + 2 = 2 + 4⊢ 1 = 2 ∧ 2 = 4 ∨ 1 = 4 ∧ 2 = 2hsum:1 + 4 = 2 + 1⊢ 1 = 2 ∧ 4 = 1 ∨ 1 = 1 ∧ 4 = 2hsum:1 + 4 = 2 + 2⊢ 1 = 2 ∧ 4 = 2 ∨ 1 = 2 ∧ 4 = 2hsum:1 + 4 = 2 + 4⊢ 1 = 2 ∧ 4 = 4 ∨ 1 = 4 ∧ 4 = 2hsum:1 + 1 = 4 + 1⊢ 1 = 4 ∧ 1 = 1 ∨ 1 = 1 ∧ 1 = 4hsum:1 + 1 = 4 + 2⊢ 1 = 4 ∧ 1 = 2 ∨ 1 = 2 ∧ 1 = 4hsum:1 + 1 = 4 + 4⊢ 1 = 4 ∧ 1 = 4 ∨ 1 = 4 ∧ 1 = 4hsum:1 + 2 = 4 + 1⊢ 1 = 4 ∧ 2 = 1 ∨ 1 = 1 ∧ 2 = 4hsum:1 + 2 = 4 + 2⊢ 1 = 4 ∧ 2 = 2 ∨ 1 = 2 ∧ 2 = 4hsum:1 + 2 = 4 + 4⊢ 1 = 4 ∧ 2 = 4 ∨ 1 = 4 ∧ 2 = 4hsum:1 + 4 = 4 + 1⊢ 1 = 4 ∧ 4 = 1 ∨ 1 = 1 ∧ 4 = 4hsum:1 + 4 = 4 + 2⊢ 1 = 4 ∧ 4 = 2 ∨ 1 = 2 ∧ 4 = 4hsum:1 + 4 = 4 + 4⊢ 1 = 4 ∧ 4 = 4 ∨ 1 = 4 ∧ 4 = 4hsum:2 + 1 = 1 + 1⊢ 2 = 1 ∧ 1 = 1 ∨ 2 = 1 ∧ 1 = 1hsum:2 + 1 = 1 + 2⊢ 2 = 1 ∧ 1 = 2 ∨ 2 = 2 ∧ 1 = 1hsum:2 + 1 = 1 + 4⊢ 2 = 1 ∧ 1 = 4 ∨ 2 = 4 ∧ 1 = 1hsum:2 + 2 = 1 + 1⊢ 2 = 1 ∧ 2 = 1 ∨ 2 = 1 ∧ 2 = 1hsum:2 + 2 = 1 + 2⊢ 2 = 1 ∧ 2 = 2 ∨ 2 = 2 ∧ 2 = 1hsum:2 + 2 = 1 + 4⊢ 2 = 1 ∧ 2 = 4 ∨ 2 = 4 ∧ 2 = 1hsum:2 + 4 = 1 + 1⊢ 2 = 1 ∧ 4 = 1 ∨ 2 = 1 ∧ 4 = 1hsum:2 + 4 = 1 + 2⊢ 2 = 1 ∧ 4 = 2 ∨ 2 = 2 ∧ 4 = 1hsum:2 + 4 = 1 + 4⊢ 2 = 1 ∧ 4 = 4 ∨ 2 = 4 ∧ 4 = 1hsum:2 + 1 = 2 + 1⊢ 2 = 2 ∧ 1 = 1 ∨ 2 = 1 ∧ 1 = 2hsum:2 + 1 = 2 + 2⊢ 2 = 2 ∧ 1 = 2 ∨ 2 = 2 ∧ 1 = 2hsum:2 + 1 = 2 + 4⊢ 2 = 2 ∧ 1 = 4 ∨ 2 = 4 ∧ 1 = 2hsum:2 + 2 = 2 + 1⊢ 2 = 2 ∧ 2 = 1 ∨ 2 = 1 ∧ 2 = 2hsum:2 + 2 = 2 + 2⊢ 2 = 2 ∧ 2 = 2 ∨ 2 = 2 ∧ 2 = 2hsum:2 + 2 = 2 + 4⊢ 2 = 2 ∧ 2 = 4 ∨ 2 = 4 ∧ 2 = 2hsum:2 + 4 = 2 + 1⊢ 2 = 2 ∧ 4 = 1 ∨ 2 = 1 ∧ 4 = 2hsum:2 + 4 = 2 + 2⊢ 2 = 2 ∧ 4 = 2 ∨ 2 = 2 ∧ 4 = 2hsum:2 + 4 = 2 + 4⊢ 2 = 2 ∧ 4 = 4 ∨ 2 = 4 ∧ 4 = 2hsum:2 + 1 = 4 + 1⊢ 2 = 4 ∧ 1 = 1 ∨ 2 = 1 ∧ 1 = 4hsum:2 + 1 = 4 + 2⊢ 2 = 4 ∧ 1 = 2 ∨ 2 = 2 ∧ 1 = 4hsum:2 + 1 = 4 + 