Learning.Algorithm.density_zero
Lemma
No docstring.
Types
-
π : Type u_1MeasurableSpace πA measurable space is a space equipped with a Ο-algebra. -
π : Type u_2MeasurableSpace πMeasurableSpace.CountablyGenerated πWe say a measurable space is countably generated if it can be generated by a countable set of sets. -
π¨ : Type u_3MeasurableSpace π¨
Given
-
alg : Algorithm π π π¨A stochastic, sequential algorithm. -
algβ : Algorithm π π π¨ -
h : Hist π π π¨ 0History ofncomplete rounds;n = 0is the empty history.
Then
alg.density algβ 0 h = 1MeasurableSpace : Type u_6 β Type u_6A measurable space is a space equipped with a Ο-algebra.
MeasurableSpace.CountablyGenerated : (Ξ± : Type u_3) β [m : MeasurableSpace Ξ±] β PropWe say a measurable space is countably generated if it can be generated by a countable set of sets.
Learning.Algorithm : (π : Type u_5) β
(π : Type u_6) β
(π¨ : Type u_7) β [MeasurableSpace π] β [MeasurableSpace π] β [MeasurableSpace π¨] β Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `π`, then takes an action in `π`, and finally receives feedback in `π¨`. The action is a random function of the past rounds and the current observation.Go to its page
Learning.Hist : Type u_5 β Type u_6 β Type u_7 β β β Type (max (max u_7 u_6) u_5)History of `n` complete rounds; `n = 0` is the empty history.Go to its page
Eq : {Ξ± : Sort u_1} β Ξ± β Ξ± β 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 (Ξ± : Type) (a b c d : Ξ±)
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 (Ξ± : Type) (a b : Ξ±) (p : Ξ± β Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (Ξ± : Type) (a b : Ξ±) (p : Ξ± β Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 βΈ 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.Algorithm.density : {π : Type u_1} β
{π : Type u_2} β
{π¨ : Type u_3} β
[inst : MeasurableSpace π] β
[inst_1 : MeasurableSpace π] β
[inst_2 : MeasurableSpace π¨] β
[MeasurableSpace.CountablyGenerated π] β
Learning.Algorithm π π π¨ β Learning.Algorithm π π π¨ β (n : β) β Learning.Hist π π π¨ n β ENNRealIf the algorithm `alg` is absolutely continuous with respect to the algorithm `algβ` and they are both interacting with the same environment, then the law of the history before time `n` under `alg` is the law of the history before time `n` under `algβ` with density `alg.density algβ n`.Go to its page
Code
lemma density_zero [MeasurableSpace.CountablyGenerated π] (alg algβ : Algorithm π π π¨)
(h : Hist π π π¨ 0) :
alg.density algβ 0 h = 1Proof
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 Β· open it in the Lean web editor.
Dependency graph
Audit surface: 7 project declarations, 44 external constants
β 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.