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Learning.IT.action_eq_eval_comp_hist๐Ÿ”—

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action_eq_eval_comp_hist๐Ÿ”—

LemmaLearning.IT.action_eq_eval_comp_hist
Details

No docstring.

๐Ÿ”—theorem
Learning.IT.action_eq_eval_comp_hist.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} (n : โ„•) :
  action n = (fun x => Prod.fst (x โŸจn, โ‹ฏโŸฉ)) โˆ˜ hist n
Learning.IT.action_eq_eval_comp_hist.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•) :
  action n =
    (fun x => Prod.fst (x โŸจn, โ‹ฏโŸฉ)) โˆ˜
      hist n

Code

lemma action_eq_eval_comp_hist (n : โ„•) :
    action (๐“
Type uses (2)
Used by (1)

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Proof
๐“) (๐“จ := ๐“จ) n = (fun x โ†ฆ (x โŸจn, by simpโŸฉ).1) โˆ˜ (hist n) := rfl

Dependency graph

Type dependencies (2)

action๐Ÿ”—

DefinitionLearning.IT.action
Details

action n is the action pulled at time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

๐Ÿ”—def
Learning.IT.action.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“
Learning.IT.action.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“

Code

def action (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : ๐“ := (h n).1
Used by (31)

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hist๐Ÿ”—

DefinitionLearning.IT.hist
Details

hist n is the history up to time n. This is a random variable on the measurable space โ„• โ†’ ๐“ ร— ๐“จ.

๐Ÿ”—def
Learning.IT.hist.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2} (n : โ„•)
  (h : โ„• โ†’ ๐“ ร— ๐“จ) : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ
Learning.IT.hist.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ

Code

def hist (n : โ„•) (h : โ„• โ†’ ๐“ ร— ๐“จ) : Iic n โ†’ ๐“ ร— ๐“จ := fun i โ†ฆ h i
Used by (23)

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