import OAI.Combinatorics.ExpectationThreshold.SelectorCountingWords
import Mathlib.Data.Fintype.EquivFin

namespace OAI

/-!
# Transport of words with a prescribed profile

Reindexing the domain of a word preserves every letter count. Consequently,
the number of words with a prescribed profile depends only on the size of
the domain, including when that domain is the subtype of a finite set.
-/

namespace TalagrandThreshold

section Profile

variable {D E H : Type*} [Fintype D] [Fintype E] [DecidableEq H]

/-- A domain equivalence preserves the profile of a word. -/
theorem wordProfile_comp_equiv (e : D ≃ E) (f : E → H) :
    wordProfile (f ∘ e) = wordProfile f := by
  funext h
  change (Finset.univ.filter (fun i : D => f (e i) = h)).card =
    (Finset.univ.filter (fun j : E => f j = h)).card
  apply Finset.card_equiv e
  intro i
  simp only [Finset.mem_filter, Finset.mem_univ, true_and]

end Profile

section Fiber

variable {D E H : Type*} [Fintype D] [Fintype E]
variable [DecidableEq D] [DecidableEq E] [Fintype H] [DecidableEq H]

/-- Reindexing gives an equivalence between words of the same profile. -/
def wordFiberEquiv (e : D ≃ E) (n : H → ℕ) :
    ↥(wordFiber D n) ≃ ↥(wordFiber E n) := by
  refine Equiv.subtypeEquiv (Equiv.arrowCongr e (Equiv.refl H)) ?_
  intro f
  change f ∈ Finset.univ.filter (fun g : D → H => wordProfile g = n) ↔
    (f ∘ e.symm) ∈ Finset.univ.filter (fun g : E → H => wordProfile g = n)
  rw [Finset.mem_filter, Finset.mem_filter]
  constructor
  · intro hf
    exact ⟨Finset.mem_univ _, (wordProfile_comp_equiv e.symm f).trans hf.2⟩
  · intro hf
    exact ⟨Finset.mem_univ _, (wordProfile_comp_equiv e.symm f).symm.trans hf.2⟩

/-- Equivalent finite domains have equally many words of each profile. -/
theorem wordFiber_card_eq_of_equiv (e : D ≃ E) (n : H → ℕ) :
    (wordFiber D n).card = (wordFiber E n).card := by
  simpa only [Fintype.card_coe] using Fintype.card_congr (wordFiberEquiv e n)

end Fiber

/-- The word count on any finite domain agrees with the count on `Fin`. -/
theorem wordFiber_card_eq_wordCount (D : Type*) {H : Type*}
    [Fintype D] [DecidableEq D] [Fintype H] [DecidableEq H] (n : H → ℕ) :
    (wordFiber D n).card = wordCount (Fintype.card D) n := by
  change (wordFiber D n).card = (wordFiber (Fin (Fintype.card D)) n).card
  exact wordFiber_card_eq_of_equiv (Fintype.equivFin D) n

/-- Labels on a finite set are counted by the word count for its cardinality. -/
theorem wordFiber_finset_card {α H : Type*} [DecidableEq α]
    [Fintype H] [DecidableEq H] (U : Finset α) (n : H → ℕ) :
    (wordFiber ↥U n).card = wordCount U.card n := by
  change (wordFiber ↥U n).card = (wordFiber (Fin U.card) n).card
  exact wordFiber_card_eq_of_equiv U.equivFin n

end TalagrandThreshold

end OAI