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Learning.snd_eval_comp_historyπŸ”—

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snd_eval_comp_historyπŸ”—

LemmaLearning.snd_eval_comp_history
Details

No docstring.

πŸ”—theorem
Learning.snd_eval_comp_history.{u_1, u_2, u_3} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {Ξ© : Type u_3} {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  (n : β„•) : (fun x => Prod.snd (x ⟨n, β‹―βŸ©)) ∘ history A Y n = Y n
Learning.snd_eval_comp_history.{u_1, u_2,
    u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} {A : β„• β†’ Ξ© β†’ 𝓐}
  {Y : β„• β†’ Ξ© β†’ 𝓨} (n : β„•) :
  (fun x => Prod.snd (x ⟨n, β‹―βŸ©)) ∘
      history A Y n =
    Y n

Code

lemma snd_eval_comp_history (n : β„•) :
    (fun x ↦ (x ⟨n, by simp⟩).2) ∘ (history A Y n) = Y n
Type uses (1)

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Proof
rfl

Dependency graph

Type dependencies (1)

historyπŸ”—

DefinitionLearning.history
Details

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

πŸ”—def
Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐)
  (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨

Code

def history (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) : Iic n β†’ 𝓐 Γ— 𝓨 :=
  fun i ↦ (A i Ο‰, Y i Ο‰)
Used by (72)

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