LeanMachineLearning exposition

Learning.action_stepsUntil๐Ÿ”—

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Minimal Lean file

action_stepsUntil๐Ÿ”—

LemmaLearning.action_stepsUntil
Details

No docstring.

๐Ÿ”—theorem
Learning.action_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} (hm : m โ‰  0)
  (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  A (ENat.toNat (stepsUntil A a m ฯ‰)) ฯ‰ = a
Learning.action_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} (hm : m โ‰  0)
  (h_exists :
    โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
  A (ENat.toNat (stepsUntil A a m ฯ‰)) ฯ‰ =
    a

Code

lemma action_stepsUntil (hm : m โ‰  0) (h_exists : โˆƒ s, pullCount A a (s + 1) ฯ‰ = m) :
    A (stepsUntil A a m ฯ‰).toNat ฯ‰ = a
Type uses (2)
Body uses (3)
Used by (1)

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Proof
by
  classical
  simp only [stepsUntil_eq_dite, h_exists, โ†“reduceDIte, ENat.toNat_coe]
  have h_spec := Nat.find_spec h_exists
  have h_spec' n := Nat.find_min h_exists (m := n)
  by_cases h_zero : Nat.find h_exists = 0
  ยท simp only [h_zero, zero_add, not_lt_zero, IsEmpty.forall_iff, implies_true] at *
    by_contra h_ne
    rw [โ† zero_add 1, pullCount_eq_pullCount_of_action_ne h_ne] at h_spec
    simp only [pullCount_zero] at h_spec
    exact hm h_spec.symm
  have h_pos : 0 < Nat.find h_exists := Nat.pos_of_ne_zero h_zero
  by_contra h_ne
  refine h_spec' (Nat.find h_exists - 1) ?_ ?_
  ยท simp [h_pos]
  rw [Nat.sub_add_cancel (by omega)]
  rwa [โ† pullCount_eq_pullCount_of_action_ne]
  exact h_ne

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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