`Learning.Environment.comap`🔗

Definition

From the authors

The environment that reads the summary F n of the past rounds and reads f a when the algorithm plays a in the current round.

This is the primitive transport operation on environments, dual to Algorithm.comap: Environment.comapAction is the special case in which F n is the round-wise map of the actions. Only the action can change type, since the observations and the feedbacks are outputs of the environment; F n can nonetheless forget or summarize the past rounds, as an environment that reads only the last round does.

Types

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

Given

env : Environment 𝓞 𝓐 𝓨A stochastic environment.
F : (n : ℕ) → Hist 𝓞 𝓐' 𝓨 n → Hist 𝓞 𝓐 𝓨 nHistory of n complete rounds; n = 0 is the empty history.
f : 𝓐' → 𝓐

Assuming

hF : ∀ (n : ℕ), Measurable (F n)A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.
hf : Measurable f

Result

Environment 𝓞 𝓐' 𝓨

Body

{ obs := fun n => (env.obs n).comap (F n) ⋯,
  feedback := fun n => (env.feedback n).comap (fun p => ((F n p.1.1, p.1.2), f 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.

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Learning.Hist : Type u_5 → Type u_6 → Type u_7 → ℕ → Type (max (max u_7 u_6) u_5)

History of `n` complete rounds; `n = 0` is the empty history.

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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.

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

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.

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

def Environment.comap (env : Environment 𝓞 𝓐 𝓨)
    (F : (n : ℕ) → Hist 𝓞 𝓐' 𝓨 n → Hist 𝓞 𝓐 𝓨 n) (hF : ∀ n, Measurable (F n))
    (f : 𝓐' → 𝓐) (hf : Measurable f) : Environment 𝓞 𝓐' 𝓨 where
  obs n := (env.obs n).comap (F n) (hF n)
  feedback n := (env.feedback n).comap (fun p ↦ ((F n p.1.1, p.1.2), f p.2)) (by fun_prop)

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: 3 project declarations, 13 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.comap🔗

`Learning.Environment.comap`🔗