Category theory · easy

Statement, exactly as the model and Lean see it

theorem lg_target {α : Type u_1} [Preorder α] {i j : α} (h : i ≤ j) : Set.initialSegIicIicOfLE h = Set.initialSegIicIicOfLE h := by

Mathlib declaration: Set.initialSegIicIicOfLE.congr_simp
Held-out module: Mathlib.CategoryTheory.SmallObject.Iteration.Basic
Banned modules (itself + downstream): 19
Reference proof premises (reachable / held-out): 0 / 0

Traces

Each configuration's full record: what was retrieved, what the model wrote, what Lean said back.

Verified2 model calls · 2 Lean checks · $0.00034

Retrieved lemmas

1. Set.principalSegIioIicOfLE_top
2. Set.initialSegIicIicOfLE_toFun_coe
3. Set.principalSegIioIicOfLE_apply
4. Set.initialSegIic_toFun
5. Set.Iic_def
6. LowerSet.Iic_le
7. Set.mem_Iic
8. Set.Iio_def

Plan

The theorem you're trying to prove is trivial because it asserts that a function is equal to itself. In other words, it's an identity statement: `Set.initialSegIicIicOfLE h = Set.initialSegIicIicOfLE h`. Since both sides are the same expression, this equality holds by reflexivity. ### Proof Strategy: This is a tautology and can be proved using `rfl` (reflexivity). However, since the theorem is presented in a way that suggests it might be part of a larger proof or has some hidden context, we can also approach it by unfolding definitions and verifying that both sides are indeed identical. But in this case, no unfolding or lemmas are needed—just reflexivity. --- ### Lean Proof Skeleton: This is the simplest possible proof for this identity. If you were to expand this into a more complex proof with intermediate steps (e.g., for educational purposes), you could use `have` statements to unfold definitions, but in this case, they are unnecessary.

Skeleton — Lean reported problems

import Mathlib.Order.LowerSet
import Mathlib.Data.Set.Iic

open Set

theorem lg_target {α : Type u_1} [Preorder α] {i j : α} (h : i ≤ j) :
  initialSegIicIicOfLE h = initialSegIicIicOfLE h :=
by
  rfl

Attempts

sample 0 · round 0Verified1.5 s model time
Proof
```
rfl
```

Verified proof

theorem lg_target {α : Type u_1} [Preorder α] {i j : α} (h : i ≤ j) : Set.initialSegIicIicOfLE h = Set.initialSegIicIicOfLE h := by
  rfl

Axioms used: propext

Reference proof

Mathlib's own source for this declaration, shown for comparison. It may use lemmas the prover is not allowed to use.