Theorems / novel_alg_02

Authored

Algebra · medium

Statement, exactly as the model and Lean see it

theorem lg_target (x : ℝ) (hx : x ≠ 1) : (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1 := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 70

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 ^ 3 - 1) / (x - 1)
is not definitionally equal to the right-hand side
  x ^ 2 + x + 1

x : ℝ
hx : x ≠ 1
⊢ (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1

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 ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1

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)
```
3:82: error: unsolved goals
x : ℝ
hx : x ≠ 1
⊢ (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1
```
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.
3:82: error: unsolved goals
x : ℝ
hx : x ≠ 1
⊢ x ^ 3 * (-1 + x)⁻¹ - (-1 + x)⁻¹ = 1 + x + x ^ 2

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
case h1
x : ℝ
hx : x ≠ 1
a✝ : (x ^ 3 - 1) / (x - 1) < x ^ 2 + x + 1
⊢ 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
case h1
x : ℝ
hx : x ≠ 1
a✝ : (x ^ 3 - 1) / (x - 1) < x ^ 2 + x + 1
⊢ 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 10Valid but mathematically wrong
Proof
```
field_simp
```
What Lean printed (this is all the model sees on repair)
```
3:82: error: unsolved goals
x : ℝ
hx : x ≠ 1
⊢ x ^ 3 - 1 = (x - 1) * (x * (x + 1) + 1)
```

attempt 11Wrong tactic

Proof

tauto

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

4:2: error: Tactic `tauto` failed

x : ℝ
hx : x ≠ 1
⊢ (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1

attempt 12Valid but mathematically wrong
Proof
```
simp_all
```
What Lean printed (this is all the model sees on repair)
```
3:82: error: unsolved goals
x : ℝ
hx : ¬x = 1
⊢ (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1
```

attempt 13Wrong tactic

Proof

aesop

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

4:2: warning: aesop: failed to prove the goal after exhaustive search.
3:82: error: unsolved goals
x : ℝ
hx : ¬x = 1
⊢ (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1

attempt 14Verified
Proof
```
grind
```

Verified proof

theorem lg_target (x : ℝ) (hx : x ≠ 1) : (x ^ 3 - 1) / (x - 1) = x ^ 2 + x + 1 := by
  grind

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

Reference proof

The proof we wrote and certified before any model ran.

have h : x - 1 ≠ 0 := sub_ne_zero.mpr hx
field_simp
ring