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measurable_argmax🔗

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Minimal Lean file

measurable_argmax🔗

Lemmameasurable_argmax
Details

No docstring.

🔗theorem
measurable_argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2}
  [LinearOrder α] [Fintype ι] [Nonempty ι] [MeasurableSpace α]
  [MeasurableSpace ι] [MeasurableEq α] [MeasurableSup₂ α] :
  Measurable fun f => argmax f
measurable_argmax.{u_1, u_2}
  {ι : Type u_1} {α : Type u_2}
  [LinearOrder α] [Fintype ι] [Nonempty ι]
  [MeasurableSpace α] [MeasurableSpace ι]
  [MeasurableEq α] [MeasurableSup₂ α] :
  Measurable fun f => argmax f

Code

lemma measurable_argmax [MeasurableSpace ι] [MeasurableEq α] [MeasurableSup₂ α] :
    Measurable fun f : ι → α ↦ argmax f
Type uses (1)
Body uses (3)
Used by (3)

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Proof
fun_prop) measurable_argmin]
lemma measurable_argmax [MeasurableSpace ι] [MeasurableEq α] [MeasurableSup₂ α] :
    Measurable fun f : ι → α ↦ argmax f := by
  refine measurable_to_countable' fun i ↦ ?_
  simp only [Set.preimage, Set.mem_singleton_iff]
  let Maximizers (f : ι → α) : Set ι := {i | f i = f.max}
  suffices {f : ι → α | argmax f = i} = ⋃ (S)
      (hS : ∀ x, Maximizers x = S → argmax x = i), {f | Maximizers f = S} by
    rw [this]
    refine MeasurableSet.iUnion fun S ↦ (.iUnion fun hS ↦ ?_)
    exact measurableSet_eq_fun (by fun_prop) measurable_const
  ext f
  simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop, exists_eq_right']
  constructor
  · intro hf x hx
    rw [← hf]
    exact Classical.choose.congr_simp hx (exists_argmax x)
  · intro h
    exact h f rfl

Dependency graph

Type dependencies (1)

argmax🔗

Definitionargmax
Details

The index of the maximum value of a tuple.

🔗def
argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι
argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι

Code

noncomputable def argmax := (exists_argmax f).choose
Body uses (2)
Used by (17)

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All dependencies, transitively (2)

max🔗

DefinitionFunction.max
Details

The maximum value of a tuple.

🔗def
Function.max.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α
Function.max.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α

Code

abbrev max : α := univ.sup' univ_nonempty f
Used by (8)

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exists_argmax🔗

Lemmaexists_argmax
Details

No docstring.

🔗theorem
exists_argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ∃ i, f i = Function.max f
exists_argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) :
  ∃ i, f i = Function.max f

Code

lemma exists_argmax : ∃ i, f i = f.max
Type uses (1)
Used by (3)

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Proof
by
  obtain ⟨i, -, hi⟩ := Finset.exists_mem_eq_sup' (by simp : Finset.univ.Nonempty) f
  exact ⟨i, hi.symm⟩