Learning.IsAlgEnvSeq.isAlgEnvSeqUntil_of_policy_eq
Lemma
From the authors
If two algorithms have the same policy before time N, then a sequence generated by the first
one interacting with an environment is a sequence generated by the second one until time N.
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 -
ฮฉ : Type u_4mฮฉ : MeasurableSpace ฮฉ
Given
-
O : โ โ ฮฉ โ ๐ -
A : โ โ ฮฉ โ ๐ -
Y : โ โ ฮฉ โ ๐จ -
alg : Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm. -
env : Environment ๐ ๐ ๐จA stochastic environment. -
P : MeasureTheory.Measure ฮฉA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsFiniteMeasure PA measureฮผis called finite ifฮผ univ < โ. -
N : โ -
alg' : Algorithm ๐ ๐ ๐จ
Assuming
-
h : IsAlgEnvSeq O A Y alg env PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment. -
h_eq : โ n < N, alg.policy n = alg'.policy n
MeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
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
MeasureTheory.IsFiniteMeasure : {ฮฑ : Type u_1} โ {m0 : MeasurableSpace ฮฑ} โ MeasureTheory.Measure ฮฑ โ PropA measure `ฮผ` is called finite if `ฮผ univ < โ`.
MeasureTheory.Measure : (ฮฑ : Type u_5) โ [MeasurableSpace ฮฑ] โ Type u_5A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. The measure of a set `s`, denoted `ฮผ s`, is an extended nonnegative real. The real-valued version is written `ฮผ.real s`.
Learning.IsAlgEnvSeq : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{ฮฉ : Type u_4} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
{mฮฉ : MeasurableSpace ฮฉ} โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐จ) โ
Learning.Algorithm ๐ ๐ ๐จ โโฆAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.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.IsAlgEnvSeqUntil : {๐ : Type u_1} โ
{๐ : Type u_2} โ
{๐จ : Type u_3} โ
{ฮฉ : Type u_4} โ
{m๐ : MeasurableSpace ๐} โ
{m๐ : MeasurableSpace ๐} โ
{m๐จ : MeasurableSpace ๐จ} โ
{mฮฉ : MeasurableSpace ฮฉ} โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐) โ
(โ โ ฮฉ โ ๐จ) โ
Learning.Algorithm ๐ ๐ ๐จ โโฆAn algorithm-environment sequence until time `N`: a sequence of observations, actions and feedbacks such that the rounds at times `n < N` are generated by an algorithm interacting with an environment. In particular, the law of `history O A Y N` is determined.Go to its page
Code
lemma IsAlgEnvSeq.isAlgEnvSeqUntil_of_policy_eq {alg' : Algorithm ๐ ๐ ๐จ}
(h : IsAlgEnvSeq O A Y alg env P) (h_eq : โ n < N, alg.policy n = alg'.policy n) :
IsAlgEnvSeqUntil O A Y alg' env P N where
measurable_obsProof
h.measurable_obs measurable_action := h.measurable_action measurable_feedback := h.measurable_feedback hasCondDistrib_obs n _ := h.hasCondDistrib_obs n hasCondDistrib_action n hn := h_eq n hn โธ h.hasCondDistrib_action n hasCondDistrib_feedback n _ := h.hasCondDistrib_feedback n
New in v4.34.0-rc2-90-gdde3322 (2026-09-16), and its meaning has not changed since.
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: 7 project declarations, 26 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.