1. FormalConjectures
  2. ErdosProblems
  3. «342»
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Erdős Problem 342

References:

open Nat Set Filteropen scoped Topology namespace Erdos342

UniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$ with $i < j < n$.

def UniqueUlamSum (a : ℕ → ℕ) (n m : ℕ) : Prop := ∃! p : ℕ × ℕ, p.1 < p.2 ∧ p.2 < n ∧ m = a p.1 + a p.2

IsUlamSequence a means that $a$ is the Ulam sequence (OEIS A002858): $a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, $a(n)$ is the least integer greater than $a(n-1)$ that has a unique representation as $a(i) + a(j)$ with $i < j < n$.

def IsUlamSequence (a : ℕ → ℕ) : Prop := a 0 = 1 ∧ a 1 = 2 ∧ ∀ n, 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ m, a (n - 1) < m → m < a n → ¬ UniqueUlamSum a n m

$a(0) = 1$ by definition.

@[category test, AMS 5 11 40] theorem erdos_342.test.a0 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 0 = 1 := ⊢ ∀ (a : ℕ → ℕ), IsUlamSequence a → a 0 = 1 intro a a:ℕ → ℕha0:a 0 = 1left✝:a 1 = 2right✝:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n m⊢ a 0 = 1; All goals completed! 🐙

$a(1) = 2$ by definition.

@[category test, AMS 5 11 40] theorem erdos_342.test.a1 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 1 = 2 := ⊢ ∀ (a : ℕ → ℕ), IsUlamSequence a → a 1 = 2 intro a a:ℕ → ℕleft✝:a 0 = 1ha1:a 1 = 2right✝:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n m⊢ a 1 = 2; All goals completed! 🐙

$a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$.

@[category test, AMS 5 11 40] theorem erdos_342.test.a2 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 2 = 3 := ⊢ ∀ (a : ℕ → ℕ), IsUlamSequence a → a 2 = 3 intro a a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n m⊢ a 2 = 3 obtain ⟨_, ⟨⟨i, j⟩, ⟨hij, hj, hsum⟩, _⟩, _⟩ := ha 2 (a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n m⊢ 2 ≤ 2 All goals completed! 🐙) a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mleft✝:a (2 - 1) < a 2right✝¹:∀ (m : ℕ), a (2 - 1) < m → m < a 2 → ¬UniqueUlamSum a 2 mi:ℕj:ℕright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 2 ∧ a 2 = a p.1 + a p.2) y → y = (i, 0)hij:(i, 0).1 < (i, 0).2hj:(i, 0).2 < 2hsum:a 2 = a (i, 0).1 + a (i, 0).2⊢ a 2 = 3a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mleft✝:a (2 - 1) < a 2right✝¹:∀ (m : ℕ), a (2 - 1) < m → m < a 2 → ¬UniqueUlamSum a 2 mi:ℕj:ℕright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 2 ∧ a 2 = a p.1 + a p.2) y → y = (i, 1)hij:(i, 1).1 < (i, 1).2hj:(i, 1).2 < 2hsum:a 2 = a (i, 1).1 + a (i, 1).2⊢ a 2 = 3 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mleft✝:a (2 - 1) < a 2right✝¹:∀ (m : ℕ), a (2 - 1) < m → m < a 2 → ¬UniqueUlamSum a 2 mi:ℕj:ℕright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 2 ∧ a 2 = a p.1 + a p.2) y → y = (i, 0)hij:(i, 0).1 < (i, 0).2hj:(i, 0).2 < 2hsum:a 2 = a (i, 0).1 + a (i, 0).2⊢ a 2 = 3a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mleft✝:a (2 - 1) < a 2right✝¹:∀ (m : ℕ), a (2 - 1) < m → m < a 2 → ¬UniqueUlamSum a 2 mi:ℕj:ℕright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 2 ∧ a 2 = a p.1 + a p.2) y → y = (i, 1)hij:(i, 1).1 < (i, 1).2hj:(i, 1).2 < 2hsum:a 2 = a (i, 1).1 + a (i, 1).2⊢ a 2 = 3 All goals completed! 🐙

$a(3) = 4$: among sums $> 3$ with a unique representation from ${1,2,3}$, the smallest is $4 = 1 + 3$. The candidate $5 = 2 + 3$ is ruled out by minimality since $4$ has a unique representation.

