Datasets:

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PAug

/

ProofNetSharp

Like 10

Cross-posted from https://github.com/zhangir-azerbayev/ProofNet/issues/27 (the same statements appear verbatim in ProofNet#).

Summary. While testing a counterexample-search pipeline on ProofNetVerif, we found that two reference formalizations (also present in ProofNet#) are false as stated, each with a one-line counterexample. Both were confirmed by the Lean kernel. Suggested fixes are below; one further label is raised as a question.

1. Artin|exercise_13_4_10

• Informal: "Prove that if a prime integer p has the form 2^r+1, then it actually has the form 2^{2^k}+1."
• Formal reference:

theorem exercise_13_4_10
    {p : ℕ} {hp : Nat.Prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) :
    ∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1 :=

• Why false: p = 2 is prime and 2 = 2^0 + 1, but 2^(2^k) + 1 ≥ 3 for every k, so the conclusion fails.
• Suggested fix: add the hypothesis p ≠ 2 (or Odd p, or r > 0), or state the conclusion as p = 2 ∨ ∃ k, p = 2^(2^k)+1. The textbook statement implicitly means an odd prime.
• Lean check (negation of the closed statement; hp written as an explicit hypothesis, logically the same):

import Mathlib
def ArtinRef : Prop := ∀ {p : ℕ}, Nat.Prime p →
  (∃ r : ℕ, p = 2 ^ r + 1) → ∃ k : ℕ, p = 2 ^ (2 ^ k) + 1

theorem artin_reference_false : ¬ ArtinRef := by
  intro h
  obtain ⟨k, hk⟩ := h (p := 2) (by norm_num) ⟨0, by norm_num⟩
  have h1 : 1 ≤ 2 ^ k := Nat.one_le_two_pow
  have h2 : 2 ≤ 2 ^ (2 ^ k) := by
    calc 2 = 2 ^ (1 : ℕ) := by norm_num
         _ ≤ 2 ^ (2 ^ k) := Nat.pow_le_pow_right (by omega) h1
  omega

2. Ireland-Rosen|exercise_5_13

• Informal: "Show that any prime divisor of x^4 − x^2 + 1 is congruent to 1 modulo 12."
• Formal reference:

theorem exercise_5_13 {p x: ℤ} (hp : Prime p)
  (hpx : p ∣ (x^4 - x^2 + 1)) : p ≡ 1 [ZMOD 12] :=

• Why false: Prime p for p : ℤ also admits negative primes. For x = 2, x^4 − x^2 + 1 = 13, and p = −13 divides 13, but −13 ≡ 11 (mod 12).
• Suggested fix: add (hp0 : 0 < p), or use p : ℕ with Nat.Prime p (natural-number x^4 − x^2 + 1 is fine there since x^4 ≥ x^2).
• Lean check:

import Mathlib
def IrelandRef : Prop := ∀ {p x : ℤ}, Prime p →
  p ∣ (x ^ 4 - x ^ 2 + 1) → p ≡ 1 [ZMOD 12]

theorem ireland_reference_false : ¬ IrelandRef := by
  intro h
  have hp : Prime (-13 : ℤ) := by norm_num [Int.prime_iff_natAbs_prime]
  have hd : (-13 : ℤ) ∣ (2 ^ 4 - 2 ^ 2 + 1) := by norm_num
  have hm := h (p := -13) (x := 2) hp hd
  change (-13 : ℤ) % 12 = 1 % 12 at hm
  norm_num at hm

3. A question on a label: test split, row 379 (Rudin|exercise_1_18b)

Reference: ∀ (x : ℝ), x ≠ 0 → ¬ ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0. The candidate quantifies over an arbitrary [Ring R]:
theorem dummy (R : Type*) [Ring R] {x : R} (hx : x ≠ 0) : ∀ {y : R}, y ≠ 0 → x * y ≠ 0, labelled correct = true. The candidate fails in ZMod 4 (x = y = 2), so it is not logically equivalent to the real-number statement. We understand "correct" may mean "a faithful-enough formalization" rather than equivalence; is this label intended?

Reproducibility

Both snippets compile unchanged on Lean 4.8.0 with Mathlib tag v4.8.0 (commit b5eba595428809e96f3ed113bc7ba776c5f801ac), the version stated on the dataset card; the negations of the two original theorem headers also check there. #print axioms shows only propext, Classical.choice and Quot.sound (no sorryAx). They also compile on Lean 4.24.0 with a recent Mathlib. Dataset revisions inspected: PAug/ProofNetVerif rev 91183e5b; the same two statements appear verbatim in PAug/ProofNetSharp. In the original Lean 3 ProofNet (hoskinson-center/proofnet, test.jsonl) the two statements read the same (nat.prime p, prime p with p x : ℤ), so the issue appears to originate there (by inspection; not machine-checked in Lean 3).

Found by an AI-agent pipeline (certified countermodels), confirmed by the Lean kernel; reviewed by a human before posting.