Theorems / novel_fun_02

Authored

Functions · easy

Statement, exactly as the model and Lean see it

theorem lg_target {α β : Type*} (f : α → β) (g : β → α) (h : ∀ x, g (f x) = x) : Function.Injective f := 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 · 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
  β
is not definitionally equal to the right-hand side
  f

α : Type u_1
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (f x) = x
⊢ Function.Injective f

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
  Function.Injective f

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)

3:105: error: unsolved goals
α : Type u_1
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (f x) = x
⊢ Function.Injective f

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
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (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
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (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
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (f x) = x
⊢ Function.Injective f

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:
  α : Type u_1
  β : Type u_2
  f : α → β
  g : β → α
  h : ∀ (x : α), g (f x) = x
  ⊢ Function.Injective f

attempt 14Other

Proof

grind

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

4:2: error: `grind` failed
case grind
α : Type u_1
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (f x) = x
h_1 : ¬Function.Injective f
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] ∀ (x : α), g (f x) = x
    [prop] ¬Function.Injective f
  [eqc] True propositions
    [prop] ∀ (x : α), g (f x) = x
  [eqc] False propositions
    [prop] Function.Injective f
  [ematch] E-matching patterns
    [thm] h: [g (f #0)]
    [thm] h: [f #0]

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
α : Type u_1
β : Type u_2
f : α → β
g : β → α
h : ∀ (x : α), g (f x) = x
⊢ False
failed

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

Verified proof

theorem lg_target {α β : Type*} (f : α → β) (g : β → α) (h : ∀ x, g (f x) = x) : Function.Injective f := by
  exact?

Axioms used: none

Reference proof

The proof we wrote and certified before any model ran.

intro a b hab
rw [← h a, ← h b, hab]