4⊢ 2 = 4 ∧ 1 = 4 ∨ 2 = 4 ∧ 1 = 4hsum:2 + 2 = 4 + 1⊢ 2 = 4 ∧ 2 = 1 ∨ 2 = 1 ∧ 2 = 4hsum:2 + 2 = 4 + 2⊢ 2 = 4 ∧ 2 = 2 ∨ 2 = 2 ∧ 2 = 4hsum:2 + 2 = 4 + 4⊢ 2 = 4 ∧ 2 = 4 ∨ 2 = 4 ∧ 2 = 4hsum:2 + 4 = 4 + 1⊢ 2 = 4 ∧ 4 = 1 ∨ 2 = 1 ∧ 4 = 4hsum:2 + 4 = 4 + 2⊢ 2 = 4 ∧ 4 = 2 ∨ 2 = 2 ∧ 4 = 4hsum:2 + 4 = 4 + 4⊢ 2 = 4 ∧ 4 = 4 ∨ 2 = 4 ∧ 4 = 4hsum:4 + 1 = 1 + 1⊢ 4 = 1 ∧ 1 = 1 ∨ 4 = 1 ∧ 1 = 1hsum:4 + 1 = 1 + 2⊢ 4 = 1 ∧ 1 = 2 ∨ 4 = 2 ∧ 1 = 1hsum:4 + 1 = 1 + 4⊢ 4 = 1 ∧ 1 = 4 ∨ 4 = 4 ∧ 1 = 1hsum:4 + 2 = 1 + 1⊢ 4 = 1 ∧ 2 = 1 ∨ 4 = 1 ∧ 2 = 1hsum:4 + 2 = 1 + 2⊢ 4 = 1 ∧ 2 = 2 ∨ 4 = 2 ∧ 2 = 1hsum:4 + 2 = 1 + 4⊢ 4 = 1 ∧ 2 = 4 ∨ 4 = 4 ∧ 2 = 1hsum:4 + 4 = 1 + 1⊢ 4 = 1 ∧ 4 = 1 ∨ 4 = 1 ∧ 4 = 1hsum:4 + 4 = 1 + 2⊢ 4 = 1 ∧ 4 = 2 ∨ 4 = 2 ∧ 4 = 1hsum:4 + 4 = 1 + 4⊢ 4 = 1 ∧ 4 = 4 ∨ 4 = 4 ∧ 4 = 1hsum:4 + 1 = 2 + 1⊢ 4 = 2 ∧ 1 = 1 ∨ 4 = 1 ∧ 1 = 2hsum:4 + 1 = 2 + 2⊢ 4 = 2 ∧ 1 = 2 ∨ 4 = 2 ∧ 1 = 2hsum:4 + 1 = 2 + 4⊢ 4 = 2 ∧ 1 = 4 ∨ 4 = 4 ∧ 1 = 2hsum:4 + 2 = 2 + 1⊢ 4 = 2 ∧ 2 = 1 ∨ 4 = 1 ∧ 2 = 2hsum:4 + 2 = 2 + 2⊢ 4 = 2 ∧ 2 = 2 ∨ 4 = 2 ∧ 2 = 2hsum:4 + 2 = 2 + 4⊢ 4 = 2 ∧ 2 = 4 ∨ 4 = 4 ∧ 2 = 2hsum:4 + 4 = 2 + 1⊢ 4 = 2 ∧ 4 = 1 ∨ 4 = 1 ∧ 4 = 2hsum:4 + 4 = 2 + 2⊢ 4 = 2 ∧ 4 = 2 ∨ 4 = 2 ∧ 4 = 2hsum:4 + 4 = 2 + 4⊢ 4 = 2 ∧ 4 = 4 ∨ 4 = 4 ∧ 4 = 2hsum:4 + 1 = 4 + 1⊢ 4 = 4 ∧ 1 = 1 ∨ 4 = 1 ∧ 1 = 4hsum:4 + 1 = 4 + 2⊢ 4 = 4 ∧ 1 = 2 ∨ 4 = 2 ∧ 1 = 4hsum:4 + 1 = 4 + 4⊢ 4 = 4 ∧ 1 = 4 ∨ 4 = 4 ∧ 1 = 4hsum:4 + 2 = 4 + 1⊢ 4 = 4 ∧ 2 = 1 ∨ 4 = 1 ∧ 2 = 4hsum:4 + 2 = 4 + 2⊢ 4 = 4 ∧ 2 = 2 ∨ 4 = 2 ∧ 2 = 4hsum:4 + 2 = 4 + 4⊢ 4 = 4 ∧ 2 = 4 ∨ 4 = 4 ∧ 2 = 4hsum:4 + 4 = 4 + 1⊢ 4 = 4 ∧ 4 = 1 ∨ 4 = 1 ∧ 4 = 4hsum:4 + 4 = 4 + 2⊢ 4 = 4 ∧ 4 = 2 ∨ 4 = 2 ∧ 4 = 4hsum:4 + 4 = 4 + 4⊢ 4 = 4 ∧ 4 = 4 ∨ 4 = 4 ∧ 4 = 4 All goals completed! 