Learning.Environment.obs0_congr
Lemma
No docstring.
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'
Then
(env.congr e𝓞 e𝓐 e𝓨).obs0 = MeasureTheory.Measure.map (⇑e𝓞) env.obs0MeasurableSpace : 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
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.
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.congr : {𝓞 : Type u_1} →
{𝓞' : Type u_2} →
{𝓐 : Type u_4} →
{𝓐' : Type u_5} →
{𝓨 : Type u_7} →
{𝓨' : Type u_8} →
{m𝓞 : MeasurableSpace 𝓞} →
{m𝓞' : MeasurableSpace 𝓞'} →
{m𝓐 : MeasurableSpace 𝓐} →
{m𝓐' : MeasurableSpace 𝓐'} →
{m𝓨 : MeasurableSpace 𝓨} →
{m𝓨' : MeasurableSpace 𝓨…Relabelling of the observations, the actions and the feedbacks of an environment along measurable equivalences. See also `Algorithm.congr`.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.map : {α : Type u_4} →
{β : Type u_5} →
[inst : MeasurableSpace α] →
[inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure βThe pushforward of a measure. If `f` is not an almost everywhere measurable function, we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way we always have `map f 0 = 0`, and the push-forward of a probability measure is always a probability measure.
Code
lemma Environment.obs0_congr (env : Environment 𝓞 𝓐 𝓨) (e𝓞 : 𝓞 ≃ᵐ 𝓞') (e𝓐 : 𝓐 ≃ᵐ 𝓐')
(e𝓨 : 𝓨 ≃ᵐ 𝓨') :
(env.congr e𝓞 e𝓐 e𝓨).obs0 = env.obs0.map e𝓞Proof
by
rw [Environment.obs0_def, obs_congr, Kernel.comap_apply, Kernel.map_apply _ e𝓞.measurable,
env.obs_zero]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: 10 project declarations, 27 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.