Number theory · hard

Statement, exactly as the model and Lean see it

theorem lg_target (n : ℕ) : ¬ 3 ∣ n ^ 2 + 1 := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 12

Traces

Each configuration's full record: what was retrieved, what the model wrote, what Lean said back.

Not verified1 model calls · 1 Lean checks · $0.00016

Attempts

sample 0 · round 0Lean 3 syntax6.4 s model time

Proof

intro h
have h1 : n ^ 2 % 3 = 0 ∨ n ^ 2 % 3 = 1 ∨ n ^ 2 % 3 = 2 := mod_self_three
cases h1 with a
· rw [a] at h
  rw [mod_eq_iff_div_eq_floor] at h
  rcases h with ⟨k, rfl⟩
  have h2 : (n ^ 2 + 1) % 3 = 1 := mod_add_right h1
  rw [h2] at h
  exact h
· rw [a] at h
  rw [mod_eq_iff_div_eq_floor] at h
  rcases h with ⟨k, rfl⟩
  have h2 : (n ^ 2 + 1) % 3 = 2 := mod_add_right h1
  rw [h2] at h
  exact h
· rw [a] at h
  rw [mod_eq_iff_div_eq_floor] at h
  rcases h with ⟨k, rfl⟩
  have h2 : (n ^ 2 + 1) % 3 = 0 := mod_add_right h1
  rw [h2] at h
  exact h

What Lean printed (this is all the model sees on repair)

6:17: error: unknown tactic
5:61: error: Unknown identifier `mod_self_three`
6:11: error: Invalid syntax for case analysis tactic: The `with` keyword must be followed by a tactic or by an alternative (e.g. `| zero =>`), but here it is followed by the identifier `a`.

Reference proof

The proof we wrote and certified before any model ran.

intro h
have hn : n % 3 < 3 := Nat.mod_lt _ (by norm_num)
interval_cases hr : n % 3 <;> simp [Nat.dvd_iff_mod_eq_zero, Nat.add_mod, Nat.pow_mod, hr] at h