lean/OAI/Combinatorics/ExpectationThreshold/SelectorCountingEncodingRows.lean

import OAI.Combinatorics.ExpectationThreshold.SelectorCountingWordsTransport
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Fintype.BigOperators

namespace OAI

/-!
# Counting row encodings

A row encoding chooses a subset of the available positions and labels those
positions with a word of a prescribed profile. The word count depends only on
the number of chosen positions, so the total count factors into a binomial
coefficient and the corresponding word count.
-/

namespace TalagrandThreshold

open scoped BigOperators

variable {α H : Type*} [Fintype α] [DecidableEq α] [Fintype H] [DecidableEq H]

/-- The number of subsets of `R` with exactly `t` elements. -/
theorem card_fixedSize_subsets (R : Finset α) (t : ℕ) :
    Fintype.card {U : Finset α // U ⊆ R ∧ U.card = t} =
      Nat.choose R.card t := by
  calc
    Fintype.card {U : Finset α // U ⊆ R ∧ U.card = t} =
        (R.powersetCard t).card :=
      Fintype.card_of_subtype (R.powersetCard t)
        (fun _ => Finset.mem_powersetCard)
    _ = Nat.choose R.card t := Finset.card_powersetCard t R

/-- A chosen set of positions together with labels of the prescribed profile. -/
def RowEncoding (R : Finset α) (t : ℕ) (n : H → ℕ) : Type _ :=
  Σ U : {U : Finset α // U ⊆ R ∧ U.card = t}, ↥(wordFiber ↥U.val n)

noncomputable instance instFintypeRowEncoding (R : Finset α) (t : ℕ) (n : H → ℕ) :
    Fintype (RowEncoding R t n) :=
  inferInstanceAs (Fintype
    (Σ U : {U : Finset α // U ⊆ R ∧ U.card = t}, ↥(wordFiber ↥U.val n)))

/-- Each chosen set admits the same number of words, so the count factors. -/
theorem card_rowEncoding (R : Finset α) (t : ℕ) (n : H → ℕ) :
    Fintype.card (RowEncoding R t n) = Nat.choose R.card t * wordCount t n := by
  classical
  change Fintype.card
    (Σ U : {U : Finset α // U ⊆ R ∧ U.card = t}, ↥(wordFiber ↥U.val n)) = _
  rw [Fintype.card_sigma]
  have hFiber (U : {U : Finset α // U ⊆ R ∧ U.card = t}) :
      Fintype.card ↥(wordFiber ↥U.val n) = wordCount t n := by
    rw [Fintype.card_coe, wordFiber_finset_card, U.property.2]
  calc
    (∑ U : {U : Finset α // U ⊆ R ∧ U.card = t},
        Fintype.card ↥(wordFiber ↥U.val n)) =
        Fintype.card {U : Finset α // U ⊆ R ∧ U.card = t} * wordCount t n := by
      simpa only [Finset.card_univ] using
        (Finset.sum_const_nat (s := Finset.univ)
          (f := fun U : {U : Finset α // U ⊆ R ∧ U.card = t} =>
            Fintype.card ↥(wordFiber ↥U.val n))
          (m := wordCount t n) (fun U _ => hFiber U))
    _ = Nat.choose R.card t * wordCount t n := by
      rw [card_fixedSize_subsets]

end TalagrandThreshold

end OAI