LeanMachineLearning exposition

Learning.stepsUntil_eq_dite๐Ÿ”—

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stepsUntil_eq_dite๐Ÿ”—

LemmaLearning.stepsUntil_eq_dite
Details

No docstring.

๐Ÿ”—theorem
Learning.stepsUntil_eq_dite.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ)
  [Decidable (โˆƒ s, pullCount A a (s + 1) ฯ‰ = m)] :
  stepsUntil A a m ฯ‰ =
    if h : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m then โ†‘(Nat.find h) else โŠค
Learning.stepsUntil_eq_dite.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} (a : ๐“)
  (m : โ„•) (ฯ‰ : ฮฉ)
  [Decidable
      (โˆƒ s,
        pullCount A a (s + 1) ฯ‰ = m)] :
  stepsUntil A a m ฯ‰ =
    if h :
        โˆƒ s,
          pullCount A a (s + 1) ฯ‰ = m then
      โ†‘(Nat.find h)
    else โŠค

Code

lemma stepsUntil_eq_dite (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ)
    [Decidable (โˆƒ s, pullCount A a (s + 1) ฯ‰ = m)] :
    stepsUntil A a m ฯ‰ =
      if h : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m then (Nat.find h : โ„•โˆž) else โŠค
Type uses (2)
Used by (7)

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Proof
by
  unfold stepsUntil
  split_ifs with h'
  ยท refine le_antisymm ?_ ?_
    ยท refine sInf_le ?_
      simpa using Nat.find_spec h'
    ยท simp only [le_sInf_iff, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,
        forall_apply_eq_imp_iffโ‚‚, Nat.cast_le, Nat.find_le_iff]
      exact fun n hn โ†ฆ โŸจn, le_rfl, hnโŸฉ
  ยท push Not at h'
    suffices {s | pullCount A a (s + 1) ฯ‰ = m} = โˆ… by simp [this]
    ext s
    simpa using (h' s)

Dependency graph

Type dependencies (2)

pullCount๐Ÿ”—

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

๐Ÿ”—def
Learning.pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„•
Learning.pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„•

Code

noncomputable
def pullCount (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„• :=
  #(filter (fun s โ†ฆ A s ฯ‰ = a) (range t))
Used by (146)

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stepsUntil๐Ÿ”—

DefinitionLearning.stepsUntil
Details

Number of steps until action a was pulled exactly m times.

๐Ÿ”—def
Learning.stepsUntil.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž
Learning.stepsUntil.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž

Code

noncomputable
def stepsUntil (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (m : โ„•) (ฯ‰ : ฮฉ) : โ„•โˆž :=
  sInf ((โ†‘) '' {s | pullCount A a (s + 1) ฯ‰ = m})
Body uses (1)
Used by (46)

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