Theorems / novel_set_02

Authored

Sets · easy

Statement, exactly as the model and Lean see it

theorem lg_target {α : Type*} (A B : Set α) (h : A ⊆ B) : A ∪ B = B := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 1

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
  A ∪ B
is not definitionally equal to the right-hand side
  B

α : Type u_1
A B : Set α
h : A ⊆ B
⊢ A ∪ B = B

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
  A ∪ B = B

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)
```
3:71: error: unsolved goals
α : Type u_1
A B : Set α
h : A ⊆ B
⊢ A ⊆ B
```
attempt 4Valid but mathematically wrong
Proof
```
simp
```
What Lean printed (this is all the model sees on repair)
```
3:71: error: unsolved goals
α : Type u_1
A B : Set α
h : A ⊆ B
⊢ A ⊆ B
```
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
A B : Set α
h : 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
α : Type u_1
A B : Set α
h : 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 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
A B : Set α
h : A ⊆ B
⊢ A ∪ B = B

attempt 12Verified
Proof
```
simp_all
```

Verified proof

theorem lg_target {α : Type*} (A B : Set α) (h : A ⊆ B) : A ∪ B = B := by
  simp_all

Axioms used: propext, Quot.sound

Reference proof

The proof we wrote and certified before any model ran.

exact Set.union_eq_right.mpr h