Theorems / novel_set_06

Authored

Sets · easy

Statement, exactly as the model and Lean see it

theorem lg_target (S : Set ℕ) (hS : S = {n | n % 2 = 0}) : 4 ∈ S ∧ 3 ∉ S := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 0

Traces

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

Verified0 model calls · 12 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
  4 ∈ S
is not definitionally equal to the right-hand side
  3 ∉ S

S : Set ℕ
hS : S = {n | n % 2 = 0}
⊢ 4 ∈ S ∧ 3 ∉ S

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
  4 ∈ S ∧ 3 ∉ S

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)
```
3:76: error: unsolved goals
S : Set ℕ
hS : S = {n | n % 2 = 0}
⊢ 4 ∈ S ∧ 3 ∉ S
```
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.
3:76: error: unsolved goals
S : Set ℕ
hS : S = {n | n % 2 = 0}
⊢ 4 ∈ S ∧ 3 ∉ S

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
S : Set ℕ
hS : S = {n | n % 2 = 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
S : Set ℕ
hS : S = {n | n % 2 = 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

S : Set ℕ
hS : S = {n | n % 2 = 0}
⊢ 4 ∈ S ∧ 3 ∉ S

attempt 12Verified
Proof
```
simp_all
```

Verified proof

theorem lg_target (S : Set ℕ) (hS : S = {n | n % 2 = 0}) : 4 ∈ S ∧ 3 ∉ S := by
  simp_all

Axioms used: propext

Reference proof

The proof we wrote and certified before any model ran.

subst hS
constructor <;> simp