🐙 ⊢ ∀ ⦃x : ℕ⦄, x ∈ Icc 1 4 → x ∉ {1, 2, 4} → ¬IsSidon ({1, 2, 4} ∪ {x}) intro x x:ℕhx:x ∈ Icc 1 4⊢ x ∉ {1, 2, 4} → ¬IsSidon ({1, 2, 4} ∪ {x}) x:ℕhx:x ∈ Icc 1 4hxA:x ∉ {1, 2, 4}⊢ ¬IsSidon ({1, 2, 4} ∪ {x}) x:ℕhxA:x ∉ {1, 2, 4}hx:1 ≤ x ∧ x ≤ 4⊢ ¬IsSidon ({1, 2, 4} ∪ {x}); x:ℕhxA:x ∉ {1, 2, 4}hx1:1 ≤ xhx2:x ≤ 4⊢ ¬IsSidon ({1, 2, 4} ∪ {x}) x:ℕhx1:1 ≤ xhx2:x ≤ 4hxA:¬x = 1 ∧ ¬x = 2 ∧ ¬x = 4⊢ ¬IsSidon ({1, 2, 4} ∪ {x}) x:ℕhx1:1 ≤ xhx2:x ≤ 4hne1:¬x = 1hne2:¬x = 2hne4:¬x = 4⊢ ¬IsSidon ({1, 2, 4} ∪ {x}) have hx3 : x = 3 := (x:ℕhx1:1 ≤ xhx2:x ≤ 4hne1:¬x = 1hne2:¬x = 2hne4:¬x = 4⊢ x = 3 All goals completed! 🐙); hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4⊢ ¬IsSidon ({1, 2, 4} ∪ {3}) hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ False have := hbad 1 (hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ 1 ∈ {1, 2, 4} ∪ {3} All goals completed! 🐙) 2 (hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ 2 ∈ {1, 2, 4} ∪ {3} All goals completed! 🐙) 3 (hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ 3 ∈ {1, 2, 4} ∪ {3} All goals completed! 🐙) 2 (hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ 2 ∈ {1, 2, 4} ∪ {3} All goals completed! 🐙) (hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})⊢ 1 + 3 = 2 + 2 All goals completed! 🐙) hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ Falsehx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ False hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ Falsehx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ False All goals completed! 🐙

The difference set of {1, 2, 4} is {0, 1, 2, 3}.

