`Learning.IsAlgEnvSeq.filtrationObs`🔗

Definition

From the authors

Filtration generated by the history before time n together with the observation at time n.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.
𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

O : ℕ → Ω → 𝓞
A : ℕ → Ω → 𝓐
Y : ℕ → Ω → 𝓨
alg : Algorithm 𝓞 𝓐 𝓨A stochastic, sequential algorithm.
env : Environment 𝓞 𝓐 𝓨A stochastic environment.
P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsFiniteMeasure PA measure μ is called finite if μ univ < ∞.

Assuming

h : IsAlgEnvSeq O A Y alg env PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.

Result

MeasureTheory.Filtration ℕ mΩ

A Filtration on a measurable space Ω with σ-algebra m is a monotone sequence of sub-σ-algebras of m.

Body

{ seq := fun n => MeasurableSpace.comap (fun ω => (history O A Y n ω, O n ω)) inferInstance, mono' := ⋯, le' := ⋯ }

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

Nat : Type

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.

Learning.Algorithm : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)

A stochastic, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation.

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Learning.Environment : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)

A stochastic environment.
At each round, an observation is drawn prior to the algorithm taking an action. Then the environment
provides feedback based on the observation and the action.

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MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called finite if `μ univ < ∞`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Learning.IsAlgEnvSeq : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated
by an algorithm interacting with an environment.

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MeasureTheory.Filtration : {Ω : Type u_1} → (ι : Type u_2) → [Preorder ι] → MeasurableSpace Ω → Type (max u_1 u_2)

A `Filtration` on a measurable space `Ω` with σ-algebra `m` is a monotone
sequence of sub-σ-algebras of `m`.

MeasureTheory.Filtration.mk : {Ω : Type u_1} →
  {ι : Type u_2} →
    [inst : Preorder ι] →
      {m : MeasurableSpace Ω} →
        (seq : ι → MeasurableSpace Ω) → Monotone seq → (∀ (i : ι), seq i ≤ m) → MeasureTheory.Filtration ι m

MeasurableSpace.comap : {α : Type u_1} → {β : Type u_2} → (α → β) → MeasurableSpace β → MeasurableSpace α

The reverse image of a measurable space under a function. `comap f m` contains the sets
`s : Set α` such that `s` is the `f`-preimage of a measurable set in `β`.

Prod.mk : {α : Type u} → {β : Type v} → α → β → α × β

Constructs a pair. This is usually written `(x, y)` instead of `Prod.mk x y`.


Conventions for notations in identifiers:

 * The recommended spelling of `(a, b)` in identifiers is `mk`.

Learning.history : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → (n : ℕ) → Ω → Learning.Hist 𝓞 𝓐 𝓨 n

History of the algorithm-environment sequence before time `n`: the rounds at
times `0, ..., n - 1`.

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inferInstance : {α : Sort u} → [i : α] → α

`inferInstance` synthesizes a value of any target type by typeclass
inference. This function has the same type signature as the identity
function, but the square brackets on the `[i : α]` argument means that it will
attempt to construct this argument by typeclass inference. (This will fail if
`α` is not a `class`.) Example:
```
#check (inferInstance : Inhabited Nat) -- Inhabited Nat

def foo : Inhabited (Nat × Nat) :=
  inferInstance

example : foo.default = (default, default) :=
  rfl
```

Code

def filtrationObs (h : IsAlgEnvSeq O A Y alg env P) :
    Filtration ℕ mΩ where
  seq n := MeasurableSpace.comap (fun ω ↦ (history O A Y n ω, O n ω)) inferInstance
  mono' n m hnm := by
    simp only
    rw [← measurable_iff_comap_le]
    rcases eq_or_lt_of_le hnm with rfl | hlt
    · exact measurable_iff_comap_le.mpr le_rfl
    have : (fun ω ↦ (history O A Y n ω, O n ω)) =
        (fun p : Hist 𝓞 𝓐 𝓨 m × 𝓞 ↦
          (fun i : Fin n ↦ p.1 (Fin.castLE hnm i), (p.1 ⟨n, hlt⟩).obs)) ∘
        (fun ω ↦ (history O A Y m ω, O m ω)) := rfl
    rw [this]
    exact measurable_comp_comap _ (by fun_prop)
  le' n := by
    rw [← measurable_iff_comap_le]
    exact (Learning.measurable_history h.measurable_obs h.measurable_action
      h.measurable_feedback n).prodMk (h.measurable_obs n)

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).

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: 6 project declarations, 28 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.IsAlgEnvSeq.filtrationObs🔗

`Learning.IsAlgEnvSeq.filtrationObs`🔗