Learning.obs0_stationaryEnv
Lemma
No docstring.
Types
-
𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra. -
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Given
-
ν : ProbabilityTheory.Kernel 𝓐 𝓨A kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
Then
(stationaryEnv ν).obs0 = MeasureTheory.Measure.dirac ()MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
{β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → PropA kernel is a Markov kernel if every measure in its image is a probability measure.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
Eq : {α : Sort u_1} → α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.Learning.stationaryEnv : {𝓐 : Type u_2} →
{𝓨 : Type u_3} →
{m𝓐 : MeasurableSpace 𝓐} →
{m𝓨 : MeasurableSpace 𝓨} →
(ν : ProbabilityTheory.Kernel 𝓐 𝓨) → [ProbabilityTheory.IsMarkovKernel ν] → Learning.Environment Unit 𝓐 𝓨A stationary environment without observations, in which the distribution of the next feedback depends only on the last action.Go to its page
Learning.Environment.obs0 : {𝓞 : Type u_1} →
{𝓐 : Type u_2} →
{𝓨 : Type u_3} →
{m𝓞 : MeasurableSpace 𝓞} →
{m𝓐 : MeasurableSpace 𝓐} → {m𝓨 : MeasurableSpace 𝓨} → Learning.Environment 𝓞 𝓐 𝓨 → MeasureTheory.Measure 𝓞Distribution of the first observation: the observation kernel at time `0` applied to the empty history.Go to its page
MeasureTheory.Measure.dirac : {α : Type u_1} → [inst : MeasurableSpace α] → α → MeasureTheory.Measure αThe dirac measure.
Unit.unit : UnitThe only element of the unit type. It can be written as an empty tuple: `()`.
Code
lemma obs0_stationaryEnv (ν : Kernel 𝓐 𝓨) [IsMarkovKernel ν] :
(stationaryEnv ν).obs0 = Measure.dirac ()Proof
rfl
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, 23 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.