@[category textbook, AMS 5 11] theorem example_difference_set : ({1, 2, 4} : Set ℕ) - {1, 2, 4} = {0, 1, 2, 3} := ⊢ {1, 2, 4} - {1, 2, 4} = {0, 1, 2, 3} x:ℕ⊢ x ∈ {1, 2, 4} - {1, 2, 4} ↔ x ∈ {0, 1, 2, 3} x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) ↔ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) → x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3x:ℕ⊢ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 → ∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) → x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 a:ℕha:a = 1 ∨ a = 2 ∨ a = 4b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ a - b = 0 ∨ a - b = 1 ∨ a - b = 2 ∨ a - b = 3 b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 1 - b = 0 ∨ 1 - b = 1 ∨ 1 - b = 2 ∨ 1 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 2 - b = 0 ∨ 2 - b = 1 ∨ 2 - b = 2 ∨ 2 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 4 - b = 0 ∨ 4 - b = 1 ∨ 4 - b = 2 ∨ 4 - b = 3 b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 1 - b = 0 ∨ 1 - b = 1 ∨ 1 - b = 2 ∨ 1 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 2 - b = 0 ∨ 2 - b = 1 ∨ 2 - b = 2 ∨ 2 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 4 - b = 0 ∨ 4 - b = 1 ∨ 4 - b = 2 ∨ 4 - b = 3 ⊢ 4 - 1 = 0 ∨ 4 - 1 = 1 ∨ 4 - 1 = 2 ∨ 4 - 1 = 3⊢ 4 - 2 = 0 ∨ 4 - 2 = 1 ∨ 4 - 2 = 2 ∨ 4 - 2 = 3⊢ 4 - 4 = 0 ∨ 4 - 4 = 1 ∨ 4 - 4 = 2 ∨ 4 - 4 = 3 ⊢ 1 - 1 = 0 ∨ 1 - 1 = 1 ∨ 1 - 1 = 2 ∨ 1 - 1 = 3⊢ 1 - 2 = 0 ∨ 1 - 2 = 1 ∨ 1 - 2 = 2 ∨ 1 - 2 = 3⊢ 1 - 4 = 0 ∨ 1 - 4 = 1 ∨ 1 - 4 = 2 ∨ 1 - 4 = 3⊢ 2 - 1 = 0 ∨ 2 - 1 = 1 ∨ 2 - 1 = 2 ∨ 2 - 1 = 3⊢ 2 - 2 = 0 ∨ 2 - 2 = 1 ∨ 2 - 2 = 2 ∨ 2 - 2 = 3⊢ 2 - 4 = 0 ∨ 2 - 4 = 1 ∨ 2 - 4 = 2 ∨ 2 - 4 = 3⊢ 4 - 1 = 0 ∨ 4 - 1 = 1 ∨ 4 - 1 = 2 ∨ 4 - 1 = 3⊢ 4 - 2 = 0 ∨ 4 - 2 = 1 ∨ 4 - 2 = 2 ∨ 4 - 2 = 3⊢ 4 - 4 = 0 ∨ 4 - 4 = 1 ∨ 4 - 4 = 2 ∨ 4 - 4 = 3 All goals completed! 🐙 x:ℕ⊢ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 → ∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 0⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 1⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 2⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 3 ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 0 exact ⟨1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 1 - 1 = 0 All goals completed! 🐙⟩ ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 1 exact ⟨2, ⊢ 2 = 1 ∨ 2 = 2 ∨ 2 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 2 - 1 = 1 All goals completed! 🐙⟩ ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 2 exact ⟨4, ⊢ 4 = 1 ∨ 4 = 2 ∨ 4 = 4 All goals completed! 🐙, 2, ⊢ 2 = 1 ∨ 2 = 2 ∨ 2 = 4 All goals completed! 🐙, ⊢ 4 - 2 = 2 All goals completed! 🐙⟩ ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 3 exact ⟨4, ⊢ 4 = 1 ∨ 4 = 2 ∨ 4 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 4 - 1 = 3 All goals completed! 🐙⟩

For any maximal Sidon set, the difference set contains 0.

