`Learning.pullCount_le_add`🔗

Lemma

No docstring.

theorem

Learning.pullCount_le_add.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (a : 𝓐) (n C : ℕ) (ω : Ω) :
  pullCount A a n ω ≤
    C + 1 +
      ∑ s ∈ Finset.range n,
        Set.indicator {s | A s ω = a ∧ C < pullCount A a s ω} 1 s
Learning.pullCount_le_add.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (a : 𝓐)
  (n C : ℕ) (ω : Ω) :
  pullCount A a n ω ≤
    C + 1 +
      ∑ s ∈ Finset.range n,
        Set.indicator
          {s |
            A s ω = a ∧
              C < pullCount A a s ω}
          1 s

Code

lemma pullCount_le_add (a : 𝓐) (n C : ℕ) (ω : Ω) :
    pullCount A a n ω ≤ C + 1 +
      ∑ s ∈ range n, {s | A s ω = a ∧ C < pullCount A a s ω}.indicator 1 s

Proof

by
  rw [pullCount_eq_sum]
  calc ∑ s ∈ range n, if A s ω = a then 1 else 0
  _ ≤ ∑ s ∈ range n, ({s | A s ω = a ∧ pullCount A a s ω ≤ C}.indicator 1 s +
      {s | A s ω = a ∧ C < pullCount A a s ω}.indicator 1 s) := by
    gcongr with s hs
    simp [Set.indicator_apply]
    grind
  _ = ∑ s ∈ range n, {s | A s ω = a ∧ pullCount A a s ω ≤ C}.indicator 1 s +
      ∑ s ∈ range n, {s | A s ω = a ∧ C < pullCount A a s ω}.indicator 1 s := by
    rw [Finset.sum_add_distrib]
  _ ≤ C + 1 + ∑ s ∈ range n, {s | A s ω = a ∧ C < pullCount A a s ω}.indicator 1 s := by
    gcongr
    have h_le n : ∑ s ∈ range n, {s | A s ω = a ∧ pullCount A a s ω ≤ C}.indicator 1 s ≤
        pullCount A a n ω := by
      rw [pullCount_eq_sum]
      gcongr with s hs
      simp only [Set.indicator_apply, Set.mem_ofPred_eq, Pi.one_apply]
      grind
    induction n with
    | zero => simp
    | succ n hn =>
      rw [Finset.sum_range_succ]
      rcases le_or_gt (pullCount A a n ω) C with h_pc | h_pc
      · have hn' : ∑ s ∈ range n, {s | A s ω = a ∧ pullCount A a s ω ≤ C}.indicator 1 s ≤ C :=
          (h_le n).trans h_pc
        grw [hn']
        gcongr
        simp only [Set.indicator_apply, Set.mem_ofPred_eq, Pi.one_apply]
        grind
      · refine le_trans ?_ hn
        simp [h_pc]

Actions: Source · Open Issue

Meaning unchanged since v4.33.0-rc1-29-gce231eb, the oldest revision on record (2026-07-30).

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 1 project declarations, 25 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.pullCount_le_add🔗

`Learning.pullCount_le_add`🔗