Learning.stepKernel_zero
Lemma
No docstring.
Types
-
๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_2m๐ : MeasurableSpace ๐ -
๐จ : Type u_3m๐จ : MeasurableSpace ๐จ
Given
-
alg : Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm. -
env : Environment ๐ ๐ ๐จA stochastic environment. -
h : Hist ๐ ๐ ๐จ 0History ofncomplete rounds;n = 0is the empty history.
Then
(stepKernel alg env 0) h = env.obs0.compProd (alg.p0.compProd env.ฮฝ0)MeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
Learning.Algorithm : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `๐`, then takes an action in `๐`, and finally receives feedback in `๐จ`. The action is a random function of the past rounds and the current observation.Go to its page
Learning.Environment : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ 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
Learning.Hist : Type u_5 โ Type u_6 โ Type u_7 โ โ โ 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 : {ฮฑ : Sort u_1} โ ฮฑ โ ฮฑ โ 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 (ฮฑ : Type) (a b c d : ฮฑ)
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 (ฮฑ : Type) (a b : ฮฑ) (p : ฮฑ โ Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (ฮฑ : Type) (a b : ฮฑ) (p : ฮฑ โ Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 โธ 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.stepKernel : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
Learning.Algorithm ๐ ๐ ๐จ โ
Learning.Environment ๐ ๐ ๐จ โ
(n : โ) โ ProbabilityTheory.Kernel (Learning.Hist ๐ ๐ ๐จ n) (Learning.Round ๐ ๐ ๐จ)Kernel describing the distribution of the round at time `n` given the history before `n`.Go to its page
Learning.Environment.obs0 : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ {m๐จ : MeasurableSpace ๐จ} โ Learning.Environment ๐ ๐ ๐จ โ MeasureTheory.Measure ๐Distribution of the first observation: the observation kernel at time `0` applied to the empty history.Go to its page
MeasureTheory.Measure.compProd : {ฮฑ : Type u_1} โ
{ฮฒ : Type u_2} โ
{mฮฑ : MeasurableSpace ฮฑ} โ
{mฮฒ : MeasurableSpace ฮฒ} โ MeasureTheory.Measure ฮฑ โ ProbabilityTheory.Kernel ฮฑ ฮฒ โ MeasureTheory.Measure (ฮฑ ร ฮฒ)The composition-product of a measure and a kernel.
Learning.Algorithm.p0 : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ {m๐จ : MeasurableSpace ๐จ} โ Learning.Algorithm ๐ ๐ ๐จ โ ProbabilityTheory.Kernel ๐ ๐Distribution of the first action given the first observation: the policy at time `0` applied to the empty history.Go to its page
ProbabilityTheory.Kernel.compProd : {ฮฑ : Type u_4} โ
{ฮฒ : Type u_5} โ
{ฮณ : Type u_6} โ
{mฮฑ : MeasurableSpace ฮฑ} โ
{mฮฒ : MeasurableSpace ฮฒ} โ
{mฮณ : MeasurableSpace ฮณ} โ
ProbabilityTheory.Kernel ฮฑ ฮฒ โ ProbabilityTheory.Kernel (ฮฑ ร ฮฒ) ฮณ โ ProbabilityTheory.Kernel ฮฑ (ฮฒ ร ฮณ)Composition-Product of kernels. For s-finite kernels, it satisfies `โซโป bc, f bc โ(compProd ฮบ ฮท a) = โซโป b, โซโป c, f (b, c) โ(ฮท (a, b)) โ(ฮบ a)` (see `ProbabilityTheory.Kernel.lintegral_compProd`). If either of the kernels is not s-finite, `compProd` is given the junk value 0.
Learning.Environment.ฮฝ0 : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ Learning.Environment ๐ ๐ ๐จ โ ProbabilityTheory.Kernel (๐ ร ๐) ๐จDistribution 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
Code
lemma stepKernel_zero (alg : Algorithm ๐ ๐ ๐จ) (env : Environment ๐ ๐ ๐จ) (h : Hist ๐ ๐ ๐จ 0) :
stepKernel alg env 0 h = env.obs0 โโ (alg.p0 โโ env.ฮฝ0)Proof
by rw [Unique.eq_default h, stepKernel, Kernel.compProd_apply_eq_compProd_sectR] congr 1 ext o s hs rw [Kernel.sectR_apply, Kernel.compProd_apply hs, Kernel.compProd_apply hs] 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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 8 project declarations, 24 external constants
โ 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.