Theorems / novel_alg_06

Authored

Algebra · medium

Statement, exactly as the model and Lean see it

theorem lg_target (x : ℝ) (h : x ^ 2 - 5 * x + 6 = 0) : x = 2 ∨ x = 3 := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 66

Traces

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

Verified0 model calls · 14 Lean checks · $0.00000

Attempts

attempt 1Wrong tactic

Proof

rfl

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

4:2: error: Tactic `rfl` failed: The left-hand side
  x = 2
is not definitionally equal to the right-hand side
  x = 3

x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ x = 2 ∨ x = 3

attempt 2Other

Proof

decide

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

4:2: error: Expected type must not contain free variables
  x = 2 ∨ x = 3

Hint: Use the `+revert` option to automatically clean up and revert free variables

attempt 3Valid but mathematically wrong
Proof
```
norm_num
```
What Lean printed (this is all the model sees on repair)
```
3:73: error: unsolved goals
x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ x = 2 ∨ x = 3
```
attempt 4Wrong tactic
Proof
```
simp
```
What Lean printed (this is all the model sees on repair)
```
4:2: error: `simp` made no progress
```

attempt 5Wrong tactic

Proof

ring

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

4:2: 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.
3:73: error: unsolved goals
x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ x = 2 ∨ x = 3

attempt 6Wrong tactic

Proof

linarith

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

4:2: error: linarith failed to find a contradiction
x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ False
failed

attempt 7Wrong tactic

Proof

nlinarith

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

4:2: error: linarith failed to find a contradiction
x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ False
failed

attempt 8Wrong tactic
Proof
```
positivity
```
What Lean printed (this is all the model sees on repair)
```
4:2: error: not a positivity goal
```

attempt 9Wrong tactic

Proof

omega

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

4:2: error: omega could not prove the goal:
No usable constraints found. You may need to unfold definitions so `omega` can see linear arithmetic facts about `Nat` and `Int`, which may also involve multiplication, division, and modular remainder by constants.

attempt 10Wrong tactic
Proof
```
field_simp
```
What Lean printed (this is all the model sees on repair)
```
4:2: error: `field_simp` made no progress on the goal
```

attempt 11Wrong tactic

Proof

tauto

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

4:2: error: Tactic `tauto` failed

x : ℝ
h : x ^ 2 - 5 * x + 6 = 0
⊢ x = 2 ∨ x = 3

attempt 12Wrong tactic
Proof
```
simp_all
```
What Lean printed (this is all the model sees on repair)
```
4:2: error: simp_all made no progress
```

attempt 13Wrong tactic

Proof

aesop

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

4:2: error: Tactic `aesop` failed, made no progress
Initial goal:
  x : ℝ
  h : x ^ 2 - 5 * x + 6 = 0
  ⊢ x = 2 ∨ x = 3

attempt 14Verified
Proof
```
grind
```

Verified proof

theorem lg_target (x : ℝ) (h : x ^ 2 - 5 * x + 6 = 0) : x = 2 ∨ x = 3 := by
  grind

Axioms used: propext, Classical.choice, Quot.sound

Reference proof

The proof we wrote and certified before any model ran.

have h' : (x - 2) * (x - 3) = 0 := by ring_nf; linarith
rcases mul_eq_zero.mp h' with h1 | h1
· left; linarith
· right; linarith