Inequalities · easy

Statement, exactly as the model and Lean see it

theorem lg_target (M : Type u_1) [Monoid M] [Preorder M] [MulLeftMono M] : Submonoid.oneLE M = Submonoid.oneLE M := by

Mathlib declaration: Submonoid.oneLE.congr_simp
Held-out module: Mathlib.Algebra.Order.Group.Cone
Banned modules (itself + downstream): 2
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 · $0.00020

Plan

### Explanation of the Proof: This theorem is a tautology—it states that the identity element of a monoid is less than or equal to itself under a preorder. Since `Submonoid.oneLE M` is defined as the property that the monoid's identity element is less than or equal to all elements in the monoid, and this is a definitional equality (i.e., it's asserting that something is equal to itself), the theorem is trivially true. The key idea is to recognize that the expression `Submonoid.oneLE M` is being compared with itself, so the conclusion follows directly from reflexivity of equality. --- ### Lean Proof Skeleton: Here: - `rfl` is used to prove the definitional equality, which is valid because both sides are the same. - The rest is just scaffolding to structure the proof.

Skeleton — Lean accepts it with only sorry steps left

have h1 : Submonoid.oneLE M = Submonoid.oneLE M := rfl
exact h1

Attempts

sample 0 · round 0Verified1.2 s model time
Proof
```
rfl
```

Verified proof

theorem lg_target (M : Type u_1) [Monoid M] [Preorder M] [MulLeftMono M] : Submonoid.oneLE M = Submonoid.oneLE M := by
  rfl

Axioms used: none

Reference proof

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