LeanMachineLearning exposition

Learning.IsAlgEnvSeq.adapted_feedback๐Ÿ”—

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

adapted_feedback๐Ÿ”—

LemmaLearning.IsAlgEnvSeq.adapted_feedback
Details

No docstring.

๐Ÿ”—theorem
Learning.IsAlgEnvSeq.adapted_feedback.{u_1, u_2, u_3} {๐“ : Type u_1}
  {๐“จ : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n)) :
  MeasureTheory.Adapted (filtration hA hY) Y
Learning.IsAlgEnvSeq.adapted_feedback.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n)) :
  MeasureTheory.Adapted (filtration hA hY)
    Y

Code

lemma IsAlgEnvSeq.adapted_feedback
    (hA : โˆ€ n, Measurable (A n)) (hY : โˆ€ n, Measurable (Y n)) :
    Adapted (filtration hA hY) Y
Type uses (1)
Body uses (2)
Used by (1)

Actions: Source ยท Open Issue

Proof
by
  intro n
  have : Y n = (fun h โ†ฆ (h โŸจn, by simpโŸฉ).2) โˆ˜ (history A Y n) := by
    ext ฯ‰ : 1
    simp [history]
  rw [this]
  exact measurable_comp_comap _ (by fun_prop)

Dependency graph

Type dependencies (1)

filtration๐Ÿ”—

DefinitionLearning.IsAlgEnvSeq.filtration
Details

Filtration generated by the history up to time n.

๐Ÿ”—def
Learning.IsAlgEnvSeq.filtration.{u_1, u_2, u_3} {๐“ : Type u_1}
  {๐“จ : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n)) : MeasureTheory.Filtration โ„• mฮฉ
Learning.IsAlgEnvSeq.filtration.{u_1, u_2,
    u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n)) :
  MeasureTheory.Filtration โ„• mฮฉ

Code

def IsAlgEnvSeq.filtration (hA : โˆ€ n, Measurable (A n)) (hY : โˆ€ n, Measurable (Y n)) :
    Filtration โ„• mฮฉ where
  seq i := MeasurableSpace.comap (history A Y i) inferInstance
  mono' i j hij := by
    simp only
    rw [โ† measurable_iff_comap_le]
    have : history A Y i = (fun h k โ†ฆ h โŸจk.1, by grindโŸฉ) โˆ˜ history A Y j := rfl
    rw [this]
    exact measurable_comp_comap _ (by fun_prop)
  le' i := by
    rw [โ† measurable_iff_comap_le]
    exact Learning.measurable_history hA hY i
Body uses (3)
Used by (18)

Actions: Source ยท Open Issue

All dependencies, transitively (3)

history๐Ÿ”—

DefinitionLearning.history
Details

History of the algorithm-environment sequence up to time n.

๐Ÿ”—def
Learning.history.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ
Learning.history.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ

Code

def history (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) : Iic n โ†’ ๐“ ร— ๐“จ :=
  fun i โ†ฆ (A i ฯ‰, Y i ฯ‰)
Used by (72)

Actions: Source ยท Open Issue

measurable_comp_comap๐Ÿ”—

LemmaMeasureTheory.measurable_comp_comap
Details

No docstring.

๐Ÿ”—theorem
MeasureTheory.measurable_comp_comap.{u_1, u_2, u_3} {ฮฑ : Type u_1}
  {ฮฒ : Type u_2} {ฮณ : Type u_3} {mฮฒ : MeasurableSpace ฮฒ}
  {mฮณ : MeasurableSpace ฮณ} (f : ฮฑ โ†’ ฮฒ) {g : ฮฒ โ†’ ฮณ} (hg : Measurable g) :
  Measurable (g โˆ˜ f)
MeasureTheory.measurable_comp_comap.{u_1,
    u_2, u_3}
  {ฮฑ : Type u_1} {ฮฒ : Type u_2}
  {ฮณ : Type u_3} {mฮฒ : MeasurableSpace ฮฒ}
  {mฮณ : MeasurableSpace ฮณ} (f : ฮฑ โ†’ ฮฒ)
  {g : ฮฒ โ†’ ฮณ} (hg : Measurable g) :
  Measurable (g โˆ˜ f)

Code

lemma measurable_comp_comap (f : ฮฑ โ†’ ฮฒ) {g : ฮฒ โ†’ ฮณ} (hg : Measurable g) :
    Measurable[mฮฒ.comap f] (g โˆ˜ f)
Used by (10)

Actions: Source ยท Open Issue

Proof
by
  rw [measurable_iff_comap_le, โ† MeasurableSpace.comap_comp]
  exact MeasurableSpace.comap_mono hg.comap_le

measurable_history๐Ÿ”—

LemmaLearning.measurable_history
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_history.{u_1, u_2, u_3} {๐“ : Type u_1}
  {๐“จ : Type u_2} {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n)) (n : โ„•) :
  Measurable (history A Y n)
Learning.measurable_history.{u_1, u_2,
    u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} {A : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hY : โˆ€ (n : โ„•), Measurable (Y n))
  (n : โ„•) : Measurable (history A Y n)

Code

lemma measurable_history (hA : โˆ€ n, Measurable (A n))
    (hY : โˆ€ n, Measurable (Y n)) (n : โ„•) :
    Measurable (history A Y n)
Type uses (1)
Used by (10)

Actions: Source ยท Open Issue

Proof
by
  unfold history
  fun_prop