Learning.Algorithm.comapFeedback_def
Lemma
No docstring.
Types
-
๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_4m๐ : MeasurableSpace ๐ -
๐จ : Type u_7m๐จ : MeasurableSpace ๐จ -
๐จ' : Type u_8m๐จ' : MeasurableSpace ๐จ'
Given
-
g : ๐จ' โ ๐จ -
alg : Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm.
Assuming
-
hg : Measurable gA functionfbetween measurable spaces is measurable if the preimage of every measurable set is measurable.
Then
alg.comapFeedback g hg = alg.comap (fun x p => (Hist.mapFeedback g p.1, p.2)) โฏMeasurableSpace : 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
Measurable : {ฮฑ : Type u_1} โ {ฮฒ : Type u_2} โ [MeasurableSpace ฮฑ] โ [MeasurableSpace ฮฒ] โ (ฮฑ โ ฮฒ) โ PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
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.comapFeedback : {๐ : Type u_1} โ
{๐ : Type u_4} โ
{๐จ : Type u_7} โ
{๐จ' : Type u_8} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
{m๐จ' : MeasurableSpace ๐จ'} โ
Learning.Algorithm ๐ ๐ ๐จ โ
(g : ๐จ' โ ๐จ) โ
autoParam (Measurable g) Learning.Algorithm.comapFeedback._auto_1 โ LโฆThe algorithm that sees `g y` when the feedback of a past round is `y`.Go to its page
Learning.Algorithm.comap : {๐ : Type u_1} โ
{๐' : Type u_2} โ
{๐ : Type u_4} โ
{๐จ : Type u_7} โ
{๐จ' : Type u_8} โ
{m๐ : MeasurableSpace ๐} โ
{m๐' : MeasurableSpace ๐'} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
{m๐จ' : MeasurableSpace ๐จ'} โ
Learning.Algorithm ๐ ๐ ๐จ โ
(F : (n : โ) โโฆThe algorithm with observations in `๐'` and feedbacks in `๐จ'` obtained from `alg : Algorithm ๐ ๐ ๐จ` by transforming the pair (past rounds, current observation) by `F n` at each round `n` before applying the policy of `alg`. This is the primitive transport operation on algorithms: `Algorithm.comapObs` and `Algorithm.comapFeedback` are the special cases in which `F n` is a round-wise map of the observation and of the feedback.Go to its page
Prod.mk : {ฮฑ : Type u} โ {ฮฒ : Type v} โ ฮฑ โ ฮฒ โ ฮฑ ร ฮฒConstructs 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.Hist.mapFeedback : {๐ : Type u_1} โ
{๐ : Type u_4} โ
{๐จ : Type u_7} โ {๐จ' : Type u_8} โ (๐จ โ ๐จ') โ {n : โ} โ Learning.Hist ๐ ๐ ๐จ n โ Learning.Hist ๐ ๐ ๐จ' nTransport the feedback of a history.Go to its page
Code
lemma Algorithm.comapFeedback_def (alg : Algorithm ๐ ๐ ๐จ) (hg : Measurable g) :
alg.comapFeedback g hg
= alg.comap (fun _ p โฆ (Hist.mapFeedback g p.1, p.2)) fun _ โฆ by fun_propProof
rfl
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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 12 project declarations, 29 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.