Theorems / novel_fun_07

Authored

Functions · medium

Statement, exactly as the model and Lean see it

theorem lg_target {α β : Type*} (f : α → β) (hf : Function.Surjective f) (g h : β → α) (hg : g ∘ f = h ∘ f) : g = h := 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
  g
is not definitionally equal to the right-hand side
  h

α : Type u_1
β : Type u_2
f : α → β
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ g = h

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
  g = h

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)

3:119: error: unsolved goals
α : Type u_1
β : Type u_2
f : α → β
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ g = h

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
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ 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
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ 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
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ g = h

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
  hf : Function.Surjective f
  g h : β → α
  hg : g ∘ f = h ∘ f
  ⊢ g = h

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
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
h_1 : ¬g = h
w : β
h_2 : ¬g w = h w
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] Function.Surjective f
    [prop] g ∘ f = h ∘ f
    [prop] ¬g = h
    [prop] ∃ x, ¬g x = h x
    [prop] ¬g w = h w
  [eqc] True propositions
    [prop] Function.Surjective f
    [prop] ∃ x, ¬g x = h x
  [eqc] False propositions
    [prop] g = h
    [prop] g w = h w
  [eqc] Equivalence classes
    [eqc] {g ∘ f, h ∘ f}
  [cases] Case analyses
    [cases] [1/1]: ∃

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
hf : Function.Surjective f
g h : β → α
hg : g ∘ f = h ∘ f
⊢ False
failed

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

Verified proof

theorem lg_target {α β : Type*} (f : α → β) (hf : Function.Surjective f) (g h : β → α) (hg : g ∘ f = h ∘ f) : g = h := by
  exact?

Axioms used: Quot.sound

Reference proof

The proof we wrote and certified before any model ran.

exact hf.injective_comp_right hg