@[category textbook, AMS 5 11] theorem maximal_sidon_contains_zero (A : Set ℕ) (N : ℕ) (hN : 1 ≤ N) (hA : IsMaximalSidonSetIn A N) : 0 ∈ A - A := A:Set ℕN:ℕhN:1 ≤ NhA:A.IsMaximalSidonSetIn N⊢ 0 ∈ A - A A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})⊢ 0 ∈ A - A A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})⊢ A.NonemptyA:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hne:A.Nonempty⊢ 0 ∈ A - A A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})⊢ A.Nonempty A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:¬A.Nonempty⊢ False; A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ False exact hAmax (Set.mem_Icc.mpr ⟨le_refl 1, hN⟩) (A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ 1 ∉ A A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ 1 ∉ ∅; All goals completed! 🐙) (A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ IsSidon (A ∪ {1}) A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ IsSidon {1} exact fun _ hi _ hj _ hk _ hl _ => A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝⁴:ℕhi:x✝⁴ ∈ {1}x✝³:ℕhj:x✝³ ∈ {1}x✝²:ℕhk:x✝² ∈ {1}x✝¹:ℕhl:x✝¹ ∈ {1}x✝:x✝⁴ + x✝² = x✝³ + x✝¹⊢ x✝⁴ = x✝³ ∧ x✝² = x✝¹ ∨ x✝⁴ = x✝¹ ∧ x✝² = x✝³ A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝⁴:ℕx✝³:ℕx✝²:ℕx✝¹:ℕx✝:x✝⁴ + x✝² = x✝³ + x✝¹hi:x✝⁴ = 1hj:x✝³ = 1hk:x✝² = 1hl:x✝¹ = 1⊢ x✝⁴ = x✝³ ∧ x✝² = x✝¹ ∨ x✝⁴ = x✝¹ ∧ x✝² = x✝³ A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝³:ℕx✝²:ℕx✝¹:ℕhj:x✝³ = 1hk:x✝² = 1hl:x✝¹ = 1x✝:1 + x✝² = x✝³ + x✝¹⊢ 1 = x✝³ ∧ x✝² = x✝¹ ∨ 1 = x✝¹ ∧ x✝² = x✝³ A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝²:ℕx✝¹:ℕhk:x✝² = 1hl:x✝¹ = 1x✝:1 + x✝² = 1 + x✝¹⊢ 1 = 1 ∧ x✝² = x✝¹ ∨ 1 = x✝¹ ∧ x✝² = 1 A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝¹:ℕhl:x✝¹ = 1x✝:1 + 1 = 1 + x✝¹⊢ 1 = 1 ∧ 1 = x✝¹ ∨ 1 = x✝¹ ∧ 1 = 1 A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝:1 + 1 = 1 + 1⊢ 1 = 1 ∧ 1 = 1 ∨ 1 = 1 ∧ 1 = 1 All goals completed! 🐙) A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ A⊢ 0 ∈ A - A A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ Athis:a - a ∈ A - A := Set.sub_mem_sub ha ha⊢ 0 ∈ A - A rwa [Nat.sub_selfA:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ Athis:0 ∈ A - A⊢ 0 ∈ A - A at this end Erdos42