openAI-math-dataset / lean /OAI /Combinatorics /ExpectationThreshold /SelectorCountingWordsTransport.lean
| 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 | |