Learning.p0_eq_deterministic
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.IsDeterministicAlg algAn algorithm is deterministic if its actions are determined by measurable functions of the history and of the current observation (and not possibly random kernels).
Then
alg.p0 = ProbabilityTheory.Kernel.deterministic (actionZero alg) โฏMeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
Learning.IsDeterministicAlg : {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 PropAn algorithm is deterministic if its actions are determined by measurable functions of the history and of the current observation (and not possibly random kernels).Go to its page
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
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.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.deterministic : {ฮฑ : Type u_1} โ
{ฮฒ : Type u_2} โ
{mฮฑ : MeasurableSpace ฮฑ} โ {mฮฒ : MeasurableSpace ฮฒ} โ (f : ฮฑ โ ฮฒ) โ Measurable f โ ProbabilityTheory.Kernel ฮฑ ฮฒKernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel.
Learning.actionZero : {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 (alg : Learning.Algorithm ๐ ๐ ๐จ) โ [Learning.IsDeterministicAlg alg] โ ๐ โ ๐The initial action of a deterministic algorithm, as a function of the first observation.Go to its page
Code
lemma p0_eq_deterministic (alg : Algorithm ๐ ๐ ๐จ) [IsDeterministicAlg alg] :
alg.p0 = Kernel.deterministic (actionZero alg) (measurable_actionZero alg)Proof
by
ext o : 1
rw [Algorithm.p0_apply, policy_eq_deterministic, Kernel.deterministic_apply,
Kernel.deterministic_apply]
rflMeaning 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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 8 project declarations, 20 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.