| import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds |
| import Mathlib.Data.Fintype.CardEmbedding |
| import Mathlib.Tactic |
| import OAI.Combinatorics.BalancedRyser.Candidates.ConstituentOffsets |
|
|
| namespace OAI |
|
|
| open Filter |
| open scoped Topology |
| open scoped Classical |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| noncomputable section |
|
|
| theorem card_distinct_pairs (K : Type*) [Fintype K] : |
| Fintype.card {p : K × K |
| Fintype.card K * (Fintype.card K - 1) := by |
| classical |
| let e : {p : K × K |
| { toFun := fun p => ⟨p.1.1, p.1.2, Ne.symm p.2⟩ |
| invFun := fun p => ⟨(p.1, p.2.1), Ne.symm p.2.2⟩ |
| left_inv := fun p => by cases p; rfl |
| right_inv := fun p => by cases p; rfl } |
| rw [Fintype.card_congr e, Fintype.card_sigma] |
| simp |
|
|
| theorem card_allConstituentPairsDistinct (D K : Type*) [Fintype D] [Fintype K] : |
| Fintype.card {M : ConstituentOffsets D K |
| (Fintype.card K * (Fintype.card K - 1)) ^ Fintype.card D := by |
| classical |
| let e : {M : ConstituentOffsets D K |
| (D → {p : K × K |
| { toFun := fun M d => ⟨M.1 d, M.2 d⟩ |
| invFun := fun M => ⟨fun d => (M d).1, fun d => (M d).2⟩ |
| left_inv := fun M => rfl |
| right_inv := fun M => rfl } |
| calc |
| _ = Fintype.card (D → {p : K × K |
| _ = _ := by |
| rw [Fintype.card_fun] |
| exact congrArg (fun n => n ^ Fintype.card D) (card_distinct_pairs K) |
|
|
| theorem finiteUniformProbability_eq_toOuterMeasure {Ω : Type*} [Fintype Ω] |
| [Nonempty Ω] (E : Ω → Prop) : |
| finiteUniformProbability E = |
| ((PMF.uniformOfFintype Ω).toOuterMeasure {ω | E ω}).toReal := by |
| rw [PMF.toOuterMeasure_uniformOfFintype_apply] |
| simp only [finiteUniformProbability, ENNReal.toReal_div, ENNReal.toReal_natCast, |
| Fintype.card_eq_nat_card] |
| rfl |
|
|
| theorem distinctConstituentProbability_eq (q : ℕ) (hq : 0 < q) : |
| distinctConstituentProbability q = (1 - 1 / (q : ℝ)) ^ (q + 1) := by |
| unfold distinctConstituentProbability finiteUniformProbability |
| have hc := card_allConstituentPairsDistinct (Fin (q + 1)) (Fin q) |
| rw [Fintype.card_eq_nat_card] at hc |
| rw [hc] |
| rw [Nat.card_eq_fintype_card] |
| simp only [Fintype.card_fun, Fintype.card_prod, Fintype.card_fin, |
| Nat.cast_pow, Nat.cast_mul] |
| rw [← div_pow] |
| congr 1 |
| rw [Nat.cast_sub (by omega : 1 ≤ q), Nat.cast_one] |
| have hq0 : (q : ℝ) ≠ 0 := by positivity |
| field_simp |
|
|
| theorem tendsto_distinctConstituentProbability : |
| Tendsto distinctConstituentProbability atTop (𝓝 (Real.exp (-1))) := by |
| have h₁ : Tendsto (fun q : ℕ => 1 - 1 / (q : ℝ)) atTop (𝓝 (1 : ℝ)) := by |
| simpa using tendsto_const_nhds.sub |
| (tendsto_one_div_atTop_nhds_zero_nat (𝕜 := ℝ)) |
| have h₂ : Tendsto (fun q : ℕ => (1 - 1 / (q : ℝ)) ^ q) |
| atTop (𝓝 (Real.exp (-1))) := by |
| simpa only [neg_div, one_div, sub_eq_add_neg] using |
| Real.tendsto_one_add_div_pow_exp (-1) |
| have h := h₂.mul h₁ |
| simp only [mul_one] at h |
| apply h.congr' |
| filter_upwards [eventually_ge_atTop 1] with q hq |
| rw [distinctConstituentProbability_eq q (by omega), pow_succ] |
| |
| end |
| end Balanced |
| |
| end OAI |
| |
