Inequalities · medium

Statement, exactly as the model and Lean see it

theorem lg_target (x : ℝ) (hx : 0 < x) : x + 1 / x ≥ 2 := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 113

Traces

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

Not verified1 model calls · 1 Lean checks · $0.00007

Attempts

sample 0 · round 0Valid but mathematically wrong3.5 s model time

Proof

have h1 : x + 1 / x = (x^2 + 1) / x := by ring
have h2 : x^2 + 1 ≥ 2 * x := by linarith [hx]
rw [h1]
apply div_le_of_le_mul_right hx
exact h2

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

4:44: info: Try this:
  [apply] ring_nf
  
  The `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.
    
  Note that `ring` works primarily in *commutative* rings. If you have a noncommutative ring, abelian group or module, consider using `noncomm_ring`, `abel` or `module` instead.
4:41: error: unsolved goals
x : ℝ
hx : 0 < x
⊢ x + x⁻¹ = x ^ 2 * x⁻¹ + x⁻¹
5:34: error: linarith failed to find a contradiction
x : ℝ
hx : 0 < x
h1 : x + 1 / x = (x ^ 2 + 1) / x
a✝ : x ^ 2 + 1 < 2 * x
⊢ False
failed
7:8: error: Unknown identifier `div_le_of_le_mul_right`
8:2: error: No goals to be solved

Reference proof

The proof we wrote and certified before any model ran.

have key : x + 1 / x - 2 = (x - 1) ^ 2 / x := by field_simp; ring
have : 0 ≤ (x - 1) ^ 2 / x := by positivity
linarith