File size: 2,823 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | 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
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