@[category test, AMS 5 11 40] theorem erdos_342.test.a3 : ∀ a : ℕ → ℕ, IsUlamSequence a → a 3 = 4 := ⊢ ∀ (a : ℕ → ℕ), IsUlamSequence a → a 3 = 4 intro a a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n m⊢ a 3 = 4 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩⊢ a 3 = 4 obtain ⟨hinc, ⟨⟨i, j⟩, ⟨hij, hj, hsum⟩, _⟩, hmin⟩ := ha 3 (a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩⊢ 2 ≤ 3 All goals completed! 🐙) a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, j)hij:(i, j).1 < (i, j).2hj:(i, j).2 < 3hsum:a 3 = a (i, j).1 + a (i, j).2hinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 m⊢ a 3 = 4 -- hinc : a 2 < a 3, hmin : ∀ m, a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 m -- hsum : a 3 = a i + a j, hij : i < j, hj : j < 3 -- Enumerate j ∈ {0, 1, 2} a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 0)hij:(i, 0).1 < (i, 0).2hj:(i, 0).2 < 3hsum:a 3 = a (i, 0).1 + a (i, 0).2⊢ a 3 = 4a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 1)hij:(i, 1).1 < (i, 1).2hj:(i, 1).2 < 3hsum:a 3 = a (i, 1).1 + a (i, 1).2⊢ a 3 = 4a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 2)hij:(i, 2).1 < (i, 2).2hj:(i, 2).2 < 3hsum:a 3 = a (i, 2).1 + a (i, 2).2⊢ a 3 = 4 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 0)hij:(i, 0).1 < (i, 0).2hj:(i, 0).2 < 3hsum:a 3 = a (i, 0).1 + a (i, 0).2⊢ a 3 = 4 -- j = 0: i < 0 impossible All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 1)hij:(i, 1).1 < (i, 1).2hj:(i, 1).2 < 3hsum:a 3 = a (i, 1).1 + a (i, 1).2⊢ a 3 = 4 -- j = 1: i = 0, so a 3 = a 0 + a 1 = 1 + 2 = 3, but a 3 > a 2 = 3 have hi : i = 0 := ⊢ ∀ (a : ℕ → ℕ), IsUlamSequence a → a 3 = 4 All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩j:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 1)hij:(0, 1).1 < (0, 1).2hj:(0, 1).2 < 3hsum:a 3 = a (0, 1).1 + a (0, 1).2⊢ a 3 = 4; a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩j:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 1)hij:(0, 1).1 < (0, 1).2hj:(0, 1).2 < 3hsum:a 3 = 1 + 2⊢ a 3 = 4; a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩j:ℕhinc:3 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 1)hij:(0, 1).1 < (0, 1).2hj:(0, 1).2 < 3hsum:a 3 = 1 + 2⊢ a 3 = 4; All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (i, 2)hij:(i, 2).1 < (i, 2).2hj:(i, 2).2 < 3hsum:a 3 = a (i, 2).1 + a (i, 2).2⊢ a 3 = 4 -- j = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 2)hij:(0, 2).1 < (0, 2).2hj:(0, 2).2 < 3hsum:a 3 = a (0, 2).1 + a (0, 2).2⊢ a 3 = 4a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = a (1, 2).1 + a (1, 2).2⊢ a 3 = 4 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 2)hij:(0, 2).1 < (0, 2).2hj:(0, 2).2 < 3hsum:a 3 = a (0, 2).1 + a (0, 2).2⊢ a 3 = 4 -- i = 0: a 3 = a 0 + a 2 = 1 + 3 = 4 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (0, 2)hij:(0, 2).1 < (0, 2).2hj:(0, 2).2 < 3hsum:a 3 = 1 + 3⊢ a 3 = 4; All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = a (1, 2).1 + a (1, 2).2⊢ a 3 = 4 -- i = 1: a 3 = a 1 + a 2 = 2 + 3 = 5 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3⊢ a 3 = 4 -- hsum : a 3 = 5. Use minimality: m = 4 has unique sum, contradiction. a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3⊢ False have h4 := hmin 4 (a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3⊢ a 2 < 4 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3⊢ 3 < 4; All goals completed! 🐙) (a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3⊢ 4 < a 3 All goals completed! 🐙) a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)⊢ UniqueUlamSum a 3 4 -- Goal: UniqueUlamSum a 3 4, i.e. ∃! (p : ℕ × ℕ), p.1 < p.2 ∧ p.2 < 3 ∧ 4 = a p.1 + a p.2 -- Witness: (0, 2) since a 0 + a 2 = 1 + 3 = 4 refine ⟨⟨0, 2⟩, ⟨a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)⊢ (0, 2).1 < (0, 2).2 All goals completed! 🐙, a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)⊢ (0, 2).2 < 3 All goals completed! 🐙, a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)⊢ 4 = a (0, 2).1 + a (0, 2).2 All goals completed! 🐙⟩, ?_⟩ -- Uniqueness: check all pairs (i', j') with i' < j' < 3 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', j').1 < (i', j').2hj':(i', j').2 < 3hsum':4 = a (i', j').1 + a (i', j').2⊢ (i', j') = (0, 2) a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', j').1 < (i', j').2hj':(i', j').2 < 3hsum':4 = a (i', j').1 + a (i', j').2⊢ i' = 0 ∧ j' = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 0).1 < (i', 0).2hj':(i', 0).2 < 3hsum':4 = a (i', 0).1 + a (i', 0).2⊢ i' = 0 ∧ 0 = 2a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 1).1 < (i', 1).2hj':(i', 1).2 < 3hsum':4 = a (i', 1).1 + a (i', 1).2⊢ i' = 0 ∧ 1 = 2a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 2).1 < (i', 2).2hj':(i', 2).2 < 3hsum':4 = a (i', 2).1 + a (i', 2).2⊢ i' = 0 ∧ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 0).1 < (i', 0).2hj':(i', 0).2 < 3hsum':4 = a (i', 0).1 + a (i', 0).2⊢ i' = 0 ∧ 0 = 2 All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 1).1 < (i', 1).2hj':(i', 1).2 < 3hsum':4 = a (i', 1).1 + a (i', 1).2⊢ i' = 0 ∧ 1 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 1).1 < (0, 1).2hj':(0, 1).2 < 3hsum':4 = a (0, 1).1 + a (0, 1).2⊢ 0 = 0 ∧ 1 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 1).1 < (0, 1).2hj':(0, 1).2 < 3hsum':4 = a (0, 1).1 + a (0, 1).2⊢ 0 = 0 ∧ 1 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 1).1 < (0, 1).2hj':(0, 1).2 < 3hsum':4 = 1 + 2⊢ 0 = 0 ∧ 1 = 2; All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(i', 2).1 < (i', 2).2hj':(i', 2).2 < 3hsum':4 = a (i', 2).1 + a (i', 2).2⊢ i' = 0 ∧ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = a (0, 2).1 + a (0, 2).2⊢ 0 = 0 ∧ 2 = 2a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(1, 2).1 < (1, 2).2hj':(1, 2).2 < 3hsum':4 = a (1, 2).1 + a (1, 2).2⊢ 1 = 0 ∧ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = a (0, 2).1 + a (0, 2).2⊢ 0 = 0 ∧ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = 1 + 3⊢ 0 = 0 ∧ 2 = 2; a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = 1 + 3⊢ 0 = 0a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = 1 + 3⊢ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = 1 + 3⊢ 0 = 0a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(0, 2).1 < (0, 2).2hj':(0, 2).2 < 3hsum':4 = 1 + 3⊢ 2 = 2 All goals completed! 🐙 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(1, 2).1 < (1, 2).2hj':(1, 2).2 < 3hsum':4 = a (1, 2).1 + a (1, 2).2⊢ 1 = 0 ∧ 2 = 2 a:ℕ → ℕha0:a 0 = 1ha1:a 1 = 2ha:∀ (n : ℕ), 2 ≤ n → a (n - 1) < a n ∧ UniqueUlamSum a n (a n) ∧ ∀ (m : ℕ), a (n - 1) < m → m < a n → ¬UniqueUlamSum a n mha2:a 2 = 3 := a2 a ⟨ha0, ⟨ha1, ha⟩⟩i:ℕj:ℕhinc:a 2 < a 3hmin:∀ (m : ℕ), a 2 < m → m < a 3 → ¬UniqueUlamSum a 3 mright✝:∀ (y : ℕ × ℕ), (fun p => p.1 < p.2 ∧ p.2 < 3 ∧ a 3 = a p.1 + a p.2) y → y = (1, 2)hij:(1, 2).1 < (1, 2).2hj:(1, 2).2 < 3hsum:a 3 = 2 + 3h4:¬UniqueUlamSum a 3 4 := hmin 4 (Eq.mpr (id (congrArg (fun _a => _a < 4) ha2)) (Decidable.byContradiction fun a_1 => a3._proof_5 a a_1)) (Decidable.byContradiction fun a_1 => a3._proof_6 a hsum a_1)i':ℕj':ℕhij':(1, 2).1 < (1, 2).2hj':(1, 2).2 < 3hsum':4 = 2 + 3⊢ 1 = 0 ∧ 2 = 2; All goals completed! 🐙

Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?

@[category research open, AMS 5 11 40] theorem declaration uses 'sorry'erdos_342.parts.i : answer(sorry) ↔ ∀ a : ℕ → ℕ, IsUlamSequence a → Set.Infinite {n : ℕ | ∃ m, a m = a n + 2} := ⊢ True ↔ ∀ (a : ℕ → ℕ), IsUlamSequence a → {n | ∃ m, a m = a n + 2}.Infinite All goals completed! 🐙

Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?

@[category research open, AMS 5 11 40] theorem declaration uses 'sorry'erdos_342.parts.ii : answer(sorry) ↔ ∀ a : ℕ → ℕ, IsUlamSequence a → let d (n : ℕ) : ℤ := a (n + 1) - a n ∃ p > 0, ∀ᶠ m in atTop, d (m + p) = d m := ⊢ True ↔ ∀ (a : ℕ → ℕ), IsUlamSequence a → let d := fun n => ↑(a (n + 1)) - ↑(a n); ∃ p > 0, ∀ᶠ (m : ℕ) in atTop, d (m + p) = d m All goals completed! 🐙

Part (iii), is the density of the sequence 0?

@[category research open, AMS 5 11 40] theorem declaration uses 'sorry'erdos_342.parts.iii : answer(sorry) ↔ ∀ a : ℕ → ℕ, IsUlamSequence a → Set.upperDensity (Set.range a) = 0 := ⊢ True ↔ ∀ (a : ℕ → ℕ), IsUlamSequence a → (range a).upperDensity = 0 All goals completed! 🐙 end Erdos342