Learning.Environment.ν0_def
Lemma
No docstring.
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
Given
-
env : Environment 𝓞 𝓐 𝓨A stochastic environment.
Then
env.ν0 = (env.feedback 0).comap (fun p => ((default, p.1), p.2)) ⋯MeasurableSpace : Type u_6 → Type u_6A 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
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.Environment.ν0 : {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 → ProbabilityTheory.Kernel (𝓞 × 𝓐) 𝓨Distribution of the first feedback given the first observation and action: the feedback kernel at time `0` applied to the empty history.Go to its page
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 βPullback 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 × βConstructs 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`.
Inhabited.default : {α : Sort u} → [self : Inhabited α] → α`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
lemma Environment.ν0_def (env : Environment 𝓞 𝓐 𝓨) :
env.ν0 = (env.feedback 0).comap (fun p ↦ ((default, p.1), p.2)) (by fun_prop)Proof
rfl
Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 2th recorded change.
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: 4 project declarations, 25 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.