Sets · medium

Statement, exactly as the model and Lean see it

theorem lg_target {α : Type u_1} {β : Type u_2} [Preorder β] {f : Finset α → β} [DecidableEq α] : StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f s < f (insert a s) := by

Mathlib declaration: Finset.strictMono_iff_forall_lt_insert
Held-out module: Mathlib.Data.Finset.Interval
Banned modules (itself + downstream): 2
Reference proof premises (reachable / held-out): 1 / 1

Traces

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

Not verified1 model calls · 1 Lean checks · $0.00009

Attempts

sample 0 · round 0Wrong tactic3.7 s model time

Proof

intro h
split
· intro s a ha
  have : a ∉ s := ha
  exact h s a this
· intro h'
  intros s t hst
  apply le_antisymm
  · intro x
    rw [finsupp.insert_le_iff]
    apply h'
    exact hst
  · intro x
    rw [finsupp.insert_le_iff]
    apply h'
    exact hst.symm
  exact le_refl _

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

4:8: error: Tactic `introN` failed: There are no additional binders or `let` bindings in the goal to introduce

α : Type u_1
β : Type u_2
inst✝¹ : Preorder β
f : Finset α → β
inst✝ : DecidableEq α
⊢ StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄, a ∉ s → f s < f (insert a s)

Reference proof

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

/-- A function `f` from `Finset α` is strictly monotone if and only if `f s < f (insert a s)` for
all `s` and `a ∉ s`. -/
lemma strictMono_iff_forall_lt_insert : StrictMono f ↔ ∀ s ⦃a⦄, a ∉ s → f s < f (insert a s) := by
  simp [strictMono_iff_forall_lt_cons]