Theorems / novel_fun_05

Authored

Functions · medium

Statement, exactly as the model and Lean see it

theorem lg_target (f : ℕ → ℕ) (hf : StrictMono f) (n : ℕ) : n ≤ f n := by

Source: Written for LeanGraph, reference proof certified
Premises used by the reference proof: 2

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
  n
is not definitionally equal to the right-hand side
  f n

f : ℕ → ℕ
hf : StrictMono f
n : ℕ
⊢ n ≤ f n

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
  n ≤ f n

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
f : ℕ → ℕ
hf : StrictMono f
n : ℕ
⊢ n ≤ f n
```
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
f : ℕ → ℕ
hf : StrictMono f
n : ℕ
a✝ : f n < n
⊢ 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
f : ℕ → ℕ
hf : StrictMono f
n : ℕ
a✝ : f n < n
⊢ 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:
a possible counterexample may satisfy the constraints
  b ≥ 0
  a ≥ 0
  a - b ≥ 1
where
 a := ↑n
 b := ↑(f n)

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

f : ℕ → ℕ
hf : StrictMono f
n : ℕ
⊢ n ≤ f n

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:
  f : ℕ → ℕ
  hf : StrictMono f
  n : ℕ
  ⊢ n ≤ f n

attempt 14Other

Proof

grind

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

4:2: error: `grind` failed
case grind
f : ℕ → ℕ
hf : StrictMono f
n : ℕ
h : f n + 1 ≤ n
⊢ False
[grind] Goal diagnostics
  [facts] Asserted facts
    [prop] StrictMono f
    [prop] f n + 1 ≤ n
  [eqc] True propositions
    [prop] f n + 1 ≤ n
    [prop] StrictMono f
  [ematch] E-matching patterns
    [thm] Set.subset_def: [@LE.le _ _ #1 #0]
  [cutsat] Assignment satisfying linear constraints
    [assign] n := 1
    [assign] f n := 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
f : ℕ → ℕ
hf : StrictMono f
n : ℕ
a✝ : f n < n
⊢ False
failed

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

Verified proof

theorem lg_target (f : ℕ → ℕ) (hf : StrictMono f) (n : ℕ) : n ≤ f n := by
  exact?

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

Reference proof

The proof we wrote and certified before any model ran.

exact hf.id_le n