`Learning.Environment.congr`🔗

Definition

From the authors

Relabelling of the observations, the actions and the feedbacks of an environment along measurable equivalences. See also Algorithm.congr.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.
𝓞' : Type u_2m𝓞' : MeasurableSpace 𝓞'
𝓐 : Type u_4m𝓐 : MeasurableSpace 𝓐
𝓐' : Type u_5m𝓐' : MeasurableSpace 𝓐'
𝓨 : Type u_7m𝓨 : MeasurableSpace 𝓨
𝓨' : Type u_8m𝓨' : MeasurableSpace 𝓨'

Given

env : Environment 𝓞 𝓐 𝓨A stochastic environment.
e𝓞 : 𝓞 ≃ᵐ 𝓞'Equivalences between measurable spaces.
e𝓐 : 𝓐 ≃ᵐ 𝓐'
e𝓨 : 𝓨 ≃ᵐ 𝓨'

Result

Environment 𝓞' 𝓐' 𝓨'

Body

{ obs := fun n => ((env.obs n).map ⇑e𝓞).comap (Hist.map ⇑e𝓞.symm ⇑e𝓐.symm ⇑e𝓨.symm) ⋯,
  feedback := fun n =>
    ((env.feedback n).map ⇑e𝓨).comap
      (fun p => ((Hist.map (⇑e𝓞.symm) (⇑e𝓐.symm) (⇑e𝓨.symm) p.1.1, e𝓞.symm p.1.2), e𝓐.symm p.2)) ⋯,
  isMarkovKernel_obs := ⋯, isMarkovKernel_feedback := ⋯ }

MeasurableSpace : Type u_6 → Type u_6

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

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.

Go to its page

MeasurableEquiv : (α : Type u_6) → (β : Type u_7) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_6 u_7)

Equivalences between measurable spaces. Main application is the simplification of measurability
statements along measurable equivalences.

ProbabilityTheory.Kernel.map : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} →
        {γ : Type u_4} →
          [inst : MeasurableSpace γ] → ProbabilityTheory.Kernel α β → (β → γ) → ProbabilityTheory.Kernel α γ

The pushforward of a kernel along a function.
If the function is not measurable, we use zero instead. This choice of junk
value ensures that typeclass inference can infer that the `map` of a kernel
satisfying `IsZeroOrMarkovKernel` again satisfies this property.

ProbabilityTheory.Kernel.comap : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} →
        {γ : Type u_4} →
          {mγ : MeasurableSpace γ} →
            ProbabilityTheory.Kernel α β → (g : γ → α) → Measurable g → ProbabilityTheory.Kernel γ β

Pullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`.
We include measurability in the assumptions instead of using junk values
to make sure that typeclass inference can infer that the `comap` of a Markov kernel
is again a Markov kernel.

Learning.Hist.map : {𝓞 : Type u_1} →
  {𝓞' : Type u_2} →
    {𝓐 : Type u_4} →
      {𝓐' : Type u_5} →
        {𝓨 : Type u_7} →
          {𝓨' : Type u_8} → (𝓞 → 𝓞') → (𝓐 → 𝓐') → (𝓨 → 𝓨') → {n : ℕ} → Learning.Hist 𝓞 𝓐 𝓨 n → Learning.Hist 𝓞' 𝓐' 𝓨' n

Transport a history round-wise.

Go to its page

MeasurableEquiv.symm : {α : Type u_1} → {β : Type u_2} → [inst : MeasurableSpace α] → [inst_1 : MeasurableSpace β] → α ≃ᵐ β → β ≃ᵐ α

The inverse of an equivalence between measurable spaces.

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`.

Code

noncomputable def Environment.congr (env : Environment 𝓞 𝓐 𝓨) (e𝓞 : 𝓞 ≃ᵐ 𝓞') (e𝓐 : 𝓐 ≃ᵐ 𝓐')
    (e𝓨 : 𝓨 ≃ᵐ 𝓨') :
    Environment 𝓞' 𝓐' 𝓨' where
  obs n := ((env.obs n).map e𝓞).comap (Hist.map e𝓞.symm e𝓐.symm e𝓨.symm) (by fun_prop)
  feedback n := ((env.feedback n).map e𝓨).comap
    (fun p ↦ ((Hist.map e𝓞.symm e𝓐.symm e𝓨.symm p.1.1, e𝓞.symm p.1.2), e𝓐.symm p.2)) (by fun_prop)
  isMarkovKernel_obs n := by
    have : IsMarkovKernel ((env.obs n).map e𝓞) := Kernel.IsMarkovKernel.map _ e𝓞.measurable
    infer_instance
  isMarkovKernel_feedback n := by
    have : IsMarkovKernel ((env.feedback n).map e𝓨) := Kernel.IsMarkovKernel.map _ e𝓨.measurable
    infer_instance

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: 8 project declarations, 18 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.Environment.congr🔗

`Learning.Environment.congr`🔗