Theorems / novel_set_05

Authored

Sets · medium

Statement, exactly as the model and Lean see it

theorem lg_target {α : Type*} (s : Set α) (f : α → α) (hf : ∀ x, f (f x) = x) : f '' (f '' s) = s := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 3

Traces

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

Verified0 model calls · 13 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
  f '' f '' s
is not definitionally equal to the right-hand side
  s

α : Type u_1
s : Set α
f : α → α
hf : ∀ (x : α), f (f x) = x
⊢ f '' f '' s = 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
  f '' f '' s = 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)

3:101: error: unsolved goals
α : Type u_1
s : Set α
f : α → α
hf : ∀ (x : α), f (f x) = x
⊢ f '' f '' s = 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: error: `ring_nf` made no progress on the goal
```

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
α : Type u_1
s : Set α
f : α → α
hf : ∀ (x : α), f (f x) = x
⊢ 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
α : Type u_1
s : Set α
f : α → α
hf : ∀ (x : α), f (f x) = x
⊢ 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

α : Type u_1
s : Set α
f : α → α
hf : ∀ (x : α), f (f x) = x
⊢ f '' f '' s = s

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 13Verified
Proof
```
aesop
```

Verified proof

theorem lg_target {α : Type*} (s : Set α) (f : α → α) (hf : ∀ x, f (f x) = x) : f '' (f '' s) = s := by
  aesop

Axioms used: propext, Quot.sound

Reference proof

The proof we wrote and certified before any model ran.

rw [Set.image_image]
simp [hf]