Theorems / novel_ineq_01

Authored

Inequalities · easy

Statement, exactly as the model and Lean see it

theorem lg_target (a b : ℝ) : 2 * a * b ≤ a ^ 2 + b ^ 2 := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 65

Traces

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

Verified0 model calls · 16 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
  2 * a * b
is not definitionally equal to the right-hand side
  a ^ 2 + b ^ 2

a b : ℝ
⊢ 2 * a * b ≤ a ^ 2 + b ^ 2

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
  2 * a * b ≤ a ^ 2 + b ^ 2

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)
```
3:59: error: unsolved goals
a b : ℝ
⊢ 2 * a * b ≤ a ^ 2 + b ^ 2
```
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.
3:59: error: unsolved goals
a b : ℝ
⊢ a * b * 2 ≤ a ^ 2 + b ^ 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
a b : ℝ
a✝ : a ^ 2 + b ^ 2 < 2 * a * b
⊢ 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
a b : ℝ
a✝ : a ^ 2 + b ^ 2 < 2 * a * b
⊢ 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)
```
3:59: error: unsolved goals
a b : ℝ
⊢ 2 * a * b ≤ a ^ 2 + b ^ 2
```
attempt 11Wrong tactic
Proof
```
tauto
```
What Lean printed (this is all the model sees on repair)
```
4:2: error: Tactic `tauto` failed

a b : ℝ
⊢ 2 * a * b ≤ a ^ 2 + b ^ 2
```
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:
  a b : ℝ
  ⊢ 2 * a * b ≤ a ^ 2 + b ^ 2

attempt 14Other

Proof

grind

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

4:2: error: `grind` failed
case grind
a b : ℝ
h : ¬2 * a * b ≤ a ^ 2 + b ^ 2
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] ¬2 * a * b ≤ a ^ 2 + b ^ 2
  [eqc] False propositions
    [prop] 2 * a * b ≤ a ^ 2 + b ^ 2
  [ematch] E-matching patterns
    [thm] Set.subset_def: [@LE.le _ _ #1 #0]
    [thm] Nat.pow_pos: [@HPow.hPow `[ℕ] `[ℕ] `[ℕ] `[instHPow] #2 #1]
    [thm] Nat.div_pow_of_pos: [@HPow.hPow `[ℕ] `[ℕ] `[ℕ] `[instHPow] #2 #1]
  [linarith] Linarith assignment for `ℝ`
    [assign] a := 2
    [assign] b := 3
    [assign] 「a ^ 2」 := 0
    [assign] 「b ^ 2」 := -1

attempt 15Wrong tactic

Proof

norm_num <;> linarith

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

4:15: error: linarith failed to find a contradiction
a b : ℝ
a✝ : a ^ 2 + b ^ 2 < 2 * a * b
⊢ False
failed

attempt 16Verified
Proof
```
exact?
```

Verified proof

theorem lg_target (a b : ℝ) : 2 * a * b ≤ a ^ 2 + b ^ 2 := by
  exact?

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

Reference proof

The proof we wrote and certified before any model ran.

nlinarith [sq_nonneg (a - b)]