`Learning.nextAction_zero`🔗

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

alg : Algorithm 𝓞 𝓐 𝓨A 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).
h : Hist 𝓞 𝓐 𝓨 0History of n complete rounds; n = 0 is the empty history.
o : 𝓞

Then

nextAction alg 0 (h, o) = actionZero alg o

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

Learning.IsDeterministicAlg : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {m𝓞 : MeasurableSpace 𝓞} → {m𝓐 : MeasurableSpace 𝓐} → {m𝓨 : MeasurableSpace 𝓨} → Learning.Algorithm 𝓞 𝓐 𝓨 → Prop

An 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 : (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, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation.

Go to its page

Learning.Hist : Type u_5 → Type u_6 → Type u_7 → ℕ → 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 : {α : Sort u_1} → α → α → Prop

The 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.nextAction : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {m𝓞 : MeasurableSpace 𝓞} →
        {m𝓐 : MeasurableSpace 𝓐} →
          {m𝓨 : MeasurableSpace 𝓨} →
            (alg : Learning.Algorithm 𝓞 𝓐 𝓨) →
              [h_det : Learning.IsDeterministicAlg alg] → (n : ℕ) → Learning.Hist 𝓞 𝓐 𝓨 n × 𝓞 → 𝓐

The action of a deterministic algorithm at step `n`, as a function of the history before `n`
and of the observation at step `n`.

Go to its page

Prod.mk : {α : Type u} → {β : Type v} → α → β → α × β

Constructs 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`.

Learning.actionZero : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {m𝓞 : MeasurableSpace 𝓞} →
        {m𝓐 : MeasurableSpace 𝓐} →
          {m𝓨 : MeasurableSpace 𝓨} → (alg : Learning.Algorithm 𝓞 𝓐 𝓨) → [Learning.IsDeterministicAlg alg] → 𝓞 → 𝓐

The initial action of a deterministic algorithm, as a function of the first observation.

Go to its page

Code

lemma nextAction_zero (alg : Algorithm 𝓞 𝓐 𝓨) [IsDeterministicAlg alg] (h : Hist 𝓞 𝓐 𝓨 0)
    (o : 𝓞) :
    nextAction alg 0 (h, o) = actionZero alg o

Proof

by
  rw [Unique.eq_default h]
  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: 6 project declarations, 19 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.

Learning.nextAction_zero🔗

`Learning.nextAction_zero`🔗