LeanMachineLearning

Learning.IsAlgEnvSeq.filtrationAction🔗

Definition

Filtration generated by the history at time n-1 together with the action at time n.

🔗def
Learning.IsAlgEnvSeq.filtrationAction.{u_1, u_2, u_3} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨} {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure P]
  (h : IsAlgEnvSeq A Y alg env P) : MeasureTheory.Filtration ℕ mΩ
Learning.IsAlgEnvSeq.filtrationAction.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} {A : ℕ → Ω → 𝓐}
  {Y : ℕ → Ω → 𝓨} {alg : Algorithm 𝓐 𝓨}
  {env : Environment 𝓐 𝓨}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure P]
  (h : IsAlgEnvSeq A Y alg env P) :
  MeasureTheory.Filtration ℕ mΩ

Code

def filtrationAction (h : IsAlgEnvSeq A Y alg env P) : Filtration ℕ mΩ where seq n := if n = 0 then MeasurableSpace.comap (A 0) inferInstance else h.filtration (n - 1) ⊔ MeasurableSpace.comap (A n) inferInstance mono' n m hnm := by simp only by_cases hn : n = 0 · by_cases hm : m = 0 · simp [hn, hm] · simp only [hn, ↓reduceIte, hm] refine le_sup_of_le_left ?_ rw [← measurable_iff_comap_le] suffices Measurable[h.filtration 0] (A 0) from this.mono ((h.filtration).mono zero_le) le_rfl exact adapted_action h 0 have hm : m ≠ 0 := by grind simp only [hn, hm, ↓reduceIte] have hnm' : n - 1 ≤ m - 1 := by grind simp only [sup_le_iff] constructor · refine le_sup_of_le_left ?_ exact (h.filtration).mono hnm' · rcases eq_or_lt_of_le hnm with rfl | hlt · exact le_sup_of_le_right le_rfl refine le_sup_of_le_left ?_ rw [← measurable_iff_comap_le] have h_le : n ≤ m - 1 := by grind suffices Measurable[h.filtration n] (A n) from this.mono ((h.filtration).mono h_le) le_rfl exact adapted_action h n le' n := by by_cases hn : n = 0 · simp only [hn, ↓reduceIte] rw [← measurable_iff_comap_le] exact h.measurable_action 0 simp only [hn, ↓reduceIte, sup_le_iff] constructor · exact (IsAlgEnvSeq.filtration h).le _ · rw [← measurable_iff_comap_le] exact h.measurable_action n

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Meaning last changed in v4.34.0-rc2-2-g4830b8c (2026-08-23).

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: 5 project declarations, 56 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.