Theorems / novel_prob_03

Authored

Probability · medium

Statement, exactly as the model and Lean see it

theorem lg_target {Ω : Type*} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (A : Set Ω) (hA : MeasurableSet A) : μ Aᶜ = 1 - μ A := 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 · 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
  μ Aᶜ
is not definitionally equal to the right-hand side
  1 - μ A

Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
⊢ μ Aᶜ = 1 - μ A

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
  μ Aᶜ = 1 - μ A

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)

3:174: error: unsolved goals
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
⊢ μ Aᶜ = 1 - μ A

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.
3:174: error: unsolved goals
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
⊢ μ Aᶜ = 1 - μ A

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
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
a✝ : μ Aᶜ < 1 - μ A
⊢ 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
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
a✝ : μ Aᶜ < 1 - μ A
⊢ 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

Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
⊢ μ Aᶜ = 1 - μ A

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:
  Ω : Type u_1
  inst✝¹ : MeasurableSpace Ω
  μ : MeasureTheory.Measure Ω
  inst✝ : MeasureTheory.IsProbabilityMeasure μ
  A : Set Ω
  hA : MeasurableSet A
  ⊢ μ Aᶜ = 1 - μ A

attempt 14Other

Proof

grind

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

4:2: error: `grind` failed
case grind
Ω : Type u_1
inst : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst_1 : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
h : ¬μ Aᶜ = 1 - μ A
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] MeasureTheory.IsProbabilityMeasure μ
    [prop] MeasurableSet A
    [prop] ¬μ Aᶜ = 1 - μ A
  [eqc] True propositions
    [prop] MeasurableSet A
    [prop] MeasureTheory.IsProbabilityMeasure μ
  [eqc] False propositions
    [prop] μ Aᶜ = 1 - μ A
  [ematch] E-matching patterns
    [thm] map_sub: [@DFunLike.coe #7 #9 _ #6 #2 (@HSub

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
case h1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
inst✝ : MeasureTheory.IsProbabilityMeasure μ
A : Set Ω
hA : MeasurableSet A
a✝ : μ Aᶜ < 1 - μ A
⊢ False
failed

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

Verified proof

theorem lg_target {Ω : Type*} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (A : Set Ω) (hA : MeasurableSet A) : μ Aᶜ = 1 - μ A := by
  exact?

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

Reference proof

The proof we wrote and certified before any model ran.

exact MeasureTheory.prob_compl_eq_one_sub hA